A Logic of Exceptions

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A Logic of Exceptions

Using The Economics Pack Applications of Mathematicafor Elementary Logic

Thomas Colignatus, March 2011

http://thomascool.eu

Applications of Mathematica



Th a C lig a i he a e f Th a C l i cie ce.

Th a C l, bli hed 1981, 1 edi i 2007, 2 d edi i 2011

P bli hed b C l, T. (C l a c & Ec e ic )

Th a C l C & E

R e da e aa 69, NL 2586 GH Sche e i ge , The Ne he la d

h :// h a c l.e

NUR 120, 738; JEL A00; MSC2010 03B01, 97E30

ISBN 978 90 804774 7 6

i a egi e ed ade a k f W lf a Re ea ch, I c.

Thi b k e e i 8.0.1

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Aims of this book when you are new to the subject

The f ll i g h ld ha e bee achie ed he fi i h hi b k.

† Y ill be e de a d he aj ic i l gic a d i fe e ce.

† Y ca check he he c i fe e ce a e alid , a d ell h .

† Y ca be e de e i e a ej i de ha he ld e acce .

† Y ca ead hi b k ,h al i h . Wi h e e i g ag a , ill ill be efi f he di c i . H e e , if ha e a c e

a d ac ice i h he g a , he e d bei g able he l gic a di fe e ce i e a d i e e e hei e l .

N e ha he f a e ca be d l aded f eel f he i e e a d be i ec ed;h e e , if a i , he eed a lice ce.

Aims of this book when you are an advanced reader

The f ll i g h ld ha e bee achie ed he fi i h hi b k.

† O e f he ai f hi b k ha bee e e l gic ha e eade ge a cleaie he bjec . Whe ead hi b k i hi a a ell, he ill

be efi f h e ai ( ee ab e) a d be able di c l gic i hi fa hi a ell.

† Y ill be a ad a ced de b igh lack i bala ce be ee ei he he

a h he hi a d hil h . Ha i g dige ed hi b k ill fi d a be elace f b h, e ha ci g he a ha lack.

† Y ill be e de a d he ai ch ice be ee (1) he he f e , (2) fhe , (3) h ee al ed l gic.

† Y ill ef c e ea ch a d i e ha a e e.

N e ha hi b k ha age da : Fi de el l gic a d i fe e ce f he b a d ed ca e de i a . Sec dl , h h he

c f i ge e a ed b R ell, Ta ki, Hilbe a d G del ca be l ed. The fibjec i e i e e a e hile he ec d bjec i e i e a ie . O cee ea che ha e ad ed he l i i hi b k, he e babl e ai li le al e

i eachi g e de ab ld c f i . I e ea i e i a j be hi ,h f l a a ag a h. Ye f he e i ill i a bjec i e:

† Y ca e lai he ha G del e bal e la a i f hi he e d a ch he a he a ic , a d, ha e de el e f he a i hh e e bal a e e be i c ec . Y ill babl ec ide ie h e he e a d a ee ha he a e a he i ele a f l gic a d i fe e ce.

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4



Abstract

† The b k gi e a E , f he ice le el G deli c le e e he e . The fi cha e ha a di ec ha d a ach ha he

ice ca di ec l lea ab he c ec i e f i i al l gic a d g a he

e f i fe e ce. Thi , a d he e e f achie e e , h ld i la e he de c i e, hile i al ide a ba i eflec ha al ead ha bee lea ed.

The la e cha e he del e i he , i h a cl i g cha e i h e f ali a i .

† Di c ed a e (1) he ba ic ele e : i i al e a , edica e a d e ; (2)he ba ic i fe e ce i : i fe e ce, ll gi , a i a ic , f he ; (3) he

ba ic e a : hi ical i , ela i he cie ific e h d, he a ad e . Thee ele e i he b k a e: (4) a , l i f a ad e , a al if c i ake , i e i .

† C f ed i h he Lia a ad ( Thi e e ce i fal e ), de l gic ec g i eh ee ai a ache : (1) he he f e , i h le el i la g age, (2) he

aba d e f h a d ad i f a he f f i h decidabili i eadf i c i e c , a d (3) h ee al ed l gic. The fi a ache ac ifice ef l

f f elf efe e ce. Al , he G del i c le e e he e a e ba ed ac i f he Lia ( Thi e e ce i able ), hich ake he e l a he

c i ci g, hile he hil hical i lica i decidabili ee la gei ela i he d bi a e f ch a a ad ical a e e . He ce h ee al edl gic i a e f i f l a ach. The l gic c i ha he i a i ab h ee

al ed l gic. The g a i le e ed i h ha i k i eicel .

† G del i c le e e he e a e a he ele f de a di g l gic a d hef da i f a he a ic . M e e , G del e bal a d e a a he a icale la a i a d i e e a i f he he e a ea a ch i a he a ic .Hi ded c i i ba ed all i g a e e efe he el e elf efe e iall b di all i g hi f he h le e i elf. The e a ach i all f

he la e a d ad he a i ( ¢ ) ( ¢ ( ¢ )) ha he ee hi g he i c ide i al e ha i e e hi g. I hi ca e he

G delia c lla e he Lia , b i gi g back he i a i 500 ea B.C.

† The ke de el e f hi c e i he .A i ellige ha dle fi f a i al a kee i i d ha he e a be e ce i he le a lied.

Thi a ach d e l all f f eed f e e i b al ake fd ea i g. A d i gi e he igh e l .

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† The g a he de el e i he b k. The g a bec e a ailable i hi a d i hi E C l (1999,2001) a ailable a h :// h a c l.e a d he e al a i g:

Needs["Economics`Pack`"]

ResetAll

Economics[Logic]

Y ca ead hi b k ,h al i h . Wi h e e i g

a g a , ill ill be efi f he di c i . See he i e e f he l gicg a .

O he ale e f E he e i a b f he U e G ide. Click he e

a d ill fi d he e i e e f hi b k a ailable he e.

Keywords

L gic, i fe e ce, i d c i , a i a ic , f he , i i al l gic, edica el gic, e he , f da i f a he a ic , i c le e e he e , lia a ad ,a i ie , i i i i , hi f l gic, ali , dali , ce ai , e i e l g ,

e h d l g f cie ce, ge e al hil h , ge e al ec ic

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Preface

I 1980 I a a g ad a e de i ec e ic a d beca e i e e ed i l gic a d hee h d l g f cie ce. Thi ee like a de a e f ec ic , b he e i e

l gic i , a h ld bec e clea bel . The d g e i l ed ha a ici a ed

a d e l ed i a 500 age d af I d c i e b k i D ch, a d i 1981 a 100 agea di c i i E gli h called I e ia Phile a f C e alii he Lia

a ad , R ell e a ad , B e i i i i , a d he G delia , The aji i 1980 1981 a , a d ill be bel , he de el e a , ea i g

ha e le a e f ee defi e c ce a d de i e c cl i a he i h, hee ha he i a c adic i ha he ha e ake e e i i . Thi

i h cie ce k i ac ice a d hi i ha l gic a d i fe e ce h ld all f .The defi i i f h ld i cl de he i ha i h ld i ge e al f che e i a he Lia G delia . L gicia ake a g ea h ab hei a ad e b ce ad ch a he ca achie e a g ea ec idi ca di g f ha he a .

I 1981 he e e e e fac ha ca ed e d he bjec agai . The l giciaha I a ached fi di g did eac l gicall . G ad a i beca e e

i a a ell. Wha a ed e e i h e da e e he e a ee The Gl bal 2000 Re he P e ide (Ba e (1981)), e i di g ha

he ld la i ld i e f 4 billi i 1975 6.4 billi i 2000. S d i gl gic had al ca ed e c e ac Ke e bi a f F a k Ra e , a d I c ld

l ag ee i h i :

If he had f ll ed he ea ie a h f e e i cli a i , I a e ha he ld

ha e e cha ged he e i g e e ci e f he f da i f h gh , he e he i die ca ch i ail, f he deligh f l a h f ag eeable b a ch f he

al cie ce , i hich he a d fac , i i i e i agi a i a d ac ical j dge e ,

a e ble ded i a a e c f able he h a i ellec . J.M. Ke e , bi a ice

f F.P. Ra e , The Ec ic J al 40, Ma ch 1930; al ed i Ha Wa g (1974:25).I 1981 a al i a ha he ld ec ic ble e e i a il ca ed i

he We . Thi a ee a ad ical i ce i e i he he c i e , e , iake ec ic e e. P lic i he We a a d i ec i e l e b

ba ie ade. S l i g e l e i he We ld c ea e c e libe a eade a d e ha ce he ibili ie f he de el i g a i . I al e a a al i

ha de el e aid i ele he he e i g d g e e a he ecei i g e d.He ce af e g ad a i i 1982 e ea ch age da beca e f c ed e l ei he We , a d he d af b k l gic e e a ide. B 1989 I had f d e la a i f e l e . P blica i a dela ed b cceeded i C lig a(1992b). E e all e i i al l gic ed i C l (1999, 2001),

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E . De ic l gic ed i C lig a (1992b) a d he i (2001, 2007),V i g he f de c ac . C lig a (2000, 2005) e e be e didac ic a e ef he e la a i f e l e i he We . C lig a (2006) i a a lica i a

Ca ibbea ec . J , i Dece be 2006 a d Ja a 2007 I fi d e i e a d

he i ec ide h e hel ed b k l gic. A g 2007 added eReadi g N e (Cha e 11).

M h e i ha he e age ill be be eficial f h e h d l gic a d hee h d l g f cie ce. The e bjec ickle he i d a d he be f b k he i e , e I ill hi k ha e eade igh be efi f he c ce f a

.A d, j he idea, ha ca be h a ed i a i gle e e ce, bal de el ed a a c e i l gic f he b .

Af e he e 25 ea a d i h hi d igh he e a ea be a d ec ic ea aell, hich I had ee i 1980 1981, a d hich ake he dig e i i l gic e e

e acce able ec ic . Si ce he Eg ia , a ki d ha bee i g l ehe ble f b ea c ac . O e f e e a ach i he le f la , a , ha a

e e la gi e defi e a le ha a b ea c ac e f ce. I i diffic l f a lah e e acc f all ki d f e ce i ha igh be c ide ed i ii le e a i . R hle e f ce e igh ell de he e i e i f hala . S e b ea c a igh ill f ch e f ce e e el la i afe ha

b d ca a ha he d d hei j b. Decade a a bef e ch de i e ala lica i i iced a d e i ed. The e i he f Ca he i e he G ea eg la l

i i i g a all a k f a e i he e ai , ha he a g a d he e; a d eh d ed ea af e he dea h eb d iced ha g a di g ha all a k had bec e ki d f ill . Whe b h la gi e a d b ea c a g e a a e f e

he he igh be e deal i h he c i ge cie f blica age e . I i a l g h hi k , f c e, b i ge e al i ld hel hee le a e l a a e f he ig f a l gical a g e le b al f he

ibili f e e ce i .

O i e ded a die ce i a ge e al blic f de . We a e ha k ea i h e ic a d e ea i g, a d ha a e illi g de el k ledgeal g ide i h ki g hi b k.

A aid, C l (1999, 2001) E al ead c ai g a i f i i al l gic. The e ha e bee da ed a d e e ded. The e g a ca be

ed a a de g ad a e le el. A ha bee b e ed bef e, i elf i

al ead a deci i achi e f he edica e calc l . Thi b k e he e g a de el l gic f he b . The f c i l gic, he g a . B if ha e he e g a a ailable, he ca ha e a ha d e e ie ce, e if he

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c cl i , ca e , a d, i e highe le el g a .

I be ack ledged ha l gic ca bec e e ab ac , ha e f he ef da e al c cl i f hi b k ill e i e a e i agai a a e ad a cedle el. If a e ch a ad a ced de he a e f ee a i h he la e

cha e . H e e , gi e he a a e l ide ead i de a di g , a e ellad i ed k h gh he b k e e iall . I ill h ac a if a e e

he bjec . I fac , hi b k f ll he a eg fi c ea e e c e e ce il gic a d l he di c he he , ha b h a i e i a clea de a e fc e i c ce i . Y ill i ha de el e if ld j

a cha e .

I ha k agai he l gicia h hel ed e i 1980 1981. S ch i e ha a ed

ha i d e ee e e i hei a e , a hi igh ca e eedlec f i . B i all ca e I ha k H a d DeL g f hi de f l 1971 b k

ha i d ced e he bjec f l gic. I ca ill ad i e i e e e (e ce ha kee a i i d hile eadi g i ).

2011 : (1) The ad e f 8.0.1 ca e a da e f hef a e. I he b k e i i g e ha e bee c ec ed. I 7

E i ale a i d ced a d he ce hi b k a d f a e ill f e e l e$E i ale ai ai he i i g de a d c i e c i h he de el e f

h ee al ed l gic. (2) I ha k Richa d Gill ( a he a ical a i ic , Leide a d R alAcade f Scie ce ) f hi ki d e ie f he fi edi i , Gill (2008). (3) Al he b k D (3 d edi i 2011) ha bee da ed. (4) Ne a eE (2009) a he a ic ed ca i , a dC (2011)

ha e e d he a ad e b di i i b e al ead e i ed bel . (5)Reflec i g L gic i (2009): i i a de f l b k b i ld be efi f a

e ha R ell Pa ad i e l ed ( ee 128 129 bel ) ha i eade ee be eha he i l a ( a l fic i i ) hi ical e i de, ee C lig a (2010).

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Brief contents

27 1. A direct hands - on introduction to logic

45 2. Logic, theory and programs

67 3. Propositional logic

91 4. Predicate logic

135 5. Inference

167 6. Applications

179 7. Three - valued propositional logic

197 8. Brouwer and intuitionism

207 9. Proof theory and the Gödeliar

231 10. Notes on formalization

237 11. Reading notes

251 Conclusion

253 Literature

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Table of contents

5 Abstract

7 Preface

17 Part I. Overall introduction17 0.1 Conditions for using this book17 0.2 Structure of the book18 0.3 A guide19 0.4 Getting started21 0.5 For the beginner 21 0.6 For the advanced reader 21 0.6 .1 Aims of this book 21 0.6 .2 Summary22 0.6 .3 The paradoxes24 0.6 .4 A logic of exceptions24 0.6 .5 A textbook with news25 0.6 .6 Empirical base25 0.6 .7 How to proceed

27 1. A direct hands - on introduction to logic27 1.1 Learning by doing

27 1.2 Not27 1.2 .1 Language27 1.2 .2 Symbols28 1.2 .3 Switch model28 1.2 .4 Truthtable28 1.2 .5 Calculation29 1.3 And29 1.3 .1 Language29 1.3 .2 Symbols29 1.3 .3 Switch model

29 1.3 .4

Truthtable30 1.3 .5 Calculation30 1.4 Or 30 1.4 .1 Language30 1.4 .2 Symbols30 1.4 .3 Switch model31 1.4 .4 Truthtable31 1.4 .5 Calculation31 1.5 Implies31 1.5 .1 Language31 1.5 .2 Symbols32 1.5 .3 Switch model32 1.5 .4 Truthtable32 1.5 .5 Calculation

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32 1.6 Equivalent32 1.6 .1 Language33 1.6 .2 Symbols33 1.6 .3 Switch model33 1.6 .4 Truthtable33 1.6 .5 Calculation33 1.7 Xor 33 1.7 .1 Language33 1.7 .2 Symbols34 1.7 .3 Switch model34 1.7 .4 Truthtable34 1.7 .5 Calculation34 1.8 Unless34 1.8 .1 Language35 1.8 .2 Symbols35 1.8 .3 Switch model

35 1.8 .4 Truthtable35 1.8 .5 Calculation35 1.9 Tautologies and contradictons36 1.10 Testing statements38 1.11 A summary of what logic is38 1.12 You can directly use a main result40 1.13 Paradoxes, antinomies and vicious circles42 1.14 A note on this introduction

43 Part II. Two - valued logic

45 2. Logic, theory and programs45 2.1 Prerequisites45 2.1 .1 A prerequisite on notation46 2.1 .2 Logic and the methodology of science46 2.1 .3 Mathematica - also as a decision support environment47 2.1 .4 Use of The Economics Pack - Relation of logic to economics48 2.2 Logic environment in Mathematica48 2.2 .1 This book has been written in Mathematica

49 2.2 .2 Notation50 2.2 .3 Input and evaluation50 2.2 .4 Full form and display51 2.2 .5 Logical routines52 2.2 .6 Getting used to Mathematica53 2.3 The subject area of logic53 2.3 .1 Aims of this book 53 2.3 .2 The subject of logic53 2.3 .3 Statements versus predicates, statements versus inference58 2.3 .4 Propositions Htwo - valued logic Land sentences Hthree - valued logic L58 2.3 .5 Truth62 2.3 .6 Sense and meaning64 2.3 .7 Symbolics and formalism65 2.3 .8 Syntax, semantics and pragmatics

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65 2.3 .9 Axiomatics and other ways of proof 66 2.3 .10 Outline conclusions

67 3. Propositional logic67 3.1 Introduction

67 3.2 Sentences and propositions67 3.2 .1 Constants and variables67 3.2 .2 Englogish68 3.2 .3 Atomic sentences69 3.3 Propositional operators69 3.3 .1 Definition70 3.3 .2 Truthtables and truth value72 3.3 .3 Singularyoperators72 3.3 .4 Binary operations74 3.3 .5 A note on not - that you might not want to read75 3.4 Transformations75 3.4 .1 Evaluation75 3.4 .2 Algebraic structure76 3.4 .3 Disjunctive normal form77 3.4 .4 Enhancement of And and Or 77 3.5 Logical laws in propositional logic77 3.5 .1 Definition78 3.5 .2 Agreement and disagreement80 3.5 .3 Methods to prove something82 3.5 .4 Contradiction, contrary and subcontrary84 3.6 The axiomatic method84 3.6 .1 The system87 3.6 .2 A systemfor 87 3.6 .3 Information and inference89 3.6 .4 Axiomatics versus deduction in general

91 4. Predicate logic91 4.1 Introduction91 4.1 .1 Reasoning and the inner structure of statements91 4.1 .2 Order of the discussion

92 4.2 Predicates and sets92 4.2 .1 Notation of set theory95 4.2 .2 Predicate calculus and set theory97 4.2 .3 Universal and existential quantifiers

100 4.2 .4 Relation to propositional logic103 4.2 .5 Review of all notations105 4.3 Axiomatic developments106 4.4 Predicate environment in Mathematica106 4.4 .1 Notations110 4.4 .2 Quantifiers111 4.4 .3 Propositional form113 4.4 .4 Element and NotElement in Mathematica114 4.5 Theorem that the Liar has no truthvalue114 4.5 .1 The theorem and its proof

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117 4.5 .2 A short history of the Liar paradox126 4.6 A logic of exceptions126 4.6 .1 The concept of exception127 4.6 .2 The liar 128 4.6 .3 Selfreference in set theory130 4.6 .4 Still requiring three - valuedness132 4.6 .5 Exceptions in computers

135 5. Inference135 5.1 Introduction135 5.1 .1 Introduction136 5.1 .2 Validity136 5.2 Induction136 5.2 .1 Introduction137 5.2 .2 Mathematical induction or recursion138 5.2 .3 Empirical induction139 5.2 .4 Definition & Reality methodology140 5.3 Aristotle’ s syllogism140 5.3 .1 Introduction141 5.3 .2 The four basic figures142 5.3 .3 Application144 5.3 .4 Comments147 5.4 Taxonomy of inference147 5.4 .1 Proof and being decided149 5.4 .2 Provable and decidable151 5.4 .3 Consistency152 5.4 .4 Decidability and consistency152 5.4 .5 Semantic interpretation158 5.5 Inference with the axiomatic method158 5.5 .1 Introduction158 5.5 .2 Pattern recognition160 5.5 .3 Infer 161 5.5 .4 Axioms and metarules162 5.5 .5 InferenceMachine162 5.5 .6 The axiomatic method and EFSQ163 5.5 .7 Expansion subroutines

164 5.5 .8 Different forms164 5.5 .9 Accounting165 5.6 A note on inference

167 6. Applications167 6.1 Introduction167 6.2 Morals and deontic logic167 6.2 .1 Introduction169 6.2 .2 Setting values manually170 6.2 .3 Using SetDeontic171 6.2 .4 Objects and Q’ s with the same structure172 6.2 .5 Universe172 6.2 .6 The difference between Is and Ought

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174 6.3 Knowledge, probability and modal logic175 6.4 Applicationin general

177 Part III. Alternativesto two - valued logic

179 7. Three - valued propositional logic179 7.1 Introduction181 7.2 Propositional operators, revalued181 7.2 .1 Definition183 7.2 .2 Singularyoperators183 7.2 .3 Binary operators : And and Or 185 7.2 .4 Implies186 7.2 .5 Equivalent186 7.2 .6 TruthValue187 7.3 Laws of logic187 7.3 .1 Basic observations187 7.3 .2 Some conventions remain187 7.3 .3 Some conventions disappear - and new ones appear 188 7.3 .4 Linguistic traps189 7.3 .5 Transformations190 7.4 Interpretation193 7.5 Application to the Liar 193 7.5 .1 The problem revisited193 7.5 .2 The fundamental tautology195 7.5 .3 Conclusion

196 7.6 Turning it off

197 8. Brouwer and intuitionism197 8.1 Introduction198 8.2 Mathematics versus logic200 8.3 Russell’ s paradox201 8.4 Unreliability of logical principles205 8.5 Concluding

207 9. Proof theory and the Gödeliar 207 9.1 Introduction211 9.2 Rejection215 9.3 Discussion215 9.3 .1 In general215 9.3 .2 Ever bigger systems ?216 9.3 .3 A proper context for questions on consistency218 9.3 .4 Axiomatic method & empirical claim218 9.3 .5 A logic of exceptions219 9.3 .6 The real problem maybe psychology219 9.3 .7 Interpretation versus the real thing220 9.3 .8 Interesting fallacies220 9.3 .9 Smorynski 1977221 9.3 .10 DeLong 1971

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223 9.3 .11 Quine 1976223 9.3 .12 Intuitionism224 9.3 .13 Finsler 225 9.3 .14 A plea for a scientific attitude

231 10. Notes on formalization231 10.1 Introduction231 10.2 General points232 10.3 Liar 234 10.4 Predicates and set theory235 10.5 Intuitionism235 10.6 Proof theoryand the Gödeliar

237 11. Reading notes237 11.1 Introduction237 11.2 Prerequisites in mathematics237 11.3 Logical paradoxes in voting theory238 11.4 Cantor' s theorem on the power set238 11.4 .1 Introduction238 11.4 .2 Cantor' s theorem and his proof 239 11.4 .3 Rejection of this proof 240 11.5 Paradoxes by division by zero243 11.6 Non - standard analysis244 11.7 Gödel' s theorems244 11.7 .1 Gödel - Rosser

244 11.7 .2 Proof - consequentness244 11.7 .3 Method of proof 247 11.7 .4 Philosophy of science249 11.8 Scientific attitude revisited

251 Conclusion

253 Literature

16



Part I. Overall introduction

0.1 Conditions for using this book

The ba ic e i e e f i g hi b k i ha ha e a lea a dece high ch lle el f a he a ic a e illi g k ha le el al g he a . We a e

ha hi b k c ld be ed i he fi ea f a c llege i e i ed ca i .Y ca ead hi b k ,h al if d ha e . E e i he e i g a g a , ill ill be efi f he di c i .

Reade e he ecific f a f a e ad i ed check he a ia eb ec i h e, i ce h e a i ill be ed.

Ye , if ha e a d a he g a , he hi b k a e ha ha e ked i h f a fe da . Y be able ,

de a d i ha dli g f i a d , a d i he ba ic le . N e ha cl el f ll a da d a he a ical a i . The e a e e diffe e ce

i h c a i h gh i ce he c e e i e e ic i c i . N eal ha c e i h a e celle Hel f c i ha a f he ba ic

Ge i g S a ed a d k he ad a ced le el . The e a e al a b kha gi e a i d c i .

Whe a heLogic` a d Inference` g a , h ld al ha e aki g c f E , a lica i f , b he a e a h

i h he f a e d l adable f he i e e .

0.2 Structure of the book

Thi b k i f b h begi e a d ad a ced eade . The a e g ided b he e ec i eec i bel .

The b k ba icall ha :

1. The ba ic ele e : i i al e a , edica e a d e . 2. The ba ic i fe e ce i : i fe e ce, ll gi , a i a ic , f he .

17





0.4 Getting started

Whe a he g a he d he f ll i g.

E bec e f ll a ailable b he i gle c a d<<EconomicsÀll` . I i g d ac ice h e e e a fe e a a e c a d li e

be e c l he ki g e i e . Th ee li e ca be ad i ed i a ic la .

0.4.1 The first line

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fi .

0.4.2 The second line

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ld all l ad CleanSlate` af e ha e l aded e ke ackage ha ld a dele e. The Re e All c a d i a ea a callCleanSlate` .

Y ad i ed ec d li e i :

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Note that if you first load CleanSlate` and then the Economics Pack, then the ResetAll will clear the Pack from your workingenvironment, and thus also remove ResetAll. If you would happen to call ResetAll again after that, then the symbol will be regardedas a Global` symbol.

0.4.3 The third line

Af e he ab e, c ld e al a e Ec ic Pack fi d he li f ackage .

EconomicsPack

Selec he ackage f i e e , l ad i , a d i e iga e ha i ca d . F e a le:

Economics[Logic]

19



Y ca e i i g b a iPrint → False . Y ca call e ha eackage i e call. If a k a he ackage a d a clea hee f ea lie ackage , i l call Re e All fi . Thi al e e heIn[] a d

Out[] label .

Economics @ xi, …D h he c e f i` a d if eeded l ad he ackage HL.I i ca be S b l S i g i h i h back-a he. T e e a e c flic ,S b l a e fi e ed. Ec ic@ D d e

eed he C l`, Va ia ed` e c. efi e

Economics @All D a ig he S b-a ib e all i e i he Pack He ce e ackage L

EconomicsPack gi e he li 8di ec Ø ackage <Note: Economics[x, Out Ø True] puts out the full name of the context loaded.

Thi b k ill e ba icall he e ackage :

Economics[Logic, Logic`Deontic, Logic`SwitchGraphics, Logic`SetGraphics, LogicÀIOE, Probability]

Al he I fe e ce` ackage i ed b l l ad ha he e i i hea ia e cha e , i ce i e he I fe e ceMachi e hich l d l adi g.

0.4.4 Using the palettes

The Pack c e i h e ale e . The e ale e ha e a e a d c e hac e d he cha e i E i elf.

† The a e ale e i TheEc ic Pack. b a d i ide he c a d ab ea d all ickl call he he ale e g he g ide de he helf c i .

† The he ale e ha e TEP_ a a f hei a e, ha he ca ea il be

ec g i ed a bel gi g he Pack. The e TEP_ ale e c ai bl e b fl adi g he ele a ackage a d g e b f a i g c a d .

† The e ce i he e i TEP_A i e. b ha l deal i h he ackage faki g a diag a .

The l gic a d i fe e ce ale e a e a f he TEP_E ha ce e ale e.

0.4.5 All in one line

Y ca al l ad he Pack b he f ll i g i gle li e.<<EconomicsÀll`

Thi e al a e Need ["Ec ic `Pack`"] a d Ec ic [All], a d e he ale e . I

20



d e call Re e All, h e e .

0.5 For the beginner

The be i d c i i l gic i a d i g i . Cha e 1 ha bee i e ha ca a igh a a .

If hi k ha a e a ad a ced de i l gic, he h ld c i e i hhe e ec i 0.6. O he i e, if a e begi i g, h ld c i e i h Cha e

1 a d k a d i cl di g Cha e 6.

PM. Y h ld ac ice a l , a d e he e a le i hi b k ha each abhe diffe e e ie f he a i l gical che e . O l af e a d , a d l if

a e illi g ceed he ad a ced le el, he c ld a eadi g he eec i 0.6, a d he c i e i h Cha e 7 a d . Kee i i d ha c ld

l k i he e b k l gic i he lib a , abl DeL g (1971), i ce i ca be f ll e cl ded ha e cha e he e e e e k ledge ha ha beef ll a ed he e. I i g d ead h e he e b k , i ce eadi g he e ill cla if

ha h e he b k he ble hile .

Si ce a i g a a e ided, ca ha e a ha de e ie ce, a d hi ill all be e de a d he i e .

OK, hi i he i he e i ch Cha e 1.

0.6 For the advanced reader

0.6.1 Aims of this book

Thi a ag a h i l e i d f he ai e he fi age f he b k.

0.6.2 Summary

The b k i a e b k ha gi e a i d c i ele e a l gic a d defi e hei f a .

I i clai ed ha he Lia i l ed a d ha he The e f G del a e i leadi g hehighe deg ee, i.e. a e l half h .

Fi a di i c i i ade be ee i le a d c le Lia . The i le Lia i l ed b al ed l gic, he c le Lia b h ee al ed l gic. I i h ha a da dl gic i d ce h ee al ed l gic i a ece a a . The a ach f R ell a d Ta kii ejec ed a ece a f a l gical i f ie a d c lica ed be ef l.

21



Al , h ee al ed l gic ca deal i h he called e g he ed Lia f h ee al edl gic. Ha i g e abli hed hi , he h ee al ed l gic i a lied he a ad fR ell, he l gic f B e , a d he he e f G del.

Sec dl , he , h ee al ed l gic a ea be behi d e he . Se ical e ca be

defi ed i al ed l gic b if e all f eed f defi i i he e icale l eed be ca al g ed a ch.

Thi dl , He i g a i a i a i f B e e f i fe e ce a ea bei ade a e.

F hl . The he e f G del a ea be e b he e f he Lia f fhe , he e called he G delia , Thi a e e i able . Whe hi G delia i

e e iall a e ical a a he Lia , he e he e a ea be b ild

e e. Di i g i hed a e eak a d g e a d i i l b eak e haG del e d e h he c adic i ha i behi d he f. Whe e eade a el g e he he e e ca be deal i h b h ee al ed l gic. I

a ic la , hile G del li i elf efe e ce l a e e , (1) e ca al all haa e e efe he e a a h le a d (2) e ca acce a a i (( ¢ ) ( ¢

( ¢ ))). The G delia c lla e he Lia a d he he e di a ea . He ce G dele a a he a ical a g e ha hi he e ld h ld f a e i i alid,

he j h ld f hi eak a i .

Fif hl , he b k c ai hi ical a d hil hical c e ele a f hea g e .

Si hl , all e i he a g e a e backed b f al e i Cha e 10.E al a i i he a g e b f c e ha e li i ed f al

a .

Se e hl , he b k ad i e he f eed f h gh a d le a ce f he i i fhe . D g a i a e he e ce i ha e ha le.

0.6.3 The paradoxes

The a e f he a he a ad e a d G del he e i a he di a i i g.The e i a g ea di e i f i i ha he e l i a ache a e. A d,

he e he e ee be ag ee e , e ac l he e a he d bi a ache a ee b aced. The ke e a le ill a e hi a e he he e f G del (1931). The e

he e a e e i h he e f a eak a ia f he Lia a ad a d he ld

h ld i ce hi a ia i a e ele a he Lia i elf, i.e. i h a T e Fal eal e ele a f e i ical a lica i . I i l beca e f eak a i , e el

a i , ha he G delia ee e. Tha he la e i i i ded eali ed

22



h e clea l f he ide ead a ecia i f h e e l , a i i e ed b he f ll i g e :

Mali (1979): (...) G del fa i c le e e he e f 1931, e f heile e f a he a ic ( 60) a d (...) e f he a di g e l f e ie h

ce a he a ic , f hi he e ca ed a f d al e a i i he ie heldab a he a ic a d cie ce i ge e al. N l ge ca a he a ic be h gh f aa ideali ed cie ce ha ca be f ali ed i g elf e ide a i a d le fi fe e ce i ch a a ha all hi g e a e able. ( 128)

K eeb e (1963): (...) i a e e b i ci i e a e , di i g i hed alike b a i hi gigi ali , f di f c ce i , a d a e f i ica e de ail, G del ca ied

e a a he a ic e a a i gle ide i i ec d a d e eflec i e ha e.

( 229)Va Heije (1967): G del a e i edia el a ac ed he i e e f l giciaa d, al h gh i ca ed e e a i e, i e l e e idelacce ed. ( 594) a d The e i e b a ch f e ea ch, e ce e ha i i i i ,

ha ha bee e aded b hi i fl e ce. ( 595)

DeL g (1971): (...) he j ifiabl fa e l f c e a l gic: G delfi a d ec d i c le e e he e . ( 160) a d G ea achie e e i cie ceha e f e e l i i ed hil hic h gh , a ca be ee i he k f Ne ,Da i , Ei ei . The hil hic i ac f e a he i like i e f d (...)( 191)

Da i (1965), Thi e a kable a e i l a i ellec al hall a k, b i ii e i h a cla i a d ig ha ake i a lea e ead. ( 4)

I hei la i a i f G del he e , Nagel & Ne a (1975) eak ab ai ellec al h . While i i i e.

I c a he e ie i i ef l c ide a e da e a ach l e hea ad e f he Lia a d G delia . The f ll i g able gi e a che e f a ad icalela i ha e l i c adic i , l he l i

a ache . The diffe e ce be ee he i le a d c le a ia f a a ad ie i el ch l gical. I he i le a ia e ca di ec l ide if he elf efe e cea d he la h he e ali ig . I he c le a ia i ake a fe e ide if

he elf efe e ce. I al ed l gic e ca i l f bid hi ki d f ega i eelf efe e ce, i h ee al ed l gic e ca decla e ch i a ce be e ele , i.e.

i h h al e T e Fal e, a d e he hi d al e I de e i a e. B h e h dac all a l each cell f he able. Ye , gi e ha he c le a ia ha beec ea ed eci el all f hi ki d f elf efe e ce, i i e ef l a cia e

23



al ed l gic i h he i le a ia a d h ee al ed l gic i h he c le a ia .

- -

= << i fal e >> The e e ce i hi bl ck i fal e.

= << i able >> N @GD i he be f he

a e e ha a ha he a e e

i h N @GDi able.

The a ach i hi able clea l c a i h he ea lie e . N l i he Lial ed b G del he e a e diag ed a c e e ce f a c i led Lia i a eak

e f c di i , aki g he half h fallacie .

Gi e ha c a , he eade i f c e i i ed d he a g e i hi b ki h ca e.

0.6.4 A logic of exceptions

Al h gh he di c i bel igi a ed f a d i l gic a d a a al i f heLia , i ha a ea ed e e e d i i h di c i he a ad f R ell, he

he ie f B e , a d he he e f G del. The ea bei g ha he e ic

a ea be ef l a lica i f ha .

0.6.5 A textbook with news

A c f a f e b k i ha he a e acce ed k ledge. Ne ielega ed he j al , a d i a e c la e he e a fe dece ia bef e i i k d

i he e b k . Thi e b k c ai acce ed k ledge a d he c i ei h he e . The e i ha i ha a ach i ce he e f ll l gicall fh e ba ic . The ai i be a a e f i ha he h le de el e i e e ed

a a e b k i deed. The ice ca a f he ba ic a d k h ee al edl gic a d f he .

The e b k i ba ed a 500 age d af e b k f 1980 i D ch a d a 100 aged af a f he ai e l f 1981 i E gli h. The a g e c ld be ecalled

i h l a li le e eadi g ha li le had bee f g e . The bjec ha beea ached h ll f e h agai , agai b ildi g he de el e f he ice

i f ie . The ba i ha bee ha E gli h a , eglec i g l f age i

D ch. Thi ced e ki he eed f e ac a la i a d c i g. Wi h hea ailable a a e ial , e i E gli h ha e bee c ea ed af e h a d h e h ld

e be didac icall clea . Hi ical e f c e eeded be ed e . The

24



li e f a g e a i f 1980 1981 ha bee cha ged h e e . The fi al ee e ed he e ha bee ce ed e ha e i e (i cl di g he 1980 1981

e i d), ha e h ld h e f e didac ic cla i .

Al , he a h ha f ll ed he li e a e i ce 1980 1981. Ye i i i e i able ha

he e ill be e de el e i he li e a e i ce he , hich hi b k cae . Thi i ac all ca e f ala , i ce, had he e ea che i l gic a dhil h i he la 25 ea e l ed he Lia a d G delia a ad e (i he a ef hi b k) he el hi ld ha e c e ge e al a e i a d al hi a h

ld ha e iced j f eadi g he e a e . Gi e he a a e ha a ef ag a i i he e bjec he a g e i e 25 ea ag i ill a f e h a i ca

be.

0.6.6 Empirical base

The b k cl e i h e e f ali a i . The a h i a ec e iciah e i e i ed ca i i cl ded he ba ic a h c e al f ll ed b def a h a d h ic a he U i e i f G i ge . A field e ai ec e ic , I

d ad i f ha i g died e l gic a d hil h beca e f bei g ca gh b ef da e al i e , b I d ha e he i e he i e e f ali e i all. The

d hil h i ha he che e k ha he ca be f ali ed if de i ed.

Wha i e f hi a h a e he g a i i e d i g he h lidaea f Ch i a 2006. Th e ea l he b d f he e . The ea lie e if E f he id 1990 al ead c ai ed i le i i al l gic a d

i fe e ce b he e ha e bee i e e a ded, f hich, gi e he able ab e, he al ac ical i le e a i f h ee al ed l gic f c e igh be j dged

didac icall c cial.

0.6.7 How to proceed

Of c e, ce ha e ked elf Cha e 9 a d 10, a d he Readi gN e i Cha e 11, he l , he ad a ced eade , ill l be efi f he

ai i ellec al e l f hi b k. Thi a gi e a f da i a d j ifica i hae ell he begi i g de . Y igh fi d ha hi e l igh be challe gi g.

The a g e i he i d c a a l k ea i le, b ha i j heh a i g ha I ch e, a d he a g e i i e ab ac . The acce ible h a i g akeha he a g e ca be f ll ed b he a e age de , b , i e e ie ce, i ake

a e ab ac l ai ed i d eall de a d he a g e . The e ill i al ei a g d ed ca i .

25



26



1. A direct hands-on introduction to logic

1.1 Learning by doing

The be i d c i i l gic i d i .

Y ha e bee fa ilia i h l gic i ce bi h b i ca be e ci i g l k a i i a e

a . The e a ic a . Thi ill gi e a e f ded c i ha ca a le .

The a i a i a d i e ill be e lai ed i e de ail i he e ai de fhe b k. The idea f hi a i ha fi de a d ha ha e . Y ca

d ha i h e c fide ce. The i a S ic l gicia Ch i (280 207 B.C.)al ead e ed ha e e e d g a e ca able f ea i g. He a a d g cha i gaf e a , c i g a a f k i h h ee a , iffi g a d, i h iffi g,

aci g he e ai i g e.

Le ake a l k a N , A d, O , I lie , E i ale ce, X , U le , a l gie a dc adic i .

1.2 Not

1.2.1 Language

C ide he e e ce J h l e Ma a d he e e ce J h d e l e Ma .

N ice a hi g a ic la ? Well, he e e ce a e i e . O e e e ce i hede ial a d ega i f he he . T ge he he f a ai ha gi e all ha i ible.If e e gge ha he e i a iddle a ch ha J h c ld l e Ma a li le bi de e c di i he hi ill ea he ega i f J h l e Ma i

ha he d e c le el l e he a he a e e e e e .

1.2.2 Symbols

We ill e he b l N e e hi i i . Th if = J h l e Mahe N [ ] = J h d e l e Ma . We igh ha e defi ed he he a

a d b ce e a ig e f i ch e he e k ha N [ ] a d f .

27



1.2.3 Switch model

Elec ical i che a e a del f l gic. The ba ic i ch la c i f hef ll i g ele e ( ee he d a i g bel ):

1. A elec ical c e fl f he ega i e le ( he lef ) a i i e le (

he igh ).

2. The e i a la ha ill b he he e i a elec ic c e a d ha ill b he he e i c e . (I i a g d la .)

3. The e i a i ch ha de e i e he he he e i a c e ha he la ill b . If he i ch i he he i e i cl ed a d he la ill b . If he

i ch i ff he he i e i e a d he la ill b .

Th he i ch i e ac l c e he la b , a d e ca e l i hi f a

del f l gic.

† The e e g a h h he ff i i , h N [ ]. Thi ill lea e f ee i e e e i a i i e ega i e a e , i a a ha c cei e f he

ld ( h = J e i ld = J e i g ).BasicSwitch @pD

- Lamp + p

1.2.4 Truthtable

We ca e lai N al b ea f a h able . Thi able li all he a e f held ha a e ible a d he gi e he c e e ce f each e a a e a e. The

ible a e f a e e a e T e Fal e. The la b i d e b .

We a e fa i de a di g N ha e i d ce a ecial b l f i .N [ ] ill be de ed i ce hi ake f ea ie eadi g.

† N [T e] gi e Fal e, a d N [Fal e] gi e T e.TruthTable @Not @pDD

p ŸTrue FalseFalse True

1.2.5 Calculation

A fi al a de a d N i e a i h e ic. Whe e h al e 1 a d 0a d he hi k f a ha i g l al e 1 0 he N [ ] gi e 1 .

28



† T h a d Fal eh d ca be a f ed i 1 0 b ea f he f c iB( a ed af e a i a l gicia ).

?Boole

Boole @expr D yields 1 if expr is True and 0 if it is False.

† B lea al e 1 a d 0 ca be a f ed back i T e a d Fal e b i ge a i . Whe he al e 1 e e e h he a e i g ea ha == 0.

ÿ p êêToAlgebra

1 - p 1

1.3 And

1.3.1 Language

C ide he e e ce = J h l e Ma , = J h li e i L d a d = J hl e Ma a d J h li e i L d . Agai , d ice a hi g a ic la ? OK,

he e a be e i b e e b ha h ld bec e a a e f i he da d , called .The e e ce ca be ee a .The c d e e ce

i l e if b h i c i e a d a e e.

1.3.2 Symbols

We ill e he b l A d ea e e e he c j c i . Th e ca i e= .

1.3.3 Switch model

† Thi elec ical ci c i e i e ha he i che a e bef e he la b .Thi del .

AndSwitch @p, q D

- Lamp + p q

1.3.4 Truthtable

The h able kee ack f h ee a e e . The e a e a e f he ld f , a el , . The e a e a e f he ld f, a el , . Wha ha e

i h de e d he 2 2 = 4 c bi a i , a el , , , , . The ligh l b he b h i che a e , h , .

29



† The a e e i l e if b h b a e e a e e.SquareTruthTable @p Ï qD

H p Ï qL q Ÿq p True False

Ÿ p False False

1.3.5 Calculation

Whe e ake he B lea al e 1 0 f a d he i he a e a Mi [ , ] ==1, he i i f b h.

† Wi h al e 1 0 e ca al e l i lica i .p Ï q êêToAlgebra

p q 1

1.4 Or

1.4.1 Language

Whe i i i e he x2 i i i e. Whe i ega i e he x2 i i i e a ell. Ifac , x2 i i i e f all e ce 0. The l i f x2> 0 ca be gi e a i i i e

i ega i e a d he ill be called .

1.4.2 Symbols

We ill e he b l O ea e e e he di j c i . Th e ca i e . The c e f La i el = . The f O i i ed he ide f

A d.

1.4.3 Switch model

† I hi elec ical ci c i l e i ch eed be bef e he la b . Thidel .

OrSwitch @p, q D

- Lamp + p

q

30



1.4.4 Truthtable

ill be Fal e l he he a e f he ld i , .

† The a e e i l e if a b a e e i e.SquareTruthTable @p Í qD

H p Í qL q Ÿq p True True

Ÿ p True False

1.4.5 Calculation

Whe e ake he B lea al e 1 0 f a d he i he a e a Ma [ , ] ==1, he a i f b h.

† Algeb a h a de e de c be ee N , A d a d O , i ce i he a e a ( ).

p Í q êêToAlgebra

1 - H1 - pL H1 - qL 1

% êêSimplify

p + q p q + 1

The algeb a e ifie ha i he a e a ( ), b e ca al e heh able , a d e ca al a e back la g age: J h ea ie Cha le

a che ele i i i he a e a I i he ca e ha J h d e ea a ie a d haCha le d e a ch ele i i .

1.5 Implies

1.5.1 Language

The e e ce If i ai he he ee a e e c ai he If ... he ... c di i ala e e called . The fi a i called he a d he ec d a he

.N e ha he ee ca be e beca e he ci ee clea e a ed b .The e e ha ake a i lica i fal e i he i ai a d he ee a e e . Iall he ca e he i lica i i e. A i a i a i i he i d e ai :

he he i lica i ill i e (a d i ha a h he ical a e).

1.5.2 Symbols

The i lica i i b li ed i h I lie ea e . Th e ca i e I lie [ , ] .

31



1.5.3 Switch model

The i ch del f i ac all he i ch del f ( ) i ce he e a ee i ale . We ca h hi i h a h able i a calc la i . We ca al ee hi ila g age, he e ab e a e e ai i e i ale I d e ai he ee

a e e . Recall he age ld li e Y e life .

1.5.4 Truthtable

† The a e e i l fal e if i e a d i fal e.SquareTruthTable @p qD

H p qL q Ÿq

p True FalseŸ p True True

† A ick i b i e f a d ee ha hi gi e he able f O . NB. The affi e he cl e i .

SquareTruthTable @ÿ p qD

HŸ p qL q Ÿq p True True

Ÿ p True False

1.5.5 Calculation

I lie ca be e e e ed i h § . B l i alg i h f i e ali ie a ediffe e f h e f e ali ie . A a la i i e ali ie i i g he e i ale ce

i h ( i g ha he affi e he cl e a iable).

p q êêToAlgebra

q p - p + 1 1

% êêSimplify

p q p

1.6 Equivalent

1.6.1 Language

Ab e, e al ead ha e bee a i g ha e e e i a e i ale a he . Be i ale ce e ea ha he e i e if a d l if he he i e. Thi ... if a d

l if ... ca al be e e ed b ... iff ... . The e i ale ce al c e e he b h ide a e fal e.

32



1.6.2 Symbols

E i ale [ , ] ill be e e ed i h ñ . Thi e e e ha a e i ale ce h ldhe b h a d h ld.

1.6.3 Switch model

Ac all he elec ical i ch f N al ead elie e i ale ce . The ligh biff he i ch i .

1.6.4 Truthtable

† The a e e ñ i l e if b h a iable ha e he a e h al e. I hica e e e he f ll e e i E i ale i ce he b lic f añ i

l defi ed f . N e he T e he diag al f e lef l e igh ,a d he Fal e el e he e.

SquareTruthTable @Equivalent @p, q DD p q q Ÿq

p True FalseŸ p False True

1.6.5 Calculation

E i ale ce i ea l e e ed b == .NB. Thi ha bee g a ed b ca beded ced.

ToAlgebra @p ~ Equivalent ~ q , All Æ True D

8H p q - p + 1L H p q - q + 1L 1, H p 1 Í p 0L, Hq 1 Í q 0L<Reduce @% , 8p, q <D

HH p 0 Ï q 0L Í H p 1 Ï q 1LL

1.7 Xor

1.7.1 Language

The e e ce Ei he I a a h e igh I ll gi e a call g e he ec ai he ei he ... ... ha e e e ha he c e e e ce a e

all e cl i e. The a e e ca be b h e b h fal e.

1.7.2 Symbols

The b l f ei he ... ... i ba ed O , called X ea e. Th e ldi e X [ , ] .

33



1.7.3 Switch model

The X elec ical ci c i i called he h el ci c i . I i ed al g ai ca e ha ca c l he ligh f d ai a ell a f ai , ha d

ha e cli b he ai j he la ff. The ligh b l he a

i gle i ch i ha he he i ff. Thi del . PM. I he g a h, he li ei he iddle c b d ch. Al , he a i ch i e , he i c ec he

e i e. The d a i g h a all ga he i e igh hi k ha he e i i ch b a e a e c ec i .

† The ei he .... ... i ch.XorSwitch @p, q D

- Lamp + p q

1.7.4 Truthtable

† N e he Fal e he diag al f e lef l e igh , a d he T e el e he e.X i he ega i f E i ale . Th X [ , ] i e i ale a i g ha ( ñ ).

SquareTruthTable @Xor @p, q DD p q q Ÿq

p False TrueŸ p True False

1.7.5 Calculation

X [ , ] ca ea l be e e e ed a ∫ ided ha a d a e 0 1. Ye , l i gi e ali ie i diffe e f l i g e ali ie , e ca be e e ã 1 ,

de a di g ha each a iable i 1 0.

ToAlgebra @Xor @p, q DD p + q 1

1.8 Unless

1.8.1 Language

The e e ce I ll ake a a le, le a , f he I ll ake a big ch c la e f dge

a e he al i a i , a a le, le a c di i a lie .

34



1.8.2 Symbols

We ca e U le [ , , ]. Whe he e i c e e ce he U le i a f f O .

1.8.3 Switch model

The e i a ic la i ch del, e ce he ai i ch d i he ba e e ff he h le e i ca e e hi g i g.

1.8.4 Truthtable

† Wi h U le i j O . A S gU le [ , ] i ( ñ ).SquareTruthTable @p ~ Unless ~ qD

HŸq pL q Ÿq p True True

Ÿ p True False

† U le k be i h h ee a iable . J like e i ale ce i a ea be ac j c i f I lie e . Beca e f he hi d a iable he a e h able

i diffe e l a d hide he , SquareTruthTable @Unless @p, q, r DD

HHŸ q pL Ï Hq r LLq Ÿq

p 8True, False < 8True, True <Ÿ p 8True, False < 8False, False <

1.8.5 Calculation

The calc la i che e i j he c bi a i f he ea lie l gical b l . Thi a be e f he ea h c b k l gic e d f ge ab U le , e e

h gh i la a i a le i l gic a e hall ee bel .

ToAlgebra @Unless @p, q, r DD

H p H1 - qL+ qL Hr q - q + 1L 1

1.9 Tautologies and contradictons

I he e diag a he e a e l e i he e e c ea ed a i ch. The ee e e c i ge a e e ha ca be e fal e. All he i a e

.

† All i i he i e i elf ca be ee a i che ha a e al a cl ed, a d h aa e e ha a e al a e . The e a e e ill be called .

35



† All i d a ca be ee a i che ha a e al a e , a d h aa e e ha a e al a fal e . The e a e e ill be called .

Whe e b ild a elec ical i chb a d ch ha he la al a b ha e ei ch e fli , he e igh a ell i a di ec c ec i g i e. Whe e b ild ai chb a d ch ha he la e e b ha e e c bi a i f i che e ,

he e igh a ell ake a a all i che a d i e .

† Thi i a a l g (al a e). T hTableF ead f igh lef .TruthTableForm @p Í ÿ pD

Or p Not p

True True FalseTrue

True FalseTrueFalse

† Thi i a c adic i i i al a fal e.TruthTableForm @p Ï ÿ pD

And p Not p

False True FalseTrue

False FalseTrueFalse

1.10 Testing statements

C bi a i f l gical e a ca be e ed b he h able , b l i g algeb aice a i b calli g a l e like L gicalE a d Si lif . I i ible c c all ki d f elec ical ci c i ha i ic c le i i , a d de e i e

hei h al e . I fac hi i ha ha e i c e . The elec ical i che

h h e e a e l ic e a d he e i e ca be c bi ed c ea e a i chb a d.

Calli g a l e di ec l i he a e b i hide all he i e edia e e . Whe a e begi i g i h he e a ic d f l gic he ld be i e e ed i

h e e check he . The h able e h d he i he be a ach. The ab eh able ed a a e f a b he a e e ha e a diffe e be f

a iable he i i be e e a f ll able f , he e all a e f he ld a e ie a a e bl ck , a d he e igh check he e i he e al a i .

We aid ha X i he ega i f e i ale . Le check ha .

36



† Thi i he a e e ha e ill check.try = Equivalent @Xor @p, q D, ÿ Equivalent @p, q DDŸH p qL p q

† Thi i he c e f he a e e .TreeForm @try D

Equivalent

Not

Equivalent

p q

Xor

p q

† Thi i he h able i e e i e f , ha ead f igh lef . Each bl ck gi ehe T e Fal e c bi a i f he a iable a d he e a ha he cc i .

The lef c l h l T e, ha he e i ale ce ha e e ed i al a

e: a a l g .TruthTableForm @try D

Equivalent NotEquivalent p q

Xor pq

True FalseTrue True True

FalseTrueTrue

TrueTrueFalse True False

TrueTrueFalse

TrueTrueFalse False True

TrueFalseTrue

True FalseTrue False False

FalseFalseFalse

37



1.11 A summary of what logic is

A i g ha i f a i i e e ed i e e ce , he e ca b e e ha e

e e i a e al a e, he al a fal e, a d agai he e i e e a de i e fal e. We ca al b e e ha he e f al a bei g e al a

bei g fal e de e d e i el he c e f he e e ce, he ac al a ef he ld. U i g he c ce f l gical ece i (a d h h ical ece i ) e

fi d he f ll i g di i c i :

1. Se e ce ha a e al a e = la f l gic = a l gie = ece a il e

2. Se e ce ha a e al a fal e = a i la f l gic = c adic i = ece a ilfal e

3. Se e ce ha a e e i e e a d e i e fal e = c i ge l e fal e

The bjec f l gic c i f i e iga i g he ded c i a e alid, he he a ei alid, a d he he a e i c cl i e. Pa f ha e i c ce de e i i g

he a e e a e ece a il e, ece a il fal e, c i ge b hei ec e a d hei ela i i fe e ce i elf.

1.12 You can directly use a main result

Si ce i ea e, ca di ec l e a ai i e a d e l i hi b k. Tha ai e l i Decide[ e ] ha ca e ac e c cl i ( he e ). A

all e a le gi e a di ec i d c i i he l gical i e a d i h h ca a l he i e . (See Ge i g S a ed i he e begi i g fi if eall

a he g a .) The i e Decide k l f i i al l gic, a dhe e be f la ed i E gl gi h , a i lified E gli h like la g age.

E gl gi h a e e ill ca ha e e ca e le e a d c a i . Nega i i l

ec g i ed he N a ea a he begi i g f a e e ce. E gl gi h al e If... he ... i ead f he e eda ic I lie .

The f ll i g e a le e he i fe e ce che e called . Whe i f a i e i , he e i fe a he e . The e i ha did

cc a a e a a e e i he li a d ha e h e e a a e f.

Decide @ x, opts D e f C cl de@Ded ce @C cl de@D , DD. Theced e a a icall A dO E ha ce

Englogish will be discussed below in more detail, and similarly the procedure of “AndOrEnhance”.

38



† The e ac i f a deci i a be ible b Decide.Decide["If it rains then the streets are wet. It rains."]

AndOrEnhance:: State : Enhanced use of And & Or is set to be True

8the streets are wet <

† Recall: he b l e e e N , ha N [ ] e e e he ega i f .Decide["If the apple falls then it hits my head. The apple falls. If it hits my head and I am Newton thenI invent gravity. If I invent gravity then I am Newton. Not I am Newton."]

8Ÿ i invent gravity, it hits my head <

† Whe e If ... he ... , , a d a d a e a he ca ef eig la g age .

Decide @"If il pleut then je t’aime. Il pleut." D

8 je t'aime <

† S c a e d e a ee Pla le Pla a ee hi . Pla d e a ee S c a e if S c a e a ee hi . Pla a ee S c a e if S c a e

d e a ee hi . Well, fi check ha ea :(Not["Socrates wants to see Plato"] ~Unless~ "Plato wants to see Socrates")

HŸ Plato wants to see Socrates Ÿ Socrates wants to see Plato L"If not Plato wants to see Socrates then not Socrates wants to see Plato. If Socrates wants to seePlato then not Plato wants to see Socrates. If not Socrates wants to see Plato then Plato wants to seeSocrates." // Decide

8Ÿ socrates wants to see plato, plato wants to see socrates <

† M ch ea i g elie c e fac al . The be a deal i h he e i hee i e f heLogic` ackage i alif ha e e . The f ll i gi a alid a g e , a ed c i ad ab d i.e. f b c adic i , hec e fac al like The ee ld be e , ld he , if i ai ed ? :

"Suppose that it rains. If suppose that it rains and our suppositions fit reality then it rains. If it rainsthen it is wet. Not it is wet." // Decide

8Ÿ it rains, Ÿ our suppositions fit reality <

† T ff he e ha ce e f A d a d O . PM. Thi ill be e lai ed bel .AndOrEnhance[False]

AndOrEnhance:: State : Enhanced use of And & Or is set to be False

39



1.13 Paradoxes, antinomies and vicious circles

Ne he bea f b lic l gic, e he i a i d l gic a e he

a ad e , a i ie a d ici ci cle ha a i e b h i dail life a d i cie ce.S e a e e ha 2000 ea ld a d he ac all hel ed i la i g a ki d de el de l gic, a he a ic a d cie ce.

Rega d he f ll i g a e e :

1. D l k he e !

2. Be a e !

3. Ge e ali a i a e bad b he el e . Y j h ld ake bad

ge e ali a i . Thi i f c e a ge e ali a i . 4. A eache , e da : 2 a d 2 i 4 , a d a he da : 2 a d 2 i 22 .

5. I lie. (The Lia a ad , G eek e d e ai . If I lie i e, he I lie. Whe Ilie i a lie, he I eak he h.) Va ia a e Thi e e ce i e Thii a c adic i .

6. S c a e : Wha Pla a i e. Pla : Wha S c a e a i e. Thed a a hi g el e.

7. She a ched ele i i a d i ed he ai i l gicall he a e a She i edhe ai a d a ched ele i i .

. H ele l i l e b c f ed: I ca l hi ki g ab b hi ki gha I d a hi k ab a e, b he I ill hi k ab ; lee hi k ha I d a hi k ab .

. Whe he e a e a e ci cle , he h ca I hi k a d a ?

10. A he de e i e ha e b e e e i icall . B a e he a d e i iced ?

11. Whe al le, h ible hi g ha e .

12. Ce a e , D Q ich e , he a age The B igde: I he La d f he J , ala i a ed ha i i i g a ge ha e a e he ea f hei i i , a d lia

ill be ha ged. The b de f he La d f he J i gi e b a i e , a d e he b idge ac i a C i e abli hed j dge i i he e la . O e da a

a ge a i e h a e : I ha e c e be ha ged.

13. S e i e ca be de c ibed b i ha a ade a e de c i i i elf ?

14. A ca al g e li a i ca e . S e i e he ca al g e ill e i i elf,e i e . Wha ab aki g a ca al g e f all ca al g e ha d e i

he el e ? Will ha e ca al g e e i i elf ? (Be a d R ell e

40



a ad .)

15. A age ld f he c c dile: A c c dile ha ake a child a d ha beeh ed d b he fa he f he child. The bea eac : I i e hechild , if e l edic he he I ill e he child .

The fa he , h i g hi i : Y ill e child e.

16. Thi e e ce c e ac l i d i e, a d h Thi e e ce d e c e ac l i d i fal e.

17. The ie f l gic a e al a c i e . Thi i e a f ll . The a e ei hec i e i c i e . Whe he a e c i e , he he a e c i e . Whe

he a e i c i e , he he ca e a hi g, al ha he a e c i e . I b h ca e he a e c i e .

1 . A ble f l gic a e he e ical e a ce f idi a d li icia . B hee idi ( eb d h d e k ab hi g ha a e ) a i d ced

igi all b he G eek f h e h did a ici a e i he di c i hea age e f he a e, i.e. h e h did a ici a e i li ic .

1 .

StringLength["The number of Berry is the smallest natural number that cannot \be defined by at most 108 numbers or letters."]

108

S e f he a ad e ab e a e i he eal f ag a ic ch l g , ee Wa la icke al. (1967). I ha ca e he f c i he ch l gical a ec f a ad icalc ica i , abl c le a d ibl c adic i a i f a i , hae e igh c ib e e al di de . Lege d ha i ha e Phile a f C e edied a a e l f hi ha d hi ki g he Lia Pa ad . I ha e ec he eade ha bee a ed. Wa la ick e al. ide d a a ic eadi g, i a ic la i ce he de ha i e he l gical l abili f a i i e .

H e e , all he e diffe e field a e ch f a i le b k l gic a d h iill be i e li i he field f a lica i .

PM 1. The l gicia , e a d g a a ia Phile a f C (ca. 340 285 BC) i e ed

ha e aid: T a elle , I a Phile a ; he a g e called he Lia a d dee c gi a i b

igh , b gh e dea h. A a e l eaki g i he af e life. See B che ki (1970:131)

a d Be h (1959:493).

PM 2. H ghe & B ech (1979) gi e a a h l g f a ad e .

PM 3. Ma e le e a ad f c adic i b he e i e e a i i a

ee i g c adic i , ea i g ha i i e e a a c adic i a fi hile

cl e e a i a i cla ifie e hidde l gical c c i . F he G eek d a

( ea i g, e ec a i ) a d a a ( e , be ide), h like bei g be ide he i ( ha i

41



e ec ed). The d a i c e f (la , c ) a d a i (agai ).

See C (1963).

1.14 A note on this introduction

Ma E a i h l ge e bef e he i d ce hel gical e a . Ab e e ha e e e ed he a he e begi i g. The idea i ha

eade a e fa ilia i h l gic i ce bi h ha he ca ea il g a b lic l gici h ch ad . L ge i d c e e d b e a d he kill i e e bb c i g a c e c ib i f b lic l gic. B begi i g i h he b lice e e a i , eade ill ee he , I lea e hi g a d ill g i e e ed

k e ab he bjec . Wha i e eciall ice ab he l gical e a i ha

he ha e del i a i e e e a i (la g age, b l , ci c i , able ,e a i ) a d ha he c bi e i i i g a . S b lic l gic i a ge f bea ,a d, he f f i i ha i g ea l e ha ce ea i g e .

Tha bei g aid, he f ll i g a idabl c i e i h l ge e . The ab e ha bee a ick i d c i b e ha e bee l a d ha e bee a i g all ki d f

hi g . Le ec ide ha e ha e bee d i g a d le a ach hea e i a e e a ic a e .

42



Part II. Two-valued logic

The fi e cha e i hi Pa a e al ed l gic. Se e ce ca l be T e Fal e. I Pa III e c ide e al e a i e a ache .

43



44



2. Logic, theory and programs

2.1 Prerequisites

2.1.1 A prerequisite on notation

Si gle e ide if a c ce a d d ble e ( a i a k ) ide if a i gf b l . Th e i he c ce f e , a d e i j he i g f

b l , , , e ce e a. The e i i di i g i h clea l be ee hei g (a e i g) a d he e i i g (li i g, i g) f a e e ce a he a cie

G eek al ead ed h a a igh ead al d a eech i h k i g ha iea . S c a e c a ed ea i g a eaki g e elf, i e l, a d ha

e ld lie e elf. I hi a e a a e i g i eaki g , clai i g hf ha i e ed. The f ll i g c e i he i ef l: (a) he a e e ce i

a e ed ed he i cc i h a i a k , (b) he a e e ce i e elefe ed he he e e ce ill be be ee a i a k . I ha ca e he i gf b l ha f he e e ce i ega ded a a bjec f i elf. Le e i alic

(e.g. ) de e a iable f e e ce . Th i j he le e a d i a e e cech a I ai . . We ca i e = I ai . a ig a c a i g f b l a

a iable; = A e [ I ai . ] he i he i i ha i ai ( i h i i ala iable a d c a ).

PM 1. Thi c e i a e ha e ill ee f he c e he he e i e d

e e e (e.g. J h aid: Be he e i i e ) he he e i h a l hi

di i c i be ee e i a d e. We al de e d he c e e e

c f i be ee efe i g e e ce a d i g i alic gi e a

e e i .

PM 2. I , e e ce a e called S i g a d ca a i la e S i g i h

a i f c i . Thi i e al a ed he e, b if d hi i he i

h a li f S i g f c i .

?*String*

PM 3. Se e ce a iable like ca be ide ified b diffe e e e i g, c e i all b

he e f . If e did d ha he e ld ha e e e i hi e e ce ,

45



ha The e e ce I ai i e. ld bec e The e e ce i e , a d

he e ld eed fi d a a e e ha ld be a a iable a he ha he

le e . I alic a e e c e ie .

PM 4. S c a e i igh f ge i g id f all he i e a d h i h a di c e a d

c ce a i g a e i e al delibe a i i e e i . The e a i def cie ce i le all a g e i k h gh i d ha ca a i e a

i e al c ic i f ha hi k i he ca e. Le e a i h a hi g

he ha a be e a g e . A d ake he i e hi k i h gh. A he ca e i hei i f a j dge i c . The j dge l e abli h he e ide ce b al

b e e i hi hi elf he elf he he hi c i ce be d ea able d b . Thi

(F e ch) i e f he legal e e i i e bef e he e ca be j dge e .

PM 5. Ye , S c a e idea igh be e el a beggi g f he e i , he he i a i

a he i ha a i d efe ea i g ab e e e. I he d , e le ca f lhe el e , a d Na e e ai he be del.

2.1.2 Logic and the methodology of science

Thi b k deal i a il i h l gic. H e e , di c i e i e e a le , a d ihi b k he e h d l g f cie ce ill f c i a he e a le a lica i . F

e a le, he cie i c e i h a h he i c jec e he i igh be ha d e i f all ci c a ce , b ha he ca d i de ig e e i e a ge ed a

ef i g i , ha , if i a e he e , he ha e e g d a ach al e i . Thiced e i g ded i l gic, a d i i ef l ee h ha i . The e a e a i

he i e i he e h d l g f cie ce ha l gic ca ill i a e a d ha a eill i a i g f l gic. I fac , l gic a d he e h d l g f cie ce a be ei i a el li ked ha e el a e a le.

A ke i e i ha he e each a c adic i he e f he hi g di c edca e i (Sch de 1890, ci ed i B che ki (1970:366)). Th l gic a d a he a icha e e i ac e i ic : c adic i a d i ibili f hel e ea ched i c i e h he e .

2.1.3 Mathematica - also as a decision support environment

i a la g age e f d i g a he a ic he c e . N e haa he a ic i elf i a la g age ha ge e a i f ge i e ha e bee de ig i g

a e hei he e a d f . Thi elega a d c ac la g age i bei gi le e ed he c e , a d hi c ea e a i c edible e h e ha ill

likel g i e f he e l i f a ki d e hi g ha ca be c a ed he i e i f he heel he al habe ; a lea , i egi e i h e like ha . N eha , ac all , i i he i e i f eci el he heel ha a e ed, i ce

46



e e b d ca ee d e like i i i e , a le i he M ; i a he a le haa he eal i e i . I he a e a e ge e a i a e likel eak ab hec e e l i , b he e e l i ld be hi i le e a i fa he a ic .

al ead i a deci i e gi e f a ki d. If e algeb aic l ii e he he e i a l f ded c i bef e he a e . H e e , ha

a e d e c e a a ea E gli h e e i a d d e ead a a c cl i ihe a ha a g d eake ld a i e hi he eech. The idea he e i , he ,

e e d e a a deci i achi e, a d e i he e i hli g i ic ded c i . N e al ha e e l ega d a a la g age i elf

, a d he ce e a e i he bjec field f be e i eg a i g la g age a d h gh .

S e ha ha e a bl ck f e , f e a le a e ha a i e e ea che l . W ld i be ice b i i c e , a d ha e i e if he he i

i l gicall d e agai l gic ? TheLogic` a d Inference` ackage a e ae i ha di ec i . Ad i edl , he e ill i a l g a g , b if e d ake

he fi e e ill ge fa a all.

2.1.4 Use of The Economics Pack - Relation of logic to economics

The Logic` a d Inference` ackage ed he e a e i cl ded i E . I

igh ac all e i e a e la a i h ec ic . The ba ic ea i ha b h bjec a e dea he a h , h hi k ee a dee c ec i e he e,a d h e j ed i i g he b k a d he g a .

The e ee be e hi ical a allel . F a ec i i i al a ice e e be ha J h Ne ille Ke e , he fa he f J h Ma a d Ke e , a gh l gic

a d e a b k he bjec . J.M. Ke e a i cli ed l gic a d ea i g, alea , a he hi elf e a ea i e babili a d ed hi a f hi ki g

ha f Ja Ti be ge a d Rag a F i ch h de el ed ec e ic a d he e h df be c chi g. Ec i h igh be i e e ed i l gic a d i fe e ce,

l f he e h d l g f cie ce, b al f bec i g be e ec i ( hde a d b h l gic a d be c chi g).

I e e i g i al hi b e a i b Sch e e (1967:103 104): N i ha e be, ha he a i ali ic a i de ha f ced i elf he h a i d i he fi lace

f he ec ic ece i ; i i f he e e da ec ic ble ha eh a de i e ele e a ai i g i a i al hi ki g a d ac i g I d he i a e

a , ha all l gic i de i ed f he a e f ec ic deci i aki g, , ea fa i e e e i f i e, ha he ec ic a e i he b f all l gic. I iki d f ha d ee ec ic a he he f he b f all l gic. Si ila l , a bab

47



e e i e i g i h If I d ea idge b ll he ablecl h he e e hi gill fall he g d d e ee be ch i l ed i he ec ic a e a da he lea i l gic f lai Ne ia h ic . B he e i a ice e.

The ke bjec f ec ic i he a al i f cial elfa e. Pa f cial elfa e i

de e i ed b he a ke echa i b hich a ac i a e c d c ed i h e .Each a ke a ac i i Pa e i i g a d a Pa e i e e i defi ed a acha ge ch ha e e le ad a ce hile b d ee hi he i ide e i a i g. Each a ke a ac i i l a , a d if he a ici a ld

ee a i e e , he ld a ake i i . Thi e i i e e i g ha eld like ee i al i he a ec f cial elfa e. O e ch he a f cial

elfa e de e d g deci i i hich i g cc . We ee i a e f hea h e le agg ega e hei efe e ce a i e a a cial i . F

e a le, i de c acie , deci i a e g e e e e di e a ei fl e ced b he ball . See C lig a (2001, 2007) V i g The f De c ac .(PM. Tha b k e a l e i g a ad e , a d he fi ed i idea

a igi all de el ed b L.E.J. B e h ge a cha e i hi c e b k.) Ahi d a ec f cial elfa e c i f lai alk, cial c e i , ch l g

e ce e a b hi e d deal i h he e ( ee h e e A (1992ab)). L gicb i l cc i all he e a ea f ec ic .

A ecial a lica i i he l gic f al , al called de ic l gic . The ec i de ic l gic i V i g The f De c ac i e d ced he e bel i ce i bel g l gic a ell a . The e a e he ec i f C lig a (2005) ha ldha e f d g d e d c i b ha a e lef f he i e bei g. The fi i hedi c i f he & , ee he a bel dei d c i . The he i he iad , , ee he a bel Cha e 7 h ee al ed l gic.

Thi aid, i h ld be clea ha hi b k ha bee i e f ec ic del . I i i e ded be ed b a e e e i g i a e i d .

2.2 Logic environment in Mathematica

2.2.1 This book has been written in Mathematica

Thi b k ha bee i e i , a e f d i g a he a ic . Thag a i b h a e edi a d a calc la a he a e i e. The e d ced he e i l ha he a h ha ed b al ha he g a ha e ge e a ed. Th eg a ha e bee i e d ce ha . Y ca cha ge he i i

48



a d ge e a e diffe e e l .

C cial a i k a e:

† = ea ha a iable ge he al e , == ea he l gical a e e haa d a e ide ical

† % efe he e l f he f e e al a i

† a f c i call ca be e e ed a [ ] a //

† /. ea ha b i i lea e a lied

I i a e j be b ca be c ed bjec . Tha he f a he d eak ab e e calc la i b ab .

2.2.2 Notation

O e c e e ce f i g i ha e ill e i a i ha i cade a d f la . The f ll i g i i al e a a d f c i a e

a ailable a da dl i . The g a al ha a e e i e Hel f c i ha ca fi d e de ail he e a d he c ce . I fac , he f ll e f hi

b k i a ailable i E , a d h ca be f d a a Add OA lica i i he Hel f c i f .

¬ p or Not[ p ] or

! p not ; here ¬ is found by Â notÂ

q … and ; also input as And[ p, q, ...] or p && q … Here is found by

Â and Â q … or ; also input as Or[ p, q, ...] or p || q … Here is found by Â or Â

p q … exclusive or (either ... or ...) ; also input as Xor[ p , q, … ] . Here is

found by Â xor ÂNand[ p , q, … ] andNor[ p , q, … ]

nand and nor (also input as and )

p q implies ; here is found by Â =>ÂIf[ p , then , else (,otherwise )]

give then if p is True , and else if p is False

Hand if none of these then otherwise LLogicalExpand[ expr ] expand out logical expressions

Above the line are “operators” and below the line are “functions”. The variables p and q are expressions in Mathematica . Thesymbol comes from Latin “vel” = “or”. The symbol & comes from Latin “et” = “and” written in a special way. The symbol “!” wasinspired by “?”, derivative of “Q.” (a Q with an abbreviation point) standing for Latin “Quaestio” = “I question”, so that the questionmark originally was a sentence by itself. In Mathematica you must enter Implies as the above and not with Â => Â (note the blankbefore the arrow) which just gives a double right arrow that merely looks the same.

N e he diffe e ce be ee N ha ha e di e i al i , I lie ha ha

di e i al i , a d A d a d O ha ca ha e e di e i al i . I lie ,A d a d O a e all called bi a e a i ce ha i h he ha e bee igi alldefi ed, hile he e e i f A d a d O i a fea e f .

49



PM 1. Whe e al a e ! i h ! a he begi i g f a li e he ld e i f

a ed i i g eadi g e file . N , ee R .

PM 2. The l gical b l A d i i h c a i f i e ca ali . I a al

la g age he e i ch a c a i b e ill eglec i . C a e She a ched

ele i i a d i ed he ai a d She i ed he ai a d a ched ele i i . O :J h a d Ma a ied a d Ma beca e eg a i h Ma beca e eg a a d

J h a d Ma a ied.

2.2.3 Input and evaluation

Ne lai e f he e edi he e a e al i a d cell f hee al a . Y e e c a d i i cell ( h i b ld e) a d he e l f hec e e al a i i i ed bel i ( h i adi i al f ).

† ca ec g i e e a e e ha ca be e al a ed di ec l be T e Fal e. O he a e e a e ai e al a ed he he d l fficie

i f a i .1 + 1 ä 2

True

p Ï q

H p Ï qL

2.2.4 Full form and display

F a e ec g i i , bjec eed ha e a fi ed f a , called hei F llF . Thif ead ea ie f c e g a b le ea f he h a eade . He ce, he

cell ca be di la ed i a diffe e f . Whe ki g i h h ld al a ha e he F llF i i d.

† I lica i ha h ee i f a : (1) he F llF , (2) a i fi f , (3) he

eÂ = > Â he c ea e a ea a ha al a d f I lie . Ihi ca e a iable a d a e defi ed a d he ela i e ai e al a ed. The

F llF i he e ca e a e all he a e. NB. D i I lie i h j a d blea .

Implies @p, q D

H p qLp ~ Implies ~ q

H p qLp q

H p qL

50



% êêFullForm

Implies @ p, qD

2.2.5 Logical routines

A he c e e ce f i g he e i e i ha e de e d ie al a i c e i . If e a c adic i i a i cell he d e i edia el c e a fal eh d i . The ea i ha ee e i a e ea be e i el ded c i e. F e a le, i e a i == 0 a d

!= 0 a fi d a e l i e a d Fal e. Si ila l , i a he a( e a) c l f c i hile bel g he l gical bjec la g age. Fa lica i f l gic, a i i e be efi f hi eak defi i i i ha hi all f a

l i al ed l gic.† E e i g a c adic i d e e al a e a fal eh d.

p Ï ÿ p

H p Ï Ÿ pL

† B he i e L gicalE a d a d Si lif fi d i .% êêLogicalExpand

False

H e e , A d a d O ca ill i lif a li le bi a d de ec c di i ha a e al af lfilled.

† A i Fal e if e c e i Fal e.And @p, q, True, s D

H p Ï q Ï sLAnd @p, q, False, s D

False

† i T e if e c e i T e.Or @p, q, True, s DTrue

Or @p, q, False, s D

H p Í q Í sL

If i e e iall a c l f c i . I k be i h h ee i ha ii be e de d a ... ... .... Ac all he be a de a d hi i

i e e g a i . If ca d he he f ll i g h ld

51



e lai i all.

† F c i "If" i a c l f c i i h a e l ha de e d i .If @1 + 1 == 2, "Wonderful" , "Too Bad" DWonderful

If @1 + 1 != 2, "Wonderful" , "Too Bad" DToo Bad

If @whatever, "Wonderful" , "Too Bad" DIf @whatever, Wonderful, Too Bad DIf @whatever, "Wonderful" , "Too Bad" , "None Of These" D

None Of These

O he he ha d i a bi a e a i h li i ed f eed i ce i ille al a e T e if he a ecede i Fal e. Y a check he la e b c ide i g IfCh i a a d Ne Yea fall e da he ll ge ha e ca hich i a

i ha i al a e ( h gh ac l ).

† I lie i a bi a e a ha gi e T e if he a ecede i Fal e. If he a ecedei T e he he c e e ce al be T e.

Implies @1 + 1 == 2, "Wonderful" DWonderful

Implies @1 + 1 != 2, "Wonderful" DTrue

Implies @whatever, "Wonderful" D

Hwhatever Wonderful LImplies @whatever, "Wonderful" , "Too Bad" D

Implies:: argrx : Implies called with 3 arguments; 2 arguments are expected. More…

Implies @whatever, Wonderful, Too Bad D

2.2.6 Getting used to Mathematica

The ab e l gi e he ba ic ece i ie ha e i e f de a di g hea i , e a d g a bel . If e c e ble bel h i e a e

i le e ed i he i i ad i able d ell a bi l ge he , i ce

di c e i g e ab i a i e e ha ca a ff i a ibjec . O e ke ad a age i ha ca i e g a ce bec e

c f able i h he la g age. A g d a l k a i ega d i a a

52



la g age i deed (a d j a c e g a ).

2.3 The subject area of logic

2.3.1 Aims of this book

Thi a ag a h i l e i d f he ai e he fi age f he b k.

2.3.2 The subject of logic

The ba ic idea i ha e le ha dle i f a i . The ba e j dge e f e j dge e , ha he e i a de e de ce be ee he e i e a d he c cl i fa a g e . We ca e ¢ de e ha e e e e a e e ,

ha e e (S c a e ) a e . Wha i i a he e i ha a f c aihe i f ha ¢ e e e ha i j ified a e he ba if hi f e a e i . I igh be c cei able eli i a ea d i cl de all h e

f e a e i , ha e l e i he e i e a d he c cl i , a d hae ca di ec l j dge he he he ded c i i c ec . I he bec e i ele ahe he e f ed he ded c i clai ed he f a d i deed a e ed he fi ale l . Ra he , e a e bec e i e e ed he he he ded c i i alid i alid. If he

ded c i i alid a d if e al acce he e i e , he e acce he c cl i. I hi b k e a e ha i deed c ld be e i ale he ele a a e e

.

L gic i a g ea libe a i g f ce, i ce e d l ge acce a e e ba i fa h i , b i fe e ce, hile he e a e bjec i e a check he alidi fi fe e ce.

Re e be ha e aid i Cha e 1: The bjec f l gic c i f i e iga i ghe ded c i a e alid a d he he a e i alid a d he i c cl i e; a d

i e iga i g he a e e a e ece a il e, ece a il fal e, c i ge bhei e c e a d hei ela i i fe e ce i elf.

2.3.3 Statements versus predicates, statements versus inference

2.3.3.1 Four combinations

A he e e , h ld be a a e f he f ll i g f bca eg ie , i h hebe h i g he de b hich he ill be di c ed i hi b ec i :

53



A S a e e P edica e

S a ic Hf la L H1L H2LD a ic Hi fe e ceL H3L H4L

A a e e i a bl ck f e e ce . A i gle e e ce i al a a e e .

The l gic ha deal i h a e e l i called i i al l gic. The l gic hadeal i h a e e a d edica e i called edica e l gic. The l gic ha a

e e ed i Cha e 1 a i i al l gic l . S a e e a e lec lec i i g f a ic a e e . A a a c i f elec a d , a a ic

a e e c i f b h he bjec f he a e e a d he edica e i he a e e .

2.3.3.2 Static statements

( 1) A e a le f a a ic a e e i If i ai he he ee a e e he i if ali ed a If P[1] he P[2] P[1] P[2] , a d h a al ed i f hede ail. He e P[1] = i ai a d P[2] = he ee a e e . The If ... he ... be ee

e i a d h ld be c f ed i h If. Th , ai i g ha he ee a e e . Clai i g ch a (h he ical) ela i diffe f

c cl di g ha he ee a e ac all e .

The a ic a e e a e j i ed f c d a e e ( lec le ) i g he

c ec i g e a f i i al l gic (A d, O , I lie , E i ale , U le , X ,...).

2.3.3.3 Statics with predicates

( 2) A e a le f a a ic a e e i P[3] = S c a e i al. I bec e aa ic edica e ca e he e c ide he bjec S c a e a d he edica e M al

a bjec b he el e . Si ila l , P[4] = All e a e al c ai he bjece a d he edica e al .

2.3.3.4 Dynamics with statements

( 3) A e a le f i i al i fe e ce i he f ll i g ( ):

If i ai he he ee a e e . If P@1D he P@2DI ai . P@1DE g The ee a e e . P@2D

The a e e i he a e e e e e ce (S a ic ) hile he e ac he a e a ded c i (D a ic ). The If ... he ... f he fi a be ee a bel gi g

54



S a ic , a d i a la ed i h (I lie ). E g i elf a be a la ed a (i) heli e i elf, (ii) he a e e I decide , (iii) Th , (i ) be ke a i i ,

eflec he f al a f he ded c i . I a a al la g age like E gli h hed Th The ef e i i cl ded i he c cl i a e e , ibl

e ha i e ha hi i he c cl i i deed. We be e a al e E g a a f c ic ce i g he h le a g e . We ca e d ce he i fe e ce ab e a f ll :

† The di e i al f a i ef l f e e a i . Thi i e d e eai elf. The e i all e a d he i e j di la .

Ergo2D @P@1D P@2D, P@1D, P@2DD1 H P 1 P 2L2 P 1Ergo ——————

3 P 2

I i al ef l ha e a li ea f a . We ca di la E g i h he¢ b l e e ha b h he a i a d he c cl i a e a e ed.

Ergo @P@1D P@2D, P@1D, P@2DD

H8H P 1 P 2L, P 1< ¢ P 2L

While lead l ge a e e (f a d e ge ) e fi d ha ¢ ca ca eh e a e e .

The i f d a ic c e f he b e a i ha e igh i cl de a i de fhe e i he a g e . S e i e e le fi a e he c cl i a d he b ildhei ca e i h he e i e , ha he e al de i ele a b he e ihe i fe e ce. N abl :

Ergo AHP@1D P@2DLt , P@1Dt+ 1 , P@2Dt+ 3E

H8H P 1 P 2Lt , H P 1Lt +1< ¢ H P 2Lt +3L

F a ele e a de el e f l gic e ill eglec he i e de a d e eldi i g i h a ic a e e e i fe e ce. The l d a ic ha e ai i he

e di i c i be ee e i e a d c cl i .

I he E g [...] bjec he c a bef e he c cl i ca be ee a a hee e i f he c j c i A d, hile he fi al c a ca be ee a a e e i

f I lie . Thi i e e a i ca be called jec i . A i fe e ce i called iff(if a d l if) he jec i f he i fe e ce i a ic i i al l gic d ce a

a l g , i.e. a ela i ha i al a e.

Ergo @P@1D P@2D, P@1D, P@2DD ê. Ergo @"Projection" DHHH P 1 P 2L Ï P 1L P 2L

55



Whe a i fe e ce i i alid he ( ¢ ). I a ea ake f be e eadi g gi ehi a e a a e b l f i elf, i di la l .

† Thi i a a e e ha he i fe e ce i i alid. N Se i i La i f i d e f ll . I di e i al di la ld c he h le che e.

NonSequitur @p, q D

H p ' qL

Ergo @ p___ , qD di la l ,i h ¢ e e i g ha he a i e l i he c cl i. I diffe f I lie @ , D i ha he i di id al a i

a e all a e ed ha he c cl i i h he ical

Ergo @ pD e e e ha i a he e . E g ca al be e e ed a Th

NonSequitur @ p___ , qD gi eŸ E g @ , D a d di la i h H e lici N L e e i g ha he

a i d he c cl i . N be c f edi h E g @ , N @DD hich i gi e b Ref ed@ , D

NonSequitur @ pD e e e ha i a ejec ed c cl i , ei he fal e e e

ErgoRules @D c ai i lifica i le E g

Ergo means thus in Latin and non sequitur means it does not follow . These terms have been chosen to emphasize the formalnature, as in a court of law. Ergo is also taken as “Proves”. Mathematica recognizes only Ergo but for texts you could use Â rT Â .

Ergo2D has the same input structure as Ergo but then prints in lines. SetOptions[Ergo2D, Label Ø Blank] leaves out line labels, anduse Label Ø {...} for your own list.

Y a ha e hea d he h a e he c a .... . The i i ha he c cl de ha d e e he i d e eed be he ca e ha e .A f f ld be a ef a i . Th e e al ha e

. The f ll i g able ha ha ea . I ha bee i e i ceedie al i e a d i i a ha d a . The diag al e e e c adic i

(a lica i f ), a d he c l gi e i lica i , abl , he lef , he i

e he hi i lie ha ld e a e. (Check he h f heigh c l .)

† A able f he c ce f a d .N e he i lica i i hiche e. The diag al a ac he h le diag al .

AffirmoNego @Ergo @p, q DDProven @qD Refuted @qDH p ¢ qL õ Contrary õ H p ¢ ŸqL

à â

‡ Not ‡á äH p ' ŸqL õ Subcontrary õ H p ' qL

NonSequitur @ŸqD NonSequitur @qD

56



Refuted @ p___ , qD gi e he c a E g@ , Ÿ DAffirmoNego @Ergo @ p___ , qDD i a able f c adic i a d c a i e

Affirmo is Latin for “I affirm” and Nego is Latin for “I deny”. To prevent a possible disappointment: Ergo, NonSequitur and Refutedare just objects and do not evaluate the validity of the argument.

2.3.3.5 Dynamics with predicates

( 4) A e a le f i fe e ce i edica e l gic i he f ll i g. I e i e ea i . The e a e h ee diffe e ea i g f he e b be , e e j e ,

a el bei g a ele e ( œ), bei g a b c llec i b e (Õ) a d bei g ide ical (=).F œ a d Õ he de i i a b = i a e ical ela i hi . The e i ala he ki d f bei g, a el bei g ide icall (ª ), hich i elf i a e ical

b hich i lie =. U i g he e b l e ca a al e he ea lie a e e .

S c a e i a a . P@3D S c a e œ MeAll e a e al. P@4D Me Õ M alE g E gS c a e i al. P@5D S c a e œ M al

I hi ca e, he i fe e ce c ea e a fif h e e i , a d e ca j dge he alidi fhe i fe e ce le e a al e he e e ce i de ail a d de e i e he ela i hi

be ee he bjec a d he edica e .Ergo2D @P@3D, P@4D, P@5DD1 P 32 P 4Ergo ———

3 P 5

F he edica e calc l , he diffe e ce be ee a ic a d d a ic ( a e ee i fe e ce) i he a e a f i i al l gic, l he i e al le diffe .

2.3.3.6 Summing up

I a , e fi d f i fe e ce a ed a ic :

1. he ded c i e e a a e e e ce

2. he h f each e e ce i a e ed

3. he e i e c ai fficie i f a i a a he c cl i

L gic a a e ea ch a ea c i b h f he a al i f he c e f e e ce( e i ed f hei i e e a i ) a d he b e e a al i f f ha ca e alid

i alid ea i g (ded c i ).

57



PM 1. While 1 + 1 = 2 i ece a il e i a i h e ic, l gic d e d hi a e e

i ce he c e f he a e e ha ela i i fe e ce i elf. B ill be

died beca e f i ela i ¢ .

PM 2. L gic die edica e ha a e ele a f i fe e ce. Wha d hi k f Thi

i fe e ce i alid. ?

2.3.4 Propositions (two-valued logic) and sentences (three-valued logic)

P i i a e ecial ki d f e e ce . A i i de c ibe e a e f held a d i i ei he e fal e. P i i a e ed i al ed l gic. A e e ce

ca be a h a e a ce a d i h al e igh be i de e i a e. F e a le, hee cla a i B h ! i ei he e fal e. Se e ce a e ed i h ee al ed l gic.Ra he ha eaki g ab e e ial l gic e ill ai ai he ge e al a e

i i al l gic .

The e e i i e f a ig a al i a e l f a ech ical a he a ical a e b al i l e he ch ice f ade a e c ce a d a i igh i he e h d l g f

cie ce. The f ll i g b ec i di c hi .

2.3.5 Truth

(a1) O i f de a e i Na e. Al h gh Na e h l e face a he

i e, he cha ge he face e i e: a d hi ha a gh ha a i a i a ha e ai e i.e. ha ha ce a he ca e eed be he ca e a a he i e. I hi

a e e each he f da e al l gical dich ha e e cc .Ha i g lea ed he idea f a i e, i d ake he f eed i agi e

i e he e he e i h ical ibili f a i e. F e a le, e ai agi e a i a i he e Ea h ha g a i , e e h gh ha i h icalli ible. I i elf ch h he ical h gh d i alida e l gic.

(a2) The f da e al a ec f (a1) i ha i c ce he ela i f he i d eali .The de f he i d a e a , a d al h gh e ea che i e iga e h e ( eeR e (1978) b fi Da a i (2003)), e a e ill fa a a f de a di g h he

i d a age ake ic e f eali , i g ha hi i a ade a ede c i i f ha he i d d e . B hi ble eed c ce he e. A

ec d i i ha he e a e e e h d ha all he be e ac i i i f ade a di g f eali ha he e h d d : he cie ific e h d ( i g hahe efe a c llec i f e h d ). See f e a le P e (1977:369).

(a3) Ha i g he dich f a d i eali a d ha i g he dich f a d i he i d a d he la e i e i ed f he

i d he i i be ca able f i agi i g eali he he f ll i g ibili ie a i e:

58



Mi d H∞Le eali H Ø L E e A cc E e A d e ccI agi e e e A ha c flic

I agi e ha e e A d e cc c flic ha

PM. We d di c he ibili ha he i d d e i agi e a hi g he ild be a i d a all (a lea f all ac ical e ).

(a4) We ha e he ha dica f e h d f c ica i : i ce e a e i gb l e ca i e e i eali , a d ei he ca e h i age i a i d.

The ef e i d ce he he e f a d e e e ha he e i ai e ela i be ee he e e ce i la g age he e e he i age ha

bel g ha e e . We ill di c he e h e le lea hei la g age ba e ha i i b e f la g age ha e a e able i e e a d i age .

(a5) We h e lace ab e able i h e i hich e e la g age. Wi h he e e ieali e a cia e he e e ce i he la g age ha de c ibe he e e . Wi h he

i age i he i d e a cia e he fac ha he i d he e e ce, ha hee e he e e ce. Le = E e A cc .

Reali H∞Le eali H Ø L N @ D i aid h fal eh d

N @ Di aid fal eh d h

(a6) The i a i f he able i (a5) i de c ibed b A i le (384 322 B.C.): T a fha i ha i i , f ha i ha i i , i fal e, hile a f ha i ha i i ,

f ha i ha i i , i e. (Ta ki (1949:54)) I i ac all j ha .A i le f ll ed S c a e i he idea ha he i d ( he l) d e lie i elf a d

a ed: (...) all ll gi [ ea i g] (...) i add e ed he ke d, b hedi c e i hi he l (...) (DeL g (1971:23)).

(a7) The S c a ic del f h a d fal eh d f a e e h gh i he l i a

g d del b e igh a ell efe di ec l a e. I ead f aki g h a dfal eh d a a de c i i alifica i f a i a i a e e , e ill ega d

h a d fal eh d a e ie f e e ce , i.e. ha e e ce ca be e fal e,he he efe a e. Thi i d ce i e, i ha e le i g ca lie he

e i g e e ce , b ha i a a ec f ag a ic ha e a eglec .

(a8) A e i be gi e he .Thi i he c e i al hi g ha a e e. Whe e e ce a e ed (i.e. cc i h a i a k

a d he ) he he a e l a ed b i i de d ha i i a e ed hahe a e e. F e a le, hi b k i j a b ch f e e ce ab hich

g e hich I belie e be e a d hich , b i i all a e ed be e.

59



(Th gh f c e I igh ake e , e e if hi e a e e e e he l e.)

(a9) Thi a e ic age all ha e le a e ile , i h he i lica i ha held be i h a i d. Whe i ai I eed a ha i ai , a d he i d e

ai he I eed a ha i d e ai . We d eed ha e e le eaki g all

he i e i g e e ha he i agi e ab eali . M e ble a ic a ei ake i a e i . I a ha e aid ha i ai ed, hile cl e i ec i h ed ha

i a he eighb kid a i g he i d . We a di i g i h a ideal l giche e e le d ake i ake a d a ag a ic l gic he e e le ake

h he e a d i hd a h e ef a i . F a ec ic i f ie i ii e e i g b e e ha e e le c a ie a egicall a age he

h al e f hei a e e .

(a10) Wi h e ec hi a e ic c e i e hi g ecial a be ed. Whe ld a e i e he acc di g ha c e i igh a ell j

a . A d c e el . Si ila l , i fal e c ld be i l e e ed a .A dc e el . Th he e e i i e i . E i ale ce, if a d lif , i e e ed, a a ecall f Cha e 1, i h he b lñ . Le di i g i h a edica e T hQ ha e h a d he Defi i i f T h hadefi e i .

† The defi i i f h f a e al a ed a iable e e i .DefinitionOfTruth @pD

HTruthQ @ pD ñ pL

† A e a le a lica i f he defi i i f h i :DefinitionOfTruth @"1 + 1 == 2"D

HTruthQ @1 + 1 == 2D ñ 1 + 1 == 2L

† I , a e i ca be delled i h he f c i T E e i , ha d

he e a d a S i g.% ê. x_String ¶ ToExpression @xDTrue

† A lica i f he defi i i f h he Lia h ha e a e f a e ha ha a check a ec i e de h, he i e e ld be l cked i fe e i . The Lia a ad ca a d b i f h.

DefinitionOfTruth @"Liar" D

HTruthQ @Liar D ñ Liar L

60



% ê. x_String ¶ ToExpression @xD

$RecursionLimit:: reclim : Recursion depth of 256 exceeded. More…

HIndeterminate ñ Not@TruthQ @Liar DDL

DefinitionOfTruth @ pD ge e a e T hQ@ Dñ f S i g . If i a S b l he T S i g@ D i

ed. E al a e i h _S i gß T E e i @DTruthQ @ pD H1L T e, Fal e,

I de e i a e if ha h e al e e ec i el ,

H2L T hQ@ _S i gD gi eIf@T E e i @ D , T e, Fal e, I de e i a eD

Liar Lia gi e he lia e e ce Thi e e ce i e , f ali ed a N@T hQ@Lia DD

The standard Mathematica function TrueQ is different. TrueQ[x] gives True iff x === True, and it gives False otherwise.

DefinitionOfTruth uses $Equivalent instead of Equivalent, see §3.3.1.

† N e he f ll i g i le e e i f he Lia a ad . The ea lie a ia i ee ligh e i g i ce i e lici l efe he i f h. Ye , c e, he Liaa i e b i g = i e ñ hich i h b hi a ia :

liar = ÿ liar

General:: spell1 : Possible spelling error: new symbol name “ liar ” is similar to existing symbol “ Liar ”. More…


ŸHold@Ÿ liar DPM 1. I e f elec ical ci c i , e ha he Lia a ad ca be e e e ed b

a achi g a e a i ch T he i ch ff he ligh ( i g he ba ic

diag a ). The e ill bec e c f ed, i ce, he he f ll he i c i a d

he i ch ff i de he ligh , he e e e ha e , i ce i g ff he i ch

ca e ha he ligh i ff. S ch a e i c f i g a d all ld be he e.

PM 2. N e ha i a ide if he elf efe e ce ha i he ca e f

he ec i . We be glad ha he g a i a e gh ice ha ec i

cc b he e a be a d g a a g a he a icia ha i e f elf efe e ce a e

i e iga ed, e e i e ha dli g . (S ch e a i c i e.)

(a11) A ag a ic he f h i di a i fied i h he idea ha he Defi i i fT h e el i d i g a i a k . The ag a ic i a i a be

f la ed a , f ll i g Q i e (1990:93): OK, if call a e e ce e i i l a e i , he h ca e ell he he a e i ? The ag a ic e i i alid, bf each e e ce i a ic la e ha e check i h he a ia e field f cie ce.

61



Whe he e e ce c ce e i he alidi f ea i g, he e check i hl gic. See Willia (2002) f a e ha e ag a ic a ach h a d

hf l e .

(a12) S a i i g, e e h ee ba ic a ec f he i f h. Fi , he ba ic

dich f he cc i g cc i g f e e a al be e e ed a hedich f h a d fal eh d. Sec dl , he e i he i age i he i d a d hec e de ce be ee ha i aid ( h gh ) a d ha i he ca e. Thic e de ce lack f i ca l be e e ie ced b a i ellige i d a d he

i f i i f da e al a d i e licable. A c a d like T E e i i i ic ha i . Thi dl , he i f h a be a ealed ideba e gi e e ha i ha i ha bee aid, b i e e ce ha l gi ee i al i ellec al e ha i , i ce a i g ha e hi g i e i e i ale j a i g i . A ag a ic ea f e ha i i ha e e le e i e lie hai igh add al e if a ha eak he h a d all b he h. I legalc , a a h a e he h a d hi g b he h i a ef l e i de f legalc e e ce f e j .

(a13) A a f i i ha a e ei he e fal e, he e a ea e i ale e ce ch a he Lia a ad ha a a e l a e i i a d ha e i e

e h ee al ed l gic. I a be ha i a e f da e al i ha .

Reali i j he e, b a ha e e ha e al e a i e eali .

2.3.6 Sense and meaning

(b1) The ea i g f a e e ce i ha i a .

(b2) The e all i i f he f e ec i a ha e e ce e e e ical, i.e.i i ha e e e a a e f he ld a d ha a e e fal e. Gi e he Lia

e e ce h e e e acc f e e ce ha a e e ical h gh hea ill ha e ea i g, i ce e e he Lia e e ce i i e i h ea i g. The

f ll i g able gi e ca eg ie . Wha i e ical i e ical a d ha i ea i gf l i ea i gle . We le a d f all i i (i.e. he fi ,

i dica i g he ld ) a d a d f all e e ce (i.e. all ). See B che ki(1970:20), F ege (1949) a d A e (1936).

E i i , j dge e , he e Ci cle a e d e ical T e Fal

h a e , c e i The Lia e a h ic- ag e - a ad ical I de e i a ei g f b l l 9 i L 5 ha ea i gle I de e i a e

PM. Bel e ill be a bi l i i g he d i i , a e e a de e ce i e cha geabl . The i e ca e i ha a e ead gl he i g l e

62





ld hei c ce ch ha hi i achie ed. L gic cc i Na e, ide f , a df ce i elf i d. Thi diffe f a a d li e a e, he e c ce a d

d igh ell be l idi e i al.

(b9) Wi h he de e ab da ce f a d f all ki d f i e e a lica i ,

de hil h igh bec e i cli ed hi k ha all ea i gle i g fb l igh ill ha e ea i g, a e a d ide ifica i . H e e , he

c e f c i e e.

2.3.7 Symbolics and formalism

(c1) We a ake he libe al f al i i ha i d e a e l gic he le a e e e ce , a l g a he a e e. I ha ca e he i f h a d

fal eh d bec e f al label .

(c2) Thi libe ali i fed b he b e a i ha e le ca hi k ab he h icali ible. Th gh e k ba ic i f de a e i he dich i Na e, i

beca e clea ha i d all a g ea e f eed i h gh . A i ilae e i e i i he i d c i f e a h ic . A e i like M heg e e d e hi g ab e ? d e e e a h ical ble b a

al e, b i ca ill be i agi ed ha e le l k f a a e i . Pe le igha e le he i e b aj i e, a d i ha e ec he e i ca be

c ide ed be e ical. I igh bec e e ical if ide l i a e gh . Be e he , e igh h ld ha e e le a e e al , ha e c ld

ceed i h l gic i a f al h he ical a e .

(c3) Th gh e ca ceed a i (c2), i be e ha i ed ha a f al a achca e e e e a a f l gic. I he f al a ach he e i hi g haf ce acce he idea ha a e e a e ei he e fal e. I i l ha ch aidea ha bee i g ai ed i b e e ie ce i h Na e ha e a l i al i belief , li ical ie , a d he like, a d al i f ali .

(c4) I fac , if e a e gi e a de c i i f l gic he e ca a : ( ) ( ) . I i b

cie ce ha e de el cce f l he ie f Na e, a d a he ca l be ce f lif i i f ee f c adic i . B d i g he c e f he e he ie e gc ci f ha e a e d i g.

(c5) We ca cce f ll edic ( i h a l gical he ) ha if a he i i c i ehe i ill be cce f l. Na el , ch a i c i e he ill edic a e eha ill e e cc , e.g. ha i ai a d d e ai a he a e i e. F hi i

f ll ha l gical he ie be c ide ed be e i ical he ie . E i ic i j a li f e e b al hei c e ( ha ca be f ali ed).

64



2.3.8 Syntax, semantics and pragmatics

We ha e bee i g he e a ( b li f ali ), e a ic , a dag a ic . I i ef l defi e he :

† ( b li f ali ): i e iga e he c e i hich b l a eed. F e a le A i he fa he f B i e a .

† : i e iga e he ea i g f bjec , a d he ela i hi f bjec ba ed hei ea i g. F e a le he A i he fa he f B he e k f he

ea i g f ha a e i ha B i he f A.

† : i e iga e he ela i f he bjec h e a la g age (f al e a ic e ) he e i e . F e a le e lai diffe e e a ic f

diffe e g , cla ifie he diffe e ce be ee i i g, gge i g,h ea he i g, c i ci g, e ce e a.

Clea l , he a i a ea ca f c i i h he he . We ca eall j dgehe he i ll i a ell f ed e e ce he e d k ha hee a d f . C e el A B fa he i f he i al habe icall ed a i fie

e a a d e ca i agi e e e a ic , e i a be clea he he A Bi he fa he f he he . U le he e i a ag a ic c e i ha if B i he fa he

he hi i e e ed i al habe ical de a A B i f he . Of c e, ea lica i f he a i field i i i e he de e de ce f he he field . The

efe ed a ach i i i e he c bi a i f he a i field .

2.3.9 Axiomatics and other ways of proof

I hei age ld ci ili a i , a 5000 ea ag , he Eg ia de el ed a e feci e ea e e . C le c c i eeded be b il , f hich he

a id e e he la ge e . The i i f he a eeded be acked. A d

he he Nile had fl ded agai a d had de ed e la d a d c ea ed e ee , e l had be ea ed f he di laced. Whe he G eek ca e i i ,

he ed hi big b d f ge e ic k ledge, a d, e ha i g all f i ,he de ed: Ca e a f hi ? E e all E clid ed hi a i , a d

hi e b k ha bee i e f a bi e ha 2200 ea , ee S ik (1977).

OK, a l g a d de f l ha bee i lified he e i e ha da ee . The di c e ie f he i f f a d f he a i a ic e h d a e ke e e

i h a hi . I i i ible d he j ice i j a fe li e . Pe ha eei he h ld l k l a he a ic a d l k f he ce i c de f la , i hha i f f. A h , he bjec f f ill ge e a e i bel . I

l gic, a ai di i c i i be ee (1) e ha d he a i a ic e h d ha elie

65



b i i g e e i i e e i , a d (2) he he ha d he ee e a i a d i e iga i f all ible ca e , hich e e a i i lie e

i f a i h e ic a d c bi a ic .

I all ca e a f e i e a de a di g i ellec ha i illi g ee, de a d

a d acce Ye , hi c i ce e . Pe ha ha de i he N , hi d e c i cee he he f fail . Of e he ice f a h i f ce e le acce all ki df a e e e e h gh he f i eak e i e . See A (1992ab)

h ee e e ca ge a e a ha h ee li e a e e al ha a e .

2.3.10 Outline conclusions

S e ea l c cl i he c e a d ele a ce f hi b k a e:

† I fe e ce cc e e he e. F al deci i aki g igh be a a e cca i , ba he ie i ha i fe e ce cc f e ha e ha dl ice i le e ee a

eed f a c ed a ach.

† L gical f a di ci li e he a ec i l ed i i fe e ce. We decide he a e e ha e acce , hei c e, ha e a k , he he che e f i fe e ce a c ec , a d he he e ca e lai he e l he .

† Of e , he aj e l f ch a ce i ha e a hi ki g ab ha e ealla a d ha he al e a i e c ld be. Of e e al ead k he c cl i b

j a ake e ha e i e a e igh . Ra he ha ge i g a i fe e ce eigh di c e ha he i a i i all diffe e ha igi all h gh .

† The e ie f he ac ical i fe e ce che e a e i e a ied. S e ea i h e ic a d h he i e edia e e ( h able ), he j h he e l(a d e a i e f a i h e ic a d b i i ), he a e lai b i i .S e e if a e e , he all he elec i f he e . A d he he e a e

he a i e f ha all all ki d f i fe e ce .

† Wi h che e he c e , ca ickl al e a i e che e , a d j dgehei e ie . Thi ill hel de e i e ha che e i e .

† The e l he e a e li i ed. Thi e ai a I d c i i Ele e a L gic. Thefield f ele e a l gic i ide , a d he he e i Ad a ced L gic, i.e. a hi g el e

called Ele e a .

† i a e celle e i e di c l gic. I ake a a he edic a i , a d i all c ce a e he a g e . I i a he e i

he he i i a g d e i e f ac al deci i aki g. Ba icall l i fe e ce. The e ill be cca i he e c ld be ed b

igh c ide j alki g a i e e .

66



3. Propositional logic

3.1 Introduction

I hi cha e e ill defi e:

1. H e e e he i i e e ce .

2. H e e e he i i al e a a d hei h al e .

3. H e e e he ded c i a d c cl i .

The e ba ic c ce a e c e ed i heLogic` ackage:

Economics[Logic, Print Æ False]

A a la e age e ill al c ide heInference` ackage.

Thi cha e ill de el al ed i i al l gic. The b e e cha e ill

deal i h he edica e calc l . La e e ill ega d h ee al ed i i al l gic.While di c i g i i al l gic he e i ef l kee he e la e de el e i

i d.

3.2 Sentences and propositions

3.2.1 Constants and variables

We ill e , , , ... a a iable ha de e e e ce i i a d A, B, C, ..., A0 , A1 , ... B0 , B1 , .... a ch c a . A a iable ca e a il bec e a c a

he i ge a a ig ed al e. A e e i i a la g age, i dica ed b le e a dhe b l be ee e , i a c a . Th If i ai he he ee a e

e . i a e e ce a d a c a .

3.2.2 Englogish

A ecial la g age i E gl gi h . Thi i he i le E gli h like la g age ha ea ic la e i e defi ed bel ca deal i h. E gl gi h a e e ill ca ha e

e ca e le e a d c a i . The e a e a f ed i e e e ce( i h fi ca i al a d fi al i ), i de all h e i e ec g i e he

67



a e e e i i diffe e lace i a a ag a h.

Sentences @ statement_String D

a f a a e e i a Li f P e Se e ce . A a e ei a S i g ha c ai E gli h e e ce ha a i h a

HBla k &L Ca i al Le e a d ha e d i h a P i H& Bla kL

S b i e h he e a e P e Se e ce, J i S a e e a d L ca eThe . Ae a le a al i i :

sents = Sentences["If it rains then the streets are wet. It rains."]

8if it rains then the streets are wet, it rains <

3.2.3 Atomic sentences

A e e ce i called a ic f he ie i f i i al l gic if i d ec ai i i al e a . The i e T P i i alL gic a al e a

a ag a h i i a ic e e ce . The c e f a a e e ca be cla ified b hee f a iable , hile kee i g ack f ha each a iable ea . The ea i g f a

e e ce i ha i a .

ToPropositionalLogic @ statement_String D

a f a E gl gi h a e e i a Li f

L gical E e i . The i e al ge e a e P i i=8P@1D , < a d P i i Mea i gR le

P P@iD i a a ic e e ce, he i h ele e i P i i .

Propositions he li f a ic e e ce P@iD. E e P i i ê.P i i Mea i gR le b i e he a i ea i g

PropositionMeaningRule Thi i a le ha gi e heea i g f he c e a ic e e ce

† F e a le:ToPropositionalLogic["If it rains then the streets are wet. It rains."]

8If @ P 1 , P 2D, P 1<Propositions

8 P 1 , P 2<PropositionMeaningRule

8 P 1 Ø it rains, P 2 Ø the streets are wet <

P i i al l gic d e f he a al e a e e . If g dee e i a al i ga a ic e e ce he a d i g edica e l gic .

68



3.3 Propositional operators

3.3.1 Definition

The L gic` ackage e e d he a da d li i i h he f ll i g.

$Equivalent @ p , qD ea H L Ï H L , if a d l if H iff LEquivalence @ p , ...D a c j c i f all ible

e i ale , a i g he e i de e de

Unless @ p , qD be ead a ~U le ~ a d a la ed aŸ Unless @ p , q, r D HŸ LÏH LImp e lace , R le Ø I lie . E a le: Ø

H Ø

L ê. I

The e f I all he ef R le HØL f l gical i eadabili

TertiumNonDatur @ xD gi e H »» N @DLNegate @ xD e f L gicalE a d@! DNotNot @ xD e f ! Nega e@D ! L gicalE a d@! DNotpOrq @ p , qD gi e N @ D »»

Thi f c i i ed e lace If@ , D a d I lie @ , DLogicalVariables @ xD ide he li f a iable f a l gical a e e .

Implications @D gi e a li f i lica i f . If j i ed b A d,he all i lica i a e e i ale he igi al

“Tertium non datur” is Latin for “There is no third possibility”. Check the Unless by “I will have an apple unless you pay.”

CounterImplies[ p, q ] = CounterImplies[ p q ] = (¬q ¬ p ).

Equivalent is new in Mathematica since version 7.0. The Logic` package uses $Equivalent to maintain consistency of the textoriginally written with 5.2, notably for 3-valued logic.

E i ale ce i a ke i f d i g l gic. The f ll i g i a e ele a f

de a di g i f c i i g i hi b k. T a i h, igh check ha ( ) ie i ale b c ide i g The a ie i e a ed da , die . A check

i g L gicalE a d d e k i ce $E i ale i a e a i he L gicàckage a d i d e bel g he i e al e . H e e , he

h al e h ha he e i ale ce al a h ld .

† L gicalE a d d e ec g i e $E i ale .$Equivalent @p q, ÿ p Í qD êêLogicalExpand

HH p qL ñ HŸ p Í qLL

69



% êêTruthValue

1

† Whe e a e e a e e i ale each he he e ca i e i h ai .

Equivalent @p, q, r, s D p q r s

† The c c i f a h able f he la e i defi ed, h gh, f he e he b acke ? I fac , e h ld c ide all ible c bi a i , i ce i igh beha l e e i ale ce d e h ld hile he he d . The i e E i ale ce

a e all ibili ie .Equivalence @p, q, r, s D

HH p ñ qL Ï H p ñ r L Ï H p ñ sL Ï Hq ñ r L Ï Hq ñ sL Ï Hr ñ sLL

† N e ca l e i .% êêToAndOrNot êêSimplify

HH p Ï q Ï r Ï sL Í HŸ p Ï Ÿq Ï Ÿ r Ï Ÿ sLL

X [ , ] i he ega i f E i ale he e c ide j a iable . Whe ea iable a e i l ed, he X , a i le e ed i , beha e i a hea ha he ega i f E i ale . F hi ea he e i i le e a i f a

ea a f a i f E i ale f e a iable .

† Thi i ge e all e fal e.Equivalent @Not @Equivalent @p, q, r DD, Xor @p, q, r D D êêTruthValue

1

2

3.3.2 Truthtables and truth value

T h able a e a i e i f Pei ce 1902 a d a a e l i de e de l Wi ge ei1921. The e e a e all ible T e Fal e c bi a i , a d he he l gical

e a a e defi ed i e f hei e l . The h al e f he c bi eda e e de e d he h al e f he c e . T h al e a e T e

Fal e b f e e a i he ca be 1 0 he ha i e c ac . If e lea e al ed l gic a d a d i g h ee al ed l gic he e a e T e, Fal e,

I de e i a e 1, 0,12 .

70



TruthValue @ xD gi e he h ha e f 0 1 Hhe a e age e all a e f he ldL

TruthTable @ xD gi e he c bi a i f T e a d Fal e f he a iable

i . O i a a i if he i O F ØTable Hdefa l L he i e a li f le

TruthTableRule @ x: Propositions D

If i a li f Hl gicalL a iable he a R le i c ea ed.If i a a e e he he

a e e i e al a ed i h ha R le

TruthTableForm @ xD e TableF f ibl e c le e e i

SquareTruthTable @ xD e e a bi a ble i a a e f a

Options[TruthTable] also apply to Decide, TruthValue, TruthTableRule and LogicalVariables. Default option is Rule Ø Implies,

meaning that ‘ Ø’ in input is read as ‘implies’. Other values disable this option. The Rule format of TruthTableRule is different fromTruthTable[x, OutputForm Ø Rule].

A a f he a e h able f a a d T hTableF i Cha e 1, he e a ehe e f a :

† Thi i he ef l f he e a e i e e ed i all de ail a d he hea e i c e ed. We e d e hi f e .

TruthTable @p qD p q H p qL

True True TrueTrue False FalseFalse True TrueFalse False True

† Thi gi e le ha ca be ed f b i i .TruthTable @p q, OutputForm Æ Rule D êêMatrixForm

8 p, q<Ø True

8 p, Ÿ q<Ø False

8Ÿ p , q

<Ø True

8Ÿ p , Ÿq<Ø True

† If T e = 1 a d Fal e = 0, a d if all ible a e f he ld a e e all likel , hehe al h al e f he i lica i i 3 f 4.

TruthValue @p qD3

4

PM 1. A e e i ha ecei e h al e 1 i called a . A e e i ha

ecei e h al e 0 i called a . A e f e e i ha c ai a

c adic i ( ibl de i ed) i called , he i e .

PM 2. The hf c i i defi ed a : Ø T e, Fal e : Ø 1, 0 a d : Ø

T e, Fal e , I de e i a e : Ø 1, 0 , 12 , i h he e e i el c le e e f

71



i i a d he e e ce . Gi e hei e la e ca i e = . S e a h

e he d hf c i f he hi g a d he i i e fe e i h he a c ec

a lica i f he d f c i .

PM 3. Wi h he hf c i e igh defi e ( ) ñ ( ( ) § ( )). N e ha he

i e e i ale ce ld ill be e al a ed i e f T e a d Fal e, ha he e f 1a d 0 d e eall eli i a e he f da e al i f a dich .

3.3.3 Singulary operators

Si ce e a e c ide i g a i g la e a O : T e, Fal eØ T e, Fal e , he e a e4 ibili ie .

BinaryTruthTables @1D1 2 3 4

p False Not TruthQ True1 0 0 1 10 0 1 0 1

The e a i al a gi e T e a d he e a i al a gi e Fal e a be d a ic be ch ef l. The e e a i h d alif f a e a a e a e

f hei . O l N gi e a cha ge a d d a icall . Si g la e a 3ha gi e T e iff i T e a d ha gi e Fal e iff i Fal e, ha al ead bee ide ified

a T hQ i he Defi i i f T h.

3.3.4 Binary operations

Whe e a e c ide i g a bi a e a O :8True, False <2 Ø T e, Fal e , he e a e 16ibili ie . The ca be li ed ea il i h hei h al e i c l a d i g

1 a d 0. We d eed a a e f all e a i i ce he ca be defi ed i e fc bi a i f he . F ea e f e e a i e e i hi able › f a dal ( ) f ( ) e e hile h e a e e l defi ed a d ca eall be

ed. N e he a i e d he iddle: d a a e ical li e be ee 8 a d 9,a d ee ha 1 8 a e N 16 9.

BinaryTruthTables @2D1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

p q False Nor Ÿ› Ÿ p Ÿ Ÿq Xor Nand And ñ q p › Or True1 1 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 11 0 0 0 0 0 1 1 1 1 0 0 0 0 1 1 1 10 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 10 0 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1

The bi a i i al e a i a d hei h able ca be de d af ll , i g a de e he li k f he d ai (i ) he a ge ( ).

72



i2 ö

I2 ö I

i

i g I = 1, 0 a d [ , ] = [ ], [ ]

Al ¢ : Ø ha c e f = 2, b he i ill diffe f . The deci if c i diffe f he e a , i ce he e a i e el a c ec hai c ea e he le g h f he a e e , hile he deci i f c i ca ca e a h e

a e e . He ce he able f 16 ibili ie efe e a a d heible f c i .

PM. The e i a ch l i l gic ha e = I ha i c ide l bi a a iable . F ege

(1949), Ch ch (1956) a d Jeff e (1967) bel g ha ch l. Ch ch (1956:25): The efi h F ege e la e ab ac bjec called h al e , e f he bei g h

(da Wah e) a d he he e fal eh d (da Fal che). A d e decla e all e e e ce

de e he h al e e, a d all fal e e e ce de e he h al e fal eh d. B

de i g ea ha e e ce a e i 1, 0 ! Wi ge ei (1921, 1976) Sa 5 a : De

Sa i ei e Wah hei f k i de Ele e a e. D (1969:42) a : age i , die

a de A agef k d ei e A age a iable be ehe de A agef ei ei e

Wah hei f c i . Agai a e e cef c i i i a ed a a hf c i .

L.E.J. B e ade a clea di i c i be ee la g age a d a he a ic , a d hec ide ed l gic he e l f he a he a ical d f la g age. Al he c cei ed

ha i he i e a d i h a he la g age (a d diffe e a he a icia ) a diffe el gic c ld ha e bee c ea ed b a he a ic a al a he a e (e e bdefi i i ). B ab e c c i h ha he e i i al e a ha e a

ece a cha ac e , a d ld ha e bee di c e ed a a i e, h gh he f fhe e e a i f c e igh be a bi a . I ffice ake he i f dich

( ai f i e ) a i i i e a d he , i h e c bi a ic , de i e he 16

ibili ie . Si ce e a e he T e Fal e dich f Na e, he i i ale a a e ece a a d e i e i f la g age.

PM 1. Le be he e f a al be . N e he e e e e dich . Le[ ] = 1 if

i e e , a d 0 he e e . The + 1 i a ega i , i ce [ + 1] = 1 [ ]. The .

ea a di j c i i ce [ . ] = 0 l he b h a d a e e e . Si ila l + i

e i ale ce [ + ] = 1 iff [ ] = [ ]. The e i a ch l gical diffe e ce i h e e ce

i ce e ha e he a e ic c e i l a e hi g hile i h be e d

ha e he c e i a c l e e be .

PM 2. We al ead e i ed ha B le[T e] gi e 1, B le[Fal e] gi e 0, a d heal e d e al a e.

73



† C e back b B le[] == 1 ha i T e, Fal e d e e al a e, e B le[]=== 1 ha i T e Fal e. Wha e de e d he he heal ead ha ade e i a e h al e ill ge e.

Boole @True D

1

Boole @whatever D ä 1

Boole @whatever D 1

Boole @whatever D === 1

False

3.3.5 A note on not - that you might not want to read

The G eek la g age ha d f : mh ( ced ea he e ea

d like i hai ) a dou,

( ced ). Ch ice ega i (ou,) e a

dile a; he ega i f e e be f hi dile a i a a he a e i fhe he e. E cl i ega i ( mh) d e e ch a dile a. He ce i i ,

like ch ice ega i , acce ible a i i e i e e a i . (Be h (1959:631)).

C ide he h able f he 16 bi a e a agai .

S e ha e e clai (c l 12) b e di ag ee a d hi k

(c l 9) . H d e ega e ha he e a ? We ca eall a ( )(c l 5) i ce hi ld c adic i h ie beca e ( ( )) ( ) iclea l fal e. A he a e i e he ie f (c l 12) ac all i f ll fal e

i ce e k ha ( ) ( ) ha he e le e a e deali g i h a e c le el g. O he he ha d, he igh be e el li e, ac all hi k e e e i e ie , b he e ( ) ( ) a d he a (

) be ag eeable . A a e l , ega i ea e hi g he effec : Wha

a i ade a e, acc a e, e e ele a , h gh i i e i a ce aie ec .

The e ble ee able he e a e eci e i ha e a . I hel e e he he e di c i gle a e f he ld ( i he able) c i e

ibili ie (c l ). We ca al e I c[ , ] = i i c ec f . The ca di i g i h:

1. Sa iff i i ea ha i e ac l fal e ( h f i c l 12 ( )i c l 5)

2. Sa p̀ ( ) iff i a A d a e e a d hi k he c a , e pè

(O he , he bc a ie ) if ha a lie

74



3. Sa I c[ , ] iff i i ea ha ecificall ha he c l a lie

4. Sa I c[ , ] iff i i ea ha a h ll he e e i eed be c ide ed.Th he e l The ld i fla ca be I d a he c ide he e i

he he elec a e d .

5. Sa N A All[ ] iff i i ea ha i e ical a d ha h al eI de e i a e ( hi e i e Cha e 7).

3.4 Transformations

3.4.1 Evaluation

I he defa l i a i ( f he ke el) he l gical e a d e al a e.O e ha call L gicalE a d Si lif achie e e al a i . I he a ea ha he ba ic hil h i al ed e .

TertiumNonDatur @pD

H p Í Ÿ pLLogicalExpand @%DTrue

3.4.2 Algebraic structure

The f ll i g i e e l i he i hi f A d, O i h Ti e , Pl .S a da d i e i like C llec a d E a d k h ha dle Ti ea d Pl a d ha e ca be ed f A d a d O .

ToLogic @f ,

input, parms Dd e @ ê. rules , D ê.

e e edrules Hrules taken from Options L

The proper truthvalue of p q is 1 - (1 - p)(1 - q) but Logic uses the isomorphism of Or with Plus for functions such as Collect andExpand.

† I he f ll i g a e e , i ake e e c llec all e a d. H e e ,he al C llec d e k a d he ce e ca e T L gic.

p & & q »»! p & & q

HH p Ï qL Í HŸ p Ï qLLCollect @% , q D

HH p Ï qL Í HŸ p Ï qLLToLogic[Collect, %%, q]

Hq Ï H p Í Ÿ pLL

75



The f ll i g a f a i a e e a bi i i ce he ac all all l ea i i i g Red ce. See Cha e 1 f a e a le. The f ll i g e el a e

ha he i e d e .

ToAlgebra @ xD i e a i b e laci g A dØ Ti e ,O @ , DØ 1 - H1- L H1- L , N @ DØ 1 - .If he i AllØ T e i e ,

he e ali ie == 1 »» == 0 a e i cl ded, ha he ca be ffe ed Red ce

ToEquationRule @var_List D gi e he le f a f i gi i i e a i

FromEquationRule @var_List D gi e he le f a f i g

e a i back i i i al l gic

3.4.3 Disjunctive normal form

The di j c i e al f i defi ed a : (1) i a lie l i i al a iablea d / c a , (2) i c ai l , , a d , (3) all c j c i a e a he l ele el a d he di j c i a e a he highe le el. The di j c i e al f ba icall

ab la e all a e f he ld i he h able f hich he e e i i e.

ToDNForm @expr D cha ge e i he di j c i e al f i h l A d,O & N

ToAndOrNot @expr D l b i e If, I lie a d E i ale b A d, O & N

ToDNForm @Hp qL qD

HH p Ï qL Í H p Ï ŸqL Í HŸ p Ï qLLLogicalExpand @%D

H p

Í q

L† A he a de a d he DNF

Equivalent @Hp qL q, ToDNForm @Hp qL qDD

HH p qL qL HH p Ï qL Í H p Ï Ÿ qL Í HŸ p Ï qLLTruthValue @%D1

† Thi f c i e el e lace he i lie a d h diffe f he DNFToAndOrNot @Hp q L qD

HŸHŸ p Í qL Í qL

76



3.4.4 Enhancement of And and Or

Y a like i ha ha e e al a e a l gical e e i i h L gicalE a d Si lif fi d a c adic i h. I ha ca e ca e ha ce A d a d O .Thi k l f h e i i al c ec i e a d f he he e . The

i e Decide e hi e ha ce e .

† E a le a e:p Ï ÿ p

H p Ï ! pLAndOrEnhance @True D

AndOrEnhance:: State : Enhanced use of And & Or is set to be True

p Ï

ÿ p

False

p Ï q Ï ÿ p Ï q

False

AndOrEnhance @False D

AndOrEnhance:: State : Enhanced use of And & Or is set to be False

AndOrEnhance @ xD i h = T e O e ha ce && a d »» Hhe i e ffLAndOrEnhance @D gi e he a e f he e

AndOrRules @D le ha e ha ce A d & O

The AndOrRules[ ] can be used for replacement in standard Mathematica. In AndOrEnhanced mode, they are added to thedefinitions of And & Or, and then are no longer available for Replacement. See also LogicState[ ].

3.5 Logical laws in propositional logic

3.5.1 Definition

A a e e i a l gical la i al ed i i al l gic iff he h able hl he al e T e he al h al e i 1. The e i ch e i . The al e

f l gic la d e lie i hei defi i i b a he lie i hei e. The f ll i g a be added h gh, f e e ec i e. The defi i i f a l gical la i cl de a

i h e ch a la . The f ll i g a ec eflec hi :

† The e e f a la a i a il c e f he f a i fe e ce

ded c i (¢ a he ha )

† A h able e he la b c ide i g all a e f he ld (i l i g ei f a i h e ic)

77



† A f ca be gi e i h he di j c i e al f ( he e i i deba able he hehi i cl de i f a i h e ic e e a i )

† A f ca be gi e i h he a i a ic e h d ha a e e a i a ea d ha ded ce he la .

We al ead ha e ed h able i de check a l gie . The f ll i ggi e e e e a le hile i g h able . I fe e ce a d he a i a ic e h da e di c ed la e .

† Na d[ , Na d[ , ]] i e i ale . I a a i a ic de el e e c lde l Na d defi e all he e a . I hi ca e i i ha d agai ha

L gicalE a d d e ec g i e $E i ale , he i e i ld j h T e.$Equivalent @Nand @p, Nand @q, q DD, p qD êêLogicalExpand

H p Hq qL ñ H p qLL% êêTruthTable

p q H p Hq qL ñ H p qLLTrue True True

True False TrueFalse True TrueFalse False True

† E a i le be l gical la . A d a d O a i f la f a cia i a d

c ica i .try = p && (q || r); try = Equivalent[try, LogicalExpand[try]]

H p Ï Hq Í r LL HH p Ï qL Í H p Ï r LL

Equivalent

And

p Or

q r

Or

And

p q

And

p r

p q r H p Ï Hq Í r LL HH p Ï qL Í H p Ï r LLT T T T T T F T T F T T T F F T

F T T T F T F T F F T T F F F T

3.5.2 Agreement and disagreement

O e g d a e l gical la i he f ll i g. S e ha J h a b Ma

a . N all Ma ld be igh f c e b i i a he e i he e J ha d Ma ag ee a d di ag ee. U i g he la f l gic e fi d ha he ag ee he ñ

a d ha he b e e l di ag ee f he he ca e , h X [, ]. L gic al

78



hel i lif a e b ed ci g c le a e e . I i l g d kha X i he i e f e i ale ce, b al k ha ha i e ac all

l k like i a a ic la i a ce.

C ide hi life h ea e i g i a i . J h : Gi e e he ca ke a e ill

ha e a e ible accide a d Ma : If fa he ca d i e he , k, I gi e heca ke if ca d i e . Whe Ma ha d J h he ca ke he hei

ie a e i ag ee e b if he gi e he he a e he he clea l ha ediffe e ie .

† The e a e he i i :ToPropositionalLogic @"If not give me the car keys then

your father cannot drive & dies and your mother cannot drive & dies." D

8If

@Ÿ P 1 ,

H P 2

Ï P 3

LD<John = % @@1DD ê. If Æ Implies

HŸ P 1 H P 2 Ï P 3LLPropositionMeaningRule

8 P 1 Ø give me the car keys, P 2 Ø your father cannot drive & dies, P 3 Ø your mother cannot drive & dies <Mary = P@2D HP@3D P@1DL

H P 2 H P 3 P 1LL

† Thi l e he e i ale ce. The e l f hi e a le i i le ha e d ha e de ha he X l k like. J h a d Ma k ha he h ldf c .

John ~ Equivalent ~ Mary êêToAndOrNot êêSimplify

P 1

† The Ag ee e i e c llec he e a i e a d al e i he i f acheck he b acke . Ag ee e elec he c di i de hich he e i ale ceh ld (a d Di ag ee e h e he i d e h ld).

Agreement @John, Mary D êêMatrixForm

1 Ø H P 1 Í H P 2 Ï P 3LL2 Ø H P 2 H P 3 P 1LL

Agreement Ø P 1Disagreement Ø Ÿ P 1

The i e Ag ee e h i a b ea c a ic ced e ha e el elec hee a a e a e e f ag ee e a d di ag ee e . F e a le i Ca l =

a d

M i e = igh clea l g e ha he ag ee a d he j he c cl i ha he di ag ee he a f . Well, hi i h a a iage

c el igh k. A l gicia , e a e i e e ed i he c di i ha ake he

79



ie ag ee a d h e ha ake he di ag ee.

† Ac all , d f he c ide a i . If Ca l a d M i e ca ca e bef lfilled he he ca li e i ag ee e .

Agreement @p Í q Í r, ÿ p Í q Í r D

81 Ø H p Í q Í r L, 2 Ø HŸ p Í q Í r L, Agreement Ø Hq Í r L, Disagreement Ø HŸq Ï Ÿ r L<

† Al e a i el , if he d f lfill ha c di i he he e i ba i fag ee e .

Agreement @p Í q Í r, ÿ p Ï ÿ Hq Í r L D

81 Ø H p Í q Í r L, 2 Ø HŸ p Ï ŸHq Í r LL, Agreement Ø False, Disagreement Ø True <

Ag ee e ca i he e l i DNF i h each O ele e a e a a e li e. Thi

e e all he he b hi f a ca be clea e f e c lea e e .

Agreement @Matrix, Hp Í q r L Í s, Hÿ p s Ï Hq Í r LL D

H p Ï Hr sLLH s Ï Hq Í r LL

Hq Ï Ÿ p Ï Ÿ r L

Agreement @ p , qD de e i e he e b h a e e ag ee, a el $E i ale@ , D ,a d he e he di ag ee, a el X @ , D. I e Si lifi ead f A dO R le . PM. label Di ag ee e i a S i g

Agreement @ Matrix , p , qD

j elec he Ag ee e a d h he e l i Ma i DNF

3.5.3 Methods to prove something

3.5.3.1 Introduction

S e ha ha e a c i ge a e e ha i l gical la a c adic i b ha ha e k be e, f ha e e ea . Y a ecif heca e ha i e, ha ca e e a i fe e ce de he e a i

hi c cl i i e . F he f ll i g a e e , f e a le, ca ide if he i he h able ha gi e h.

try = p Hr Ï qL

H p Hr Ï qLL

80





3.5.3.4 LogicalExpand

try êêLogicalExpand

HHq Ï r L ÍŸ pL

The e i , i he d , hi g e de he . All he e a ache a ee e iall he a e a he h able elec i e h d. The Ag ee e i e e ecall Si lif (L gicalE a d) ha he be he a e ece a il . The ldiffe e ce i he e e h d i ha hei a f e e a i diffe : able , li e , 1 0,a a f a i f he i .

3.5.4 Contradiction, contrary and subcontrary

L gicia ha e a a e i e f h ee e . The ba ic dich f T e Fal e e d

be le hel f ll he he e a e e a iable i l ed. I ead f 1, 2, 3, 4, ... 1, 2,a , l gicia a he c ce a e 1, 2, 3 . Thi gi e a .

Th , i ead f f c i g he i di id al a e e , le l k af he , a d i a ic la be ee h e c bi a i . T ea e

de a di g, e e a il e ca i al i dica e c bi a i , h = [ , , ...]f e l gical e a .

We ha e al ead ee c a i e i Cha e 2. The e i a he a lica i , a

diffe e ki d. Thi ki d f c a i e i l ible f A d a e e a dbc a i e h ld l f O a e e .

† The d a e e e i lica i . The he a ea ha he label a l .Contrary @Table, p Ï qD êêSimplify

H p Ï qL õ Contrary õ HŸ p Ï ŸqLà â

‡ Not ‡á ä

H p Í qL õ Subcontrary õ Ÿ H p Ï qLThe able c e he e defi i i :

† T A d a e e a d P `

a e called he he e cl de each he , hilehe igh b h be fal e.

† T O a e e a d P `

a e called he b h igh be e, hilehe ca b h be fal e.

C a i e i a e e e ega i f a e e. I ca be de ed a P `. While N [ ]

ill lea e hi g g e , P ` ecificall a ge ha i he ca e ha de ie.

The f A d a e e i ha a e e P ` ∫ ch ha P

` . F ( P

`

) i al f ll ( P `). The ega ed a e e a d P

` a e he called each

82



he .

Gi e a d P `

he e i a e ai de O he [ ] ha gi e he e , a d ha ca be de ed

a P è. The a i e b P

` P

è ñ .

† Thi gi e a ee e ide . I i e i ale Te i N Da .Contrary @3, P D

I P P ` P

èM

† The ide h i e he he e a e e a e e i l ed.= elec he fi , P

` elec he la (e e e ega i ), hile P

è c llec all

eglec ed i be ee . The e e e gi e ea i e e able i a i hile heiddle gi e a j ble f T e Fal e.

TruthTable @p Ï q Ï r D p q r H p Ï q Ï r L

True True True True

True True False FalseTrue False True FalseTrue False False FalseFalse True True FalseFalse True False FalseFalse False True FalseFalse False False False

Contrary @Table, p Ï q Ï r D êêSimplify

H p Ï q Ï r L õ Contrary õ HŸ p Ï Ÿq Ï Ÿ r Là â

‡ Not ‡á ä

H p Í q Í r L õ Subcontrary õ Ÿ H p Ï q Ï r LY a ha e ee he l gical e a N . The f ll i g able babl e lai i

ch be e ha a h able. I i he a e a he able ab e b i h he c l

e cha ged.† Thi cla ifie ha he e a a d f .

Contrary @Table, Nor @p, q, r DD êêSimplify

p q r õ Contrary õ H p Ï q Ï r Là â

‡ Not ‡á ä

ŸH p Ï q Ï r L õ Subcontrary õ H p Í q Í r L

83



Contrary @ pD de e i e he , ch ha & i Fal e,hile ibl N @D V N @D;

j i i h he ^ HO e Ha L if i d e e al a e

Contrary@

3, pD gi e he h ee ibili ie i h O he@D a he e ai de

Contrary @Matrix, pD gi e 88, C a @D< , 8!C a @D , ! <<Contrary @Table , p H , LD e TableF he e a a e

i cl ded f c adic i a d bc a ;label fi he a e a i f a

Other @ pD c e f he l gical la O @, C a @D , O he @DD

3.6 The axiomatic method

3.6.1 The system

B ab ac i f eali e ge a f al e , ha diffe f i (i e ded)i e e a i i ha l ge he e a ic a l b l he a . The ad a age

f a f al e i ha e a e l ge di ac ed b hidde a i f de a di g f he ble a ea. All ha i ele a ake e hi g k i

i che e ha a e ca e a e, e e e e h d e de a d he i e(like a c e ). Le e ake he bjec f i i al l gic i h all i e a ic adi c ed ab e a d le c ea e a f al a i a ic e f i . We he ge ae f al c e ha e igh i e e e i a i he a . I hea i a ic e h d e l ide a i a d le f ded c i b e al

a e a li f b l a d f a i le f hich he a i h ld. The ela i j di c ed a e de ic ed i he f ll i g diag a . The i a i ac all i ligh l

e c le , i ce ha i d a i ha e di c he e ela i i a e ala g age.

A

HI e dedLi e e a i F al e

A f ll e f i i al l gic ca be ake f DeL g (1971:107). Thei i i e ba e i defi ed a f ll . We e , a d a a iable i

e ala g age i dica e c a , a iable a d f la i he bjec la g age f hee . We a e ha i i f he b i h he a e he e be ed

84



(DeL g e i cl de h e b he i le eadable). Al , a 1 ld ac alle i e a i a k , e.g. , b he e ha e bee dele ed f eadabili .

The i i i e ba e ( i a l gic i i al calc l ) f i :

1) Li f b l

) T l gical b l : ,

) T a e he e : (, )

) A i fi i e li f i i al c a , A , B , C , A1 , B1 , ...

) A i fi i e li f i i al a iable , , , , p1 , q1 , ...

2) F a i le

) A c a a di g al e i a f la

) A a iable a di g al e i a f la

) If i a f la he i ) If a d a e f la he i

3) A li f i i ial f la che a a (a i )

) ( )

) ( ( )) (( ) ( ))

) ( ) ( )

4) T a f a i le

) F a d , i feThi i i .

† Y a check ha he a i a e a l gie . If he e ld be a l gie hehe i hi e c ld be i e e ed a he i la g age.

TruthValue êû 8P HQ PL, HP HQ RLL HHP QL HP RLL, Hÿ P ÿ QL HQ PL<

81, 1, 1<

N ha e ha e c ea ed hi f al e he e a all a i e a be f

e i ch a he he i eall i a g d e . A i a ic he ha de el eda be f c i e ia j dge ha g d e .

The adi i al e h d e he ade ac f a a i a ic e i ide ae i i g e a le i he eal ld ha f a del f he e . Si ce he ld ia ed be c i e ( he e i l e eali ), a g d fi ld h ha e ha ef d a g d f al del. I a ea be e ligh e i g a al e ha e ac all

ea b a g d fi , i ce ha ge e a e all ki d f e ie f e ha e a

ha e bee a a e f bef e.DeL g (1971): O ai a f ali a i ill be achie ed if he i f al he

e e ed ab e i a i e e a i f he f al e . ( 106) a d The

85



i i al calc l i c i e , c ec , i de e de , e e i el a dded c i el c le e, a d decidable. I i ca eg ical, b a be ade ca eg icalif e de i e. ( 141).

Th e e ie a e e a e ic ha ca be e e ed i i elf. The efe a

e a ic c ce ( h i h e ec a i e ded a lica i ) a d a f al c ce(a i a d he e e i hi he e ).

† : O l h a e able. : The a i a e a l gie , hea f a i le e e e he e, a d a l gie a e e. (PM. Thi ill all

ha e h ca be e b he e .)

† : The e i ch ha a d ca be de i ed . :If ld bei c i e he e e hi g ca be e . B e he a l gie ha a e

e e fi d e e i like ha a e a l gie a d ha he ce a e e .

† : A a i i i de e de if ei he i i ega i ca be ef he he a i . :If i i c i e he he ega i ca be e ,e e i h he hel f he a i i elf. He ce i ffice h ha each a ica be de i ed f he he a i . Thi ca be d e b h i g del ch

ha he he a i h ld b he e de a ge , f all a ge . See DeL g(1971:135 136).

† : F i i al calc l hi e ea ha all 16 bi a e a i ca be e e ed. :DeL g (1971:137 138) e lici l a la eall 16 c l i c bi a i f a d . I igh be b i al ead f hedi j c i e al f a d he a la i f A d a d O .

† : All l gical h i he e ( de he i e dedi e e a i , he e a l gie ) a e he e . The e ca be e la ged

i h e i i g a cha ge i he i e ded i e e a i . :All h ca be

e e ed i a di j c i e al f . All he e ca be e e ed i di j c i eal f . ca e (i.e. ) f a i gle a iable. S e ha h

Ak c ai a iable c a he i ca be i e a Ak - 1 ( ) f e .All he a d he f f he i gle a iable.

† The f ll i g i ic he a i a ic ced e, a i g i h a h.

H! p Hp qLL êêToDNForm

HH p Ï qL Í H p Ï ŸqL Í HŸ p Ï qL Í HŸ p Ï ŸqLL

ToLogic @Collect, % , 8p, ÿ p<DHH p Ï Hq Í ŸqLL Í HŸ p Ï Hq Í Ÿ qLLL

86



% ê. Hq Í ÿ q L Æ True

H p Í Ÿ pL

† : The e i a ced e d decide i a fi i e be f e he he aa bi a f la i a he e . The ced e ac all gi e a f. :Thi igi e b c ec e a d ded c i e c le e e . The f ed he e ic c i e ha a f ca be ge e a ed he eeded. (C a ed he

e he e ch a f igh be acce ed b ld e i e e ha a i fi i ebe f e .)

† : All del f he e be i hic. (1) a ea be ca eg ical. :I i ible i e e e he e ial ed l gic ( h ghl i g e f he he e ie ab e). A he i i ha he e a e c i ge

a e e , i.e. f hich he h al e igh be k . (2) ca be adeca eg ic. :U e l c a T e a d Fal e.

3.6.2 A system for

Ha i g e i ed i i ef l di ec l de el a ide e f . A fficie lich e f al ed i i al l gic* c ai , c ai , l he

e a , , , , ñ i h hei h able gi e ab e, he a ig e (=) a dide i ela i (==) a d he Defi i i f T h. Ne he a f a i le fal he h able e h d i all ed. We al all he h he ical de he e ah he i i c jec ed a d ca be e ac ed ce a c adic i i a i ed a ; i haca e he i lica i i acce ed ( i l ).

† H he ical ea i g. S e 1 gi e .Ergo2D @p q, Hyp : p Ï ÿ q, q Ï ÿ q, Retract : p Ï ÿ q, Not @p Ï ÿ q D D1 H p qL2 Hyp : H p Ï Ÿ qL3 Hq Ï Ÿ qL4 Retract : H p Ï ŸqLErgo ————— 5 ŸH p Ï ŸqL

3.6.3 Information and inference

The a i a ic e h d diffe f he h able e h d. The fi e l le fb i i , e a i a d c ac i , ha ca be a lied a libe a d ha ca

ded ce i di id al a e e . The h able e h d f ll a a da d alg i h haecificall ide ifie a l gie . Ne e hele , he e e ai a (hidde ) c al

ide i be ee he e e h d , abl he e he alg i h e he a e ki df le . The e a be a diffe e ce h gh i h e ec fi di g e h . I

87



a ea be i c i e c a e he e he a i a ic e h d i h he h ablee h d he bjec f he ha dli g f i f a i .

N e ha a a e e c ai i f a i if i h able ha T e . A l gicalla ca ie i f a i i ce ld al ead k ha i i al a e. A

c adic i ca ie he i f a i ha h ld a id i .

The a f a i ea i g ced e F a d , i fe ca bec a ed he h able f i jec i . C ide he h able bel , a d he bl ck i i , ha each e l i a h. Whe e a e he e ake he fi bl ck f he able. Whe e a e he e ake he fi , hi d a d f h bl ck.Whe e j i he i h he A d he l he fi bl ck e ai . The he a efal e a d l ge ele a . Th i f a i ca be a i ed a a d i

highl i f a i e. A e a k he h al e f, ab hich e ha e a e ed a i g e , he , i ce , e fi d ha be e (b hi gi e

agai , hich i le i f a i e).

† The jec i f .TruthTableForm @HHp qL Ï pL qD

ImpliesAndImplies p q p

q

True

True

True True TrueTrue True

TrueFalseFalse True FalseTrue

False

TrueFalseTrue False TrueFalse

True

TrueFalseTrue False FalseFalse

False

Wi h a i fe e ce p1 , ..., pn ¢ he e he a e diffe e e f c cl di g: . eake i g: he e he c cl i c ai le i f a i ha he e i e

. de e i i g: he e a i gle a e e ecie e a h al e hich il he a k

. e g he i g: he e he c cl i c ai e ac l a ch i f a i a hee i e , ha he e l i e i ale a d i f a i i l , b he e he

i f a i igh be e a ed e ha i a ea ie f

The e f i fe e ce a al de i e f ha e le ge e i f a i , hieed be ce ed, a d f he e i f a i e idbi a e ed ie a e e ei he a a . Thi ce f i f a i ce i g igh

88



be e c le ha el he ¢ f a ( ha i hi ca e lackc ehe i i bl cked b i ade a e a i ). We ill e he i fi f a i ce i g i Cha e 5.

3.6.4 Axiomatics versus deduction in general

Gi e he (hidde ) c al ide i f he a i a ic e h d a d he e h d f heh able , i bec e a alid e i h e i i g he a i a ic e h d a all.

The i i ha he a i a ic e h d ill i he a da d i a he a ic f a edefi i i f a e . E e he h able e h d a be a al ed a bei g ba ed ia i a ic , a e ba ed * .

Tha bei g aid, hi b k ake a ela ed a i de a d a i a ic . I a ea hahe diffe e ce be ee he a i a ic e h d a d a e ha le f al b ill

ded c i e e bec e e ha f . If e ee he a i a ic e h d a e elb i i f h i h acc di g a h c e i g le he e a e igh

c i ici e hi f eglec i g l i a egie ha ed ce he i e f a f.Ma he a ical f ali a a g al i i elf ha li le al e a ell. The bjec i e f a

f i c i ce a c i ical e a d i a ffice ha he he ec g i e hef, a l g a he e h d e ai alid. I he e h d l g f cie ce i a ea

ha a i i g be f i e a e f ll defi ed. A i a i a i f h e i e

ee e d e, h gh a bi e f ali e i e hel . A ef l ded c i ee , e e f ll a i i ed, ill ha he ai e ie f a a i a ic e , i

ha i e a d a f a i le be defi ed eh .

He ce, hile a e f al, * a al ead le , a d i he e ai de e igh be e e e. Ye he e ill e ai a ded c i e backb e a d a he a e i e e ill be i g a a l gical e al a .

89





4. Predicate logic

4.1 Introduction

4.1.1 Reasoning and the inner structure of statements

P edica e l gic deal i h i fe e ce ch a : S c a e i a a , e a e al, he ceS c a e i al. I i ef l fi d f al a i f he e ela i i ce hiall bec e eci e. If e d f ali e he e he i k f i eci ia d aki g g a i . The f ll i g i a g d e a le h e igh g

g (DeL g (1971:238)):

1. E e d g i al.

2. E e a i al i al.

3. The ef e e e d g i a a i al.

The e i e a e e a d he c cl i i . B i i al a g d i fe e ce che e ?S b i e la f d g a d ll fi d he che e i alid beca e he e i ea e ill e b he c cl i l ge i . B a he ha i g all ki d f che ea d d i g all ki d f b i i , le g f a c al a al i .

4.1.2 Order of the discussion

We ill a i h e he a d he diag a igi a ed b J. Ve i ce he e ide

he elega i d c i i edica e l gic.

A i le ll gi al ead e edica e b e be e di c i de i fe e ce, fhich i igi all a c ea ed. I hi cha e e a de e i e i h able h gh.

Le l ad he all a lica i ackage a d di c he Ve diag a . PM. Thei e i hi ackage a e l ea h he diag a , i i e c bi e

h e i de f a g a hical e e e a i f a l gical a g e .

Economics @Logic̀ SetGraphics, Print Æ False D

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4.2 Predicates and sets

4.2.1 Notation of set theory

S c a e i a a , e a e al, he ce S c a e i al ca be a al ed i hia : S c a e i a f he f all e , ha e a e a f all al bei g ,

a d he ce S c a e i a f he f all al bei g .

O he d f e a e c llec i , agg ega e, cla . Se a be de ed a =S c a e , J h , Cha le , ... . Thi j li he ele e f he e . We ca ega d e a

, ha e d eed efe he ele e he ha e. Le A, B, C, ... be e .The f ll i g c ce he a l :

1. The b l ill be ed f e

2. The b l « ill be ed f he

3. The diffe e ce f e A a d B ill be de ed b A \ B

4. The c le e f A (all he i i he i e e ha d bel g A) ill beed i h a ba ac i , h A = \ A

The Ve diag a a e a i e e, a d he e c ai ele e f ha i e e.The e a be d a i a hade f g a b e i e e j d a a b da , a di he ca e he e a c i f a ea a d d ha a e c ec ed.

† The ba ic diag a gi e a del f e A a d he c le e f A. N e hei hi i h N [ ]. N e ha a d « a e c le e . (A a e l«i i he d a i g, l ca e i .)

BasicSetGO @"Set A" D

Complement of Set A

Universe

Set A

5. The i f e A a d B ill be de ed b A ‹ B

92



† B d a i g B a d C i he a e hade f g e e e ha i e ha he a e i ed. N e he i hi i h . N e ha ifB a d C ld e la he

ill c ld be i ed, b he ld be di c ec ed.UnionGO @"Set B" , "Set C" D

Universe

Set B

Set C

6. A i a b e fB he all ele e f A a e c ai ed i B, a d he A Œ B

7. Se A a d B a e ide ical iff he c ai each he , ( A = B) ñ (( A Œ B) (B Œ A))

. A i a ic b e fB he all ele e f A a e c ai ed i B b A ∫ B, a dhe A Õ B

† Thi h D Œ E. We d a a b da a d D he i e e ld eehe diffe e ce i he a ed a ea . N e he i hi i h a d § .

SubsetGO @"Set D" , "Set E" D

Universe

Set E

Set D

. The i e ec i f e A a d B ill be de ed b A › B

10. Whe A B i a b e f he he , he A \ B ca be ee a A \ ( A › B)

93





† The f A i he e f all i b e , h Œ A . If A ha ele e hehe e e ha 2n ele e , d e he c e i ha« Œ A. F e a le, he

e e f« i « , i ce l « Œ « . He ce ha e e ha20 = 1 ele e .

† R ell e a ad i he f ll i g. We ca b e e ha a e d c aihe el e a a ele e . R ell e a le i he e f ea ha i aea i elf. Al = 1, 2, 3 d e c ai i elf, a d l ckil , he i e e

ld ge a i fi i e eg e i = 1, 2, 3, if e e e b i e i i elf. I ia al ega d a e ha d e c ai i elf a a e . Le be he e f

al e a d h e fi d he defi i i : ª – . F e a le he e eha ele e , h « – « , a d h « œ . Thi i i fac a e i e ce f f ,i.e. ha ∫ « . B œ gi e – , gi e œ a d . C adic i ! S l i :He ce he defi i i f ca eall be ade. Th he e i ch ha = – , al e a i el , f a e e h ld ha ∫ – . I i a bi i i g

ha ch a i a a d l g di c i i he hi f hil h a da he a ic ca be a i ed i ch a fe li e . If i ca . N e ha a i g = «

i a a da d a a ha he e d e e i . I he a e a he e f alla e ci cle i he e e . B e h ld ha he e ca defi ehe e

ei he ca a ha = « , i ce he la e ld e e hai ac ce . If ld be a ell defi ed c ce he , agai e.g. « – « , a d he a ad

e e agai . Th , i i gge ed, he e a e e e i f e f hich eca e e a ha ch e a e e a d h d e i . We eed a heca eg , a el ha ch a i .I e i l g i ill ha ea i g,

i ce e de a d he c e e , ele e , e ce e a; b eci el hiea i g ake e ele . We ill e he i e bel .

4.2.2 Predicate calculus and set theory

P edica e l gic a be e a ed e he . O i i diffe he he hi ca be eall

d e. A c a d ef l eff a di i c i i he f ll i g:† Se he i e e i e, h e e e a i f he ele e ha bel g a e . A

e i f e a le de ed b = x1 , ..., xn a d e e be hi i de ed b x1 œ .

† The edica e calc l i i e i e, h e e ie de e i e he he aele e bel g a e . F e i i de ed b [ ] ha e ha he

e . Th [ ] i e f iff a i fie he e .

The di i c i ca be h i h he a e e S c a e i al . Se he ld

c ea e he e f all al bei g b a li f he all. Thi ld i cl de all bac e ia f 5 billi ea i he a a d ibl e di c e ed alie , h gha e i he e he i ld a d a k he he he li a l g e gh.

95



O ce he e ha bee de e i ed he e ca a e S c a eœ M al e e haS c a e i f ha e . I he edica e calc l e ld ha he

i f ali i fficie l ell k ha he i e ca be j dged hee ie f ali a d bei g S c a e . We di ec l i e d [S c a e ],

he e [ ] i e iff ha he e f bei g al.I ac ice he e a ache f e e ge. F all he e ge e e , hee e a i ca be ac ical. The li 12, 23, 56, 528 d e ha e a if i g e

ha e a be a i fied i h he e e i e e h d f e el c ide i g all heli ed ele e . The I e al Re e e Se ice f he USA e abli he a li f all e

h a e ed a a e ; he a f c e b c ide i g hei e ie , bal , gi e he c le i f he e i e , b kee i g ack f he a e a d add e e

ce he a e i cl ded. O he he ha d, i i a ci e a a e ad i ed heh i g he e f ha i g a icke , a d kee e a he ga e all d aii h ad i a ce ill he ha e f ll li ed all ele e .

The ea ake he di i c i be ee he e e i e a d i e i e e h d i l f ac ical ea b al beca e f hil h . If e a e ha e le ca

ake j dge e e ie he i i a alid e i he e he k ledge abh e e ie c e f . A e he e i a e, ha babie ha e ca aci ie be dhei e ec a i , ca he e be ac i ed, a d, if , h ? Phil he ha e all

ki d f idea ha .A ke i e i he i fi i e. F i fi i e e f be , i i ge e , a d ,

a he a icia c ea ed e h d defi e e . The e e e a he a ical edica e ,like i a e e be . The idea i ha ch a he a ical edica e ld be

eh diffe e f edica e like ali . S ch a he a ical edica e ca bei ha he b ild f all acce able c ce e c le i . O

he he ha d, ali i a k a d check all acce able check.

F he i f l gic he e a e i e e i g hil hical e i , b , e a ef c ed alid ea i g a d deci i aki g. He ce e c ce a e c alf . T i lif i e , e ill all f he i a i ha a e ca be defi ed b a

edica e, a d c e el .

† We ca c ea e e i h f ll e e a i g he . Na el , a e ca be b a i g ha i ele e a i f a edica e. The c di i i e e ed i h a ba a i , a d hi i ced a he e f gi e .Mortal @xD ~ $Equivalent ~ Hx Œ 8y "y is Mortal" <L

HMortal H xL ñ x œ8 y y is Mortal <L

96



† C e el e ca defi e a edica e a a ic la e . E.g. defi e A = 12, 23, 56,528 a d he e:

BelongsToSet @AD@xD ~ $Equivalent ~ Hx Œ AL

HBelongsToSet H ALH xL ñ x œ AL

He cef h e ill e i e e edica e a d e i e e e , a i g ha heab e a la i k .

4.2.3 Universal and existential quantifiers

P e ie h edica e e ca be a i fied b a be f bjec . H a ?

O e he age , a hil hical a d a he a ical c e i ha g ake adi i c i be ee , a d .If e c ide he hai Pla head, he e

e d di i g i h bald e , e hai , a head f hai . O e he age he e ha g a e de c d a he li e a 1258 hai ha e e he ecific be . F

a he a ical ble , i i c ide ed ele a he he a ble (i) i a a l gha a lie all, (ii) ha e l i , (iii) i a c adic i a d h ha

l i . A d i i g e a le i ha li e ei he e la (a e he a e), c , a e a allel, ha hi ble ha li i ed ibili ie i deed. P i g ha a blei l ble (ha a lea e l i ) ca al be called a f. A he

d i i g e a le i he i fi i e. We ca k he i fi i f all i a li e, b b e e h d e ill ca de e i e he he all h e i a i f a c i e i .

U i g he ega i N , he h ee e , a d ca be ed ced a ai a d. The e a e called .All ill be called he a ifie , a d

S e he a ifie .

† The i e al a ifie All i de ed a" (i F All Â fa Â )

† The i e al a ifie S e i de ed a$ (i E i Â e Â )

PM. " i he le e A ed ide d . I e e i gl , he le e A igi all a

c cei ed i ha e a e a , a el a a i age f a c i h i h .

Si ila l he le e B ga e he la f a h e i h .

Le c ide bjec , bjec ele e ha a a i f a e . Fe a le, he ele e a e he a ed illi e e Pla head ( a a hi 50 h bi hda ) hile he e i bei g i h hai (ha i g a lea e hai ). The f ll i gdiag a ha bee ed i ce edie al i e :

97



All a e õ C õ All a e Ÿå ç

C -

ã éS e a e õ õ S e a e Ÿ

N e ha e ha e ee he a e ki d f able al ead i Cha e 2 a d 3, b he ea e he a ifie All a d S e. Thi i ac all he igi al edie al a lica i ( i h j he d All a d S e, he b l ).

† The able c e ab b he f c bi a i f All, All a d , . Theh a i d i e e e a d ble ega i agai i a i i e a e , e e he he

ega i d ble i diffe e lace .88all@x, P D, all @x, ÿ PD<, 8ÿ all@x, ÿ PD, ÿ all@x, P D<<

allH x, P L allH x, Ÿ P LŸallH x, Ÿ P L ŸallH x, P L

The able h ha " a d $ ha e he ice e ha he i each he bega i . Th " [ ] ñ $ [ ] hich a e a ed agai i ega i e f a a $

[ ] ñ " [ ].

A i le di c ed he e ca e , h gh did d a ha diag a (DeL g (1971:16).The Sch la ic ed he e aid f f he lef c l (La i f I affi ,

i h el A a d I) a d f he igh c l ( I de , i h el E a d O), a d

he a i ed he a e i

. The a i dica e he f ll i g

ela i hi .

† õ : Y ha e c adic ed ha he e i a head f hai iff fi d e laceha a e bald.

† õ : Y ha e c adic ed bald e ( All lace a e i h hai . ) iff fi de hai .

† õ : A head f hai i c a bald e . The e a e e e e i e . If e ie he k ha he he i fal e. Whe de e he ce ai l he

eake ed i e h ld , a d e ha e e he c a he .

† õ : S e lace ha e hai i bc a S e lace ha e hai .The e ca be c le e a . N e h gh ha i . S e hai

d e cl de ha he e i a f ll head f hai , le i ha bee e lici l a ed hahi i a i i e cl ded. Si ila l , he i i a ed ha e lace a e bald, he

bald e i ill ible le ha ha bee e cl ded b fi di g e lace i h

98



hai . The e i e a e ca ed b he diffe e ce be ee (Œ) a d (Õ).

The f ll i g e i e hi ackage:

Economics @LogicÀIOE, Print Æ False D

Wi h 4 c e a d 2 h al e e ge 8 ha a c l . Thef ll i g able li he . The able gi e he a e i f a i a al ead c ai ed i

he ea lie che e, b i i a he e e a i a d i igh hel de a di g. N ehe e be ee h a d fal eh d, a d he a i e ac he

bdiag al . We al di i g i h de e i ed (U, c i ge , T e Fal e b de i able f he i ).

AffirmoNego @TruthTable D

Legend:

H1LTrue »False are input, and appear on de subdiagonals

H2LC@1Dand C @0Dare truth and falsehood by contradiction

H3L1 and 0 are truth and falsehood by weakening

H4LU indicates contigent truth or falsehood Hundetermined LPM. A row input is False iff its contradiction is True

A EI O

T »F A I O E

All x are P A True True 1 c0 0Some x are P I True U True U c0

Some x are not P O True c0 U True U No x are P E True 0 c0 1 TrueAll x are P A False False U c1 USome x are P I False 0 False 1 c1

Some x are not P O False c1 1 False 0 No x are P E False U c1 U False

PM 1. The ab e ( l ec i 5.3) eh he hi f l gic f A i le B le, F ege a d Pea . I i e e iall c ai ed i he i i g f A i le. DeL g

(1971:23): All i all, A i le l gic i a ag ifice achie e e : he a ed i h

i all edece a d i e ed a he hich da i c ide ed i ae ec igh a d e e c le e. If f da a age i i al ee li i ed, i

be e e be ed ha he di c e f i li i a i i a a he ece achie e e ,

a d ha i a 2000 ea bef e a e be ide he S ic ade b a ial g e i

f al l gic. A i le b ea clai ed ha hi ll gi ic he c e ed all ki d f

a g e . He a a a e f he . (....) T de a d hi achie e e , e: (1)A i le ed , a d ,a d b l " , $ a d a e j fa c e ic f

e le la hi k f he el e , (2) A i le c ce a ed i fe e ce, i.e. he

99



a i la i f c bi a i f ab e A, I, E, O a e e , a i e a c cl i .

M de e he i a bi e b le, i h he diffe e ce be eeœ , Õ a d =. Ye

A i le che e de c ibe h e le a i e a c cl i , a d, if a ked he a , he

babl ld ha e k e ell he diffe e ce be ee a i gle e a d a g .

(3) A i le al had c h gh he f e f li g i ic c e i , a f e a le Alla e i c l e e ed a N a e , a d . Pe ha he a i e G eek

la g age a e e e. (4) A i le had ake a li i g . He a he de ig a ed

eache Ale a de he G ea , had i e e he b k , had he Pe i a eic

Acade , a d .

PM 2. S a i h a head f hai . P ll e hai . I he head bald ? N ? P ll a he

hai . I he head bald ? N ? Th , lli g a i gle hai d e ake a diffe e ce a d

d e ca e bald e . He ce, c i e hi ced e a d he e ill ill

bec e bald. Thi ble i called he .PM 3. A de a ach i e.g. f l gic. Thi d he a i f bi a ( e

) e be hi a d all f a g ad al e be hi . A e a le i a fa il ela i hihe e he fa il e be l k alike, b i diffe e g ade .

PM 4. Phil he ed i g de e i e he e ac be f hai Pla head

he hi ca ed ch hai li i g. The a e al a i e.

AffirmoNego @Table D e e he edie al a e

AffirmoNego @Ergo @S , pDD c ea e he a e f E g@, DAffirmoNego @f @S , pDD ide f f = U decided, U able,

U decidable, C i e

AffirmoNego @Truthtable D e al a e c l f he edie al che e. E ec i

f he i e gi e a lege d ha e lai all,e i h P i Ø Fal e

AffirmoNego @Truthtable, f @S , pDD

e.g. f f= E g , U decided, U able,U decidable, C i e . Be a e hac i e c i f ll c ec beca e f EFSQ

4.2.4 Relation to propositional logic

4.2.4.1 Sets and quantifiers

[ ] l e e e ha a i fie edica e, b he h le e e i i al a

e e ce ha i i bjec i i al l gic. A edica e ( a Big f J h i big ) ca be ee a a i ha i a lica i Big[] e e e a

e e ce i Big . Al e a i el , a e e ca be f c i (a d he a e ed

100



he he a e a lied eali ).

A fi c e e ce i ha he a ifie ca be defi ed a c j c i a ddi j c i i i i al l gic.

1. (" : [ ]) ª ( [1] [2] [3] ... ) i h e de ig a ed de i g

2. ($ : [ ]) ª ( [1] [2] [3] ... )

NB. A a iable de a a ifie i called a b d a iable .

Ne , ha i he edica e i elf ? I i ha f c i . F c i a e defi ed f ad ai a a ge, : D Ø . The i f he f c i i ha a i g i elf. We ca

e he b l A [ ] i dica e ha e ab ac f a d c ide i i he ec he h le f he d ai a d a ge, a d he a i g i l ed. Thi h ld

b defa l f he i e al a ifie , All. B e ca d he a e f he a , S e.

1. Ha i g a head f hai ª A [" : [ ]] i h be f he ili e e g id, i hica e fi i e.

2. Ha i g a bald head ª A [" : [ ]]

3. Ha i g e f he eª A [ ($ : [ ]) ($ : [ ]) ]

P edica e ca h be li ked i i he i e ha a i f he . C j c if edica e , like bei g big a d ed, ca h al be e e ed i c j c i f he

c i e . Si ila l f he di j c i .

Al e ca be defi ed i g he e a f i i al l gic. O e f a i :

" : ( œ ( A ‹ B)) ñ ( œ A œ B)

U i g he a i e eed e i he a ifie a d l keeacc f he c di i . The i e al a e e a e T e Fal e b he e l i a e(a d h T e Fal e):

† A ‹ B ª œ A œ B

† A › B ª œ A œ B † A \ B ª œ A – B

† A ª œ – A

Th gh he la e a e e , a d T e Fal e, he ill ca be ed i a e e haa e T e Fal e, e.g. i H A › BL ∫ « hich ea ha he e i a ele e a i f i g b h e (a d he e ie ha defi e he e).

4.2.4.2 Laws of substitution

The i e h ed e h d d he c e f a e e i diffe eb i i . The f ll i g e a le i ake f Q i e (1981:71).

101



† A e a e e i dec ed i a c j c i f e a e e ."London is big and noisy" ï H"London is big" Ï "London is noisy" L;

Le b i e i i h all . The e i ale ce e ai e. B i g a i

f ch b i i e fi d ha hi a f b i i i l gicall alid.† A fal e a e e i dec ed i a c j c i f i e , agai fal e.

"London is big and small" ï H"London is big" Ï "London is small" L;

Le b i e e hi g f L d . The e i ale ce b eak d hahe b i i i i alid.

† A fal e a e e i dec ed i a c j c i f e a e e ."Something is big and small" ï H"Something is big" Ï "Something is small" L;

The la e h ha e hi g i a d j like L d . U i g hea ifie a d edica e l gic e ca e i e he la e e i a d diag e h he

e i ale ce b eak d .

† The e i ale ce i ill a fal eh d b e be e de a d ha hec j c i e edica e a d e he i i .

"$ x: HBig @xD Small @xDL" ï H"$ x: Big @xD" Ï "$ x: Small @xD"L;

O c cl i i ha Big a d S all a e e e , b he e i ele e ihei i e ec i .

Le b i e i back.

† O l he i lica i i e i ge e al, b beca e f L d he i e ec i f Biga d N i i e .

"$ x: HBig @xD Noisy @xDL" ï H"$ x: Big @xD" Ï "$ x: Noisy @xD"L;

The b e e c cl i i ha he la e e i ale ce i a l gical la , i ce e

ha e f d a c e e a le i Big a d S all. The ela i a h ld c i ge l fe edica e b a h ld f he .

PM 1. We k hi e a le f Q i e (1981). He al e a k : Th e h a e ha

a he a ic i ge e al i ed cible l gic a e c i g œ i he cab la f l gic a d

h eck i g e he l gic. Thi e de c ha bee e c aged b a c f i f

heF[ ] f l gic i h he œ A f e he . P e l c ide ed,F i a a ifiable

a iable efe i g a e a ib e a hi g f i elf. The i a ce f hi c a

be ee he che a ic edica eF a d he a ifable e a iable A i e hel i g,

ce e c ide ha a ifica i e e gh . ( 125, a i ada ed).C e : (1) L gic i ab i fe e ce a d a he a ic ab all he f ali , ha

ed c i i a i e. (2) P edica e efe he el e , like = [

102



] hich i l i g ha a e l i g. Thi i a ac defi i i i ce he e i

a edica e ha a i fie hi e a edica e,$ : [ ], a el i elf.

(3) We h ld edica e a d e a e i ale . See 4.6.3 f a ad e i e he .

† Thi i af e 256 b i i . ( check ec i elf efe e ce.)

Lying @x_ D:= Lying @ÿ Lying D@xD

Lying @Lying D;



PM 2. Q i e (1981:124): A celeb a ed he e f G del a ha f ced e cae c a all he h f be he , he e cl i f fal eh d . Si ce e ca

e e be he i e he , i f ll ha he e i h e f a c le e e

f e he . Thi i a i ce if a eak e ( be he a.k.a.

a i h e ic) i e c a ed i a ge e ( e he ) he he ge e

defi i i el ca add fea e . See Cha e 9.

4.2.5 Review of all notations

We ha e a ab da ce f a i f he a e i e . Bef e e c c he able fall he e i ale a i e a b e e:

† A ke b e a i f Jaakk Hi ikka i ha a a ifie al a e i e a d ai .Ab e e ha e bee l a d i l a ed All a e hile e h ld ha eide ified e d ai D a d he ha e aid All i D a e . A c e i i

akeD = ha œ bec e a bi ( i ce e e hi g i i he i e ale ) ha e le e d e i. H e e , hi i i e a i le i e

a d he ce i i be e i cl deD. A a iable a e b d b a a ifie , hea ifie i b d b i d ai . Al a e e i like œ D ca be called a

e a e e a d eed a a ifie a b i i b a c a f cl e.

† Whe e a acc f he d ai he e a ge a i e be eeedica e a d e he . F edica e e a ch e be ee i i g" œ D:

[ ] a d " : D[ ] [ ]. If i i clea haD = he e igh a ell i e " : [ ].B ifD ∫ he he i lica i e f a i i e c e i al b ill efe able.

† O e ad a age f i d ci g a d ai i ha e ca c ide he,f All D a e E e i D he e e i ch

d ai . Thi i a e e f e i ha l gicia a d edie al k del ed

i . H e e , i g e he , he a e i lied b a da d f la a dhe cef h d e i e ecial a e i .

† If D a d a e e he all ele e fD a e ac l al ele e f, haD Œ . F A D ld like c cl de ha al D , i.e. he

103



eake i g f i he edie al a e. H e e , he la e i f ali ed a

HD › L ∫ « i l i g ha he i e ec i f he e e i h i elf i e .

He ce he f ll i g T e Fal e a e e h ld f e d aiD:

All i D a e " : D@D @D " œ D : œ D Œ P

S e i D a e $ : D@D @D $ œ D : œ HD › L ∫ «

S e i D a e Ÿ $ : D@D Ÿ @D $ œ D : – ID › M ∫ «

All i D a e Ÿ " : D@D Ÿ @D " œ D : – D Œ

N e ha he e a e a e e a d e . The a e eH A › BL ∫ « i al a a hegl a d ci c a ial e e i f e hi g ha i a he i le. I i ki d f

a ge ha e he i he la h d ed ea ha c e i h a aigh f a db l e e ha e e la . H e e , i i a ef l i e e d

he e f A d a d O e . Ge e all e ill bec e c f ed be eea e e a d e , a d e al ead ha e a ecede ce i he a lica i f he

i ical e a edica e . He ce e ad he f ll i g defi i i :

† A B ñ (A › B) ∫ « i he a e e ha e e la

† A B ñ (A ‹ B) = i he a e e ha e f he i e e ( hile ece a il bei g i ie )

† ( A B) ñ ( A B) : N e ha A › B = A ‹ B a d h (A › B) = ( A ‹ B).

He ce ( ( A B)) ñ ((A › B) =« ) ñ ( A ‹ B = « ) ñ (( \ ( A ‹ B)) = « ) ñ (( A ‹ B) = ) ñ ( A B)

† ( A B) ñ ( A B) : b i e he c le e i he f e ela i a dhe e ( ñ ) ñ ( ñ )

Recall ha e i e A = B he h e e e la .

We h ld be a e f e i e ce i e . Whe e e la he he e eall h lde i a ele e ha he ha e. Whe e k ha A Œ B he i i e i g c cl de A B , b gi e he c e i ha« Œ B e ld ge « B , hich i ac adic i . He ce he e f la i i :

† ( A Œ B A ∫ « ) A B

Le e A Õ B ñ ( A Œ B A ∫ « ) e e ha e e a e i l ed. Thia i i c e i al h gh a d i ld be clea e i l a e ha he e

a e e .

† We defi ed f a e A ha « Œ A, hich e igh a la e i " : œ « œ

A. F eake i g e ge $ : œ « œ A. Thi ee like a g clai ,

104





ca be dec ed i i lica i :

† T hi g a e ide ical each he if a d l if he ha e all e ie :" " ((= ) ñ (" ( [ ] ñ [ ]))

† The i ci le f i di ce ibili f ide ical : If hi g a e ide ical each hehe he ha e all e ie :" " (( = ) (" ( [ ] ñ [ ]))

† The i ci le f ide i f i di ce ible : If hi g ha e all e ie he hea e ide ical:" " (( = ) › (" ( [ ] ñ [ ])).

The f de f de b lic l gic F ege de ed ab he e ali f heM i g S a a d he E e i g S a . The a ea a diffe e e i he k he ce

hei a e . B a h ha i i j he la e Ve . If c ide i ai a e h a bjec i he k a ea he face f he la e ,

he igh h ld ha he M i g S a i e al he E e i g S a . T e edi c i i h a e igh ag ee ha Ve i ide ical i elf i h he

e f ha i g diffe e a a a i i .

4.4 Predicate environment in Mathematica

4.4.1 Notations

Si ce e e , a di c i f e a i ill be ef l de a decific a e e i hi b k.

See Sec i 1.8 f a l ge di c i f he Li bjec a i ha beei le e ed he e. A Li i a de ed bjec . If a a al habe ical de he

e S U i .

† A li ca be a hi g, e.g. be , b l , i g , e e i , he li , ... Thee li i hile« i e el he he e ical b l f he e e . Th , 1

c ai b h a d 1.lis = 80, b, John^2, “Amsterdamwill befloodedin 2060”, 8<, ∆ , 88<, 1<<

90, b , John 2 , Amsterdam will be flooded in 2060, 8<, « , 88<, 1<=

† The f ll i g defi i i ca e a i fi i e e al a i a d agai a e f ai i g ill e e i .

selfreferent = 81, 2, 3, selfreferent <;


106



† Thi c ea e he li f he i ege ill 20. T ake i a bi e i e e i g, e a ehe e e be a d ake he a e f he e e be .

Table @If @EvenQ @iD, i ^ 2, Sqrt @iDD, 8i, 1, 20 <D

:1, 4, 3 , 16, 5 , 36, 7 , 64, 3, 100, 11 , 144, 13 , 196, 15 , 256, 17 , 324, 19 , 400 >

† al ec f he i fi i e ced e .Table @i, 8i, 1, Infinity <D

Table:: iterb : Iterator 8i, 1, ¶ < does not have appropriate bounds. More…

Table:: iterb : Iterator 8i, 1, ¶ < does not have appropriate bounds. More…

Table @i, 8i, 1, ¶ <D

† The ced e J i j li k he li e e he he a e ele e cc . Thi ief l he he ele e a e e el ele e b e e e e c e c de

f hich he i i a d / he f e e c f cc e ce i i a .Join @81, 2, 3 <, 83, 2, 1 <D

81, 2, 3, 3, 2, 1 <

Mathematica al ha f c i ha ea ch a d e f ele e f li . See Sec i 2.3 f le elec i a d a e ec g i i .

† Me be hi i T e Fal e. Wi h e ec he defi ed ab e:MemberQ @lis, ∆DTrue

† E i e ce ca be e ed i h F eeQ. F eeQ[ ,_I ege ] gi e T e iff i f ee fi ege . If i f ee f he , he he e d e e i a i ege i. The e i a lea

e i ege i iff F eeQ gi e Fal e.ÿ FreeQ @lis, _Integer D

True

† Whe he i e ca be e ed b Le g h b a di ec e .

8Length @lis D === 0, lis === 8<<

8False, False <

107



8e1 , e2 , < a de ed li f he ele e ; ca al be e e ed a Li@e1 , e2 , DTable @expr , 8i, imax <D ake a li f he al e f expr i h i i g f 1 imaxJoin @list 1 , list 2 , D c ca e a e liMemberQ[ list , form ] e he he form i a ele e f list FreeQ[ list ,

form ]

e he he form

d e cc ilist.

F eeQ[ , , ]e l h e a f le el ecified b Length[ list ] c h a ele e cc i list Position[ list , form ] gi e he i i a hich form cc i list

Cases[ list , form ] gi e he ele e flist ha a ch form Select[ expr , f ] elec he ele e iexpr f hich he f c i f gi e True Count[ list , form ] gi e he be f i e form a ea a a ele e flist

Definition of the List object, creation, testing and searching for elements of lists.

c ai e e he e ical f c i ha a l li . The d ece a il a l he i f b e ha c ld e e d he i e

elf. The ced e a e f ll e he e ic i ce add efea e . F i a ce he i e e a be a g d he e ical c ce b f ac ical

e i be a ed.

† U i l eli i a e d ble cc e ce b al he e l . U ic ide he li a a e ha he de d e a e , he ce i i ea ie li

he ele e a habe icall ha i ead ea ie .Union @82, 3, 1 <, 8c, 3, b, 2, a, 1 <, 8b, c, a <, 811, 10, 3 <D

81, 2, 3, 10, 11, a , b , c<

† A e i h 3 ele e ha a e e i h23= 8 ele e .Subsets @8A, B, C <D

88<, 8 A<, 8 B<, 8C <, 8 A, B<, 8 A, C <, 8 B, C <, 8 A, B, C <<

Union[ list 1 , list 2 , ] a i e li f he di i c ele e i helist Intersection[ list 1 , list 2 ,

] li he ele e ha cc i all helist i

Complement[ universal , list 1 , ]

li he ele e ha a e iuniversal , b i a f helist

Subsets[ list ] lists the powerset, all subsets of list

Set theoretical functions.

E add e f c i . E i Q diffe f F eeQ i : (1) i i e

hi ki g a he ha ega i e hi ki g a i F eeQ, (2) i e T e Fal e e a heha a e ec g i i ha he i d e ai i he e i g de, (3) he defa l ihe fi le el a d all le el .

108



† C a e E i Q a d F eeQ.ExistQ @8a, b, c, 81<<, IntegerQ DFalse

ÿFreeQ @8a, b, c, 81<<, _Integer D

True

† O de edU i i a ice i e i f Ca l W ll i e e b kee i g he a ede . PM. The i e J i Ne e C le e ide if he e , b

C le e ha e .OrderedUnion @82, 3, 1 <, 8c, 3, b, 2, a, 1 <, 8b, c, a <, 811, 10, 3 <D

82, 3, 1, c , b , a , 11, 10 <

JoinNews @82, 3, 1 <, 8c, 3, b, 2, a, 1 <, 8b, c, a <, 811, 10, 3 <D82, 3, 1, a , b , c , 10, 11 <

AllQ @list , crit , levelspec_ : 1D gi e T e if i a e Li a d all iele e ca e c i be T e. The i e eMa Le el. Le el ec Hdefa l 1L a d i a e f Le el

ExistQ @list ,crit , levelspec_ : 1D

gi e T e if i a e Li a d e fi ele e ca e c i be T e. The i e e

Ma Le el. Le el ec Hdefa l 1L a d i a e fLe el. U e le el ec I fi i he c i i a l a le el

OrderedUnion @list 1 , list 2 , D

gi e he i f helist i i h i g a U i ld d ; i f Sa eTe ; alg i h f Ca l W ll

ForList @list , e, var_List , body D e al a e b d f all ele e i i g f al ele e . F e

Some additions in The Economics Pack . See ShowPrivate for the specific implementations.

A lica i f f c i edica e he ele e f li ca be d e i hef ll i g a e .

† F he e a i " œ A : [ ], ha he ced e Ma [ , A] a d ihe li [e1], [e2], ... f he ele e i A. Thi ca al be de ed a /@ A. Whe

i a e he he e l i a li f T e Fal e al e .NumberQ êû lis

8True, False, False, False, False, False, False <

† F Li ha a f a ha e i e i e ag eeable e he e ic a i , a dha a he a e i e all e c lica ed f c i ha e a ed i e ala iable . If ld e he i e al f c i e f e he f c e i ld

109



be ad i able e a a el defi e i a d e Ma di ec l . B f i gle a lica ihi igh k ell.

ForList @lis, x, 8y, z <, y = If @StringQ @xD, 100, x D; z = If @ListQ @yD, 200, y DD

90, b , John 2 , 100, 200, « , 200=

4.4.2 Quantifiers

Mathematica ge e all he a da d a f a he a ical l gic. The Mathematica e i e f ce ecific ch ice h gh. N abl , a iable haa ea i he a ifie " a d $ a ea a b c i he i e he e c ld be ac flic i h l i lica i . Whe c c i g f la i ee ge e all ea ie e

e al i a d le d all he la .

† Thi a e ha e h ld f all ai ha a i f c di i C.ForAll @8x1, x2 <, C@x1, x2 D, P@x1, x2 DD" 8x1,x2 <,C @x1,x2D P Hx1, x2 L

† If he e ie a e de e di g he e ge a e e i lica i .ForAll @x, C, P D

HC P L

ForAll[ x, expr ] " xexpr expr h ld f all al e f x

ForAll[{ x1 , x2 , }, expr ]

" 8 x1 , x2 , <expr expr h ld f all al e f all he x

ForAll[ x, cond , expr ] " x ,cond expr expr h ld f all x a i f i gcond

ForAll @8 x1 , x2 , <, cond , expr D

" 8 x1 , x2 , < ,cond

expr expr h ld f all xi a i f i g cond

Exists @ x, expr D $ xexpr he e e i a al e f x f hich expr h ldExists @8 x1 , x2 , <, expr D

$8 x1 , x2 , <expr he e e i al e f he xi f hich expr h ld

Exists[ x, cond , expr ] $ x ,cond expr he e e i al e f x

a i f i g cond f hich expr h ld

Exists @8 x1 , <, cond , expr D

$8 x1 , x2 , < ,cond

expr he e e i al e f he xi

a i f i g cond f hich expr h ld

Two forms for quantifiers in Mathematica . NB. AllQ and ExistQ in The Economics Pack concern only Lists.

The g ide a ha i ca e he a ifie ill i edia ele al a e. B he ca be i lified b i ed i eReduce Resolve .

110



F e hi , ee Sec i 3.4.11.

† Thi a e ha all a e a e i i e. Thi ce ai l ld h ld f all ega i ebe . The e e i h e e d e e al a e e .

ForAll[x, x^2 > 0]

" x x2 > 0

† The e e i gi eFalse i ce he i e ali fail he x i e .FullSimplify[%]

False

† Re l e ca eli i a e a ifie f h a e e . I he e a a ig e hea iable ha ake he e e i e ?

Exists @8p, q <, p »» Hq && ! qLD$8 p,q< H p Í Hq Ï Ÿ qLLResolve[%, Booleans]

True

† N e ha ki dl e l e e i f a i g f . I he f ll i ge e i he e igh ee a ge e he i he i e $ x e hec e i i ha a b d a iable l k f i cl e b d.

Exists @x, x ^2 > 0 && Exists @x, x ^2 ä 0DD$ x J x2 > 0 Ì $ x x2 0NFullSimplify[%]

True

4.4.3 Propositional form

We a be able ec g i e a i i al f e e he a ifiecc . Thi h e e e i e e i ge i i ce ch c c ca be c le ( eehe f e ec i ).

A a e e like bei g e i a edica e, i a a iable, a d f all , a i fie hiedica e i e ha d bi , i ce cc b h f eel a d a a b d a iable,

hile edica e ha diffe e le . Si l i g e lace e i ighca e ha c. We ill a be able a i la e a a e e like ha , if l e e hi d bi e . B i ad a ce i i clea ha ki d f edica e e ill

e. A ag a ic l i i kee a li f edica e ha e ca e f , a d, if ee a e e, he e add ha he li .

111



† Thi e a le i a d bi a e e ha ha all cc i g all e . The i eS a e e k ab F All a d E i a d h ca fi d he a iable f

i i al l gic.example = p Ï ForAll @p, P @pDD Íq ÿ ForAll @x, P @xDD

III p Ï " p P pM Í qM $ x HŸ P xLMStatements @example D

9 p, q , $ x HŸ P xL, " p P p=

† The i i al f ca bec e clea e b aki g b i i i h f ala a e e ( hile kee i g he e ha d ca e ble ).

PropositionalForm @example D

HHH p Ï P 2L Í qL P 1L

† The P i i Mea i gR le i hi ca e l li he e ha a e ha d ec g i e.

PropositionMeaningRule

9 P 1 Ø $ x HŸ P xL, P 2 Ø " p P p=

Statements @ xD gi e he li f a iable i a i ghe ibili f edica e . Whe Sh Ø

Fal e he j f al P@iD a e h , ee P i i Mea i gR lePropositionalForm @ xD gi e he f f he i i he e all edica e k i

P edica e Q@Li D a e e laced i h f al a a e e P@iD. Thei e e P i i a d he P i i Mea i gR le,

ee al Re l @P i i alF D. The i ee Add i O i @P edica e D e la ge he li

PredicatesQ @ xD e T e he a lea e f hePredicatesQ @List D cc i , a a le el

H b a a head f a e e i a d j a a b lL. Thei i a e ed i Re l@P edica e QD. I eAdd i O i @P edica e QD e la ge he li

PredicatesQ @List D c ai a li f edica e e f i S a e e

Tha k he e i e , e ca ake ef l h able . N e ha T hTable i a ec g i e he" a d $ igh be c adic i g.

112





NoteElement @expr D fi e lace Ele e i h Bel g ,a d N Ele e i h ! Bel g , he e al a e ,

he b i e back. F e a le N eEle e@ œ P Ï – P êêL gicalE a dD ill e al a e Fal e

Belongs @ x, yD like Ele e , ha N @Bel g @, DD ca be ec g i ed bL gicalE a d. I T adi i alF i i e ac l he a e

ToBelongs @expr D e lace Ele e i h Bel g , a d N Ele e i h N @Bel g @DDFromBelongs @expr D e lace Bel g i h Ele e

4.5 Theorem that the Liar has no truthvalue

4.5.1 The theorem and its proof

: I a fficie l ich e f al ed l gic ( ee ab e f**) he e d e e i a Lia . See f ha e he Lia ha h al e (T e

Fal e).

: The f ha he f ll i g c e. Fi e e he h he i ha a Liae i . The e de i e he c adic i , ca i g e ac he h he i . Tha a

f he ded c i i ai ai ed a a i lica i , h e e . S b e e l , e e hai lica i de i e he ai e l .

Le ha he e i a a i g ha i i e. The he f ll i g ded c ica be ade.

Ergo2D @"L = Not @TruthQ @LDD", L ÿ L, ÿ L L, L Ï ÿ L D1 L = Not@TruthQ @LDD2 H L Ÿ LL3 HŸ L LLErgo —————

4 H L Ÿ LL† Check hi a f he ded c i

HL ÿ L L Ï Hÿ L LL HL Ï ÿ LL êêTruthValue

1

The fi h he i ca e a c adic i ha e e ac he h he i , hilekee i g he i lica i f he e .

114



† He ce = N [T hQ[]] (L N [L]). H e e L L.Ergo2D @"L = Not @TruthQ @LDD" HL Ï Not @LDL, L »»Not @LD, "L π Not @TruthQ @LDD"D1 HL = Not@TruthQ @LDD H L Ÿ LLL2 H L Ÿ LLErgo ————— 3 L ∫ Not@TruthQ @LDD

† Check hi a f he ded c i

HHp qL Ï ÿ qL ÿ p êêTruthValue

1

O e l ki g he h le, e fi d ha e ha e eached a defi i e i ha e hehe e .

† Fi al c cl iErgo @"L π Not @TruthQ @LDD"D

H ¢ L ∫ Not@TruthQ @LDDL

. . .

PM. Thi f ee e he c e fac al S e he e i ... .** all f

c e fac al ea i g a d l all he b i i f h i h , i ce*

d e . H e e , ab e f ill k i ceca be ed i he e e f All ha he

a i h ld f all a iable . F ** i i deed i i a e ial he he e i e

f he Lia c c i . The f h al h ha f all ch c c i ca be gi e . Tha bei g aid, i ill i ef l a e he f i e f S e he e i ...

i ce ha ead be e f ch l gical ea ( hile i al i alid f e ha

acce h he ical ea i g).

B i h f ll ha he e i e e ce i ha a i fie hedefi i i f he Lia . I e i ale d :

(1) The e i i ch ha = N [T hQ[ ]](2) F all i i h ld ha ∫ N [T hQ[ ]]

I he e h d l g f cie ce, Na e i he (i e ded) i e e a i f a d e cab e e ha he i e e a i ill a d , i ce e d b e e ha Na e f

a e e ch a he Lia .

The l ble ha e ha e i e l i e e e hi e l . We k a ac llec i ch ha e fel ha a e e ce c ld be c c ed b he

ca be ead a a i g d f he Lia . I be ad i ed ha if he he ei added he f a i le f he hi i a a a e e he cc e ce fhe Lia igh i ead f ha i g fi d hi b ea f ded c i . H e e , e

115



feel f a ed b hi i ce e ca f a Lia i a al la g age a d e hadh ed ha ld c e la g age.

We c cl de ha a al la g age ed f h a c ica i i e la geha . Bel e ill de el h ee al ed l gic all f e e ce like he Lia

i h a I de e i a e h al e. Th e ha e i h al a f E gli h a de le l gic de e i e he he a e e ce i i \ , i h el i g f a i le f .

Th ee al ed l gic i e i ed f a al la g age . : The Lia i ae e ce ha ca be f ed i h T e Fal e a d i ce e a be able f

i , i ha e a hi d al e. Q.E.D.

F la g age a h ee al ed l gic i al fficie .

See i i el , he Lia ide a e i e ce f f h ee al ed l gic. I i like hei d c i f he 0, he e e le igi all did k ha fi ge e f i .The e i l gical di i c i be ee e ic i g a al ed l gic a d hei d c i f a hi d al e (i.e. he ∫ ).

We ill a N A All[ ] iff i i ea ha i e ical a d ha h al eI de e i a e.

NotAtAll @Liar D

† Not@TruthQ @Liar DDPM 1. i called dagge b e (i deed i he fficial li f b l ) a d

b he i i ed i blica i a a c i dica e a e decea e. The e i h

a c i e a a al dea h i l ed b a h i h he gh f li ge .

PM 2. We d eed ( = ) i ce e al ead ha e ( ∫ ). Th he l a lie a

if i e e defi ed ch.

PM 3. i a ef l ge e al i . C ide he a e e All d a e a e d.

Th gh hi i e ical igh h ld ha i i ac icall c ec i g All X Y a e

X . If a e igh he f c e al All X Y a e Y he ce All d a e a e

a e . Th All d a e a e d a d all d a e a e d :

c adic i ! A l i igh be a i he fi lace ha All d a e a e

a all d a e i ce he d e i i he fi lace .

116





a d e: b i a ea be a ble e e i beca e i i ea ee ihich f he c ec i he d ab l el i be e de ed i h T e i h Fal e. The e i , h e e , hi g e e i f bei g Fal e ab l el , h gh

T e i e a ic la e ec (...) O e igh i e e e hi a h ee al ed l gic. B

a a e l ha eed e de el e bec e acce able.The la e a ach a ce ai l acce ed b A i le c e a ie , a d hea a e i l abili f he ble a ac ed he a e i f a . The h a , adi ci le f A i le, e h ee b k he bjec , a d Ch i f S li e ha28. We al ead e e be ed he ad a i g f Phile a f C .

B i e e he S ic , a d a g he Ch i f S li (ca. 281 208 BC) h i deed b gh he ble e l i . T a i h, he acce ed a ba ic i f

A i le ab he bjec a e f l gic: Ye e e e e ce a e e hi g, bl h e i hich he e i h fal i , a d all a e f ha ki d. Th a a e i

a e e ce, b i ei he e fal e ... he e e he i c ce ed i h che e ce a a e a e e . (B che ki (1970:49))

The e ( cab la ) , acc di g A i le, a eal he l ( heigi f ea ): Ce ai l a igh i i he b d , i ea i he l (...)

(B che ki (1970:48)) a d All ll gi a d he ef e de a i iadd e ed a d eech b ha i hi he l. (B che ki (1970:46)).

A i le a ia Pla a g a d il f S c a e .

I d e ee i c ec a ha he l ( i d) i he ce e f he ea i g fc ce a d h a he a e i e he fac l ha di i g i he e e ( i i )f e e ( e el e e ce ). (See B che ki (1970:46) a d Be h (1959:24))

A i le h e e did i e all a l hi i e i . I a e he S ic h ake aa i h i ge e al a lica i . Ch i a : The (fallac ) ab he heake a d i ila e a e be ( l ed i a i ila a ). O e h ld a ha

he a e a d (al ) fal e; h ld e c jec e i a he a , ha he a e( a e e ) i e e i e f e a d fal e i l a e l ,

. A d he ejec he af e e i ed i i a d al he i i ha e caee e a d fal e i l a e l , a d ha i all ch ( a e ) he e e ce i

e i e i le, e i e e e i e f e. (B che ki (1970:133), i alic )The e i al a he f ag e f Ch i : he ( h a e he Lia ) c le el

a f d ea i g; he l d ce d , b he d e e a hi g(Be h (1959: 24)).

Wha ha bee a la ed he e a ea i gle ld be a la ed i cab laa e ele . We ee ha a clea di i c i be ee i i a d e e ce i

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bei g ade, i ch a a e ha l i i a e (ei he ) e fal e b i ef he fac ha l he ca e e; a d i ce he Lia ca e a c adic i i bee ele a d he cef h i be e el a e e ce a d a i i .

Be h (1959:24) gge ha he aj i f he S ic did acce Ch i ie

b he I ega d Be h he ab ha aj i he I a i cli ed c cl de hahe i fac did acce i . Fi he e i he ge e al S ic defi i i f a i i

( a e e ) a a e e ce ha i ei he e fal e ; ec dl he e all ge e allacce ed defi i i f h a d fal eh d: e i ha hich ha e be a d i

i e e hi g, a d fal e i ha hich d e ha e be a d i i e e hi g . The Lia ha clea i e ( likel I eak he h ) a d

he ce i f ll ha he Lia i a i i ( a e e ). Thi a ach iide ical ha f Ch i ce a l k i he e f bei g

i e e hi g a d ee ha hi i lici l a eal e ea i g e e. Ica be be a la ed a ha i g e e i ce i i l f e ical a e e ha eca i agi e a i e.

N , c ce i g he e hi ical e a k , a d f ca i i e i ed. F e hi g,l li i ed i f a i ab he i i f A i le a d he e i , a d he

a e ial ha e i i f e f ag e a a d l a iall e e ed. E e adi ag ee ab a la i a d filli g f he ga i f ag e . F e a le, he e a e

a ible d f he c ce f e e ce , e.g. a e e , i i , a e i ,e e i , he e , b i a he e ical de el e he e d ge a eciali ed

ea i g, a d he a la i bec e a c i ical i e, i h he i k f i i g la ehe ea lie i e . Readi g Be h (1959) a d B che ki (1970) e a dehe he he e a h ha e a lied he a e a la i c de. We fi d he a e

ble i h e ec he a la i f ea i g e e . The e c cl ii ha he e hi ical e , h al able he a be, ca be ed each a fi al j dge e he i i f he a cie G eek he bjec f he Lia . The l

ge e a e e a i e e i a d a e , a d h ide e hi ical i igh a da a i g i f f he a al i .

C i i g hi h hi ical e ie , i a be e i ed ha he Lia al cc ihe Bible, a el he e Pa l a : O e f he el e , a he f hei , aid:

C e a a e al a lia , il bea , la gl . (Ti I:12 13, ee DeL g(1971:30)) I i ge e all a ed ha hi he a E i e ide , h li ed a he begi i g f he 6 h ce BC. A a e l he Lia a ad i e i e a c ibed

E i e ide : b hi be g, i ce E i e ide ce ai l ld ha e beei g ab he l gical ble . If he ld ha e e ed ha h a e, i c ld l

ha e ha e ed beca e he did ee ha i h ld be fal e b i e f he fac ha he

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hi elf a a C e a a d h a lia , il bea a d la gl hi elf. Al a lePa l did ee hi , i ce he add he c e : Thi e i i e. (Ti , ide )

The Ch ch Fa he A g i e ca be e i ed ha e died he a ad .

4.5.2.3 Middle Ages and Renaissance

B he i e f he Middle Age he k f he S ic e e c le el f g e a dl he k f A i le beca e g ad all k , he e he la e e e all ca ed

he Re ai a ce. The Medie al Sch la ic h had e ake he a e de el ef A i le a he S ic had.

Pe ha e a e a l gical he i ca e a d fall f ci ili a i a d heb e e ha he e a ill i e . Ab Sch la ic l gic B che ki e : he

ble f e a ical a i ie a faced i eall e ea i e . N ea i ie f hi ki d e e i ed a d (...) e ha a d e diffe e l ia e ed. Be ee he he c ai ea l e e e e ial fea e f ha e k

da hi bjec . (B che ki (1970:251))

I deed, Pa l f Ve ice ( b. 1429) gi e ab 15 l i , f hich a e i e e i g(B che ki (1970:242)): The fif h i i a e ha he S c a e a ha he hi elf

a ha i fal e, he a hi g ... ( e i di g f Ch i ), a d The i hi i a e ha he i l ble i ei he e fal e b e hi g i e edia e,

i diffe e each (a h ee al ed l gic). I i ef l he e e ea h Pa l fVe ice h gh he e a ache be i ade a e, a d h he h gh he be

. (O de el e ake he e a bei g he a e.) I ead i i ef l eh ee i :

1. I ead f i g he e e ce I lie he Sch la ic a i g he f a dable he e e ce Thi e e ce i fal e . B ida e e e The e e ce

i e hi age i fal e hich e e ce be i e a age ha i

f he bla k. The la e ced e hel eli i a e ag e e ab d likelie a d hi . The d lie a el a be el l gical b igh ha e

c a i f decei , h e lai bad e , hich igh c f ehe el l gical i e. Al he elf efe e ce i i le a e b fficie l

c le c i ce acce i .

2. The Sch la ic a ici g he c i i f he Lia . Whe= Thi e e ce i fal e he e e he cc e ce f ide , a d he e i i ed b

P e d Sc : I i be aid ha a a a a a ca a d f he h le

i i . (B che ki (1970:238)) Elab a i he la e b Willia f Ockha( b. 1349) al k f Occa a ( e e ialia l i lica da )

b i g e cl e a l i . A e e i e e f S a hki (1969:44) i

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ele a , a d i dica e Ockha a a e e f he di i c i be ee e i ale ce(ñ ) a d ide i defi i i (=).

Acc di g Ockha , he ce f he a i lie i he fac ha he e e i ed

f a i f i i a e e i e ed f a i f he a e i i i

hich he a e ed a c i e . M e lai l , Ockha ea ha a f ai i ( he edica e i fal e ) (i de eli i a e a a i ) efe

he e i e a i i hich i a ea . (The ble f hi e f e ic i did a ea i c ec i i h he edica e i e , beca e, i defi i g h, Ockha ,

f ll i g A i le, a ed ha he i i (1) a d (2) i e a e e i ale .)

Ockha ie he ef ed ce a i e dic f ci c la defi i i . I he d , i

i e i ible e i e li g i ic c c i i hich, f e a le, a gi e

i i a eal di ec l fal e e e . (...) B eli i a i g ci c la a g e i

hi die f a ad e , Ockha a led hi fa a ( e e ial h ld bel i lied e ha ece a ). He belie ed ha hi l i he ble a he

ge e al ible (i he e e ha if hi c i i i f ll ed, a i ie d

a ea .

3. The e i he de el e f e f e e ce ha gi e i e a ad e .S a hki e ha B ida , il f Ockha , f d a c e e a le i

hich Ockha c i i i i la ed ( he e i di ec a eal fal e e ), b a a i e hele a ea . The e S c i f he f ll i g

a e e : (A) a i a a i al , (B) l A i e , (C) A a d B a e he la ailable a e e . (...) The e S he ef e c ai a a ad ical a e e ,B. I ie f hi e f e e i , B ida di i g i hed e f a ad ical

a e e : The fi e c ai di ec efe e ce (...) hile he ec d ec ai i di ec efe e ce . (S a hki (1969:44 45)) Ha i g e abli hed hec ce f i di ec efe e ce, B ida did i l f bid i e (a Occa hadd e f di ec efe e ce), hich i e ible i ce ch i di ec efe e ce igh beca ed b e e le a d i i a he diffic l c l hei ac i . B ida

ed l i eed be c ide ed he e, h e e .

I he fl e i g f he Re ai a ce f he k f he Sch la ic e e a ide,a ld a d d , i cl di g he k l gic, al h gh he cle g e f ed a allg i hich he idea e e ke ali e.

4.5.2.4 Since Leibniz, Boole, Frege, Peano and Russell

Wi h he k f Leib i (1646 1716), B le (1815 1864), F ege (1848 1925) a d Pea

(1858 1932) a e i e e i L gic a ke, i hich l gic de el ed di i ca he a ical fea e . The i e e i he Lia a ad h e e l e ed af e

1900 he R ell li ked he Lia hi a ad e f e he . He ed a

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ible di c he ela i f la g age eali , he la g age i e edhe e di c e a hi g, ee Wi ge ei (1921, 1976:8). R ell ed he

f ll i g l i . The di i c i f e i a d e f a e e ce al c ea ed hedi i c i be ee a bjec la g age a d a e a la g age. S ch a e ca di c

D ch a d Ge a i E gli h. All e e ce f he a e e ca be aid ha e a a ele el, ha he e a e le el i a la g age, i h a l e le el bei g he bjec f a e ale el. Th he ela i f a bjec la g age eali ca be di c ed i a e ala g age ha Wi ge ei ble a e i e .

Th gh Wi ge ei did acce hi l i ( h babl e cei ed ha he c ldal e Ge a di c E gli h), a he did, abl Ta ki (1949). I fac ,acc di g Be h (1959:493): he fi al di e a gli g f he k i d e Ta ki . Le

e ha a defi i i f h i ible f a la g age, a d f c e e e i ef ch a defi i i ha i d e lead c adic i . The h e e he Lia if ed, e l i g i a c adic i . The ef , ch defi i i f h i ible.Be h (1959:511): I f ll ha , if a ade a e defi i i ca be gi e f h a dfal eh d f e e ce i a f al e A (...) he hi defi i i ca be

e d ced i hi ha e A i elf; f he i e e c ld agai de i e he liaa ad i hi he e A. Thi la e a k h ha e ac l i g i h

di a la g age f e e i f ie ; f i ill be clea ha , if a ade a e

defi i i f h a d fal eh d c ld be gi e f e e ce c ai ed i di ala g age, he hi defi i i c ld al a be e a ed i hi di a la g age.

Th he ie f R ell a d Ta ki e ge he e, h ldi g ha a ade a e defi i i fh i ible i di a la g age a e c l de a d i b l i a

f ali ed e c ed la g age, c ea ed b l gical a d a he a ical ea , i h aell defi ed c e f e : i hich he l e le el l di c e a e ial bjec

a d i hich he c ce f h f e a ic la le el l a lie e e ce f hele el j bel ( e ha all l e le el ). U e f di a la g age igh be

a a e f hi , j a a e igh ide a bike i h bei g able e lai h . I fa a di a la g age a ea ef l ha ca l be i ce he e igh be hidde

ce e i he i d ha e a e a a e f ha ca e e ec he le el fa lica i .

Be h (1959:510): The Lia a e e (...) be acce ed a e i i g; i i i ed ihe e e b k a d he eade a ead i al d, if he h ld i h. Si ce he Lia

e i (i a al la g age) he a lied c ce f h be i ade a e. The ef i

i ece a f la e a he (a ificial) la g age ( i h e ) i hich a ade a edefi i i f h ca be gi e . A d he e e ha he e i a Lia a e e

he i ca be h ha he e i a .

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PM. Pa ad ical i a ce a a e l ca be e l ed. A le el Lia[ ] = [ 1] f le el

1 i fal e igh be e e ed b h ldi g ha he e le el ha di c e eali ha

l e la g age le el. Whe a i e le efe each he ha he le el i

de e i ed he he be e i ed e he a ificial la g age a d ecif he

a ia e le el.The ie f R ell a d Ta ki c i cide i h he S ic ie , f cef ll e e ed b Ch i , ha he Lia i a e ical a e e , f ei he i i la e he e

c e i e i e a i ade a e defi i i f h. C a he S ic R ela d Ta ki gi e e ecific ea h he e e i ca ed. The S ic ga e e

ea b R ell a d Ta ki a e e ecific.

PM. The S ic ie i e i el i h ble ei he . If he e e ce = Thi

e e ce i fal e i e ical he i i ei he e fal e. Tha i lie ha i be

e, a lea . B a ha i i e. He ce i i e a d h e ical.

E ce e a. See DeL g (1971:242). B al ee Cha e 7 e l e hi .

R ell a d Ta ki l ed he c adic i f he Lia , b a e c :

† The decla a i f a e c e ba icall ea he e cl i f elf efe e ce. Ae e ce ca efe i elf i ce i ca be a le el a he a e i e. Wh

i l f bid elf efe e ce ? U i g le el b c e he de c i f elfefe e ce. T de elf efe e ce i a high ice i ce i igh be a ice e . I

fac , la g age i elf efe e a i e , ha he he f e i i ade a e fa al la g age.

PM. Wi ge ei ble ca be j dged e i e beca e f elf efe e ce. Whe a

e e ce di c e he ela i be ee i elf a d eali he i i ble ha

la g age i e ed he ha la g age ca be elf efe e .

† Wi h R ell a d Ta ki e a ad he i ha di a a al la g aged e igidl ec g i e a he f e . B he c cl i ha a defi i i f

h he bec e i ible f a he a ical l gic a be g. S ch ac cl i i la e ba ic i i i ab he ge e ali f he c ce f h.K i ke (1975:694 695) e e e ha ea e: Th fa he l a ach he

e a ic a ad e ha ha bee ked i a de ail i ha I ill call heh d a ach , hich lead he celeb a ed hie a ch f la g age f Ta ki.

(...) Phil he ha e bee ici f he h d a ach a a a al i f i i i . S el la g age c ai j e d e , a e e ce fdi i c h a e true n , a l i g e e ce f highe a d highe le el .

The he f e h ld ha he Lia a ad i ca ed b a i ade a e defi i i fh. S e ejec ha ie , ee Ma i e al. (1970) a d abl K i ke (1975:698):

Al all he e e i e ece li e a e eeki g al e a i e he h d a ach

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I ld e i e eciall he i i g f Ba a F aa e e R be L. Ma i (1970) ag ee a i gle ba ic idea: he e i be l e h edica e, a licable e e ce c ai i g he edica e i elf; b a ad i be a ided b all i g hal e ga a d b decla i g ha a ad ical e e ce i a ic la ffe f ch a

ga .Th a hi d al e ( ga a ad ical U de e i ed I de e i a e) ii d ced, ff e i g he dich f T e Fal e. I deed, if he Lia ca bef la ed i a he f e , b e f la e i i di a la g age, he ed lea e i ? N all ld ha e a b he a ic e ff d e.

H e e , f e a le K i ke agai acce e e a d li i a i he eb , eeK i ke (1975:714): (...) he e a e a e i e ca ake ab he bjec la g age

hich e ca ake i he bjec la g age. F e a le, Lia e e ce a e ihe bjec la g age, i he e e ha he i d c i e ce e e ake he e; be a e ecl ded f a i g hi i he bjec la g age b i e e a i fega i a d he h edica e. (...) The ece i a ce a e ala g age a be

e f he eak e e f he e e he . The gh f he Ta ki hie a ch i illi h . K i ke al efe a al la g age i i i i e i , bef ehil he eflec i e a ic (i a ic la , he e a ic a ad e ) a d h

gge ha a al la g age igh e e be able di c a d l e he

a ad e ...Q i e (1990:80) a e : A c i i f h j ca cel he a i a k . T h idi a i . (I he i e ced e a e T S i g a d T E e i .)H e e , f a la g age he al e i e : I h edica e (...) be i c le eldi a i al. S ecificall , i di e all he e e ce ha c ai i . ( 83)a d A d e e he e e cl ded a lica i ca be acc da ed b a hie a ch f h

edica e . (...) defi abili f h f a la g age i hi he la g age ld be ae ba a e . Th he he f e i li i ed e e ce efe i g h.A e e ce Thi e e ce i true n ei he c ld be f ed a be j dgedfalse n+1 .

4.5.2.5 Summing up

The f ll i g i a be ed:

† The le el ( e) c e a a l he h le la g age i ce he e e i ca ef elf efe e ce ( all a ad ical).

† The di i c i be ee le el i la g age a be f a l gical b a li g i ica e, ea i g ha igh d i , f e a le a di c i g E gli h i Chi e e,

b i all likelih d he e i l gical ece i clai le el .

125



† The ad i f a h ee al ed l gic ee ge e al. Ye , e h hac adic i ca be a ided. I i elf hi i a diffic l c ce i ce a

c adic i ee l defi ed i al ed l gic, ha i igh be cleaha e h ld a id he e ad h ee al ed l gic.

The e i i a e i he e i he i i g f A i le. Ob i l he a c ci feglec i g ea i gle a d di i f a i e e e ce he he da ed a ha alle e ce a e e fal e. Reg e abl he a e lici a d c le e he

a i a d e e hi f ll e . The ad i f a h ee al ed l gic he ce ca be e e ie ced a e A i elia . B i eed eall be .

Thi hi ical li e ha bee ef l e he ble , i d ce e c ce ,a d a i e a e a i e a e a d e ea ch e i . We de ha he

defi i i f h eall i , h h ee al ed l gic l k like, a d . I ceedi g,e h ld heed he a i g b Be h (1959:518): a i di e ge i i hi

ic ha e bee defe ded b a be f a h ; he e he e d i i , h ghf e e l e ac e, e l i ca e f a de e i a i f he

e h d l gical i a ce f he a ad e a d f a eglec f he a da d fig hich a e cha ac e i ic f c e a f al l gic.

4.6 A logic of exceptions

4.6.1 The concept of exception

H a hi ab d i h e ce i le . Rega d he f ll i g e a le(e ce he f h):

† A la like All de e ill be ha ged . Well, f c e, e ce ha he a a ka d .

† The e ca be e ce i a le, ha e a le.

† O life i led b d c i e . E ce ha life i ge ha a d c i e.

† Ca ch 22: S ldie e ai i ba le, e ce h e h l hei a i . If a l fhe e ce i he ha i ake a a f f a i .

The e a e diffe e a e e e ce i . I he f a f e he e ighdefi e f a e A:

A == 8

x H

conditions@xDL

~ Unless ~ H

exceptions@xDL<

A 8 x HŸ exceptions H xL conditions H xLL<

126



† F e a le:A == 8x Even @Sqrt @xDD ~ Unless ~ Hx £ 0L<

A : x J x > 0 EvenJ x NN>

† I ca e a e ied ab a ea he e A i defi ed, ca e he le ,he c c i , like: I ll ake a a le, le a , f he I ll ake a big

ch c la e f dge .A == 8x Unless @conditions @xD, exceptions @xD, then @xDD< A 8 x HHŸexceptions H xL conditions H xLL Ï Hexceptions H xL thenH xLLL<

The diffe e ce be ee c di i a d e ce i i f a ch l gical a e, i hahe e ce i e d cc le f e e l , e e ch le ha e le e d f ge

ab he . B i a f al e e he a e c di i like all he he . E ce , hae ce i e d d i a e he he c di i . N al c di i d a l he

he e ce i ike .

ha ell de el ed fea e f a e e i g a d kee i g acc fle f e ce i .

† The f c i C di i ( b l /;) e he i f a f c i a d l all i e al a e he he c di i i a i fied. The f c i h d e e al a e e ce

he he c di i a e a i fied.testedSqrt @x_ D:= Sqrt @xD ê; x > 0

8testedSqrt @10D, testedSqrt @- 10D<

: 10 , testedSqrt H- 10L>

† The e b l E ce e e e a a e bjec ha fi a hi g e ce h e haa ch i .

Cases @8a, b, c, d, e, f, 1 <, Except @ _Integer DD

8a , b , c , d , e , f <We a ake he c ce f e ce i be ell e abli hed.

4.6.2 The liar

O hi ical e ie f he Lia ga e a i ca e , i h A i le, he S ic , heedie al , he e a f he ed l i a i d ce a e ce i . The ai

e ce i ed a ha Te i N Da h ld le a e e ce i ai i .

127



Hp Í ÿ pL ~ Unless ~ NotAProposition @pD

HŸ NotAProposition H pL H p Í Ÿ pLL

We ill de el hi i Cha e 7 a h ee al ed l gic .

4.6.3 Selfreference in set theory

The ble f he Lia a d R ell e a ad hi ge f eed defi ec ce a libe . The e i hi g i he c ce f li g i ic agg ega i ha

gge , a fi gla ce, ha e ca c e all ki d f c bi a i f hec ce a d b l , a d he e hei h al e. E ce ha e ha e i a cef hich e ca fi d ch a h al e. The a ach i ce R ell ha bee hi

he f e , ch ha ell defi ed c ce a i f a hie a chical de . S a he

e lace i fe e ce. Se A e e ceB l a l e e e ce f a l e le el. Able i h hi a ach i ha i c e he a al feeli g ha la g age

a d agg ega i a e e fle ible. Le el e cl de elf efe e ce hile i i a a ac i ee .

I ca al be h ha e a he eed elf efe e ce ha e h ld all i .

The igi al defi i i f a e ca e f Ca : U e ei e Me ge e ehe i jede Z a e fa g be i e hl de chiede e Objec e e

A cha g de e e De ke (...) ei e Ga e (Ze el (1908:200)). Recall a i f bef e, i h ª – he R ell e f al e ha d

c ai he el e . I ha e i d i ee ed ha a i fied Ca defi i i . Beca ef hi a ad , R ell e ic ed he ki d f e be hi . Ze el , h ked i hea e field, a d ac all di c e ed he a ad i de e de l , decided e i

Ca defi i i . He gge ed he i f : A e i a e i (...)i aid be if he f da e al ela i f he d ai , b ea f he a ia d he i e all alid la f l gic, de e i e i h a bi a i e he he i h ld

. ( 201). S b e e l he de ig ed he a i f e he ha a e ba icall illi e da . The e a e diffe e c e hi : (a) he i f defi i e e ac alli lie a h ee al ed l gic, a el f e e ce i h e ha a e i defi i e, (b) e

a de he he defi i e e eall diffe f Ca hl de chiede , be i ha i d e , (c) Ze el a i eflec a a ia f he he f e ,

h gh he d e call i ha a , (d) e ill ee l e elf efe e ce e he e.

Hi icall , e a b e e ha R ell ggle a diffe e ha Ze el . Aha i e he ee ha e had a Pla ic c ce i f a he a ic , i hich i deed ah gh ld ha e e i i e a he , a d h ld b i elf be

hl e chiede (defi i e) i ead f ibl e ical. Hi de el e f he

128



he f e a a f hi ggle ed ce hi Pla ic e de cie . A he e d fhe P i ci ia Ma he a ica he i e : i a ea ed ha i ac ice he d c i e f e ie e ele a e ce he e e i e ce he e a e c ce ed he e a lica i a e

be ade e a ic la ca e (R ell & Whi ehead (1964:182)). F i i le

i g ha e ical hi g ca e i i a b i he a ic he e he a e l geed.

Rec ide he R ell e a ad , a d c ide ha e igh all f al e ,ha d c ai he el e , be c ai ed i e , e e if he a e f highe

e ha .

A c i f i , called he Ze el e f al e , ha i cl de a allc i e c c di i ha ac all i e a e ef l f f elf efe e ce.

= – œ

Clea l , if e c ide he e f ea he hi h ld i h ble . Whe e œ he he c di i f e be hi a ea be a i fied (a c adic i )

he ce e c cl de ha i a e be f i elf. Thi a ach e i d f he ef Pa l f Ve ice l e he Lia a ad ( ee B che ki (1970) a d Be h (1959)).

Whe hi k ha e i i g œ f he e f ea i e d e, he e cad , hich i deed i a e ge e al a ach:

† If ∫ he – † If = he – b f c i e c e e i e œ .

Thi i f la ed j i l a ha – a lie le = , f he he c i e ce i e e i e lici l i ed. Of c e, c i e c h ld i ge e al, b he e

li i a e i a i gle a e e he e a la e e all c i e c hea ic la e i ed f he e e i de c i .

† S l i g R ell a ad i h U le .

def = $Equivalent @y Œ Z, Unless @Hy œ yL, y == Z, Hy œ yL Ï Hy Œ Z LDDH y œ Z ñ HH y ∫ Z y – yL Ï H y Z H y – y Ï y œ Z LLLL

† Re lace e b l ha L gicalE a d d e ec g i e.def2 = % êêToBelongs êêToAndOrNot êê LogicalExpand

HHH y œ yL Ï H y – Z LL Í HH y œ Z L Ï H y – yLL Í H Z y Ï H y – Z LLL

† F he e f ea ha i a e be f i elf, e fi d ha i bel g hi Z.

def2 ê. HBelongs @y , yD Æ False L ê. HZ ä yL Æ FalseH y œ Z L

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† A lica i i elf gi e he c cl i ha Z– Z, a d hi h ld i hc adic i .

def2 ê. y Æ Z

HHH Z œ Z L Ï H Z – Z LL Í HH Z œ Z L Ï H Z – Z LL Í H Z – Z LL% êêLogicalExpand

H Z – Z L

Th he e i a a f R ell a ad i h e i g a he f e .Whe a ce i e ed he 0, hi ca ed a ad e f he , a d e f he

eall had a ha d i e, b e e all a dece e f c ce a f d. I he a ea he e f all al e a a a l be defi ed a bi a e . The gh fa ad i ill i h beca e f he eed add a c i e c e i e e . A de ha he e i e i el ha ai el had i i d f a e f all ale , b a lea c i e c ha bee a i ed a .

A he e a le a hel . The defi i i f a e fec bike i ha i ha d heel ide a d a d a e heel a k i . Si ce he e a e d a e , hi

e fec bike d e e i . I ha e hi g ab ? I l h ha idea f bike e fec i a i c i e i he fi lace, ha e c c a bike

i h a a da d. Thi he bike i e fec i ha i k . I he a e a e e

h eall a k i h a e f all al e ill be ha i h ab edefi i i f , e e he i igh be a igi all c cei ed.

4.6.4 Still requiring three-valuedness

Tha bei g aid, a d ha i g i dica ed ( e f c e) he ible c i e c fa e he i h elf efe e ce, he e e ai he he b e a i haha g ea a . I i ea c cl de d f i ( ai f i c i e c ) b ha illca e e ea i e . R le e ic f eed he e be a g d ea f

he .

The c i : h all f eed f defi i i , a d le l gic ell ha dhe e each a c adic i ? The ba ic hi g l gic h ld ell i : e i e da efe a hi g, a d, e i e ci c a ce a e like e h gh .

B e e he h ld f a i le , he ill ld eed a lace all he hi g ha d f . Sa i g d defi e he Lia a ad ie i ale a i g he d ha e a al e i T e Fal e 1 0 . If e a ig a

al e 1/2 he e ca c c he f ll i g able:

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œ – N A All@ œ D1 0 00 1 012

12 1

Whe e b i ded c i e facili ie , e de i e a c adic i , he ce ec cl de ha i e e, a d he e a ig a h al e 1/2. O , h ha a ee i , e l ha e h ha i ha a ele e ( hich ca al a be d e bEFSQ), b al ha i defi i i i i c i e , ha he c ce a e elldefi ed i he fi lace.

A aid, he ke i e he e i f eed f e e i . The ld i T e Fal e b c ce ca be e ical. I i ef l, l ha e he f eed be c ea i e, b

al ha e he f eed e i e e. The be e a le he e i he he f e : ic ea e a igid c e ha f bid he elega c c i f he Ze el c i f

. Rigidi ca be high a ice a . The Ze el i l a elegac c i b i al c ai all ki d f ele e ha he i e ld bec ai ed i ha a .

F a i le a hel e ic e e, b he a al c ea e e e, aee f a l gical i f ie .

PM 1. I i ef l e lace he al e 1/2 b 0 i ce N [– ] ld ge e a e a 1 i

he l e , a d i ila l f i g 1. Th h ee al e a e e i ed. Pe ha

d like 1/2 a d efe U f . I a ea ha he be ch ice i

.

PM 2. A he f e igh ill be ef l f e . Ca ed ha he e e i

al a g ea e ha he e i elf. A e f all e ld ha e c ai i e e a a

b e , hich i h ca d . He ce ch a e f all e d e e i . S e

hie a chie i deed igh e i . See Cha e 11, Readi g N e , f a di c i f hi

a g e .

PM 3. M ch f hi di c i de i e f a he a icia h a e e hi g be

eall ea . We ca l e ec hi a d ac all be ha ha he e a e ch e le. The

achie e e f e le like E clid a e i d agge i g. Ye a bi f fle ibili ld hel .

Ma he a icia e d be i ed b he de i e de a d hi g a d d he

e d eli i a e all e ce i , hile de a d he ld all f

e ce i . Ma ki d ie l e he ble f b ea c ac i ce he i e f hea id a d i ld hel he he b ea c a ic a i de fi d fe ile g d i he

eal f he i d.

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4.6.5 Exceptions in computers

C e g a ha e deal i h defi ed i a i . Whe he ha e a igidc e he hi igh ca e a l f e a d h a g ea deal f e ha dli g.

P g a ld be e fle ible he he c ld g e ha a i e ded (

e ha i l lea e hi g a he a e).

† Thi e e i d e i lif T e Fal e he he a iable d ha eal e . Wha i i h al e ?

p Ï q Í r êêLogicalExpand

Hr Í H p Ï qLL

† A d ha i he h al e f hi a e e ?p Ï q Í

Syntax:: tsntxi : "\" p Ï q Í \" is incomplete; more input is needed. More… "

A he g d e a le i i g c ca e a i . I e i e i g a d a lea fhe . F ge i g e e e diffe f f ge i g e e . I he

fi ca e he c ic e age i j he age a e ki g i h a d i heec d ca e he e al a ea a e a a e e age eb k i h a he c ice age.

† A a f g e i g he e , he he e i i ha i h al e i ."ab” <> “cde

Syntax:: tsntxi : "\" ab <> \" cde” is incomplete; more input is needed. More… "

"ab” <> cde

StringJoin:: string : String expected at position 2 in ab <> cde . More…

ab <> cde

N e he diffe e ce i ( e a d de c e). The diffe e ce be ee a d i i he a f ha he fi ill i ai i g be ce ed f

bec i g e i f he ke el hile he la e ha al ead a ed ha e hahe ke el a e ha ld be a a iable. B ha i e e a i a e f he

fi ca e ha he a i a k i a i g e . Whe he a i a ki a a i g e , i he e e ha he e ied f ll he a a d ied

i a a iable , he he e g he g e age. NB. The e cell a e l ckedi ce la e e i f igh ha dle he i a i diffe e l . Al , i he

fi e e age a i gle e ha bee cha ged i \ i ce he i e gi e a hi d e e age (a e a e e age) he e i g hi eb ka hi lace.

I b h ca e he c e f a el d e c a h a d e e e d ce he i . Id ce he i ell ha e e a hi k ha i h ld be able g e

132



ha a i e ded. Reall , he c e e al ead ed i g a d hef c ca e a i . Sh ld a a c e be able g e (a a defa l

l i ) i he fi ca e ha c ca e a i f abc a d cde i i e ded ? (I ighe he he cde i a a iable.) Wha i g a ed ee i efficie a d

c f i g, a d a g d le f g a e i ha h ld g a f heha ld a be g a ed elf. A e e f ie dl a ach

ld be c ea e ha c ca e a i a d e OK, i a all a da d a e e cei e he , b e ade a g e : if i i igh , lea e c ec i ,

a d, if f ll a he d ge e age like hi .

Ac all , i i al ead a e ahead i li e e ha he e e age a e called ee age . A e i hi ca e i l a diffe e ce f i i ha a i

acce able.

Thi i j a e a le. Wha c i he hil h . (a) Deali g i h e ce i e ha de ig . (b) T h al e f e e (gi e e a ).

Of c e, i he c ca e a i e a le, e e i e c e de e de c . If hi e e i gf c de a la a he a d ha f g f ba ki g acc , hee ha ld be ha i h e deg ee f ic e . A d f c e, h i g f

e f ie dli e i al a f f ic e . I ld e ha be ch de a dha he e be h le c a de a e affed i h e le i g g e ha

ld be he hidde ea i g f i e (i.e. ca ed b a f hede a e ). (Thi bec e c le . Pe ha i i be e bec e c le el c icala d ca e ab a hi g a e. Le ickl e .)

133



134



5. Inference

5.1 Introduction

5.1.1 Introduction

I fe e ce f c e he ce f ha dli g i f a i . A c llec i f i di id alb e a i gi e i e, ia , he c jec e f a ge e al ela i hi . A

c llec i f ge e al ela i hi gi e , ia , e ge e al ela i hi , ,i h e a ic la , he a ic la c cl i . The i f a i ha i ce ed

be he e he i e i ca be ce ed. I fe e ce e e iall a e heha dica ha he i f a i i ec g i ed i edia el . I fi be ce ed bef e i i ec g i ed i he f ha i ca be ed. A d he fi al e e h ld be

Ah, hi i .

L gic a d a he a ic a e he ded c i e cie ce . The e a f i d c i a ege e all lef he he cie ce . Ye i ill be ef l i cl de a ec i i d c i

cla if he e he ge e al la c e f ha a e ed i l gic a d a he a ic .O he b k a i h elf e ide h b i i ef l be ecific.

The a al i f i fe e ce e i e a a f h a e al ce e . F he l ef he edie al a e hi a e e a La d f C ckaig e.

† Fi l ad he ackage.Economics @LogicÀIOE, Print Æ False D

PM. The a i ¢ f alid a d ' f i alid i fe e ce igh gge he T hTablei e ha he e a e diffe e f i l ed, aa d a i de e de .

H e e , e ca ec e he e i i al f ha h hei c ec i .

† T hTable ec gi e e i i al f .Ergo @S, p D Ï NonSequitur @S, p D

JHS ¢ p

L Ì JS p

NN

135



PropositionalForm @%D

H P 1 Ï Ÿ P 1L

5.1.2 Validity

S e ded c i a be f all alid b f i ele a ea . F a fal eh d ca de i e a hi g a , hich i La i i he e fal e i dlibe

(EFSQ) a g e . A e a le i he a e e ( h): If i ai a d d e ai ahe a e i e, he ll ge a e ca . He ce he f ll i g di i c i i ef l:

. A a g e i f all alid if i a ic jec i i a a l g

. A a g e i a e iall alid if i i f all alid a d if i e i e a e f ee fc adic i a d (k ) fal eh d

† The EFSQ a g e c eTruthTableForm @Hÿ p Hp qLLD

Implies Not p

Implies pq

True FalseTrue

TrueTrueTrue

True FalseTrue

FalseTrue

False

TrueTrueFalse

TrueFalseTrue

TrueTrueFalse

TrueFalseFalse

† A e d a ic EFSQ e a leTruthTable @Hp Ï ÿ pL qD

p q HH p Ï Ÿ pL qLTrue True True

True False TrueFalse True TrueFalse False True

5.2 Induction

5.2.1 Introduction

Pa f life i he ce i g f i f a i . O ga i a a e l ha e fe ie . S e e i b il i he ha d a e a d he called i i c . The e

136



c le ga i a ha e e ba k a d le e i f a i . B e elifele bjec ha e e e , a a c e d i e b i ha e e e be ha ica ha dle l e e f c e . S a i i g a li f e e i a ge e al le icalled i d c i a ed ded c i ha g e f he ge e al le he

a ic la .† Ma he a ical i d c i ec i i a f al che e c e f he a ic la

he ge e al, i h a a e l efe i g eali .

† E i ical i d c i a e he a f e i ical cie ce, de e i e e i ical la .

† The defi i i & eali e h d l g i he a ach e i ical la ih b defi i i .

Ded c i elie i d c i . Wi h i d c i he e ld be ge e al le , a d

i h ge e al le he e ca be ded c i . Ded c i a a i h elf e ideh b he e ca e ab f i d c i defi i i .

I ld ha e bee efe able i d ce i d c i i he i d c i , b i e i ee a i ha i l de el ed i he c e f hi b k.

5.2.2 Mathematical induction or recursion

We ill ake ec i a d a he a ical i d c i a e i ale e . The e

ec i f a el d e e lici l efe i d c i .Le A[ ] be he e de c ide a i a d le be he e f a al be ,

h = 0, 1, 2, 3, ... . We a k he he e A h ld f all a albe . Ma he a ical i d c i c i f hi a g e :

Ergo2D @"Mathematical Induction" , AD1 It can be shown that property A holds for n = 0, thus A @0D2 It can be shown for arbitrary n that if A @nDthen also A @n + 1DErgo —————

3 A

@n

Dfor all n œ

The ab e a i h = 0, b i igh al a i h a he be a d e ae f all be f he e a d . Y igh ake bigge e ha e,

c ide i g l a b e , a d igh e e a iable .

Thi ki d f che e i called ced al c c i e . Y d ha e kh gh all a al be b e el i e a deci i ced e alg i h

check hi g , a d, ca check ha beca e he ced e i . A ef l eh i effec i e fi i e ced e . Of c e, f c le i e c a i i e

e e f fi i e ca e igh be l g.

S e i e e ie a e defi ed i hi eci e a e . P eci el he e f a albe ca be defi ed b i g he idea f he cce he e . Le 0 be he

137



fi ele e , 1 i e 0, 2 e 1, e ce e a. The be a de a da he a ical i d c i i ee ha i i ic hi ki d f defi i i . The f all i

3 f he che e ab e ac all d e efe all a al be a a c le ede , b e ha if a e a be he ca ake he e e (ac all defi i g

f all ).A e a le f a defi i i b ea f ec i i : he e k ha addi i i ,

he e ca defi e l i lica i a f ll . Le e he d l i lica ie b a ag e e e a . The defi e: (1) 0 = 0, (2) ( + 1) =

+ . He cef h e ill call hi i e .

O he e a le f ec i a e he ced e F ldLi Fi edP i i :

FoldList @f, x, 8a, b, c <D

8 x, f H x, aL, f H f H x, a L, bL, f H f H f H x, aL, bL, cL<The i ci le f a he a ical i d c i ld h ld f all ible edica e . I hica e he f all igh be ag e i ce he e d e ee be a a al de f

edica e .

† Defi i i f ec i . i e F All d e di ec l ec g i ehe i i ed l a a edica e. The ef e e i i e lici l .

ForAll @P, Ergo @P, P @0D, ForAll @n, HP@nD P@n + 1DLD, ForAll @n, P @nDDDD

" P H8 P , P 0 , " n H P n P n+1L< ¢ " n P nLPM 1. Ma he a ical i d c i i defi ed f he a al be . Real be e i e

he c i ( ee e.g. L.E.J. B e ).

PM 2. Th gh a he a ical i d c i ha bee e e ed he e i he c e f i d c i ,

he e i be ca al g ed a i h, la e e ill ee i i he c e f ded c i

agai . Whe c ce ha e bee defi ed i e f ec i he e ld e

ec i agai f f .

5.2.3 Empirical induction

Economics @Probability, Print Æ False D

S e ha e ha e i e i ai a d e i e i d e ai . Y alb e e ha e i e he ee a e e a d e i e he a e . Y bec e

i e e ed i he c bi ed cc e ce . Y c llec e 100 ca e . I all he 25 ca eha i ai he ee a e e . The e a e 3 ca e he e i did ai b he ee a e

ill e (e.g. beca e he ad clea i g ca ca e b ). The da a ca be c llec ed i he

f ll i g able.

† The b e a i a i ha 9 cell b l 4 deg ee f f eed , beca e f hei g f a d c l . I he Di ea e Te a i i e ide i l g he

138



di ea e (ca e) a da dl i i he c l a d he e e l (effec ) i he .We a e ha ai i he ca e. I he b e a i ha he ee a e e a g d

edic f ha i he ca e ?

mat =

"Observation count" "It rains" "It doesn’t rain" "Total""The streets are wet" 25 3 É

"The streets are not wet" 0 É É"Total" É É 100

;

Headed2DTableSolve @mat DObservation count It rains It doesn't rain TotalThe streets are wet 25 3 28The streets are not wet 0 72 72Total 25 75 100

I ead f e e be i g all he e 100 ca e ei he i di id all b f e e c

di ib i , he e ce i g i igh a e age a d e ie al c bad i g a ge e al le ha If i ai he he ee a e e .

Thi ca bec e a ge e al le f hich e ca e a h able. The h able ehe c di i he he he f e e c i e e .

† The S a eT hTable ced e he e i al habe ical de , a a e f h he i ca ali . Th e a e i e l .

SquareTruthTable @"It rains" "The streets are wet" D êêTranspose

HIt rains The streets are wet LIt rains ŸIt rains

The streets are wet True TrueŸThe streets are wet False True

Clea l , ch ge e al le be ea ed i h ca e i ce igh i ca ehe e he le i ef ed. F e a le, a ee i g de a b idge h gh a

c e ed h i g all igh a ea be d e e hile i i ai i g.

A hi i i i ef l e ha e.g. a i lica i i be a bi a e ahile i elie he b e ed a iable a e e . If i d eai he a he e e e lai he ee ible e e , ee C lig a

(2007e).

5.2.4 Definition & Reality methodology

O e a i la e a le agai ef a i i i i a defi i i . The c cef I ai a d The ee a e e ca be defi ed ch ha If i ai he he

ee a e e bec e a ce ai b defi i i . F he dd ca e he e c ea d ee he i ai he l k f he ca e f he e ce i , a d decide

ha hi i a eal ee . I de e d he c f e a age e ha

139



i he chea e l i . Y igh decide ad he le If i ai he ee i hee ai a e e ha he ge e al i f ee i ada ed, igh kee he

ge e al i f ee a d c ea e he i f he ee a c e eda h a . A hi g g e a l g a e ai c i e a d ick he fac i

igi al li f 100 b e a i . N e ha ch c e ed a h a a ill be e hei ai , beca e f i l a e clea i g.

The ced e i la e i agai ef a i b i g he i defi i ica be called he defi i i & eali e h d l g . While eali i c i ge a de e e i e , i i a c effec i e e h d f e a age e edefi i i ha all ce ai (a ed babili i ic) c cl i .

See C lig a (2005) f a cha e ha e lai e a d a a lica i f hi

e h d l g ec ic .PM 1. The e i he e a le f Wi ge ei h ke ea chi g f a hi a i

he cla beca e b d had gi e a ade a e defi i i ha he c ld ide if

ha he e a e. Thi igh be he e a i de f a he a ic b i al

cie ce defi i i igh be a he ag a ic.

PM 2. Y ld ill be efi f a c e babili a d a lied a i ic .

5.3 Aristotle’s syllogism

5.3.1 Introduction

A i le l ec g i ed he A, I, O a d E f a d ha he all ac cl i j b he el e , b he al c ea ed f f i fe e ce he e

e i e a e c bi ed he c cl i .

G eece (Hella ) a ha i e ffe ed f hi (la e a d c l a ) hde ig ed all ki d f a g e a i a d ad cac ha eeked f e i . (Thi

i e diffe e ada . Ma ki d ha ade eal g e .) DeL g (1971:12) ela e :a e a le f el e bal c adic i a d ibbli g a e gi e i Pla

dial g , e eciall i he E . He gi e hi e a le f Di d( al le e ) a d C e i (i alic ):

.... Y a ha ha e a d g. , , aid C e i . A d ha he ie? , . A d he d g i he fa he f he ? , he aid,

. A d i he ?

. The he i a fa he , a d he i ; e g , he i fa he , a d he ie a e b he .

A he ble a ge e , he e e a h i l begged he e i (ifa d a d he ), he e A a ed ha B ed a d he he a a d,

140



c ea i g a i ellec al a he e ha de a i a i ible ha c lda e a hi g .

A i le a bi i l e de e i e f ha e e alid.

5.3.2 The four basic figures

Whe e c ide a c cl i like Th , S c a e i al he he e i a bjec(S c a e ) a d a edica e (M al), ab ac l a d . A i le did e he label

a d , b ed he label a d . O each e e cai d ce a e a a e e i . The e i he aj e i called he aj a d

a d fi . The e i he i e i he i a d c e e . Thec cl i f ll a hi d. A i le de d he a g e a If aj a d i ,e g c cl i , i.e. a a li ea i fe e ce a d a 2 di e i al able. Of c e, he

e i e be c ec a e e a d h e i e a he e , called he . A i le he c ide ed aki g, , f he 4 ibili ie f he

Affi Neg (AIEO) che e. The e a e called ca eg ical ll gi .

Mai ai i g he de f fi , he , he e a e ibili ie f heaj e i : M, P P, M . The e a e ibili ie f he i e i : S,

M M, S . He ce he e a e f c bi a i i al, gi e b he f ll i g able.A i le ed he e f he e c bi a i , h gh did c ide he f h

e. NB. Mi a d ai a e La i .Syllogism @All, P, M, S D

Maior H M , P L Maior H P , M L Maior H M , P L Maior H P , M LMinor HS , M L Minor HS , M L Minor H M , S L Minor H M , S LConclusion HS , P LConclusion HS , P LConclusion HS , P LConclusion HS , P L

F a ca eg ical ll gi he e a e 4 fig e a d 4 ibili ie f each li e, h 4 * (4* 4 * 4) = 256 ibili ie . O l 24 c bi a i a e alid che e f i fe e ce, eeDeL g (1971:19 24).

The i a e ie he a e:

† Y ha e a e e hi g. S e i a e i h de a i a d a e elfe ide h . A i le: O d c i e i ha all k ledge ide a i e. (DeL g (1971:20))

† Y e he alid i fe e ce che e . Thi ake f a alid i fe e ce.

† B al ake e ha he a i a e T e. Th EFSQ i e cl ded.

O l he he e i a e .A i le: S ll gi h ld be di c ed bef e de a i beca e ll gi i he

e ge e al: he de a i i a f ll gi , b e e ll gi i a

141



de a i (DeL g (1971:20)).

He e ha i ed ha i fe e ce c ce i e ha ake e e, a diffe e f e ee e i : (...) all ll gi , a d he ef de a i , i add e ed

he ke d, b he di c e i hi he l (...) (DeL g (1971:23)).

Fi all (f ) he e gi i g e defi i i , i de ide if h e elfe ide h . Acc di g A i le S c a e a he fi ai e he ble f

i e al defi i i a d: i a a al ha S c a e h ld be eeki g he e e ce,f he a eeki g ll gi e, a d ha a hi g i i he a i g i f ll gi(...) (DeL g (1971:23)). A e e defi i i d e ca e ha he defi ed e ealle i . Al , a g d defi i i e i e ha i e a e be e de d ha ha idefi ed.

PM. Thi i he l l gical che e ha A i le ed. I hi i i g he f e a leal ed , , e g , a lea i d h gh e ha i f la . Al all

he a h a d ge e f he G eek de e de ea i g. Pe le ge e all e all

ki d f che e . A i e he e i he f ali a i f e h d .

5.3.3 Application

The f ll i g gi e e ca eg ical ll gi .

† Thi gi e he AIOE che e agai , i ca e f g .Syllogism @AIOE, S, P, Text D

All S are P No S are PSome S are P Some S are not P

† Thi gi e he a da d ll gi , fig e 1, A, I, I .Syllogism @1, 8A,I , I<, 8mortal, humans, Socrates, Text <DA All humans are mortalI Some Socrates are humansErgo —————

I Some Socrates are mortal

† Defa l i he e f e e be hi f I.Syllogism @1, 8A,I , I<DA M ŒPI S œ MErgo —————

I S œ P

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† Whe he c cl i c ce e he e e he defi i i A B ñ (A › B) ∫« he e e i g ha e e la .

Syllogism @1, 8A,I , I<, 8Mortal, Human, Men, And <DA Human ŒMortal

I HMen Human LErgo —————

I HMen Ï Mortal L

† Whe efe he a ifie he e i he a iable ha i b ded b he .Syllogism @AIOE, S, P, x D

" x HS H xL P xL " x HS H xL Ÿ P xL$ x HS H xL Ï P xL $ x HS H xL ÏŸ P xL

Syllogism @1, 8A,I , I<, 8Mortal, Human, Socrates, x <DA " x HHuman H xL Mortal H xLLI $ x HSocrates H xL ÏHuman H xLLErgo —————

I $ x HSocrates H xL ÏMortal H xLL

† O he la e he i i e ide ha i i he d ai :% ê. Socrates @xD Æ True

A " x HHuman H xL Mortal H xLLI $ x Human H xLErgo ————— I $ x Mortal H xL

The ll gi i e al ca gi e li ea , he ecif ha E g i beed i ead f E g 2D.

Syllogism @1, 8A,I , I<, Ergo D

H8M ŒP, S œ M< ¢ S œ PL

Syllogism @ , 8 , , < , 8 , , H , L< , :E g 2DDi he ll gi f , he e he e i e be i A,

I, E O, he e , a d a e he e e i f he edica e,iddle a d bjec ; a be eeded f he a ifie , b a be Te ,

ca be i ed f e e be hi ca be A d f e i e ec i ; ca al be E g f c i

Syllogism A , XYZ , 8 , , H , L< , :E g 2DE

a e a ab e b i h le e X, Y, Z a le e A, I, E O i ead f b l

143





ha a alid i fe e ce che e i . I a ell be ha he d ill igi a ed fhe al f lk i he illage e hea i g he k d i g hei ll gi .

Syllogism @4, 8A, E, O <, 8dogs, fathers, puppies, Text <DA All dogs are fathers

E No fathers are puppiesErgo —————

O Some puppies are not dogs

I i a gge i g he lib a , ge DeL g (1971) a d l k age 21 f heable f A i le 24 alid che e . Y igh a ead e e i hi g d

b k. Al e a i el , igh d a Ve diag a , a d fi d ha a i che e ca be ed ced he . The a i a ic e h d ill hel fi d he i de e de a i

ha a e fficie f he e ded c i .

PM. Si ce e ca fi d a d g ha i a fa he e.g. a i elf he i fe e ce i e b EFSQ, ha e lai h A i le i i ed ha he e i e h ld be e. Thi

che e i ac all alid b ld e i e a he e a le i h e i e ha a e e.

† O e di ec a ackle he AIOE che e i c ide he E i a e e f heele e ha ake i e, a d b i e hi a e ele e i he F All a e e ,

he e he a ifie d , a d he ble ca be ackled b i i all gic.

Syllogism@4,

8A, E, O

<, 8

dogs, fathers, puppies, x<, Ergo

D ê. Ergo

@"Projection"

DHH" x HdogsH xL fathers H xLL Ï" x Hfathers H xL Ÿ puppies H xLLL $ x H puppies H xL ÏŸdogsH xLLLSubstituteQuantors @% , x Æ case D

HHHdogsHcaseL fathers HcaseLL Ï Hfathers HcaseL Ÿ puppies HcaseLLL H puppies HcaseL ÏŸdogsHcaseLLL% êêLogicalExpand êêMatrixDNForm

HdogsHcaseL ÏŸ fathers HcaseLLHfathers HcaseL Ï puppies HcaseLLHŸdogsHcaseL Ï puppies HcaseLL

SubstituteQuantors @expr , rules List of rules D

i a ick a d di i e e lace a i h e i gle i a ce ha e caa l i i al l gic. The e ake e ha he e lace e i c ec ,

he e i f al e ha . Re l@S b i eQ a D gi e ediag ic : he c llec i f he b d a iable i he F All a d E i e ec i el ,

he i f h e a iable , he i e ec i f h e f E i i h h e ha a ee laced. Be a eg i b i e he b d a iable f he E i i he

F All. B he e ca be li f a iable , a d c di i , i i a i k

145







A e a hel decide a a e e like = The e e heHi ala a . The a e e i i hi he e iff he e e i

e . Whe he e e b h he i e able. Whe he e all (¢ ) ( ¢ ) ( ¢ ) he he e i a f f a c adic i . We e he e

f , a d he e f he ch ha ( ¢ ) ( ¢ ).

† C a i e f U decided.AffirmoNego @Undecided @S 0 , pDDUndecided @ pD Untenable @ pDJJS 0 pN Ì JS 0 Ÿ pNN õ Contrary õ HHS 0 ¢ pL Ï HS 0 ¢ Ÿ pLL

à â‡ Not ‡

á ä

JJS 0 pN Î JS 0 Ÿ pNN õ Subcontrary õ HHS 0 ¢ pL Í HS 0 ¢ Ÿ pLLTenable @ pD Decided @ pD

Decided @S :, y___ , pD E g @, , D Í E g @, , N @DDUndecided @S :, y___ , pD N @Decided @H, ,L DDTenable @S :, y___ , pD N @U e able@H, ,L DDUntenable @S :, y___ , pD E g @, , D Ï E g @, , N @ DD

To prevent a possible disappointment: these are just objects and do not evaluate the truth.

Tha e a e e i decided i a cca i ha e e c e f e e l i life.The e e i ab e f decided i a bi ieldl . Gi e he i a ce fded c i f l gic i igh al be efe able i d ce a e a a e a i ch a

HS pL he e e e Ne e ide . We ill call hi ide e l ed , a d e

i a el e i ale i h he he e e i . I i j a label, h gh.

Unresolved @Definition, S, p D

JHS p

L ñ

HS ¢ p

Lè

ñ

JJS p

N Ì JS Ÿ p

NN ñ Ÿ

HHS ¢ p

L Í HS ¢ Ÿ p

LLNUnresolved AH,L D

h a Ne e ide eflec ha he ded c ii decided. Thi i a al e a i e a h O he@E g @S, DD ,gi e ha ~ igh ead ell f E g , e eciall i e c le a e e ,

hile i al a be ec e dable ha he e i a e a a e b lf he a lica i f O he E g . U e l ed d e e al a e,a ei he O he

@E g

@DD d e . See U decided

@S,

DUnresolved ADefinition , H,L D h he e defi i i

148



Unresolved @Rule, a , bD

gi e le f e lace e f a b,f a a d b i 8U e l ed, O he , O he@E g @ÒÒDD&, U e l ed< ,

he e j O he ca be ed a h ha d f he e e c le e e i

Independent @H82 ___< ,L ___, De e e ha a a i i i de e de f S he i i

decided f S. Bei g a a i ea E g @HS2,L a i D

5.4.2 Provable and decidable

S 0 i a ecific e a e e . Le e ha e ha e a i e ,

f e a le he b e f S 0 , i h h a ele e f he e e fS 0 . If a df a a e e i a b k he e ca ake a ag a h cha e . IfS 0 i a lib a he

ca be a c llec i f b k . Whe e e he h a e$ he e bec eable a d decide bec e decidable .

S a e e i able iff e fi d a lea e ha i . U i g a ifie e f ll ad he AIOE che e. The i e a d c a ie d eed a ch

b i i g d ha e he ee ce, if l beca e igh de he he hed a ch.

† C a i e f able .AffirmoNego @Unprovable @S, p DD

Unprovable @ pDAxiomatic @ pDor Inconsistent @S0D

" S JS pN õ Contrary õ " S HS ¢ pLà â

‡ Not ‡á ä

$S JS pN õ Subcontrary õ $ S HS ¢ pLSome S are consistent for p Provable @ pD

A a e e i decidable if he e i i S 0 ha decide i . C e il i idecidable, ha a e able. The e l i he l e lef c e igh al be

ee a i de e de ce h gh ha bee ake he e a a c i e a e e .

149



† C a i e f decidable .AffirmoNego @Undecidable @S, p DD ê. Unresolved @Rule, Undecided, Unresolved D

Undecidable @ pDAxiomatic @ pDor Inconsistent @S0D

" S HS pL õ Contrary õ " S HHS ¢ pL Í HS ¢ Ÿ pLLà â‡ Not ‡

á ä$S HS pL õ Subcontrary õ $ S HHS ¢ pL Í HS ¢ Ÿ pLLSome S are consistent for p Decidable @ pD

The a e f abili a d decidabili c ai a a e e i he e igh ha dc e ha h ha c i e c i e . If he e i e i c i e a e e

he he h le e h ld bec e i c i e . The label gi e he e i i ec i e c , i ce he a ifie e hile he c ce f c i e c ha a

a ifie i g e . Thi b i g he e ec i .

C e i al e b k e d e e he e c ce a defi i i i h li f fh he ela e. The AIOE a e h e e ee a ch be e e la a i h hea e ela ed. The c ce a e a he elf e ide , a d e igh a g a

a la e he e i he he .

Decidable @HS , y___ , L p D E i @8S, < , Decided @S, , DDUndecidable @HS , y___ , L pD F All@8S, < , U decided @S, , DDProvable @HS , y___ , L pD E i @8S, < , E g @S, , DDUnprovable @HS , y___ , L pD N @P able@Ha e e ,L DD

To prevent a possible disappointment: these are just objects and do not evaluate the truth. Ergo is the label for Proven.

ContraryLike @Last D a li f edica e f hich C a Like lega e he la ele e He.g. F AllL. Cha gehe li b C a Like

@La

D=

8.

<ContraryLike @f @ p , …, qDD gi e f@! , , ! D le f i i C a Like@La D ,

i hich ca e f @, , ! D Hch a F AllL

150



5.4.3 Consistency

We i ch f c , a e le i e e ed i i di id al c cl i , a d de aball c cl i . Thi ca e ha he a ifie e i ead f .

† A ded c i e e i i c i e he i b h e a d ef e a a e e .Beca e f EFSQ, e i c i e c i lie ha all a e e ge e .

AffirmoNego @Consistent @S0 , pDDConsistent @S@0DD Inconsistent @S@0DD" p JJS 0 pN Î JS 0 Ÿ pNN õ Contrary õ " p HHS 0 ¢ pL Ï HS 0 ¢ Ÿ pLL

à â‡ Not ‡

á ä$ p JJS 0 pN Î JS 0 Ÿ pNN õ Subcontrary õ $ p HHS 0 ¢ pL Ï HS 0 ¢ Ÿ pLLConsistent @ pD Inconsistent for p

but EFSQ implies inconsistent S @0D

We igh e he Affi Neg T h able h h he h al e a e ela ed.H e e , hi i e he able i c ec . If he e i a i c i e c he e cade i e a e e ha i hi ca e O E. I he a e a , ha i g a i glec i e a e e ( ei he i i i e e ) i lie ha all a e c i e

i ce if he e a a i c i e c he c ld e a hi g.

† Thi able i i c ec .AffirmoNego @TruthTable, Consistent @S, p D, Print Æ False D

A EI O

T »F A I O E

" p JJS pN Î JS Ÿ pNNA True True 1 c0 0

$ p JJS pN Î JS Ÿ pNNI True U True U c0

$ p HHS ¢ pL Ï HS ¢ Ÿ pLL O True c0 U True U

" p HHS ¢ pL Ï HS ¢ Ÿ pLL E True 0 c0 1 True

" p JJS pN Î JS Ÿ pNNA False False U c1 U

$ p JJS pN Î JS Ÿ pNNI False 0 False 1 c1

$ p HHS ¢ pL Ï HS ¢ Ÿ pLL O False c1 1 False 0

" p HHS ¢ pL Ï HS ¢ Ÿ pLL E False U c1 U False

Consistent @S_ :, y___ , p_ D

F All@8< , N Se i @S, , D »» N Se i @S, , N @DDD

Inconsistent@S_ :, y___ , p_ D

E i

@8< , U e able

@S, ,

DD

151



5.4.4 Decidability and consistency

I igh be ef l defi e i h X . If e had defi ed aldecidabili i h X i ead f O he e ld ha e e i ed c i e c , i ce

hi i i lied b X . C ide i g j e a iable, he a e e e cl de a d h i e acc a e ha . The ba le c h e e i

a la ed i h e el beca e O a ea ef l f he bi a i a i f a d e . The i lied eak e f he c ce f decidabili ( ha ali cl de he ibili ha a e i i c i e a d h e a hi g) ic ec ed b al e i i g c i e c . The ch ice f c ce f he i i a e ial

i ce all ha be e abli hed a a .

5.4.5 Semantic interpretation

5.4.5.1 Relation of a system to the world

Whe e b ild ded c i e e , e all ha e a i e ded a lica i . P ea he a icia igh d i h , j g i g f he j a d e he ic f a e ,h gh e ha ch ac i i igh be called a a lica i . The i e ded

a lica i i called he f he e , i ce, e he b la d f al ela i i hi he e , e i g h e all hi k ab hei e f hei ea i g a d h he de a d he f a lica i he eal

ld.

Th he e ha e a e ha all a i e a c cl i (¢ ) he e ac a e hi he ld ( he ac al ). We a hi al ead i h he defi i i fT h, he e T h i a edica e, a d e c a ed T h[ ] i h . We ca d he

a e i h a ded c i e e . Thi a be e ge e al, ch a f e a le adiag ic e , like aki g a bl d a le de e i e he he a a ie ha a di ea e

. The e e l ba icall de i e f a ded c i e e ba ed h a

k ledge. U i g he h ee e , P ` a d P è ha e e c e ed e a e ih ded c i ela e he ld. T d hi e ca c c a able i h all hec bi a i f he a e f he ld i h he ible deci i a e .

152



† The ld i e e e ed i he c l a d deci i a e i he. I ld be a k a d if d d ha . The ea f hi e e a i al

c e i i ha he leadi g i e a e ea ie ec g i ed i c l . (Pe ha beca e f a a e like efe e ce f ee .) I he f ll i g able he c e i gi e b

c l f E g , , hile he fi h ee c l he e la a i .SemanticTable[All, Ergo[S, p]]

P Pè

P`

Ergo p Ÿ p

HS ¢ pL ŸHS pL JS Ÿ pN HS ¢ pL Right @1D Error @2DJS pN HS pL JS Ÿ pN HS pL Error @1Dand Unresolved @1DError @2Dand Unresolved @2DJS pN ŸHS pL HS ¢ Ÿ pL HS ¢ Ÿ pLError @1D Right@2D

The able all he f ll i g b e a i :

1. The e c f eali he (1) if i he e i a i , i decide ,h ( ¢ ), , (2) if i he e i a i , i decide , h ( ¢ ). Thi i

like i h he Defi i i f T h.

2. The e ake lai e he (1) if i he e i a i , i decide ,h ( ¢ ), , (2) if i he e i a i , i decide , h ( ¢ ).

3. The e lea e e i a k he i ca decide he a e f he ld.The i e ded a lica i fail beca e f decidabili .

4. Rejec i g a deci i , like (S' ), ill lea e ible c e f ac i , ei he hec a he decided e .

) T ejec hile ac all i he ca e, i called he .

) T ejec hile ac all i he ca e, i called he .Thi ca

be h a ed a faili g ejec a fal e ll h he i .

F a f al i f ie he e i diffe e ce be ee h e defi e . Weigh ake = The hi e = I i cl ded . The ef e he e i he c e i

ha h ld eflec he a ( he c e i a i ( he e cl e i )). Ihi a he i e l i f he e 1 a d 2 ge e ea i g. Whe a ge ha

decide e a i g M . A M . B, a d = Wi h a e a i M . A ha e cha ce i e l ge a d be e ha M . B , he i ake a diffe e ce h ge he be efi

f he d b he ha e i ca be f ll e led. Medical e hic f e e l ied ced a ba e, be e ge i he h i al a ickl a

ible.

153





† Ge e all e e hi i le f . N e ha" al c e i a ce f .SemanticallyCorrect @S, p D" p HHS ¢ pL pL

N all he i e ded a lica i f i clea . F e a le i l gic, a a i a ice c ld be i e ded c e i i al l gic. O a a i a ic e c ld be

i e ded c e he e f e . Whe he e i clea i e e a i he he f alla ifiie a ha e a d ai a d he e i he i k ha e a e g ided b

k ledge f a ac ical i a i . I ha ca e i hel i cl de he c di i ha hef all efe all a e e iha al be i e e ed.

† The f ll defi i i f e a icall c ec e i :SemanticallyCorrect @8Interpretation <, S, p D" p , pœS H p œ Interpretation Ï HHS ¢ pL pLL

† Whe " : ( ¢ ) h ld he e ca ded ce ha a f f ea ha he e i f f . Whe e gi e a del i eali , he he e ill be c i e

( ecall ha defi i i ).Ergo2D @Grounded @S, p D, Grounded @S, ÿ pD, CounterImplies @Grounded @S, ÿ pDD,

Ergo @S, p D NonSequitur @S, ÿ pDD1 HHS ¢ pL pL2 HHS ¢ Ÿ pL Ÿ pL3 J p JS Ÿ pNNErgo —————

4 JHS ¢ pL JS Ÿ pNN

Consistent @S, p D

" p JJS pN Î JS Ÿ pNN

Whe " : ( ¢ ) h ld he e ca ded ce he f ll i g i e e h able ,he e e a ig h al e he deci i a e a d he ee ha hi i lie

f eali . I (¢ ) he a d he e ca be a f f , i ce f ( ¢ ) i ldf ll ha , he ce ld f ll a c adic i . Si ila l f ( ¢ ). Fi all , ica be ha i decided a d f h le decidable, a d i ha ca e . N e

ha he lef ha d ide (S¢ ) (S ) (S ¢ ) i e i ale he igh ha d ide ha e ca ead he able b h a .

155



TruthTable @SemanticallyCorrect, S, p D

HS ¢ pL HS pL HS ¢ Ÿ pL p Ÿ p

1 0 0 1 00 1 0 1 00 1 0 0 1

0 0 1 0 1

† Thi able de i e f ha b h a d a e g ded.Grounded @S, p D&& Grounded @S, ÿ pD êêLogicalExpand

JJ p Ì JS Ÿ pNN Î JŸ p Ì JS pNN Î JJS pN Ì JS Ÿ pNNN

† P e e i g he a i he O a e e e a a e li e ake f ea ie eadi g.Grounded @S, p D&& Grounded @S, ÿ pD êêMatrixDNForm

J p Ì JS Ÿ pNNJŸ p Ì JS pNN

JJS pN Ì JS Ÿ pNN

PM. The e i a i fall ha e h ld fall i . C ide he a e e ha e i

decided. If e i e e e he" : ( ¢ ) a he ibili i d

e lace e , he decided e ca e a c adic i (i.e. i c bi a i i h hi

i e e a i f e a ical c le e e ).

† Thi i h a c adic i ld be ca ed.Undecided @S, p D

JJS pN Ì JS Ÿ pNN

% ê. Ergo @S, p_ D ¶ p

HŸ p Ï pL% êêLogicalExpand

False

The a e hi i ha" : ( ¢ ) i i e a che e f e lace e . I

a ic la , ch a e lace e le a a e l e lace (¢ ) i h . The e i ha

hi a e e i ale ce (ñ ) i ead f j i lica i . Y ca e lace b h ( ¢ ) Ø

a d ( ¢ ) Ø he i h ld ha (( ¢ ) ( ' )). S if b i e ai c i e c he h ld be i ed ha e .

156



Grounded @S :, y___ , pD i Ergo @, , D

SemanticallyCorrect @S :, y___ , pD

e e e ha f all : G ded

@, ,

DSemanticallyCorrect @8S2, y2___ <, S :, y___ , pD

a e hi f all i 8S, < i h he cla ified c di i ha al be ihe i e ded i e e a i 8S2, 2<. Th 8S, < al be a b e f 8S2, 2<

TruthTable @SemanticallyCorrect, S , pD

gi e he h able he E g@, D h ld

5.4.5.4 Deductive completeness

Thi ec i c ide ( ¢ ). S e i ded c i el ca able f e iff, he i e i he i e ded i e e a i ca e i . A he i e he ab e

g ded e ca e he e lif ed he e. Whe all a e lif ed he e icalled .

† Agai e ill efe he fi e e i ha i i le .

DeductivelyComplete @S, p D" p H p HS ¢ pLLDeductivelyComplete @8Interpretation <, S, p D" p , pœInterpretation H p œ S Ï H p HS ¢ pLLL

† I a ded c i el c le e e b h he a e e a d i i e ha e bee lif ed ha i i l ge decided.

Ergo2D[Lifted[S, p], Lifted[S, ¬p], p ¬p, Decided[S, p]]

1 H p HS ¢ pLL2 HŸ p HS ¢ Ÿ pLL3 H p Í Ÿ pLErgo ———

4 HHS ¢ pL Í HS ¢ Ÿ pLLHe ce:

ForAll @p, Decided @S, p DD" p HHS ¢ pL Í HS ¢ Ÿ pLL

Ded c i e c le e e e el i lie ha all a e e a e decidable b i ale cl de a j bled e ( he e all a e e a e decidable b e i i

157



f ):

ForAll @p, Decided @S, p DD ÏExists @p, p Ergo @S, ÿ pDD

I" p HHS ¢ pL Í HS ¢ Ÿ pLL Ï$ p H p HS ¢ Ÿ pLLM

The f ll i g h able ba icall i eeded i ce i i he a e a ha f . J he i a e ical a e e ch ha i ha a i de e i a e h al e, heded c i e c le e e all ha he e de i e a i c i e c . T e e hi ,

e eed h ee al ed l gic.

TruthTable @DeductivelyComplete, S, p D p Ÿ p HS ¢ pL HS ¢ Ÿ pL HHS ¢ pL Ï HS ¢ Ÿ pLL1 0 1 0 0Indeterminate Indeterminate 1 1 10 1 0 1 0

† The e a e he ible i a i he b h a d a e lif ed. Pa i he Oa e e a e e a a e li e .

Lifted @S, p D&& Lifted @S, ÿ pD êêMatrixDNForm

H p Ï HS ¢ pLLHHS ¢ pL Ï HS ¢ Ÿ pLLHHS ¢ Ÿ pL ÏŸ pL

DeductivelyComplete @S :, y___ , pD

e e e ha all ha a e e de he i e ded i e e a i a e he e

DeductivelyComplete @8S2, y2___ <, S :, y___ , pD

cla ifie ha hi ac all ea ha f all i he i e e a i

8S2, 2< i h ld ha he a e al i 8S, < a d ca be ded cedLifted @S :, y___ , pD i E g @, , D

5.5 Inference with the axiomatic method

5.5.1 Introduction

Thi ec i i e iga e le ba ed i fe e ce f i i al l gic.i a la gea a f le ha ha bee c ed a a icall b I fe e ceMachi e. Re ea ed

e lace e i g I fe gi e a l gical c cl i . The e h d ill ffe f hege e al eak e f he a i a ic e h d, ha a h ca be de i ed f fal e

e i e .

Economics[Inference, Print Æ False]

158



5.5.2 Pattern recognition

We i e iga e he a i a ic e h d. S e ela i a e ed a a i , a dc jec e a be e b e ea ed a lica i f he e a i . The h able

e h d ha e di c ed ab e i g f ca e he e e ha a defi i e idea ab

he ela i be e . The a i a ic e h d he he ha d ha he ad a ageha f a be c ea ed ha e had h gh f bef e.

I he Logic` ackage e al ead e c e ed he e f le . He e e

ee , a la ge a a f le ha ha bee c ed a a icall bI fe e ceMachi e.

Th gh i a le ba ed g a b i elf a d ha e a kable ca abili ie ia e ec g i i , i ill i a i le a i igh ee de el a i fe e ce

achi e. The ble e ide i he fac ha a e ec g i i e ai a c lica edaffai .

F e a le i ha N [ ] i al a ec g i ed a a i i a ai h .

† Whe e ha e a a i ( _ ! _) :> ! , he h ld a d f a a d h al: b e lace e d e ec g i e ha .

axiom = (p_ !p_) :> !p

H p_ Ÿ p_ Lß Ÿ p

q !q /. axiom

Ÿq

(!q q) /. axiom

HŸ q qL

Si ila a e ec g i i ble ee e i f he ba ic e ie , like hea i e f If.

Gi e hi ble i h a e ec g i i , e be di i g i h be ee (1) he a ie , i.e. he a i a e i h he h ld, a e e ed b a e lace e a e ,

a d (2) he e a le , ha all ec g i e a d a l he a i .

Wi h e ec he e e a le , e ill ha e le el f c le i . The fi le el fe a le i a e ec g i i i gi e f e a le b a e defi ed b ha d, a ihe ca e f A dO R le SelfI lica i . A ec d le el f e a le a i e he e

e e ie like e a d e beddi g i hi Bla kN llSe e ce , a d e b e f ce b ha i g ge e a e all c bi a i . F aigh f a d

a e b h a ache a c e d he a e. Of c e i i elega ahi d le el he a ge e al a e ca be defi ed, f e a le ha N [ ] c e a

159



a i h , b he he de el e ha e l ed hi a e ble , b e f cea be a g d al e a i e.

The I fe e ceMachi e ide e b e f ce, a d all f la e le a d e a d he .

A ble ha i i he e i he a i a ic e h d i ha a h ca be de i ed ffal e e i e . I La i , hi i he E Fal Se i Q dlibe i a i . Thi EFSQ

ble i l ed he e.

5.5.3 Infer

I fe i c ed b I fe e ceMachi e he heInference` ackage i l aded b . I fe i a da dl ba ed he a i SelfI lica i , M d P e ,

EFSQ, IfT a i i e, a d IfT A d, hile i g e a le A dS e ic, O S e ica d IfA iS e ic.

x êê. Infer ie l e a l gical a e e

InferQ gi e gge i he e f he ackageInferAndOr A dO R le ~ J i ~ I fe

InferQ

The will here have the function of implication within the object language.On terms: Object language constants are P @1D, P@2D, ... and variables are p1, p2, .... Commonly, the term

’axiom’ is used for an Hobject language Lexpression like p1 p1 which one accepts as true, for variable p1. In other words H p1 p1L:> True Hconcludes to True Lfor every substitution of a constant. Commonly,next, there are rules that allow manipulation of those axioms. Here however, those replacement rulesare much more in focus. It becomes rather natural to use the term ’axiom’ for p_ p_ : > True. Our somewhat deviant use of terms is likely caused by the fact that patterns and variables are different.

aLTo allow better manipulation of patterns, define axioms in terms of symbols p, q, ... Of course, use logical operators And, Or, Xor, Not, HImplies L,True and False. Use RuleDelayed for the result. E.g. H p pL:> True.

You can use patterns, so that your axioms are replacement rules. Note that H p_ p_ Lgives a

warning message. Then use LogicalPattern @ p pD. Perhaps, for the distinction with the metarules,you don’t want to use patterns for the axioms, and have these automatically inserted later.

Note: since you would like to use repeated replacement, your axioms should simplify rather than expand. bLGive metarules He.g. on symmetry Lwith normal patterns x_, y_ ,

... in the premisses. These metarules must be single, named and in MetaRuleForm.cLEach axiom can be inputted in ExpandAxiom with the metarules for its operators. Options

allow replacement of symbols p, q, ... with patterns Hin the premiss L, and If with Rule.dLJoining these expansions gives YourSetOfRules.eLAnalysis of object language logical expressions like x = H p1 q1Lworks like this:

x ê. YourSetOfRules or x êê. YourSetOfRulesf

LThe default inference rule is Infer. It presumes AndOrEnhancement. Otherwise use InferAndOr.

gLInfer has a preference for both using and solving by substitution.

160



† The i a i ha e a e e e i :Ergo2D[(p q), (q r), p, "find this"]

1 H p qL2 Hq r L3 pErgo ———

4 find this

† A i le cce i :(p q) (q r) p //. Infer

r

† Pe ha e c le i :

(!r !q) (!r p) (p q) //. Infer r

† B c a i , Si lif c ec l c llec all i f a i . B hi d e e e ehe i a i ha e a e e e .

(p q) (q r) p // Simplify

H p Ï q Ï r L

5.5.4 Axioms and metarules

I a he a ic , all a i a e ea ed he a e. f ce be eecific. Me a le ill he e be le ha ill be a lied he a i i de

i c ea e he likelih d ha he a i a ic a e i ec g i ed.

ModusPonens a i HH_ _L && _L :>IfToIf a i H_ H_ _ LL :> HH L H LLIfToAnd a i HH_ _L _ L :> HH! L && H LLEFSQ a i H! _ H_ _LL :> T eIfTransitive a i HH_ _L && H_ _LL :> H LSelfImplication a i H! _ _L :> a d H_ _L :> T e

Note: EFSQ = Ex Falso Sequitur Quodlibet: From False follows anything you want.

AndSymmetric e a le H_ && _L :> H && LOrSymmetric e a le H_ »» _L :> H »» LIfAntiSymmetric e a le H _ _L :> H! ! L

The f ll i g i e a e ed ge N [ ] a a i h :

161



Deny @ xD gi e a le f de ial f : Ø N @D; ca be a li

DenyPattern @ pattern , x, prin: True D

e lace _ i h !H_L i hi a e a d i h ! Hf c cl i L. Defa l i

5.5.5 InferenceMachine

The i e ha e e hi g ge he i I fe e ceMachi e. B e a di g aa i e ill ea he a lica i f he e a le ha a a i i e lica ed i alli a e .

InferenceMachine @ D e a d he a i i g he e a le ,

c ea i g a li f e lace e le

I fe e ceMachi e i di ec ed b a i i . The i i a e: H ldAllØ

a i ake a he a e , H ldØ a i ha al ead c cl de T e; e a dable ,A i T T e Ø a i ha d c cl de T e, a d a add a i :>T e ; e a dable , E a dA i Ø a i hich ill be e a ded ,PlacedP e ie Ø A dØ , IfØ , a e he e a le (e e ed a S i g

ha he ill e al a e a he call) ha h ld h ld f a i a ce f he e a .

† A e a le i he defi i i f I fe a a :Infer = InferenceMachine[

HoldAll Æ {SelfImplication},Hold Æ {EFSQ},AxiomToTrue Æ {ModusPonens, IfTransitive, IfToAnd},ExpandAxiom Æ {ModusPonens, IfTransitive, IfToAnd},PlacedProperties Æ

{Or Æ "OrSymmetric", And Æ "AndSymmetric", Implies Æ "IfAntiSymmetric"}];

5.5.6 The axiomatic method and EFSQ

I ha I fe i i h a a ad . We ca ace hi a ad he E FalSe i Q dlibe i a i , , ha f a fal eh d a hi g ca be de i ed. I i

ef l be a a e f hi , f he i e e igh c cl de ha he e i a e i I fe e ceMachi e. The f ll i g i a c cial e a le.

† The f ll i g a lie he a i i i f If:(x !x) (!x x) /. Infer

H x xL

162



% /. Infer

True

† While SelfI lica i gi e ( i g //. a he ha /.):

(x

!x) (!x

x) //. SelfImplicationHŸ x xL% //. InferAndOr

False

The a al i f he i a i i a f ll . The e a e i ided bSelfI lica i , a d he a e e i Fal e. Gi e ha he e i a fal eh d, a hi g ca be de i ed, ch a he e l T e ( he fi a ach).

Ab e ble ee be ela ed he fac ha IfT a i i e d i a eSelfI lica i . B he e ble i f a e ge e al a e. If e ld cha ge

he de , e ld i a he ca e he e EFSQ ca e ble .

I h ha I fe i a he i k f a lica i eal ld e i . I age e a e T e, b ded c i , hile he e a e i Fal e. F hi ea he eha bee eff e e d Decide[ ] i h I fe , a d c ea e a I fe E ha ce[ ] dea i h A dO E ha ce[ ]. Thi d e ee bad, a e ca al a fall back

he e h d i h he h able .I fac , if e i he a e he a i a ic e h d, he he e a ach ld

ee be ha e a i ded c i a allel each he . E e all , hi a he a e a he h able e h d.

5.5.7 Expansion subroutines

AndEmbedding @r D e bed _ && _ i hi Bla kN llSe e ce ,

hile he a e - a e a e added he la a Ha ed be he c cl i LAxiomToTrue @ x, y: Implies, adjustable: True D

gi e H ê. R leDela ed Ø L :> T e. C c he le: h le a i :> T e.If adj able, he he c cl i i a e ed if he e i a

ExpandAxiom @axiom , opts D

a lie he e a le f I lie , A d, O & X ,

f all e a a e cc e ce f he e e a . The e a le be gi e a e a Ø a e e a Ø 8a e1, a e2, <

he e a ei a e S i g . H$ a ei ill be he c f he cc e ce .L

163



Defa l i f E a dA i a e: Machi eF Ø T e a lie Machi eF he a i fi , Fi Ø T e ea ha he e a le l a l he e i ,

L gicalPa e Ø T e gi e a e i , R leØ T e e lace I lie i h R le,A dE beddi g Ø T e e bed && i hi Bla kN llSe e ce , U i Ø T e

fla e he e l a d he eb e e ible d lica i . P ible i a edefa l le f I lie , A d, O & X .

5.5.8 Different forms

The a al i gi e i e a i f , a el Machi eF , Objec F a dMe aR leF . Ac all , R le a d Pa e a e e lici l called F , b ill a ef f a l gical a e e .

MachineForm @ xD d e U Pa e @D ê. R le Ø I lie .A i a e i e all e Machi eF , facili a e a che

ObjectForm @ xD d e U Pa e @D ê. R leDela ed Ø R le.T a f i bjec la g age f , i cl di g :>

MetaRuleForm @ xD d e L gicalPa e @Machi eF @DDToRule @ xD d e L gicalPa e @D ê. I lie Ø R le.

T a f bjec la g age e e i i le

LogicalPattern @D cha ge a l gical e e i i a l gical a e . Defa li Fi Ø T e a lie a e l he e i

5.5.9 Accounting

The I fe e ceMachi e k b a ig i g a a ke each cc e ce f a l gicale a . The e a le he a e a lied each e a i di id all . Af e hi , he

a ke a e e ed agai .

164



PlacedMetaRule @name_String D

b i e f E a dA i a d PlaceP e ie ,de i e a laced e a le f he ge e al e a le

PlacedProperties @len , xD b i e f E a dA i ,

gi e each laced e a i li f e a le ;le i he be f cc e ce f he e a ,a d gi e he li f ge e al e a le f ha e a

Place @Operator , iD a k he lace f he i h e a

PlaceOperator @expr D gi e each e a i lace a k

PlaceUndo = 8 Place @ x_ , y_ DØ x< hich e e Place a k

5.6 A note on inference

The i ila i f , a d i e ec i el p1 , ..., pn ¢ a d ( p1 ... pn) i

i c b e ca al b e e a i ila i be ee¢ a d ha i e e li g.

† Ea lie e e:Ergo AHP@1D P@2DLt , P@1Dt+ 1 , P@2Dt+ 3E

H8H P 1 P 2Lt , H P 1Lt +1< ¢ H P 2Lt +3L

† B i i a e a ali i g i e he f ll i g i ce i cla ifie ha ¢ ca al bef ali ed i h t ha i i l a all lea f h gh ab ac f.

Implies AHP@1D P@2DLt Ï P@1Dt+ 1 , P@2Dt+ 3E

HHH P 1 P 2Lt Ï H P 1Lt +1L H P 2Lt +3L

I he d , he diffe e ce be ee a a e e a d a a g e ee l beh he a d b l a e ead a d i e e ed. Readi g a icall gi e a a e e ,

eadi g d a icall gi e a a g e .The d a ic a i igh al hel f ali e h he ical ea i g, i.e. he

e e e hi g b la e ha ejec i a fal e. I a da d i fe e ce a iha bec e ( ) ¢ e i ac ice e ead a p t ; HŸ pL t +1 ; h HŸ pL t +2

he e he d h l ha he ch l gical al e f e ha i .

Ha i g e abli hed he diffe e ce be ee i lica i a d i fe e ce, he e i a i ehe he a f he i e iga i i i fe e ce i ef l. I ee ha cha ac e i ic

f i fe e ce ca be ca ed b he d f a e e . The d a ic a ec f i fe e ced e add a hi g i alidi ( a l gical cha ac e f he a cia ed

a e e ). B de c ibi g i fe e ce, l gic ha ade f he e h a ac i i f

165



ded c i , b he a lica i f ¢ i elf d e ee lead ge e al e l haca be de c ibed he i e.

Th , e i lef i h a feeli g f ea e he he he diffe e ce (a d e e di i c i ) be ee Decide ( ¢ ) a d I lie ( ), h e e l ge e e i , i eall ha

ele a (e ce f ch l g ).

† PM. I a ea ha al ha a b l The ef e, ha d e e al a e b h i h h ee d . I i c cei able e ic he e f¢ e

e a d he hi \ S c a e , i.e. h a i fe e ce. B gi e hec al ide i hi ca e a l i de f b l i h eall addi g ch,

e ki hi .Therefore @p, p Ï qD p \ H p Ï qLBecause @p Í q, p Ï qD

H p Í qL ‹ H p Ï qL

166



6. Applications

6.1 Introduction

The i e f al , k ledge, babili a d dal l gic a e a lica i f b hedica e a d i fe e ce.

I he e a ea f a lica i he i f c a i e i ch ed. The f ll i ge ie he e c ce , he e he e de ic l gic i ed f he l gic f ali .

@ D P O gh K Ce ai Nece a

Pè

F eed Ig a ce U ce ai C i ge

P`

N All ed K Ÿ Ce ai Ÿ I ible

I e e i g b e e i h he e c ce ela e, a d al ha he igh bel g

he d ai f a he a ic e i ic . Ma he a ical h a e called ece a il e.The a e f k ledge ab a e i ical a e igh be c a ed he deci i a e

f a f al e . P babili ge e all c ce e i ical a e , b he a e e i100% ce ai he e igh call i ece a . A 0% babili f ea ha i100% ce ai . U ce ai i he ke i , i ce he a e ce ai ab ,

he i ill igh be . Ce ai a be alified a ha i i ha i ce ai .E e ha a e 100% ce ai igh al be called ece a .

A fi al a lica i ha be e i ed he e i he a lica i f l gic a d i fe e ce la ge i e i he ld.

6.2 Morals and deontic logic

6.2.1 Introduction

M al ha e he a e c al f a P efe e ce :Preferences Morals

Better OughtIndifference Freedom

Worse Not Allowed

167



M al a d efe e ce a e a e a le a lica i f he edica e calc l . M al a eefe e ce i a g f , ch ha e le a e illi g c ide he a ec

bef e e i ci le ha e bee acce ed fi . S ch a de i g i al called ake f he a al g f a dic i a he e d a e de ed ch ha

f e a le a i al a bef e a .

The i e bjec f he he f al i ha he e i a ga be ee I a d O gh .Thi i ci le i elf e ide . Pe le e d c f e eali i h ha h ld be.O ce a e a a e f he di i c i , i ee e b i e c f i c ee a e ec ed e a a . S e c ie i he ld f e a le ha e a dea h

e al , a d he ci i e f h e a e a e ed he idea hich a ca e e fhe hi k ha hi i h i be. B a h ld ca e e be de i ed f a

i .The l gic f al i called de ic l gic. The i a a i i ha if

e hi g i all i e a i e, he al all i i lica i a e all i e a i e. Ifa e d , a d if accide al dea h gh be e e ed, he e h ld

a e ha e .

Economics[Logic`Deontic]

Cool`Logic`Deontic`

Allowed FreedomQ NotAllowedQ SetDeontic ToOught

AllowedQ MoralConclude NotOught ToAllowed

DeonticAxiom MoralSelect Ought ToFreedom

Freedom NotAllowed OughtQ ToNotAllowed

SetDeontic["Explain"]

SetDeontic @u, o, na Dsymplifies the following steps:

The user has to setUniverse @D = 8 p, Ÿ p, q, ... <where each p has a Ÿ p Hnot pLOught @D = Ought @8...<Dwith a selection from the universe, for the Op

NotAllowed @D = NotAllowed @8...<Dwith another selection, for the ŸApThen ToAllowed @Dand ToFreedom @Dgive what is allowed and what is free to chooseThe crucial idea is that Op ñ Ÿ AŸ p.The universe consists of three disjoint sets: Ought, Freedom and NotAllowed. The

Universe, Allowed and Freedom objects read as Or @ D, the Ought and NotAllowed objects readas And @ D. Ought, Freedom and NotAllowed may also be seen as Better, Indifferent and Worse.

SetDeontic @Universe Dcreates the universe from the binary states, and selects the Ought @Universe Dcases

168



6.2.2 Setting values manually

B fi e i g e al e a all , e ill be e de a d he c e .

† Re i ed a e e decla ed S b l . Each e e e e a e e , like =Thi e d ,= I hel .

symbs = {p, q, r, s, t, v}

8 p, q , r , s , t , v<

† The ele e f he i e e h ld al c ai he ega i like = Thie d e d .

u = Universe[] = Flatten[FromEvent /@ symbs]

8 p, Ÿ p , q , Ÿ q, r , Ÿ r , s , Ÿ s, t , Ÿ t , v, Ÿ v<

† ea Thi e h ld d . Le al akef e .o = Ought[] = Ought[{p, r}]

Ought H8 p, r <L

† Le decla e ha a d a e all ed: A & A .na = NotAllowed[] = NotAllowed[{t, v}]

NotAllowed H8t , v<L

The ke c ce i ( )ñ

A . F e a le: Y h ld keñ

I i all ed ha ke. (A e hical i ci le i ge ha a heal h a i g !)

† I ha e did e l a e ha gh ha e . We f g a d .too = ToOught[na]

Ought H8Ÿ t , Ÿ v<L

† A d ei he e e e ecific ha i all ed. We f g a d .ToNotAllowed[o]

NotAllowed H8Ÿ p , Ÿ r <L

ToAllowed @D de i e ha i all ed f ha i all ed

ToFreedom @D de i e ha i bjec f eech ice f O gh @D a d N All ed @D

ToNotAllowed @ x_Ought D de i e ha i all ed if O gh

ToOught @ x_NotAllowed D de i e ha O gh if i N All ed

Note that only the last two require an input. They must be called before the first two can be called.

169



6.2.3 Using SetDeontic

The i e Se De ic hel c i e l defi e he eal f he di c i .He ce, e l edefi i g O gh a d N All ed.

SetDeontic[symbs, {p, r}, {t, v}]

88 p, Ÿ p , q , Ÿ q, r , Ÿ r , s , Ÿ s, t , Ÿ t , v, Ÿv<, Ought H8 p, r , Ÿ t , Ÿ v<L, NotAllowed H8Ÿ p , Ÿ r , t , v<L, Allowed H8 p, q , r , s , Ÿq, Ÿ s, Ÿ t , Ÿ v<L, Freedom H8q, s , Ÿ q, Ÿ s<L<

SetDeontic @U_List , O_List , NA_List D

The i e e ele e a e defi ed a he ele ei U a d hei ega i . Wha gh i defi ed a heele e i O a d he ega i i NA. Wha i N All edi defi ed f he ele e i NA a d he ega i f O

SetDeontic @Universe D

e U i e e@U i e eD he e d c f 8, Ÿ < f he ele e i U,a d e O gh@U i e eD he li f ibili ie ha a i f ha gh

Se De ic ha al defi ed he bjec All ed a d F eed .

† All ed i ha i N All ed. Wha gh , i al all ed. (I ld bea ge a Y gh hel , b a e all ed hel . )

Allowed[]

Allowed H8 p, q , r , s , Ÿ q, Ÿ s, Ÿ t , Ÿ v<L

† F eed e i he e e a e all ed d hi g ha e d gh d .Freedom[]

Freedom H8q, s , Ÿ q, Ÿ s<L

170



6.2.4 Objects and Q’s with the same structure

Allowed @D h ld efe a All ed@8...<D bjec

Allowed @8…<D i he bjec ha c ai ha i all ed

AllowedQ @ pD i T e iff i a ele e f All ed@DAllowedQ @ p_List D i T e iff all ele e i a e i All ed@DAllowedQ @Universe, p_List D i he a e a All edQ

Freedom @D h ld efe a F eed @8...<D bjec

Freedom @8…<D i he bjec ha c ai ha i f ee ch e f

FreedomQ @ pD i T e iff i a ele e f F eed @DFreedomQ @ p_List D i T e iff all ele e i a e i F eed @DFreedomQ @Universe, p_List D i T e iff all ele e i ha a e -

gh a e i F eed @D

NotAllowed @D h ld efe a N All ed@8...<D bjec

NotAllowed @8…<D i he bjec ha c ai ha i all ed

NotAllowedQ @ pD i T e iff i a ele e f N All ed@DNotAllowedQ @ p_List D i T e iff all ele e i a e i N All ed@DNotAllowedQ @Universe, p_List D i T e iff e ele e

i N All ed@D al cc i

Ought @D h ld efe a O gh@8...<D bjec

Ought @8…<D i he bjec ha c ai ha gh

OughtQ @ pD i T e iff i a ele e f O gh@DOughtQ @ p_List D i T e iff all ele e i a e i O gh@DOughtQ @Universe, p_List D i T e iff all ele e i O gh@D al cc i

N e: Al defi ed ha bee N O gh , i ce e i e he e i li g i ic c f ii h O gh N ( he e le a e ha i e e hi g, f e a le). N O gh

( ) = F eed N All ed (j he c le e ).

NotOught[]

NotOught H8q, s , t , v, Ÿ p , Ÿq, Ÿ r , Ÿ s<L

171



Freedom[] || NotAllowed[]

HFreedom H8q, s , Ÿq, Ÿ s<L Í NotAllowed H8Ÿ p , Ÿ r , t , v<LL

NotOught @D de i e f hich i i aid ha i gh HF eed N All edLNotOught @8…<D i he bjec ha c ai ha gh

Other functions for NotOught are not available.

6.2.5 Universe

Ab e gi e j he ele e f he i e e. The eal i e e i a l gical c bi a if e if i ele e . P ible a e f he ld a e f e a le & & , b al

& & . Gi e ele e , e ake all ible c bi a i f , , , ,

e ce e a. Ra he ha i g he b l & e ill e li . Th a li , , i hea e a he a e i ha & & , i h all he e he e a cc i g a he a ei e. The i e e f all ch ible c bi a i i U i e e[U i e e].

Se De ic[U i e e] ill c ea e hi i e e. H e e , ai l i e e i g iO gh [U i e e] ha gi e he li f ible a e ha a i f ha gh . The la ehe ce i al b Se De ic[U i e e].

† Thi gi e he ible c bi a i ha a i f ha gh .SetDeontic[Universe]

p q r s Ÿ t Ÿv p q r Ÿ s Ÿ t Ÿv p Ÿq r s Ÿ t Ÿv p Ÿq r Ÿ s Ÿ t Ÿv

MoralSelect @lis_List ?MatrixQ, qD

elec f he a i i g c i e i . The la e bedefi ed f @U i e e, D- hich i he ca e f = All edQ,

F eed Q, N All edQ a d O gh Q

MoralSelect @qD e U i e e@U i e eD , a d f = O gh Q i gi e O gh@U i e eDNote that the q[Universe, ...] criteria have different meanings for elements or a state of the universe.

6.2.6 The difference between Is and Ought

Ab e e k = Thi e d ,= I hel . Ab e i e e gge ha iill ld be all ed ha a e d b i hel ed. The de ic a i

h e e gge : If e e i d i g a d ca babl be a ed b hel i g, a d if c ide ha hi e h ld d , he h ld a e hi he .

172



The e a e a a i la e l gical a e e ha c ai O gh . O e a i e a e lace e le, he he i e he M alC cl de[ ] c a d. B h a e

eak i e , b he fi i eake .

MoralConclude @argument D le e C cl de i hhe De ic A i HO & L O

DeonticAxiom gi e he De ic A i i lef a HO gh @_D & _ _L :> O gh @D

MoralConclude can best be used in combination with the function Conclude of The Economics Pack . Conclude is further notexplained here. The DeonticAxiom can be combined with Infer, idem.

Le f he de el he i e b clea d a he ha a d . Le c ide a e e . The fi i hil hical i ce i e ac l c ie he c e f he a i .

† A i a ce f he a i .stat1 = Ought[¬drown] && (¬drown help)

HOught HŸ drown L Ï HŸdrown helpLL

† U i g a e lace e le i fa a d igh a ge .stat1 /. DeonticAxiom

Ought HhelpL

The ec d a e e i e ac ical a d e e he ea c e f hehil hical a g e . (1) I a e he c cl i he e ld hel . S ee le a e l d a a c cl i he e a d i he ea i e. (2) I

cla ifie ha hel i g i lie ge i g e e elf. A d e ha he e i da ge ha ed e elf. (3) The idea ha he ic i f he accide h ld d c e

l a a la e eali a i .

† Wha d ?

stat2 = (¬help drown) && (help getwet) && Ought[¬drown]

HHŸ help drown L Ï Hhelp getwet L ÏOught HŸ drown LL

† Re lace e ge he e. See he di c i ab e he diffic l f i ge laci g le ( he a i a ic e h d).

stat2 /. DeonticAxiom

HHŸ help drown L Ï Hhelp getwet L ÏOught HŸ drown LL

Le e he C cl de a d M alC cl de i e .

173



† N e ha e fi i i iali e C cl de[] hi e C cl i = . S b e e callgi e l he e . The , he l gical c cl i f he fi a e e a e i e i e.

Conclude[]; Conclude[stat1]

8Hdrown Í helpL, Ought HŸ drown L<† Ne c cl i f he ec d a e e a e ei he i e i e. N e ha A d

a d O a e O de le .Conclude[stat2]

8Hhelp Í drown L, HŸ help Í getwet L<

† Thi ld be he al c cl i h e e .MoralConclude[stat2]

8Ought Hgetwet L, Ought HhelpL<

S e hil he a g e ha , i ce ge i g e ca be a g al i e a i e,he de ic a i l ha li i ed a lica i . Ye i hi ca e i ell ha h ld

be d e.

6.3 Knowledge, probability and modal logic

Thi ec i i a he b ief i ce i e e l i he ide f de ic l gic e el b i e he he label . The gge i i ha he e ide i ca eg ical f

all ch ide .

† ld all b i i like hi :Options @ToModalLogic D =

Thread @8Ought, Freedom, NotAllowed < Æ 8Necessary, Contingent, Impossible <D;

ToModalLogic @x_, opts___Rule D:= x ê. 8opts < ê. Options @ToModalLogic D

† F e a le:Necessary @D = ToModalLogic @Ought @8p, r <DD

Necessary H8 p, r <L

A al e a i e e h d i i l defi e Nece a = O gh , C i ge = F eed ,N All ed = I ible. Thi igh ell fi d a hil hical i e e a i (i) ha

gh f a e i ece a , (ii) ha all f eed f a e i c i ge , a d (iii)

ha i all ed f a e i i ible. (Thi c ea e a b idge be ee I a dO gh b i g e e hi g i al . I hi ca e i i e el a a f a

g a i g ble .)

174



A hi d al e a i e i k i h de ic di ec l a d e el a la e hec cl i .

A f h ibili i c he de ic l gic ackage a d c ea e ackage f he ea lica i . E e all hi igh be d e, if l f ch l gical ea .

6.4 Application in general

6.4.1 Introduction

Thi ec i i l e i d ha he e i a la ge ld ide. A lica i fl gic a d i fe e ce ca be d e b i gle e g , i h i h he

f l gical achi e . G deci i aki g ha i a ad e , eeC lig a (2007b), V i g he f de c ac . Wha i ef l i he c e f hi b k a e he i e a ailable f . R a Maede ce bli hed e

i e ha e e i fe e ial, I l he efe e ce. I he e a e e ackage l gic a d e he b ch de el ed. The e i al a ackage ha e he

hi ica ed e i e f g a a e i le a d gl l ki gki d f i fe e ce, hich lea e e de i g. The f ll i g b ec i e i

ha el e i i E (TEP).

6.4.2 Analysis of longer texts

I a al i g l ge e , i a ea be ef l ha e e e a age efacili ie . I a ic la , a ag a h ca be a al ed b he el e , a d i e edia el gic e ca be eli i a ed i ce he eed be ele a f he fi al c cl i . Ae a le f a l ge a al i i gi e i L gicE a le. b, ha di c e he Fi a cialTi e edi ial f F ida J l 26 1991.

See TEP 4.3 a d Financial Times Editorial a d FT Editorial Analysis .

6.4.3 Logic laboratory and inference

Y a i h e e i e i h a i le a i l gical a e e . Fe a le, a i h e e d he A dO R le [ ] ed i E ha ce e . I ha ca e

he ced e L gicLab ca be ef l. See TEP4.4 a d Logic laboratory .

See here f a l ge di c i f i fe e ce i h he I fe e ce` ackage.

175



176



Part III. Alternatives to two-valued logic

A al e a i e al ed l gic e ill c ide :

1. h ee al ed l gic ha i b i l a al e a i e

2. i i i i ha i aid be a al e a i e b he ac all i i de d

3. f he , ha i clai ed be al ed, b ha ed ce c adic i

beca e f he Lia .The e a e al h ee a ache deal i h he Lia a ad . Ne e had e e

ch a e i f e i e ea che a ha e ed i h he Lia . We al ead l edi i al ed l gic b le cli ch i f he e ai i g bi f ea i g i ic .

177



178





de e i i a d j acce ha e d k e e hi g. E ce f he C fc e he e e a e ali . He ce, A i le a g e c cee i l g a d al , a d d e eall e i e a ada a i f he T e Fal edich .

I de e de f ha , he e ill ee be a i hi be ee he h ee e¢

, ¢ a d S p a a , ha he de el e f h ee al ed l gic ca be a bi

e d e i ce he e al ead igh be a del e e i . Ye i i g d ha e eeh ee al ed l gic. U de e i ed decided diffe f .

The ea c ide h ee al ed l gic he e i ha e a acc da e fe e ce ha a e e ical, ch a he Lia a ad . F al ed l gic i h ldha he e e i ca be al ed a all b e ld like ha e a e ge e al

e ha acc da e f he ich e f la g age. The i ade ac f al edl gic h f aki g he h able f (N A All). O e f he f i g la

e a al a gi e a fal eh d ha e e a e f he ld. B ha ill ifal eh d a d ill all f he Lia .

TruthTable @NotAtAll @pDD p † pTrue FalseFalse False

He ce e add a hi d . a e he a e e i al ed l gic a d a e hee e ce ha a al be h ee al ed. He e Õ hich ea ha i hi h eeal ed l gic e ill ha e a c e f al ed e ha all di c i a T e

Fal e a e he h ee al ed e f he e e ce . The hf c i: Ø 1, 0 i

a bf c i f : Ø 1, 0, 12 , a d a e e like [ ] = 1 a e ill al ed. Thi

i a di i c i i le el b i .S a e e i igh al be called elldefi ed e ac . Si ce i d e a e e ca i e = .

P e l de ed, he e e ce i T e I de e i a e Fal e 1,12 , 0 . Thi igh

ake e e i e e ec b i a ea e ef l e el a e d I de e i a e he igi al de , i ce hi all f a icke e ifica i f he al ed a .

Ne e hele , e h ld kee i i d ha T e ha he a i al al e a d Fal e hei i al e.

180



7.2 Propositional operators, revalued

7.2.1 Definition

Whe e a e e ha e d k he h ab he e igh i la l , hich i e, a d lea e i a ha . H e e , if e i h be able

defi e a Lia e e ce he e eck ha ch e e ce a e ei he e fal e,he ce i de e i a e. Thi i ca e c ide a h ee al ed l gic.

Th ee al ed l gic c ai a c e he e al ed e ill k . Th , ae ical a e e ha h al e I de e i a e, b i i T e Fal e a ha i

ha T e Fal e al e. I ha e ch a e e ca bec e c f i g

( ee bel ), ha i , he a e ed he bjec .

I h ee al ed l gic e e T e, Fal e, I de e i a e a d 1, 0,12 . I e e ec he

e f h ee h al e cha ge he defi i i f he l gical e a i ce h ee ie ha . I a he e ec he e i li le cha ge i ce e kee a cia i g N [ ]

i h 1 , A i h Mi , i h Ma , i h § a d $E i h =.

NB. Si ce $E i ale ha l a iable e ill ada i f h ee al ed l gic.Si ce E i ale ca ha e e a iable e ill lea e i cha ged.

† I he defa l e i g f e fi d ha A d a d O al ead f c i a Mia d Ma al i h e ec I de e i a e

8True Ï Indeterminate, True Í Indeterminate <

8Indeterminate, True <8False Ï Indeterminate, False Í Indeterminate <

8False, Indeterminate <

† B N a d I lie d i e e al a e, e e i h L gicalE a d. The e e ai a I de e i a e e .

Not @Indeterminate D êêLogicalExpand

Ÿ Indeterminate

Indeterminate False êêLogicalExpand

Ÿ Indeterminate

† I lie e al a e he i i j he e O a e e .Indeterminate True

True

181



† Thi i al acce able.Indeterminate Ï Indeterminate Ï Indeterminate

HIndeterminate Ï Indeterminate Ï Indeterminate L%

êêLogicalExpandIndeterminate

We he i a e e L gicalE a d Si lif i ce he e f ce al ed e . B ife d e L gicalE a d a d Si lif he e e e i d e al a e ell,

e e h gh e k ha he c ld. He ce e de el a e a a e e e al a e e e i i h ee al ed l gic.

Th ee al ed l gic i ed i h he c a d Th eeVal edL gic[T e O ]. I

di able A dO E ha ce i ce he la e e al ed l gic. If a e ce aiab ha a e he e i i , ca e al a e L gicS a e[ ] ha check Th eeVal edL gic[ ] a d A dO E ha ce[ ]. N e ha L gicalE a d a d Si lifal a e ai al ed a he e ai i e al he e . If d a cha ge he defi i i f he l gical e a b e el a e e i e i h e e lace e le , he c ld e he Th eeVal edR le [...]; b he e ca be ed a e ce he e f h ee al ed l gic ha bee ed .

ThreeValuedLogic @True D

ThreeValuedLogic:: State : The use of three - valued logic set to be True

NB. The e ie ab e I lie ha e bee cha ged ee bel .

ThreeValuedLogic @x D f =T e O he e f h eeal ed l gic. O he e ie

i ff a d e he defa l e

ThreeValuedLogic @D i T e if h eeal ed l gic i ed , he i e Fal e

LogicState@D

check Th eeVal edL gic

@D a d A dO E ha ce

@D ,

hich ca b h be T e

ThreeValuedRules @D he i Bla k he e al J i @Th eeVal edR le @T eD , A dO R le @T eDD

he e Th eeVal edR le @T eD i le e h ee-al ed l gic. Whe Th eeVal edL gic ha bee

ed he he e le ca be ed

NotAtAll @ D e e e ha ei he Ÿ a l , b a hi d al e i h eeal ed l gic, de ed a

i h a h al e f I de e i a e Hi e e able a 1ê2LLogicalExpand and Simplify are internal to Mathematica and use two-valued logic. Enter NotAtAll as such otherwise Mathematica will not recognize it. For texts you could use Â dgÂ . Paradox: ThreeValuedLogic[x] recognizes only two states for itself.

182



7.2.2 Singulary operators

† The i de e i ac f I de e i a e ca be acce ed.TruthTable @Not @pDD

Ÿ p

True FalseFalse TrueIndeterminate Indeterminate

We d eed e b lic i ce e ca a T hQ[I de e i a e]ã I de e i a e.Th i i a i le N A All e ai j a label a d a e a . A e a

igh gi e he gge i ha i e ical a e e i e hi g e ical,hile a label d e ha e ha e a d e el i dica e e hi g. Le c idehe i h e e . The e a e 3 a d each ha 3 ible c e . Th

he e a e33 = 27 ible i g la e a . Defi i el he e i e ha ld ha e aT e he b . C ide i g hi , i ake e e acce a eci el he

a la i f T hQ[ ] ã I de e i a e ( hich i i al ed l gic).

† The defi i i f h f al ed l gic l k al e , b f hee e al ead all ed he hi d al e.

TruthQ @Indeterminate DIndeterminate

† The label ca ef ll be i e e ed a a e a . The h able eflec ee f he label. NB. I be N A All[ ] a d , i ce he la e l

di la a ch ( ee h i i i h $E i ale ).TruthTable @NotAtAll @pDD

p † pTrue FalseFalse False

Indeterminate True

PM. I al ed l gic he e e e 4 i g la e a . We ha e ake he fie (Fal e) a d i cl ded a al e he hi d . Ea lie e h gh e had e f

hi c l b i a ea he e i e e.

PM. Wi h 27 i g la e a , e ha e ed l T hQ, N a d N A All .D e eed e ? I a ea ha e d eed he he . Wi h h ee e a eca acce all , a d ha ffice .

7.2.3 Binary operators: And and Or

I al ed l gic e had 4 ai i h each 2 i gi i g24 = 16 bi a e a i b e ha e 9 ai i a h able i h each 3 al e , gi i g39 = 19683 ible

183



bi a e a . Thi i l be ilde i g b a he c e e ce i ha ae a ill ickl ha e e c i ha eh l k like i . F a el , e

eed l acce i e e a a e , a d, e ha e e g c e i d .

F A d e ca ai ai he Mi c di i ha he h al e[A d[ , ]] =

Mi [ [ ], [ ]] ha f e a le Mi [1,12 ] = 12 a d ha Mi [1, I de e i a e]

gi e I de e i a e. I Cha e 1 e a la ed i algeb aic e a Mi [ , ] ==1. We l e he e i ale ce ha all ed l he 0 al e a i e. If Mi [ , ] ∫ 1 he

al e 0 12 a e ible.

† F A d he adj ed i i c di i gi e :TruthTable @p Ï ÿ p D

p H p Ï Ÿ pLTrue FalseFalse FalseIndeterminate Indeterminate

† F A d he adj ed i i c di i gi e a ell:SquareTruthTable @p Ï qD

H p Ï qL q Ÿq † q p True False IndeterminateŸ p False False False† p Indeterminate False Indeterminate

† F O a i ila l adj ed a i c di i gi e :TruthTable @p Í ÿ p D

p H p Í Ÿ pLTrue True

False TrueIndeterminate Indeterminate

SquareTruthTable @p Í qD

H p Í qL q Ÿq † q p True True True

Ÿ p True False Indeterminate† p True Indeterminate Indeterminate

† The c e i be ee A d a d O ia N i ai ai ed.SquareTruthTable @Not @ÿ p Í ÿ qDD

ŸHŸ p Í Ÿ qL q Ÿq † q p True False Indeterminate

Ÿ p False False False† p Indeterminate False Indeterminate

184



7.2.4 Implies

F I lie , e kee he i e e a i ha e e e § . U i g § i h al e 12

ea ha he i lica i l e T e Fal e a d h e al ed l gic di c h ee al ed c e . The ice i ha e l e he i e e a i ha

ill e h ee al e . F hi , e ca e N O .

† F e a le, he l e igh c e12 § 12 h ld gi e T e.

SquareTruthTable @p qD

H p qL q Ÿq † q p True False FalseŸ p True True True

† p True False True

† S e eade a efe he li e f a f hi i a e .TruthTable @p qD

q H p qLTrue True True

True False FalseTrue Indeterminate FalseFalse True TrueFalse False TrueFalse Indeterminate TrueIndeterminate True TrueIndeterminate False FalseIndeterminate Indeterminate True

While he i lica i e ai EFSQ f e ical a e e , e ha e a e li i ede e . F W ll b ca de i e ha H ga

h ld a ack I dia, ided ha a e illi g a e he a ecede , a dided ha H ga i deed a ack I dia. If H ga d e a ack I dia he a

i lica i f a e ical a e e i fal e. S ca l c f e he

H ga ia i h e ical a e e i lica i g ha a ack ce i ide a .

Indeterminate False

False

185



† The ela i e ai i ac f N O , b defi i i . Whe I de e i a ea d f 1

2 he he N O ela i ca be§ . See he b igh c e .

SquareTruthTable @NotpOrq @p, q DD

HŸ p Í qL q Ÿq † q

p True False IndeterminateŸ p True True True

† p True Indeterminate Indeterminate

7.2.5 Equivalent

† $E i ale ill e = a d e ai e i ale e eciall i ce ead ed ha § i e e a i f i lica i . B h ha e al ed c e l .$E i ale h ill ca a e ha e e i a e e i ale (i.e. ha e he a e

h al e ).SquareTruthTable @$Equivalent @p, q DD

H p ñ qL q Ÿq † q p True False FalseŸ p False True False† p False False True

† E i ale a d N O d ee e ef l.SquareTruthTable @NotpOrq @p, q D Ï NotpOrq @q, p DD

HHŸ p Í qL Ï HŸq Í pLLq Ÿq † q

p True False IndeterminateŸ p False True Indeterminate† p Indeterminate Indeterminate Indeterminate

SquareTruthTable @Equivalent @p, q DD p q q Ÿq † q

p True False IndeterminateŸ p False True Indeterminate† p Indeterminate Indeterminate True

7.2.6 TruthValue

† N he e a e hal e eigh ed b a la ge be f ible a e f he ld.TruthValue @p qD2

3

186





† A d hi fi he ld f . TND h ld i ge e al i h he e ce if e e. (We ea lie ed N AP i i b N A All i a be e ge e al

f .)TruthTable @Hp Í ÿ pL ~ Unless ~ NotAtAll @pDD

p HŸ† p H p Í Ÿ pLLTrue True

False TrueIndeterminate True

† Thi i aigh f a d. We ha e acce all h ee .TruthTable @p Í ÿ p Í NotAtAll @pDD

p H p Í Ÿ p Í † pLTrue True


7.3.4 Linguistic traps

We ca acce ha i e ical iff i e ical . B ca e al acce ie ical iff i e ical ? F , e igh al c ide be ab d. I

a ea de e d ha ea b iff . Whe if i a N O , he igh acce he fi e i ale ce l f a ic b ha e ejec i f ch

ab di . I ca e, e ha e ake he ge If a d E i ale ce, a d e ca acce b h e i ale ce . Thi e g h all f i e al a lica i , hich i j ha e

eed. Thi al ea ha he e e f e ide i he i ha i iece a il I de e i a e. F he Lia i l ee ha e ded ce Lia Lia b

ac all e fi d Lia . N , clea l , he a e ed he e defi i i a d a e ed c ide ha i ea ha e hi g i N A All ... (a Ke e e

i he bi a f Ra e : he e i g e e ci e f he f da i f h gh ,he e he i d ie ca ch i ail e e be Phile a f C h ac all

died f hi c gi a i ) ... he a be g a ef l ha he e i a l gical achi eha ca check hi g .

† Thi a la e i e ical iff i e ical .$Equivalent @Not @NotAtAll @pDD, Not @NotAtAll @ÿ pDDD

HŸ † p ñ Ÿ † HŸ pLL% êêTruthTable

p HŸ† p ñ Ÿ † HŸ pLLTrue TrueFalse True

Indeterminate True

188



† Thi a la e i e ical iff i e ical"$Equivalent @Not @NotAtAll @p Í ÿ p DD, Not @NotAtAll @p Ï ÿ pDDD

HŸ † H p Í Ÿ pL ñ Ÿ † H p Ï Ÿ pLL%

êêTruthTable p HŸ† H p Í Ÿ pL ñ Ÿ † H p Ï Ÿ pLL

True True


7.3.5 Transformations

The c e i le e a i f h ee al ed l gic i cl de e ada a i i A da d O , diffe e f he f ll e ha ce e f al ed l gic. Thi a liee eciall he ca e f e ha a iable , i h he idea ake hi g e

ac able. A a c e e ce L gicalE a d d e k a i ed k.

† E a i a e c ac ed agai .try = p (q r);

try2 = $Equivalent[try, LogicalExpand[try]]

HH p Ï Hq Í r LL ñ H p Ï Hq Í r LLL

† Ab e a l g i ec g i ed a ch.TruthValue @try2 D1

The e i al a he de ig fea e ha i ece a il al f h ee al ed l gic b ha de e d e e e ch ice . The DNF i e d e h he

e DNF a e, h gh i ill h all he a e i he ld ha a e c ai edi ha e i ale ce.

† Gi e hi e i e c e. The e l i he e DNF beca e f heada ed A d a d O .

try2 êêToDNForm

HHr Í Ÿ r Í † r L Ï Hq Í Ÿ q Í † qL Ï H p Í Ÿ p Í † pLL

† B he h al e f he e i ale ce i OK.% êêTruthValue

1

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7.4 Interpretation

The ab e e he f a ha ha e e idea ha h ee

al ed l gic e ail . N i a g d e de ha i ac all ea .E e h gh la g age al ead c ai a di c e e, h ee al ed e illca e c f i g a ad ical a e e . S e e igh a g e ha if W ll

b i e fal e b i de e i a e he a lea i ca be aid W ll b i i de e i a e ( e he e i hi e ) ha i d e ha e e

ea i g, he ce e e ( d ), he ce be e fal e. S ch a e ighclai ha i ce he he ca ea ab he a e e i ha e e i he e

h fal eh d. O a he e igh gge ha if e hi g i i de e i a ehe i ce ai l i fal e f he ld, a d he ce fal e. We ha e gi e defi i i fe ha e cl de ha ki d f ea i g b e ha h e ha e k i . O ee le i l d a f ll h e defi i i . Ne e hele , cla i i e e ial.

I de e i a e a d f e ical a e e a d T e Fal e f a e e hac ce a e, l gic a d a he a ic . Thi ea ha i de e i a e a e ed cc i a e. I de e i a e ac all d e bel g he e ie T e Fal e

hich e a cia e i h a e b a he i h a e ie e fal e i de e i a e

hich e a cia e i h la g age. We al ha e c ide he a e ic c e i ,he ¢ c c c i . The f ll i g gi e a ef l able, he e a d f S c a e .

AT e e ¢ Fal e fal e ¢ Ÿ

i de e i a e S p

Th ee al ed e d e ee bel g a e a d ei he he a e icc e i . The a e ic c e i f al ed l gic a : a ha c ide

e, be ile he i e. Wi h h ee al ed l gic hi bec e c lica ed. Wi h l de f eaki g a d h ee al e , e hi g ha gi e. O e i i a e a

ed flag he a e eaki g e e, b a e le e d f ge b i g al ghei ed flag. I a ea h e e ha la g age ha c ea ed l f e e i

i dica e he de ia i f he e a e ic e, h e beddi g h ee al ed ei al ed e , ch a I hi k ... S e ... I hea d ... . I a ala g age h e e he al ed e f di c i f h ee al ed a e e .B b i l hi li g i ic c e i i fail afe, a e e igh a I lie .

The e i e ch l f h gh ha all f a e l c a e e i . Ae a le i A0= O Ja a 3 2007 i ai ed i H lla d . Clea l hi e cl de he e

190





fal e a e i e e ed i a ab l e a e , i.e. efe i g eali a d l gicalc i e c . Thi d a ic ea i g f he e h ld ala f i i j he

e lace e f a al ed l gic i h a e la i g h ee al ed che e. Thedefi i i f h f al ed l gic i c ai ed i ha f h ee al ed l gic (i i

). A h , e ca e l defi e he Lia i al ed l gic ha ied defi i i bel g he eal f h ee al ed l gic. Af e he h he ical

age ha bee lef , he c ce fal e i Thi e e ce i fal e i i e e ed a e f ah ee e, h gh i ill ca be ea ed i al ed l gic ha he e a e h eeal e . The e i ha he di c i ceed i e f he h al e , a d he a e e he el e , i dica e he ible ch ice f h ee al ed l gic. Si ce he

hf c i ill i defi ed i al ed l gic, he di c i ca ceed i ga e ic al ed l gic b he bjec i h ee al ed l gic.

Ca ¢ e lace h ee al ed l gic ? Th ee al ed l gic ha added al e if a d lif he h ee e f ¢ ca be ed i h ch a i e e a i f d a ich he ical ea i g. The e ld ha e a i hi . Whe he la e i heca e he i e ai ef l ai ai he del ha ¢ i a il c ceded c i T e Fal e e i , i h e a e e decided h gh ill

e ical. I ha ca e e kee a ef l di i c i be ee decidable a d e e.

The e i i i h ee al ed l gic. Q i e (1990:92): O e igh acc di gl

eli i h he la f e cl ded iddle a d a he f a h ee al ed l gic,ec g i i g a li b be ee h a d fal i a a hi d h al e. (...) B a ice iaid i he c be e e f h ee al ed l gic. (..) life a i a k. I ca

ill be ha dled, b he e i a e ide e i i le ea li ed al edl gic. We ca adhe e he la e , i he face a a f he h ea f e i g la

e , b i l di e i g i h i g la e (..). Ca el i fai bec e $ ( iCa el a d i fai ) . I d e g i li b ; i i l g e fal e if i i fal e ha$

( i Ca el ). The edica e i Ca el i ee a a i h i fai , a a edica e

i ed cibl .

I he a e a igh a la e Thi a e e i fal e i h Thi a e e ifal e le i i a a e e , a d c cl de ha i i a a e e a all. B haa ach f ce he i e. The la e c di i al e e ce ha all he e f a

ake i acce able a a a e e ha he c di i be j dged be e. The be e le f e ce i i Thi a e e i fal e le i i a a e e , hichi acce i g h ee al ed l gic. The c be e e i e a ed. I i e

diffic l ha he a da d e i ai e i h ibili ie , a d D . O ,he cl e a file, a i d a ki g he he a cl e i i ha i g cl e i i h a i g, . I a be c be e he l gicia h

192



ha he adi i ad i f l ibili ie , b e ca ge ed i , a dac all i i ki d f ha d . Be ide , hile he e a e a la ge a f ible bi a

e a i , e l eed i e (i cl de he i g la e ) acce he e a a e .Fi all , h ee al ed e i l ed a a g a e a d f e e, af e hich e

e al ed l gic f e ible ea i g.

7.5 Application to the Liar

7.5.1 The problem revisited

C ide he ea lie a g e : If he e e ce = Thi e e ce i fal e i e elehe i i ei he e fal e. If ha i he ca e he a lea i i e. Ba

ha i i e. He ce i i e. A a e l i i e ele . E ce e a. Thi i l e. A a f he f ha ca be defi ed e icall i hi al ed l gic

(i defi i i i fal e, h i ha he e e ha i ca be defi ed) e ca al

fi d he f ll i g. Le he Lia be defi ed a = [ ] = 0 . I i c ec ha [] = 12

[ ] ∫ 1 b f [ ] ∫ 1 e ca de i e ha [] = 0 . . A aid i hef e ec i , hi e i e i f h he ical ea i g a d e hif f f c be ee al ed e a d h ee al ed e .

The e i h e e a i c ide . Na el , he e ec i f he Lia i hi h eeal ed l gic.

7.5.2 The fundamental tautology

T de a d hi , i i g d l k agai a he Defi i i f T h: T h[ ] ñ hich a e f he f i g la e a . The ch e defi i i f h i he

l i beca e e al igh ha e defi ed he Al e a i e Defi i i f T h(ADOT) T h[ ] ñ hich ld be he a i a e ic c e i , l a ha

di ag ee i h. The i a hi g b e e i ha he defi i i f he Lia i ai a ce f he ADOT: = . N clea l e ca defi e DOT a d ADOT a he

a e i e, ha i a c adic i . S i i ell e lai ed ha a c adic i a i e .

PM. Whe ha e ble i agi i g a a i c e i , c ide a c he e,

he b e hi g, d ha e a b a e bei g aid. Th ge heca l i ice. F c i e c , ha i g e ha e a l cial a . Y a e

al bliged ca all he e i h , hich i hea . A ba ke ill be glad hel i g

ca i , b he e i he i e e , hich ea ha he ba ke ill gi e e

e e ca . E e all a e i a ed ge id f e ha a e e b e hi g f . If a e l e i illi g ha e , a

hi age.

193



A he a de a d he al ed Lia i l k i he h able f , fi d c l ha a e each he ega i , a d he defi e a e e ce ha e l i ha .

Well, he a e ca be d e i hi h ee al ed l gic. We fi d ha a d d a ch. He ce e defi e L3 = L3 L3 .

TruthTable @ÿ p Í NotAtAll @pDD p HŸ p Í † pLTrue FalseFalse TrueIndeterminate True

We ca e l e hi b ea i g agai i al ed l gic ab a h ee al ed

i a i . D i g he a k a d ffi 3: = [ ] = 0 [ ] = 12 . The f ll i g

able c ide he ibili ie a d h a c cle.S e @D = 1 @D = 1

2 @D = 0

Fi d @Ÿ D = 0 @D = 1 @Ÿ D = 1

@D = 0 @Ÿ Í D = 1 @Ÿ Í D = 1@Ÿ Í D = 0 @D = 1 @D = 1

@D = 0

C adic i ! We ca c cl de: F all i : ∫ [ ] = 0 [ ] = 12 A b le i

i ha hi i a l gical c cl i a d a c cl i f a i . We ca kee L3

i hi . I l ea ha he al ed a e e i he e d a l , a d he e c cl i i ( L3) ( i g he ffi agai ).

† We igh add a f h i h I de e i a e f L3 a d T e f L3 .TruthTable @NotAtAll @NotAtAll @L3DDD

L3 †H† L3LTrue FalseFalse FalseIndeterminate False

† Thi i he be e e e i . The igh ha d ide f L3 ca l be ed if i ie ed ha he e i a all defi ed. The c adic i h ha ha

c di i d e h ld ha he defi i i ca be ade.TruthTable @Hp Í ÿ p Í NotAtAll @pDL ~ Unless ~ NotAtAll @NotAtAll @pDDD

p HŸ†H† pL H p Í Ÿ p Í † pLLTrue True


The c c i a d l i f Lia bec e a b i g ga e. Y defi e L4 = L4

L4 L4 a d I e l ( L4). E ce e a. I i a hil hical a e if call

194



a lica i f a e h al e ( i ce i e e d he) .

N e ha he igi al l i f he Lia a i ha e ec i le i ha i aid id e e i d f i . Tha i a clea i le l i a d i e ai

al ed l gic. O l b i i e ce be able f he Lia a d he c ea i f h ee

al ed l gic e fi d ch a i fi i e ga e . I he e d he l i a e i ila , f ia be h ed ha ce aki g h e lia .

N e al ha e d e le el i la g age he e b e el e ea ed a lica i fh al e e a ic la e e i . The e ca be he ch i fi i e i k i

la g age a d l gic, b , edl he ei he e i e g e ic i f a i , j a a e e ha a e falli g d a i k a d ha e c a l .

The ge e al l i i bec e a a e ha ab e ga e ha a c c i e f a ha

ca be e l i ed. Ra he ha f biddi g he f a i f e e ce i h ic le le el a d he like, e ca gi e a defi i i f h f a e e ce i f . Le†n

a d f a li f h e dagge . a d le N =†0 . The e ca f la e:

: If c ai †n he ... †n+1 .

Which al e abli he he f h ee al ed l gic f la g age.

N e ha e be e ef ai f i d ci g e h al e . Tha i , he lia a e a

i fficie ea d . I he c e ca e, he e e i b ildi g lia , e caa k h he he i c i l i g bec e defi ed. I ead, if e ldall he i d c i f e h al e he hi b ilde igh a e h, b he ea e a diffe e h al e , i eall i c le a d i e e i g . O l he ch

al e a e i d ced f he ea ha lia he h e ea igh be d.

7.5.3 Conclusion

Ha i g eached he l i f he Lia , e igh a l k back a he a e fe i hi . S ch a hi d igh be ece a il h he e. We ca l i dica e

e i b i ca be b e ed ha hi ical e ea che e i ed e a ec fhe l i . A i le di i c i be ee a ic la a d ab l e a ec f h a d

fal eh d e i d f he di i c i be ee h he ical a d e ical h a dfal eh d. We d k if he e i a a ch i ce A i le did elab a e a dc ce a ed e i a a ache like he ll gi . F he S ic e e

ha he ade a clea di i c i be ee ea i g a d e e. Thi a agai be a

a la i i e b e e if he ac all did ake ha di i c i he a lea heia ach ca be a ecia ed i ce i e ail h ee al ed e . Th gh he didde el i fficie l f c e. Of he Middle Age , he c ib i f Willia f

195



Ockha i he a di g, e i i ce f c cl i ha e h ld f he Lia . B e decided ega d ha a ach a ic i ce i f bid allki d f elf efe e ce ha ee ef l. We l f bid he e leadi g c adic i a d ac all eall f bid he b gi e a le h he

e ical.The ece a ache f R ell a d Ta ki i h he he f e al f bid elf

efe e ce. Thi a ach a ea ece a a d a al be ec g i ed a beggi g he e i , i ce he e i ece a ea h ld e i e e .The e i e hie a ch i e he , i h Ca he e ha a e al a ca e ala ge e e , b e d ha e a i ila he e f la g age, i ce he e e

f i la g age a e. The c ce f a e a la g age i al ick . O eigh e i i a f al la g age ha h e e , b bei g f al, i i e e ed

a d h ca be ed i a g e . Th a f al la g age ld be ealla g age i ha e ec . Na al la g age ca be e a a hi g a l g a hec i de a d i . The i f a e a la g age ee l l call defi eda d ab l el . K i ke ha ake a iddle i i , gge i g ad h ee

al ed l gic a d ill e b aci g he gh f Ta ki. Self efe e ce ld ecei e ah al e agai h gh, aki g hi a ach a ac i e.

A ed he e hi ic eff he ab e ha b h e lai ed a d l ed he Lia

a d i a i c i . i al ed d e a e a d i h ee al ed d e h a b lic . The Lia i c i e f al ed l gic. Tha c cl i ie i ale a i g ha e a di c i h he icall . Tha c cl i i e i aleagai i g a hi d h al e, I de e i a e. S b e e l e ca de ig a le deal i h all he Lia c i .

7.6 Turning it off

Th ee al ed l gic ha g d al e j f h i g ha i i . B i a lica i eeli i ed. T e e c f i , d f ge Th eeVal edL gic ff.

† T i g ff he e f h ee al ed l gic.ThreeValuedLogic @False D

ThreeValuedLogic:: State : The use of three - valued logic set to be False

196



8. Brouwer and intuitionism

8.1 Introduction

Like he g ea K ecke ade Ca life diffic l b ef i g hi a i e ii i ( beca e K ecke did like Ca idea a fi i e ), he g ea

Hilbe ade B e life diffic l b ha i g hi ed f he b a d f edi fhe A al ( beca e Hilbe did like B e idea e h d f f).

B e i he a he a icia h i e ed he fi ed i a d he c ce il g . E e head i h hai ha a c f hich he hai ee fl , beca e

he e be a fi ed i f c i l a i g all i f he head hehead i elf. Hi c e a Hilbe i k ha e bee able de i e i a da haEi ei had bee ki g f e ea e (Hilbe c ld ha e d e i chea lie b j did hi k f i a a ele a e i ).

B e be e i ed he e i ce he a he fi e i a he a icia e i he A i elia i ci le f Te i N Da , i ce he ided

ece a ele e i Hilbe de el e f f he , hich i g i g la ale i he e cha e he G delia . The a he a icia H. We l a e

B e i a ce:

L.E.J. B e b hi i i i i had e ed e e a d ade ee h fa

ge e all acce ed a he a ic g e be d ch a e e a ca clai eal ea i g

a d h f ded e ide ce. I eg e ha i hi i i B e , Hilbe e ee l ack ledged he f d deb he, a ell a all he a he a icia , e

B e f hi e ela i . (Q ed i He i g (1980:778))

Hilbe i i a :

I i i i ha e a d a i a e challe ge i he e i fli g a he alidi f

he i ci le f he e cl ded iddle (...) The i ci le (...) ha e e e ca ed he

ligh e e . I i , e e , clea a d c ehe ible ha i e i ecl ded. I

a ic la , he i ci le (...) i be bla ed i he lea f he cc e ce f he ell

k a ad e f e he ; a he , he e a ad e a e d e e el he i d c i

f i ad i ible a d ea i gle i , hich a e a a icall e cl ded f f

he . (Hilbe (1927:475))

197



The de el e f h ee al ed l gic ab e a a ef l e e i al ed l gici ce i all gi e a hi d al e he a e e ha Hilbe called e e

( ea i gle ). F hi l i f he e a ad e Hilbe d a he he fe a d ha a ach e dee ed i ade a e. Th Hilbe b a e he ge e al

i ci le b hich he a ad e ca be ega ded a e e, a e di c ed ab e,a d, d i g , he e cl ded i f elf efe e ce ha igh be a he a icall ef l.

B e hi elf ab Hilbe gge i ha c i e c a he hall a k f aa he a ical e :

We eed b ea de ai f eachi g hi g al, b hi g f a he a ical al e

ill h be gai ed: a i c ec he , e e if i ca be i hibi ed b a c adic i

ha ld ef e i , i e he le i c ec . B e (1967:336)

A Hilbe ga e he d i a ie ha e ha e bee di c i g al ead , hi cha eill f c B e a d hi i i i i . We ca lea a bi e he d

a d he i f f.

A e a be e i ed ha hi a h d e ega d hi elf ela ed hii i i i . S e a h call i a baffli g e f h gh ha he ca ge ag i , a d e ha e e e gge ed ha he e be a hidde c le i i heD ch la g age i elf. Si ce D ch i a diffic l la g age lea , he a g ed, he

ld be e f ge ab e e i g de a d i i i i . I d f ll hai e e i g a ach h e e a d la h ha B e , defi i el a b illia

i d, al a a bi i ed . A Hilbe .

The ba ic b e a i i ha B e a a a he a icia a d l gicia a d a al cie i . He i g, B e l al a i a , de i ed a e f a i f

i i i i ha ( l ) ed ce he a da d i i al l gic he ali cl de . Thi i i le a de c i i f ha B e i e ded i h hi

i f f. He i g (1980:779) ac all a i e a ha a e c cl i . He i g

al e a k , i e c f i gl , a d if a ad ical he i c i e l , a de i i ce f ich beha i :

I i g d a id ega i he e i i ible. (He i g (1980:747))

T a la e hi a : !

8.2 Mathematics versus logic

The i a e f B e f he e age i he f ll i g:

f a a he a ical a e i a he ca e f e l e cl i el ad i ed e e e laced

b he f ll i g f : 1.a ha bee ;2. a ha bee , . .

;3. a ha ei he bee ed be e be ab d, b a alg i h i k

198



leadi g a deci i ei he haa i e haa i ab d; 4.a ha ei he bee ed

be e be ab d, a

a .I he fi a d ec d (fi , ec d a d hi d) ca ea i aid be

( ). (B e (1975:552))

B e i e ed be e ( i g h ) b f hi a la e a je . Selec i g he diffe e dich ie hich a e i l ed i hi e e a i

e fi d:

1. ( i le) h e fal eh d

2. ece i ( a l g ) e c i ge c

3. f e ab e ce f f ( ¢ , ¢ , )

4. e e e e e (ab di , c adic i , i ibili )

Ea l a he a ic a a i fied i h ha he ega i f a a l g i a c adi i ,hile he ega i f a c adi i gi e a a l g . B e d e ejec ha b

e e d efi e i . Clea l he i i e e ed i c i ge h fal eh d. Fh he eglec i le h b e i e ece i , he e hi ece i

be e ( hile he ee a e ( ¢ ) ). Acc di g hi , afal eh d i j he i e f a a he a ical h.

The a i h ee e ha e di c ed ab e a e a f ed he e i ai a d

B e elec i f he c bi a i ake e e f hi field f e ea ch. B ee ai he a he a icia . He add he i f he alg i h a d ecificall efe

fi i e e h d , i.e. ha Sk le la e called ec i e e h d . El e he e hegge ha i e i he ba ic a he a ical i i i , b h f he e e f de

a d he e e f c i i . He al called a he a ic a a ic i e ic c i al ac i i (1975:551).

A i a e l f B e fe i al backg d i he acc la i i hii i g f i ica e de ail e e i l gic: b i ead f de el i g a ge e al l gical

he (e ce he ab e) he e e e hi a e i he de ail f he a g e ( fhich he ge e ali igh e ha be clea hi elf). He igh ha e h gh I each

b e a le . The e l h e e i ha ch f hi i i g i he field f l gic eea i ele a a d a lea a i acce ible a f he age f R ell & Whi eheadP i ci ia Ma he a ica. If he e i a ge e al he he i e ai i he f g (e ce fab e 4 i ).

A d, he e i e i l g , he e hi h i a l g a d c i ge h. I

he e ab e B e e lace he ca e f e l e cl i el ad i ed a dHilbe i e e e hi a he ejec i f he Te i N Da , he i ci le f hee cl ded iddle. B B e did d ha ! A a a he a icia , a d i g ha

199



a he a ic deal i h ,B e e laced he ai f a l g c adi ii h ( ¢ , ¢ , ) f i a l g c adi i . He j did c ide

f c i ge cie ce, he ba ic dich i a e. Which ake ch f hi l gicle ef l f al cie ce.

P ed i i el , hi ejec i f he c e i i he e i d c ai ed ele e haa e defi i el i e e i g: (a) a c i ici f he e h d f ea i g, (b) a c i ici fe.g. R ell a i a d f he e a ad e , (c) a hil h ab he a ,

ibili ie , li i a d bjec i e f a he a ic . Th e bjec a e ce e ed a dhe c ce f a he a ical h, c i ge e i ical h a ed i he i le

c ce f h i de l gic. He a e lici he di i c i , a i e , ba a e l f e e e gh f Hilbe ice ( e ha f e e ).

U de a di g B e k i hi a e ake hi k ch clea e a dc ec . F e a le, he e A e (1936) e ded call e a h ic e ical (a eal ad ed a e i l g ha a cia e i h he Vie a Ci cle), B e i eelega a d a I i d he e i l gic (B e (1975:111)).

8.3 Russell’s paradox

B e (1975:89 90) di c e R ell a ad . I a : (i) he Te i

N Da f a he a ic (if Hilbe l k e !), (ii) he ake he ef l di i c i be ee e e ( a he a ic ) a d e e li g i ic e , (iii) he e cl de

elf efe e ce ( a ic la l he e ali i i l ed), (i ) he c cl de a he ha . The c adic i l a i e b i g decide e hi gha i decidable.

RUSSELL gge a i e h d e ca e f he c adic i , b he e d b

ejec i g he ; he belie e ha a dee ea chi g ec c i f l gic ill be eeded f

he l i . He i i cli ed he i i ha a he i e i ed hich d e ad i

e e cla , c ide ed a e, be ade i a l gical bjec . A he gge i , hea , ld be de he i f , b i a ca e he i f

be e ai ed, f he e a e h , i . he l gical i ci le , hich h ld f e e

bjec .

B hi i i ake : he l gical i ci le h ld e cl i el f d i h a a he a ical

c e . A d e ac l beca eRUSSELL l gic i e ha a li g i ic e ,

de i ed f a e ed a he a ical e hich i ld be ela ed, he e i

ea h c adic i ld a ea .

F ha a e , i i e ide c e e a hich i he ea i g, hich lead

he c adic i , cea e be ali e a d c e e l i l ge eliable; i i e e

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ece a gi e he ill i f he chi e ical e e hi g . F le e ha I

k a e e hi g i h a ali f ela i e i i g be ee he bjec , a d a

e f i i hich a h ld f he bjec . The , gi e a i i al

f c i , I ca decide f b ea f i gi e ela i he he i

a i fie he f c i , i he d , hich f he cla e defi ed b he f c i i bel g .

B he I i h decide he he he bjec hich i he cla i l ed i hec adic i , a i fie he gi e i i al f c i , he I ee ha he deci i i l

ible de he c di i ha i ha al ead bee c le ed. C e e l he deci i

, a d he eb he c adic i i e lai ed. We ha e he e a i i al

f c i hich defi e c le e a cla e hich d a i f he e i da . Thi i i i g, f he l gical i ci le h ld l f he la g age f

a he a ic ; f he li g i ic e , h e e aki ha f a he a ic he a be, he i ci le eed h ld.

B e fall back i i i ha a he a ic gi e e e he a d l he hec ce a e c c i e. The e be a che e a i h he all a d b ild

he big. The defi i i f R ell e fail ha . B e a ach ca be ee ai i e i ce i b h gi e a e la a i a d a eci e f c ea i g ef l hi g . B

he ice he a i he l f elf efe e ce. Ab e e h ed ha e ca all fa ad ical a e e , al i hi a he a ic , b all i g f a hi d al e.

8.4 Unreliability of logical principles

B e De be baa heid de l gi che i ci e (1908) be ega ded a b illia e e h gh e d f ll acce i . We a gi e e e e i e e

ha e ca e if a e e ab B e a d be efi f hi i igh . I hef ll i g, B e e i d f he de el e f E clidea ge e a d hec cl i f cie i ha i de e d he fac ha a he a ical ede c ibe eali . S b e e l B e e e d hi e i ici l gic:

A d, like a eligi c ci e , cie ce ha ei he eligi eliabili

eliabili i i elf. I a ic la , a a he a ical e f e i ie ca e e e ai

eliable a a g ide al g e ce i , he i i i defi i el e e ded be d he

e ce i hich i ade de a dable.

C e e l l gical ded c i hich a e ade i de e de l f e ce i , bei g

a he a ical a f a i i he a he a ical e , a lead f cie ificall

acce ed e i e a i ad i able c cl i .The cla ical a ach, ba ed he e e ie ce ha i ge e l gical ea i g

ded ced l di able e l f acce ed e i e , c cl ded ha l gical

201



ea i g i a e h d f he c c i f cie ce a d ha he l gical i c le e able

a c c cie ce.

B ge e ical ea i g i l ald f a a he a ical e hich ca be e all

c c ed i h efe e ce a e e ie ce ha e e , a d he fac ha ch a la

field f e e ie ce a ge e c f la i gl he c e di g a he a icale gh be di ed, like a cce f l a f cie ce.

Beca e e de a d ha l gical ded c i a e eliable i cie ce, A i lec cl i he c e f a e d c i ce i h ac ical e ifica i ; e

feel he i d ha bl i S i a k a c le el i de e de f hi l gical

e ; e a e l ge a ed b Ka a i ie b he ab e ce f h ical

h he e hich ca be ca ied h gh i hei e e e c e e ce .

M e e , he f c i f he l gical i ci le i g ide a g e c ce i g

e e ie ce b e ded b a he a ical e , b de c ibe eg la i ie hich a eb e e l b e ed i he la g age f he a g e . T f ll ch eg la i ie i

eech, i de e de l f a a he a ical e , i he i k f a ad e like haf E i e ide .

3. The e i e ai he he he l gical i ci le a e fi a lea f a he a ical

e e e f li i g e a i , i.e. e c c ed f he ab ac i f

e ea able he e a, f he i i i f i e, id f li i g c e , f he ba ici i i f a he a ic . Th gh he age l gic ha bee a lied i a he a ic i h

c fide ce; e le ha e e e he i a ed acce c cl i ded ced b ea f l gicf alid la e . H e e , ece l a ad e ha e bee c c ed hich a ea

be a he a ical a ad e a d hich a e di agai he f ee e f l gic i

a he a ic . The ef e e a he a icia aba d he idea ha l gic i e ed

i a he a ic . The b ild l gic a d a he a ic ge he , i g he e h d f

he ch l f ,f ded b Pea . B i ca be h ha he e a ad e i e

f he a e e a ha f E i e ide , i ha he igi a e he e eg la i ie i

he la g age hich acc a ie a he a ic a e e e ded a la g age f a he a ical

d hich i c ec ed i h a he a ic . F he e ee ha l gi ic i alc ce ed i h he la g age f a he a ic i ead f i h a he a ic i elf,

c e e l i ca h ligh a he a ic . Fi all all he a ad e a i h hee c fi e el e eaki g ab e hich ca be b il e lici l f he

ba ic i i i , i he d , he e c ide a he a ic a e ed i l gic,

i ead f l gic i a he a ic . (B e (1975:107+))

PM 1. Wha i fi , l gic a he a ic , i f c ce .

PM 2. A de a e f he ch l f Pea i b lic l gic. We ld a he ee F ege

a he f di g fa he f b lic l gic, f ll i g B che ki (1970). L gi ic i he b a ch i ec ic he e he bjec i e i ge a d c i he igh f , a d a he

igh i e i he igh lace. (C i l f B e , i e e e he e .)

202



PM 3. B a f e i l g , B e a a a l e i e f a he a ic hich ca be

e all c c ed i h efe e ce a e e ie ce ha e e ha i al i

c c ed f he ab ac i f e ea able he e a, f he i i i f i e,

id f li i g c e , f he ba ic i i i f a he a ic . Thi a d

i c i e b he ke d i ab ac i , ea i g ha he gh f a e e ie ceha e e ill ca be e ai ed, a el i ha ab ac i , called i i i .

Acc di g B e he f da e al i ci le f a he a ic a e ba ic ai ellige i d, ca be f he e lai ed. La g age a d b lic l gic eflece e ie ce b fficie l ab ac a d he ce he ca lead c adic i . Thef ll i g diag a gi e a che e he e he h i al li e eflec e e da

l i e a d he e ical e e eflec he ig f a he a ic a d iab ac i .

eali ö la g age ö l gicé ç

a he a ic

Pe all , I a ch i e ed b hi a al i . (1) The a be ee la g age,l gic a d a he a ic ca b h a . The d de c ibe he a e elldefi ed. I i e ha ab ac i c e bef e e e ie ce. Ra he ha e e e ie ce a d lea e hi g bef e ca ab ac f i . Ab ac l

f da e al igh ead a e i e l gicall lea f da e al . If ab ac i ie , a d l gic ba ed he a he a ical a al i ( f he c e f cie ific

he ie ) he l gic ld be e e e ab ac ha a he a ic . B e a achi e he ch l g f a he a ic ha l gic a d h. (2) Tha Pea e al.a ached i diffe e l d e di alif l gic i i elf. M de l gic i a d

f la g age b deal i h he ece i ie f i fe e ce. (3) I i clea h hei i i f i e h ld be f da e al a he a ic a d h he i i i fT e Fal e. Thi igh al ca e a di c i he he ace i i a , i h

i i ch e a di ec i . Clea l ace i ick , i h E clideai e e a i , b d e ha di alif he ab ac i f ace ?

Wha i i a i hi e a e he , he he he a igh g.Leib i , B le, F ege a d Pea e al. al ead a ed ake la g age c . hela g age f a he a ic e ig . B e gge ha i i f ile , i ce

a he a ic i d e i i d a d e c le ha a he a ic e el i gla g age. He d he ie ha a he a ic i e c le ha l gic i h iT e Fal e dich , ha he l gic f hi i e ca h a ligh

a he a ic . B e e i e , j h. The e a ec e eel eglec ed i he di c i l gic i hi i e.

203



He c i e he bjec f f. He i e e e h a ( ¢ ) a d fal eh d a ( ¢

) b h ld ha f he a he a ic f he i fi i e i i ible ha ( ). Hece ai l acce d ake a c adic i b ad i e ha ea i g i i fi i e

a he a ic ca ill be eliable e e he ch c adic i :

N c ide he i ci i : I clai ha e e i i i ei he e

fal e; i a he a ic hi ea ha f e e ed i beddi g f a e i

a he , a i f i g ce ai gi e c di i , e ca ei he acc li h ch a i beddi g

b c c i , e ca a i e b a c c i a he a e e f he ce hich

ld lead he e beddi g. I f ll ha he e i f he alidi f he i ci i

e ii e lc i i e i ale he e i

The e i a h ed f a f f he c ic i , hich ha e i e bee f a d,

he he e e i l able a he a ical ble .

I fa a l fi i e di c e e e a e i d ced, he i e iga i he he a

i beddi g i ible , ca al a be ca ied a d ad i a defi i e e l , i

hi ca e he i ci i e ii e cl i i eliable a a i ci le f ea i g.

We c cl de ha i i fi i e e he i ci i e ii e cl i i a e eliable. S ill

e hall e e , b a j ified a lica i f he i ci le, c e agai a

c adic i a d di c e ha ea i g e e badl f ded. F he i

ld be c adic ha a i beddi g a e f ed, a d a he a e i e i ld

be c adic ha i e e c adic , a d hi i hibi ed b he i ci i

c adic i i .

I ha e del ed i ha ha la c l ed e e ce eci el ea , i i i a he e.

I i i hi c e ha he c i e :

A d i like i e e ai ce ai he he he e ge e al a he a ical ble D

? i l able.

The e like i e i i eci e, i a ic la he e he a h al ead l he

defi ed i ci i c adic i i . A d he he he h ldi a he a a e f defi i i a he ha e hi g ha be e . Ye , B ehe e fa l ca d b he e l gical i ci le ha e ea i h. A d,f he ec d, he e he d i h e ce i . The i ca i b ke . B ed e ide he c i ci g al e a i e e , b e ca a ecia e hi d , he

e defi e h e l gical i ci le h ld e ce he e i ca be e ha e e e d h e defi i i i h h ee al ed e f e ele a e e like he Lia .

204





Na @ D Na @Na @ DD1 0 012 1 012

12 1

... ... ...

Whe e e N , he i k a a ga e kee e , a i g f e dle i g f ....

Whe e f ge ab N a d j e Na he e d ha e ch a ga e kee e ,ca i g c le di c i a d he di e able, a d a k a d e i h he ki che .

PM 2. A he a i e e e he i a i i e he T e Fal e dich

a d di all N b e Fal e, he fi f he i g la e a . Fal e he i j a

al e b al a e a , e e i g al de ial:

Fal e@ D

Fal e@Fal e

@ DD1 0 0

0 0 0

The c cl i i ha i i l a eak e a d i i e ad a age e ge

a i ( i ce e eed fea f a Lia a ad ).

206



9. Proof theory and the Gödeliar

9.1 Introduction

A i h e ic ha bee a d i ce he da f ci ili a i , e 15,000 ea ag , if ea lie . A d 1900 he e e e a i e a i i e i . Thi ca ed he e i

ha e elec , a d, ha d if ha a i a ic e ed bei c i e ? Y ld ge 1 + 1 = 3 a d cie i ki g i h ha a ic laa i a ic e ld ge e a e l f e e. Ad i edl , ch e e igh be

ickl iced b , d ha , ld el k ledge f he ld, a d ld el he f al e g h f a he a ic i elf. Gi e he e f

a he a ic , i h ld be he he a a d, Da id Hilbe h gh . H ae i ical k ledge f he ld i f agile a d e le be e el d a h.He ce, e f he bjec i e f Da id Hilbe beca e e he c i e c f

a i h e ic, , i deed, e i ge e al.

The e i ee ed a ic la l i a i ce i ee ed ha all effec i ec able (fi i e) e h d f f e e ec i e, i.e. ed a he a ical i d c i ,

hich a he e h d f a i h e ic. Tha e h d gh be c i e a ell, a dhe f f he c i e c f a i h e hic ld e ha e e h d.

Hilbe e a de K G del, ki g a he i . Hilbe gge ed hi aall b l f a i h e ic he i ege he el e , ha e e i ab

a i h e ic c ld be e e e ed i a i h e ic. De e di g he a ch e , aa al be igh be i e e ed a a c de f a a e e a i h e ic i elf. Iha a , a ki d f elf efe e ce a e abli hed ha a acce able he a he a icali d, c a ed he l i f al a f elf efe e ce i a al la g age. O ef h e a al be ld be he c de f A i h e ic i c i e . Hilbe

a ked G del he he he c ld fi d a f f ha .

G del c ide ed ha a e f a i h e ic A i c i e he i c ai a e eha a e e .

Consistent @A, p D

" p JJ A pN Î J A Ÿ pNN

207



S b e e l he ca e i h he elf efe e ial a e e Thi a e e i e . I f la = ( A ' ), ea i g ha a e ha A d e e i . I ed

ha hi i a a ad ical a e e . Whe e a l A e i e ded a lica ihe e ca j dge he h a d fal eh d f a e e , a d i a ic la , ee he

able bel . We he ca e ha A e d e e i he c l a dc ide he effec he h fal eh d f i he . A d eadi g back a d ,

e fi d he e i ale ce ñ ( A ' ).

¢ g

Fal e T eŸ T e Fal e

The a e e clea l i a c i f he Lia a d he ce e ill call i he G delia .

N e ha i l a ad ical he e a l a i e e a i he f al ech ha i a e e ca be j dged be e fal e. G del called hi ae a a he a ical i igh b e a ee i a j a a lica i . If e d ch e

a a lica i he e j ha e a f al e a d a ad . PM. We i e ( A ' ),ea i g ha A d e e , hich i i le ha ha dli g abili a i Thi

a e e i able . P i igh a ee a a ifie ha he e i a e e i A ha e , l a c c i e f e h d. We ill di c bel

ha hi i eedle l c le .

The b e e a h i he Ph. D. he i i edi h gh aigh f a d. F e i i fficie ake a h c . The h c i ha e d eed he

e ical c di g a d j e b l f he edica e calc l . La g age al eadha ea a e e elf efe e ce a d h gh i i la dable ha a he a icia

ake hi fficie l e ac f hei e , e d eed f ll ha ack fi d. We ake a fficie l g e a d ake i G delia = ( ' ). ,hile G del fi g e h gh hi edi a d af e a d a lie hi

( e a a he a ical ) i e e a i , e hi a d a d a i h a i e deda lica i . We a e i e e ed i f ali j f i elf a d e a a l hie . (A d e e all G del ake ha a e e .) Th , e a be

e a icall c ec ( cie ific), ha e e a i e e a i a lica i f chha all he e f a e al e i hei a lica i . Y a ecall he defi i i

" p HHS ¢ pL pL a d he a cia ed h able. I fac , e ca e d ce ha able a d

di ec l a l i i ce hi i he c e a e e ha e a e i e e ed i :

208



TruthTable @SemanticallyCorrect, S, g D

HS ¢ g L HS g L HS ¢ Ÿ g L g Ÿ g

1 0 0 1 00 1 0 1 00 1 0 0 1

0 0 1 0 1

C i g h a l f ded c i , i ffice e al a e he i ha la able.The fi b i l fall (check able ab e A). The la e i e hef ll i g ded c i , i h (1) e a ic c ec e , (2) defi i i f , (3)

i a lica i :

Ergo2D @Ergo @S, g D g , ÿ g g, g Í ÿ g, g D1 HHS ¢ g L g L2

HŸ g g

L3 H g Í Ÿ g LErgo —————

4 g

Th , i he e a ic h able 1, 3 a d 4 fall . All ha e ai i ( ) .Re e be ded c i e c le e e a d ee h hi gi e a c e e a le ha

he e i i c le e:

DeductivelyComplete @S, p D

" p H p HS ¢ pLLG del Each fficie l ell de el ed e ha ic i e a d (a lea ) e he e h d f a i h e ic i ded c i el i c le e. The ei a e e i ha i e b ha ca be e i he e .

The ec d e ha G del k i e he igi al e i , c i e c . Lea d f he a e e hai c i e . N e ha a i c i e e c ld e

a hi g ha ( ¢ ) .Si ce e al ead ha e e ca de i e he

c i e c f a e a e a ic le el. Thi fi he i ha e ha e ided aa lica i f i . The b e e e i i he he he e ca e abli h i c i e c , ( ¢ ).

Whe he e i elf c ld e b ha d he he e ld be a f f ,aki g he e i c i e . Th , if e ca h ha i e i he ehe a c i e e ca ha e a f f i c i e c .

209



† We ca e he label H : i dica e a h he i ha i b i ed ai fe e ial e . I hi ca e e la ch he h he i ab haigh be able d .

Ergo2D @Hyp : Ergo @S, c D, Hyp : Ergo @S, c gD, Hyp : Ergo @S, g DD

1 Hyp : HS ¢ cL2 Hyp : HS ¢ Hc g LLErgo ————— 3 Hyp : HS ¢ g L

We ad he a i ( ¢ ) ( ¢ ( ¢ )) ha he e e hi g he ic ide i al e ha i e e hi g. Thi e ca be called fc e e i ce he i f f i a lied c e e l . Ne , e all

ea i a h he ical de a ell, he e a e e hi g a d la e e ac i .

I e 1 c jec e ha i h he icall e a d he de e i e ha i ld bec e i c i e . We a e ha c ide a he e f i c i e ci acce able. Th i e ac he a i a d j ake he de i ed i lica i i

e 8:

† I hi ca e i d ce a d e ac he h he i .Ergo2D[Ergo[S, Hyp: ¬g], Ergo[S, Ergo[S, g]], Ergo[S, Ergo[S, Ergo[S, g]]],

Ergo[S, Ergo[S, ¬g]], Ergo[S, Ergo[S, g ¬g]], Ergo[S, ¬c],Ergo[S, Retract: ¬g], Ergo[S, Implies[¬g, ¬c]], Ergo[S, Implies[c, g]]]

1 HS ¢ Hyp: Ÿ g L2 HS ¢ HS ¢ g LL3 HS ¢ HS ¢ HS ¢ g LLL4 HS ¢ HS ¢ Ÿ g LL5 HS ¢ HS ¢ H g Ï Ÿ g LLL6 HS ¢ ŸcL7 HS ¢ Retract : Ÿ g L8 HS ¢ HŸ g Ÿ cLLErgo —————

9 HS ¢ Hc g LLHe ce e ca ec d:

Ergo2D @Ergo @S, Implies @c, g DD,Implies @Ergo @S, c D, Ergo @S, g DD, NonSequitur @S, g D, NonSequitur @S, c D D

1 HS ¢ Hc g LL2 HHS ¢ cL HS ¢ g LL3 JS g NErgo —————

4 JS cN

G del Each fficie l ell de el ed eha i c i e a d (a lea ) e he e h d f a i h e ic ca e i

210



c i e c . I fac ,ñ ( ' ), ha ac all i e i ale he G delia .

G del S e ha e i cl de he e a e f he a i f , i aeff ake he e c le e. The , h e e , e ge a e eS 1 = , = ,( ' ) a d he e ill fi d a g 1 a d he h le ce e ea . N e ha g 1 ∫ i ce

he efe diffe e e ,S 1 ∫ . Th , e a i deed a i h a i h e ic a dk a he bigge e . Thi addi g f a i al h ld f = [ ].

He ce, ab e he e c ai ed he d a lea be ee b acke , a d e d he e b acke .

G del: i ca be ed ig l ha i c i e f al e ha c ai ace ai a f fi i a be he he e e i decidable a i h e ic

i i a d ha , e e , he c i e c f a ch e ca be ed

i he e (1931:616).Hilbe d ea b ke d .

G del hi elf de ed f he e f hi life he he he had eall e e hi g j c ea ed e f he c i f he Lia . He e e k e he he he h ld dl

alk he Ea h hide i ha e.

Gödeliar = NonSequitur @Gödeliar D

a a la i f . Thi f a e e ha ee S i ed i ali ha a a ifie ca be a ided. O he i e

ecif elf G= F All@, i S, N Se i @, GDDEnter the ö in Mathematica using Â o"Â . Gödeliar uses a String on the right hand side to prevent recursion.

9.2 Rejection

G del e a li i ed ki d f elf efe e ce. S a e e i hi a e all ed efe

he el e b he e a a h le i all ed efe i elf. The defi i i fc i e c ha a a ifie all i b a gel i i all ed ake [ ] i .The e i like lea i g ha c e a d he e ha a e d e i .

Whe e ake he e a ha i ca e e he c di i f c i e c ,a d b e e l a ha i k e hi g ba ic ab c i e c ( a el

ha ), he e ca fi d:

211



† Whe a e i i c i e he i ca e a hi g. Al ha i i c i e .The e ca al e ha hi i . He ce i ill e ha i ic i e .

Ergo2D @Ergo @S, ÿ c cD, Ergo @S, c ÿ cD, Ergo @S, c DD

1 HS ¢ HŸ c cLL2 HS ¢ Hc ŸcLLErgo ————— 3 HS ¢ cL

Th he ba ic ca e f G del e l i a . N b a i g e hi g b b a i g e hi g. He e a li i ed ki d f elf efe e ce, he i ehe ld ha e had a he e . He ld j ha e f d he Lia agai . O e ighh ld ha ch elf efe e ce a ( ¢ ( [ ] [ ])) i l ge a acce able ec i e

e h d . The e e l ld be ha i ill e ai dece l gic. Pe ha heec i i a e illi g acce a l gical e ce i hei che e (if i eall iec i e).

Ac all i i a bi c i ha a e e i c i e c , ha e a k chf a e , hile c i e c e d be e ed. The h le i e f c i e c

f ee be i laced, a fallac f c i i , ee bel .

The f ll i g i a he e a le f a all e e i G del a i ha

h ha he G delia , e l ea ed, c lla e he Lia . We all h he icalea i g ake he a g e ac able, b ee he Readi g N e i Cha e 11 f a

e c le f la i :

† A e he a i ( ¢ ) ( ¢ ( ¢ )) ha he e e hi g he i ale ha i e e hi g. S e 2 i h he ical, i e ac ed i e 5 i he

i lica i i 6. The 1 a d 6 e l i a f f . Tha i , i decidable.Ergo2D[Ergo[S, g ¬ g], Ergo[S, Hyp: g], Ergo[S, Ergo[S, g]],

Ergo[S, ¬ g], Ergo[S, Retract: g], Ergo[S, g ¬ g], Ergo[S, ¬ g]]

1 HS ¢ H g Í Ÿ g LL2 HS ¢ Hyp: g L3 HS ¢ HS ¢ g LL4 HS ¢ Ÿ g L5 HS ¢ Retract : g L6 HS ¢ H g Ÿ g LLErgo —————

7 HS ¢ Ÿ g LWhe e i e e e he e , i.e. fi d a del a e e a ical c ec e , he

hi e l c adic he decidabili f ( ee 208 209), a d e ca al de i e .The e ld ill be c i e b del ca e a c adic i . Th

e h ld i e e e he e , aki g i e ha ele i a licable.

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N e ha e ed ( ¢ ) ( ¢ ( ¢ )) al ead i he f e ec i h e f f ( ¢ ( )). G del did hi f diffe e l ha he e ab e a l f

didac ic e . N , h e e , e a de h e d i cl de ha a ia a , ha he e bec e g e gh ef e he G delia . I a be

ed ha a i g f c e e e i eake ha ded c i e c le e e .

ProofConsequent @HS , y, L pD

e e e ha e i c e e i i f edica e, i.e. ha ,f all , if i e , he i i al e ha i e

A ke e f ea i g i he li e a e G del e l i : If al ead i lea i h e ic ca e i c i e c , , he ce ai l he e c le eca ei he . He ce he e a e f da e all decidable e i , a d a ki d ifaced i h da i g e i ha ca l be i i ed b he a a d e ie (

e). Thi ea i g i a alid a if a bab ca alk he ce ai l a ad lca alk . I deed, he e a be da i g e i , b beca e f a e icalG delia a eak e ha ake c i e c e i ale i h ha e icalG delia .

C ide G del a : i ca be ed ig l ha i c i ef al e ha c ai a ce ai a f fi i a be he he e e i

decidable a i h e ic i i a d ha , e e , he c i e c f a ch

e ca be ed i he e (1931:616). Thi i i acc a e beca e he ake he e ge , he e l di a ea . Y ca e cl de ha

a e e i he e efe he e .

The i h i ha G del ha a ed ake he hidde c adic ic e i he e . A d i ce ai l d e ee a i al ba e

hil hical i i he e f ch eak e . Ma he a icia igh bjecha he a e c ce ed i h hil h a d e el e he he e ha ch

e (called eak ha e e ) ha e ch e ie . Thi i e fac . Hilbea d G del did i e d he f al e be i e e a i e f a i h e ic a d hei lied hil hical i f he .

A e ade a e l i i : ei he d f ch lia e h ee al ed l gic.Hilbe c di g f f la i be a a che e all f elf efe e ce b ,

ch c di g d e c e i l gic i elf. A d i i i ed if i e e e hec di g ch ha elf efe e ce he h le e i e cl ded. O he a c ea e

elf efe e ce i i e a e e ce a all ha efe ha e e ce he all. Y

ca all f ch elf efe e ce, e ce f h e i a ce he e ch c di g e li i c i e cie i ie ie i he i e e a i , f hich ca e he

ake a e d (e.g. h ee al ed l gic).

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The c di g f a e e i be a i e ded all f elf efe e ce. H bala ced i ha ? Ge e a i f l gicia i ce Ta ki a d G del ha e a gh : dc c he Lia beca e f he le el i h, b a e all ed c c heG delia i ce he e e d eed le el . If G del had bee all ed c c hi

e e ce, i.e. he a e i e ha le el al a l he i f f, he held ha e had a he e . The e a e fe l gicia h a e f ll a a e f hi , a d

ha i i a ge de a d le el f h b f f.

A fi al ba e f ejec i i e he i ha e he f al e . Whehe e a e j be he he e i a ad . O ce e ide a i e e a i ha i bec e e a icall c le e, bec e a cie ific e , he e al eed a

i e e a i f he G delia i eali . The l ef l i e e a i i he Lia .G del ca a ha , ca ake ha i e e a i , beca e he call he G delia

hile i hi al ed l gic he ca f he Lia ai f c adic i . Aa ach f hi a i i ide if b h c c a e ical, a d a l a

h ee al ed l gic f he ake f h e h de ha e ical ea .

B i elf i ld be a g d c e i a a ega d e hi g a e ical i aalle e a ell, if i be e ical i a la ge e c a i g e .

T cl e, a e i e i ed Ch ch he i . DeL g (1971): I 1936 AlCh ch ed ha e ide if he i i i e i f a effec i el calc lablef c i [ decidable edica e] i h he a he a icall e ac i f a ge e al

ec i e f c i [ edica e] ( 186) a d Ch ch he i i a e i ical he i , a a he a ical he e . I i a clai hich i bjec c fi a i ef a i be i ical e h d . ( 195). The : Si ila l , he G del e ha he e a e

decidable f la , Ch ch ha he e i deci i ced e, (...) all hi c e i ha [ hei c cl i f ll ] (j a E clid

elec ed a aigh edge a d c a ) (...) i j gge ha a he a icia a dl gicia l k f he ea . (...) Thi bjec i ld ha e a g ea deal f i g

e e i f e ci c a ce: . ( 193) Oe l ld be, ell, , he e a e he ea . All a e e i he e

di c he e a a h le a d i cl de a i ( ¢ ) ( ¢ ( ¢ )). I i i a e ialhe he hi ld ill be ge e al ec i e ; i e i i ha i babl

c ld be ide ified be , b , if , e igh e a e ge e al ec i e le hi g . A cie i e ca defi i el ag ee i h Si ila l a h a c e , c e , i bjec he a e li i a i (DeL g (1971:199)). The a g e i

ch li i a i . The i e el i ha he c e h ld be c i led ai ha bee .

Al e a i el , igh ega d all hi a a e i e ce f ha h ee al ed l gic ie i ed. A l g a d c f e decidable i h e i ab e

214



c ehe i b ick he i ha he G delia i j a c i f he Lia , a d j a e ical ha i d e ide g d f hil hical i . Thee i e ce f ac all ha al ead bee gi e , ee ea e f he Lia f

al ed l gic. The f al a he a ical i i ha h d e e i , l a i

f f , i ha dl acce able f he i i f a cie i h deal i h he ld a e e da ba i . B he he a he a icia eed hei e i e ce f f

h ee al ed l gic he he ha e i .

9.3 Discussion

9.3.1 In general

Ma he a icia ha e f e e e ie ced ha h a k ledge i li i ed. F e a lei he field f a i h e ic a ble ca be ed, b b d i able l e he

a he e . Thi a gi e i e he e i : d he e e i ble ha a ei l ble ? A e i g a e i like hi e i e a g d defi i i f ha

e h d a e all ed, f e i e e le clai k a e hich a e acce able i a he a ic cie ce. B e e f e hi g like he acle f Del hi

e igh de ig a ced e check he fac j e if ha he acle ieliable. S ed b Hilbe a d B e , a he a icia a ed be e defi e hei

a f f. The ec i e e h d a he a ical i d c i beca e ali e all acce ed a d he a fi i e e h d a ill be ejec ed b e.

9.3.2 Ever bigger systems ?

G del (1931) e a ecific ki d f i f f: i i e de e de . Hi e h delie a e ha ha a f edica e, a d he i l bili l ea

l able i h e ec he f edica e f. Thi di ec l ca e hee i ab a la ge . A f f c i e c f b i elf igh be le

c i ci g, a d e igh a he a ee a ge* ha l h hec i e c f b ha igh al e e f i l able ble . Ge ei deed e e ed a f f c i e c f a i h e ic i g a fi i e e h d .

A d f hi * agai a ge e c a i g e , e ce e a. I deed, h d ee he c i e c f Ge e e h d ? U i g a ge e d e , i i elf,

i l elf efe e ce f ha e . We igh e d i h a i cie , a d he ee i e e a ic a d ded c i e c le e e he e e d i h h agai , i ead

f abili , he e e ca ega d eali a he big calc la i achi e hac i l e he e ; a d he e he e ld be elf efe e ce le e ha e bee i d ced e he e al g ha e c a i g ce . E e all e e

215



i ha e ca e a d ha e acce a a i . I igh be c cei able.I all hi , he fall back i i f a i e al edica e f f ill i he f ll i g fc cl i f e i e hich i ge e ic i all.

Ra he , h gh, e a fi d ea able a i hich all de e i e

he hidde c adic i i he G delia a al i b hich a i a e ahea a he a i f i cie ce. I a ic la , e a be , b be a g a e a e ( he be ide f , f ee f c adic i ), ha ha e e eca a e ab ca al be a e ed b . The ea f hi gge i i b i :

he i a e a e, he e ca e a i age, a d b d i g e ca ga hek ledge ab ha e ca d el e . O e c e e ce f hi a ach i : all

¢ ( [ ] [ ]) he ce ¢ he ce he G delia c lla e he Lia a d e ha e ac adic i . We ca all f hi i a i b ide if i g he G delia Thi

a e e i able a e ical, i h ee al ed l gic.

O e ea ad elf efe e ce i hi a e ( ha al ¢ ( [ ] [ ]) iall ed) i ha he i e e c ea e l gic , e elf efe e ial i hi he

e a d e ide he e ha clea l i elf efe e . Self efe e ce d e i l i cie ia a d ca be li i ed ha i k ca be acce ed.

Thi a al i i ha G del a al ed l be a d ha he h ld ha e a al edh a ki g i h be . I i j ha e e i e elf efe e ce, a G del

gge ed ha he ld e b did e.

9.3.3 A proper context for questions on consistency

Ob e e ha a ki d i i e ed hi k i e f G del che e. Pe le di cha he e le di c . Ge ge: J h aid ... La e J h aid ... The he aid ... Allhe hile Ge ge d e a e ch hi elf, l e ha J h a ed. Pe le

a e al a a e ha (J h ¢ ) i i e e i ale (J h¢ ) (J h ¢ ), i.e. J h ad i g al ed l gic d e i l ha he ha a i i e e hi g.Q e i f c i e c f ha J h a ca be ele a . B e l ake i a

ble if he e i ea d .

I i elf i igh be a bi a ge e c i e c he a e i c c ed bec i e . F a i h e ic e ha e: (1) he a i f A a e a l gie , (2) he ded c i

le e e e ha e , (3) a d e e e ha e a e a le ha d e ha e hae f bei g a a l g (e.g. he ega i f e a i ). Hilbe ac all

e i ed hi ki d f f hich igh be e be called .

The c ce f c i e c i a he ed f he cca i ha a c adic i i f d, ha e ca a ha he a i e e i c i e . The i e e f e

a he a icia f c i e c f de i e ai l f Hilbe deci i i

216



hi li , i a e i d l g ag he he e igh ha e bee ca e f e .(N ada e ld ab e l b c e .)

The e a e a i he a f a c i e c f, l aki g ¢ . If eall he e a e ha i acce (1) e i da , (2) he a i ( ¢ ) ( ¢

( ¢ )), a d (3) ha i ejec a c cl i ible i c i e c a fa a ac adic i , he he e ca e i c i e c . (Th gh e a al e ili le he a e a ha f i elf.)

Whe he e acce TND he i ca acce TND i ce hi ld ca e ac adic i , a d c e el . Thi ha a c e e ce f i g c i e c .C i e c i e i ale he f he f ( e , h ee

i e ).

† Whe he e acce TND.Ergo @S, TND D ~ $Equivalent ~ Ergo @S, ForAll @p, TertiumNonDatur @pDDD

IHS ¢ TNDL ñ IS ¢ " p H p Í Ÿ pLMM

† The ke i i ha $ .. a d ... he e i he a e a $ ... a d ... hee .

$Equivalent @Ergo @S, ÿ TNDD, Ergo @S, Exists @p, p Ï ! pDD, ÿ Consistent @S, p D ê. ErgoRules @DD

IHS ¢ ŸTNDL ñ IS ¢ $ p H p Ï Ÿ pLM ñ $ p HHS ¢ pL Ï HS ¢ Ÿ pLLM

† Taki g he ega i f b h ide f he e i ale ce h ha c i e c ie i ale .

HNonSequitur @S, ÿ TNDDL ~ $Equivalent ~ Consistent @S, p D

JJS ŸTNDN ñ " p JJS pN Î JS Ÿ pNNN

The a i all a f f c i e c i he e . (See he Readi g N e iCha e 11 f a f i a he f a .)

217



† He ce, e 1 i TND, e 2 i ( ¢ ) ( ¢ ( ¢ )), e 3 i d ce he h he iha i e e e ¢ TND. S e 4 a d 5 di c e ha hi ld ca e a

i c i e c . S e 6 e ac he h he i a d 7 a i e he i fe e ce i ai lica i ha TND i lie ha he e d e e ¢ TND. S e 8

c bi e 1 a d 6, a d 9 a la e .Ergo2D @Ergo @S, TND D, Ergo @S, Ergo @S, TND DD, Ergo @S, Hyp : Ergo @S, ÿ TNDDD,

Ergo @S, Ergo @S, TND D Ï Ergo @S, ÿ TNDDD, Ergo @S, ÿ cD, Ergo @S, Retract : Ergo @S, ÿ TNDDD,Ergo @S, Ergo @S, TND D NonSequitur @S , ÿ TNDDD,Ergo @S, NonSequitur @S, ÿ TNDDD, Ergo @S, c DD

1 HS ¢ TNDL2 HS ¢ HS ¢ TNDLL3 HS ¢ Hyp: HS ¢ ŸTNDLL4 HS ¢ HHS ¢ TNDL Ï HS ¢ ŸTNDLLL5 HS ¢ ŸcL6 HS ¢ Retract : HS ¢ ŸTNDLL7 JS ¢ JHS ¢ TNDL JS ŸTNDNNN8 JS ¢ JS ŸTNDNNErgo ————— 9 HS ¢ cL

9.3.4 Axiomatic method & empirical claim

A a ce f c f i e e i ce L bache k , a he a icia e d ide ifc i e c i h e i e ce . The e i e ce f c e i i e e ilib i f e a le

e el ea ha e a i e l i e e ie (b e ai c i e ); a dhi d e ea ha ac al ld a i fie he e a i . Thi a he a ical age

i i ad i able. Valid i l ha e i e ce i lie c i e c , c e el hai c i e c i lie e i e ce.

9.3.5 A logic of exceptions

We a f la e he le ha e hi g i ell defi ed if i all he ded c if a c adic i . F cla i e add he a i ha hi le i ell defi ed i elf. A de ell defi ed c ce e eed ell defi ed e (if c ide ed c ce )

f hich he c di i h ld .

A e ai ed i fficial d c i e i c f ed i h he a ad ha he h lei e f c i e c ee a i h. F , if e ee a c adic i he i ffice decla e ha he c ce e e ell defi ed a d ha i all. Whe e e e

k i h e ha a e g e gh defe d he el e he i bec e a a ef i e gi e c i e c f a d he h le ble a i he i ce he e e

a e c i e (b c c i ).

218



N i i gl , hi le eflec he ge e al i a i i cie ific e ea ch. We al eadk ha a , a d e al e i e i f he f al e ha e b ild i l gic

a d a he a ic .

Thi igh al cla if he eed f c i e c . The e e da cie i ee hi elf he elf f ced e c ce ha , if i c e i , a ac all be ha d defi e. The

i h i e a ch e e a ible a d el al ed l gic a ch aible. He ce he a a e e f he ele a ce f c i e c ha e he i e

ld ake f g a ed if he e e e ell defi ed begi i h.

B che ki (1970:407) e i ha Waj be g 1931 ga e a a i a i a i f h eeal ed l gic. Had l gicia a d ec i ed hei bjec i h e

i ellec al di ci li e he ld ha e ade a bigge diffe e ce i i g e e

W ld Wa II.

9.3.6 The real problem may be psychology

Ma he a icia e d a ha ch c e e ce f G del he e a e e elhil hical. B hi i c ec i ce he l e he a h. B i a be haa he a icia ac all belie e a a dee e le el ha hei e a e decidable a dha he e el e he G delia ake hi belief e f al. Thei ad i f

G del he e he a i fie a dee e ch l gical eed, aki g he le a fhe c e e ce .

L gicia ha e i i ed i ce 1931 ha he a g e i ic l l gical, ha i ha aece a cha ac e a d i ef able c e e ce . B e ha hei i e i e e

diffe e . A he did ad h ee al ed l gic, he c ld e decidable a aca ch all f e e i h e i i g he d e e . Pe ha he h ledi c i i l ab hi d.

The e ca be g ch l gical de c e . Hilbe ga e a g e dic

B e a d ed hi f he A al . Thi i j a e e e ca e f calli ga e each he b i a he h he e d. Wh did G del i cl de i

hi e ? Si ce i ld ake he G delia i he Lia . Wh d l gicia d i ?Beca e igh ha e c cl de h ee al ed l gic. Wh d he a ake

ha e ? Beca e i de ia e f adi i a d c e i .

A d acade ic a h a ha e a c e cial i e e i a i g e a de a i hei b k .

9.3.7 Interpretation versus the real thing

The Hilbe G del c di g a ha e bee hel f l f a he a icia e abli h haelf efe e ce igh be acce able a a a he a ical c ce . A i h e ical a e e

219



c ce be a d he he e a e e ca be c ded he a i h e ic di c ei elf. (T be eci e: a e e di c he el e b he e i elf i all ed

d i ce he e i a i f f c e e e .) Wha e e hi hi icale i , he c di g i i h ble . If I gi e a a bi a be like

19540805 he i i k he he i i , e.g. a aeci e f da e d e e e . Th , c de a e ed i c e a d a c de i

a be , j a bijec i ( e e ela i ) i hi a c e . E e hee c a e e he e di i g i h he ff a d he beha e c . The e i a c ce al diffe e ce be ee a e e a d be ,hich diffe e ce ca be bli e a ed b a c di g. The di i c i i e e e e ialhe e ake a i e e a i f he f al e , f , eci el beca e e fi

i e e e he f al e a be , e b e e l c de he a e e i h

be . Wha e ai c cial i ha he i e e a i c ce a e e , hichhe a e be j dged f a l gical i f ie . Acc di gl , ha e e he hi ical

c e , he c di g i j e i ale i i g a blackb a d ha he a e e he blackb a d i fal e (c . able he he e i eak ha i l ha a

i f f a d e f h). All ha i e i ed ide if ha he a e ea i e e all c e al i f a i ha ake he c de ecific f ha c e

(a e abl ie k ell) ha he e a e e e ai .

9.3.8 Interesting fallacies

G del i i c i e i ce he c ea e a del ha d e fi hia ed e . Hi e ai c i e i ce he ( aci l ) i d ce a i

ha lea e he c cial e .

Whe i i a ed ha G del e i i c i e he hi igh ca e he e iIf Pea a i h e ic i i c i e , h d hi k ha be e i ? F , G del

ed a i h e ic, a d a i c i e c f d ld a l i a ell.

B hi i a fallac . A i c i e a lica i f a i h e ic d e ake a i h e icfal e. J i i g he Lia i h e hi g e ical d e ake he e ical a

e ical .

The e a e illi f ch fallacie ha ake di c i G del he e h iblehe e le l d a li e .

9.3.9 Smorynski 1977

Ab e e d c i f G del e l i ba ed DeL g (1971) a d S ki(1977), he Ha db k . The la e di c i eglec he ibili f h ee al edl gic. M e e , he e a e e eedle l a h i i ha di c i . Whe e

220



B e f e a le held ha i i d he e i l gic S ki i l eglecha B e aid a d did, a d ell ha B e i e ded a he a ic ieligi . S ki a e ha G del he e ill i d ce igh a e a g he

i fi ( 825). Al h gh I d belie e ha all h e h had a d ha e hei d b

ab he e he e a e a i al, S ki e a k i a i le ablege e ali a i , a d i ee be a eff ile ce c i ici b hee i i ida i .Thi h ld ha e i cie ce. M e e , i be ha S ki d e i e

ee he ea i g f G del he e . Th e h acce he he e ei hehe c i e c f a i h e ic i fi i e e h d , a d i b h ca e he a e j

belie i g. I ha e ec he a e ce ai l e a i al ha h e h belie e ihe hidde i c i e c f G del e h d: f i all j ee belie i g. I i ad ha

S ki d e e ed a d c ec ed b he edi .

9.3.10 DeLong 1971

DeL g (1971:225): The a al g f a ga e i ef l i e lai i g he i f heec i e. I f e ha e ha a ga e i i e ed (a d he le a e laid d hich

defi e ha ga e), b a a la e i e a ci c a ce cc f hich he le gi e g ida ce. A hi i a ha be ade a ha ill be he lec ce i g ha ci c a ce. The deci i igh be ade he ba i f fai e ,

he he i ake a be e ec a , he he i i c ea e he da ge , e c. H e ei ca be ade he ba i f he le f he ga e beca e he a e i c le eldefi ed. N a f he i ac f he li i a i e he e i ha he le b hich edi c e a he a ical h l a e, b be, i c le el defi ed. We a e

h f ced defi e he i f a i h e ical h hi icall ; ha i , i ca bee lica ed ce a d f all b be c i all edefi ed. We ha e ee h b hGa a d L bache k ca e he c cl i ha he ble f h i E clidea ge e e i ed ha he g be d he da a f e ge e . I a

a al g a e g be d he da a f a he a ic defi e a he a icalh. Ma ha i e ed a ga e f a he a ic hich i i c le e a a e l beca e

f he i c e abili f a ideal a d hi abili ie .

PM 1. A hi ical, e e i e cha gi g, a i h e ic h i high a ice. The G delia

a ad e be e be di ca ded.

PM 2. The e ca be ch a ga e f e e i c ea i g k ledge, b beca e f he e

a ad e . Ma ideal a d abili ie a a ch, b beca e f he e a ad e .

PM 3. g be d he da a f a he a ic defi e a he a ical h i ag e.

PM 4. The i e f E clidea ge e ca be c a ed he e icala ad e . The e ble ha e a e i el diffe e c e. I a al be ed ha

e ca ega d E clid e a a defi i i f h e de a d ace, ha a

221



E clidea i e e a i (e.g. a gl be) i ill be e lai ed i hi a e c a i g

E clidea e. Ei ei i e e a i f E clidea ace i e igh i ai il be a

a deal i h b e a i ble .

PM 5. The c f DeL g i i i ha a ha li i ed abili ie , b he

li i a i he ee de i ed f he a e f a he a ic i elf. Thi i a big ag e idea,i ce he a e f a he a ic i ell defi ed he e, b i c e ac a a

bjec i able idea a a , i ce e ld h e ha l gic a d a he a ic ld be

e h gh a d h e ld be f ee f a li i a i he ha c ea i i a d hec e f he bjec a e .

PM 6. DeL g i f ie i e al able i ce i ake e lici ha he l gicia

a d a he a icia f e a e i lici l , l eglec i g he c e e ce f heii e e a i f G del he e . F e hi g, hi e e b k ld ha e e i ed

if he e e a h had be efi ed g ea l f DeL g he i e e celle b k a dif DeL g e lici a e e had ca ed a ai f h gh .

DeL g (1971:227): I he d , a i a ece a c le e cie ce a d l gic.(...) e igh ecall ha i f ll f Ta ki h he e ha f al e i

ich e gh a e i e a ic . B ha i he diffe e ce be ee a i e e edf al e a d a di a cie ific he ? The l a a e e i ha f ig .The ef i ee e ha hi e l a lie all c ehe i e he ie

ha e e . A fi ed c ehe i e acc f eali hich a e i h

c di i c ld ibl be e, b l hical fic i al. N e icacc f he ali f hich e a e a a ca be ade a e. (...) Wha icha ac e i ic f e ic di c e ab e hi g i ha c flic i g a d c adic i gacc ab i a e e i ible.

PM 1. Thi a be e, b beca e f he a ad e f Ta ki a d G del. Ta ki

e l i ba ed a he f e beca e he did i d l i g elf efe e ce a d

he did f h ee al ed l gic. Ta ki a ach i c i ci g a d i i ible

defi e h i hi he a e la g age if all f he e ce i f he a ad e

ha eed a i k.

PM 2. A e d e lie i h hi he bjec i e ha e a i ac . The h le che e

a be a lie, b i chi g e le. O he i e i i a bad e .

PM 3. DeL g c ec l i dica e he delli g e h d f ha i g a f al e a d a

e a ic i e e a i . Thi ac all ca e a e i . The i e ded a lica i f eali

a d h a e e ie ce i h ge ha e a de he he e ca c ea e all h e

f al e ha b e e l ca be i e e ed. The delli g e h d igh be

ef l f ke he e a, b , f he h le f e i ical eali e babl ill k

di ec l i h ha eali (a e cei ed b b ai / i d/b d c le ) i ead f i h

he e f ch a f al backb e.

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9.3.11 Quine 1976

Q i e (1976): Le e, i cl i g, ch a la e da a ad ha i b ea aa i b i ic l a e dical a ad , a d e i c a able he a i ie i

he a e f i f, i he i i g e f he e l a d e e i i ca aci

eci i a e a c i e . Thi i G del f f he i c le abili f be he . WhaK G del ed, i ha g ea a e f 1931, a ha ded c i e e , i ha i h e e a bi a , i ca able f e b aci g a g i he e all he h f

he ele e a a i h e ic f i i e be le i de c edi i elf b le i g lie f he fal eh d . G del h ed h , f a gi e ded c i e e , he c ld

c c a e e ce f ele e a be he ha ld be e if a d l if able i ha e . E e ch e i he ef e ei he i c le e, i ha i

i e a ele a h, el e ba k , i ha i e a fal eh d. (...) Tha he e ca be d a d c le e ded c i e e a i a i f ele e a be he ,

ch le f e a he a ic ge e all , i e. I i decidedl a ad ical, i hee e ha i e c cial ec ce i . We ed hi k ha a he a ical h

c i ed i abili . ( 16+)

I hi ba e le Q i e a g e ha if a bab ca alk he el a g eca . B f a el he c adic hi elf a li le bi : I d ha e i e e laih a li g h . B d e i i a e able e i ? I d e . N

h f ele e a be he a e e a a b G del he e a able.Ra he , each a i e f ced e ill i e f h e h ; he

f ced e ca c e h e, e f he , a d i he . A I al eadgge ed i c ec i i h he c i h he i , la ibili c ide a i ca

a g e e i i g c difica i f acce ed a he a ical la . G del he e hha ch a g e a i ca e e ield a e fi i hed e i hich e e h f

ele e a be he ad i f f. B i d e h ha a e h fele e a be he i f e e i acce ible. ( 66) Q i e h ad i ha he e

ca be i acle ch ha a child ca alk e e he bei g able d a a bab . We ld el la ibili c ide a i ide f a he a ic e , a

.

Q i e i c ide ed b a be a e i l gicia .

F fac . DeL g (1971:277) Q i e b k j ed: Read ge he , he i e l i h Q i e fa elega le, b al i h hi c l g d

j dg e .

9.3.12 Intuitionism

DeL g (1971:271) call S e he C le Klee e Ma he a ical l gic J h Wile 1967:I i i , hi i he be i gle i d c i he ech i e f a he a ical

223



l gic . Klee e al i i ed he i i i i ch l i A e da a d e ag a h, Klee e (1965), he e la a d diffe e ce f cla ical a he a ic ,

i i i i , a d he ec i e e h d a ed b G del. Thi i f a e i ce iall a ie i i i i , l b a a he a icia i h di ed adi i al

b al i h a a ha i d a d gif ed e .Klee e b e e ha he ec c i f cla ical a he a ic al ead a ed i hK ecke a d 1880 1890. The : B , al h gh i i i i bega e fi e ea bef e he he f ge e al ec i e f c i a d c ib ed he cli a e f

e ea ch i he f da i i hich he he f ge e al ec i e f c i a e (...),he la e he i i de ail de el ed i e i de e de l f he i i i ia he a ic . O he he ide, i he e e fi e ea af e he he f ge e alec i e f c i had a ea ed, i i i i de B e c i ed i ai h aki g e lici ice f he he f ge e al ec i e f c i . ( 4) Th ,

li i g a a ge he , a ag ee e di ag ee.

The ke e: B e k he i i hil hical g d ha heibili ie f c c i ca be c fi ed i hi he b d f a gi e f ale , l g bef e hi i i ecei ed c fi a i i he fa f b G del

1931 ha f ali ade a e f a ce ai i f be he a e i c le e. Si i be de d f he e ha f al e f i i i i ic

a he a ic i c le e (...) I , a i i i i icall c ec e e i f i , c ld be e e ded b he G del ce ( hich i alid i i i i icall ), i.e. b addi g af all decidable b e f la (...) ( 5).

PM 1. Q i e a e f a i i i i ha he igh ha e eali ed.

PM 2. Th gh Hilbe ejec ed B e , hi Ph. D. de ed B e e h d a d

ed B e hil h e. (Th gh, hi b e a i a ha e li i ed al e,

G del e a a e i i i i i 1933 hich ld c ai hi ie .)

PM 3. B e ejec ha Ph. D. The i ha B e i i e ai lhil hical.

9.3.13 Finsler

Fi le (1967) j dged, ee he c e f he edi Va Heije (1967:440): Thef al a g e b ea f hich G del decidable i i i ec g i ed

a e, Fi le c ide be f al , i ce i ca be e e ed a d fai l i le aha i a la g age. He ce, acc di g Fi le , G del ha e hibi ed a f all

decidable a e e a all (...) The edi efe he e G del c cl i :Th he i i ha i decidable , ill a decided be a a he a ical c ide a i (G del (1931:599).

224



F cla i , ha a g e i :

Ergo2D @NonSequitur @S, g D g, NonSequitur @S, g D, gD

1 JJS g N g N2

JS g

NErgo —————

3 g

A i ha a g e i ha he ec d li e i l alid if he e i c i e .O he i e e e hi g ge e . He ce:

Ergo2D @NonSequitur @S, g D g, c NonSequitur @S, g D, c gD

1 JJS g N g N

2 Jc JS g NNErgo ————— 3 Hc g L

Thi c lica i a i e i ce di c i ab e k he h c f i ge a ical c le e e hich ld be acce able if e ld ick he de

i hich ha G del e e ed hi g .

Ye , Fi le i i c ec ha he e a e e e iall f al ded c i . We ded cehe e a e e , he d fall j f hea e . B G del f bid hadi c e

i elf, hi f ali ca be e e ed i. Self efe e ce f a e e i all ed b f .

9.3.14 A plea for a scientific attitude

The idea f a d he e e i h fie ce di dai a d c e f l gicia I ei 1981. I 1982 I a ed fe i al life a a ec e icia a d hel ed he d af b k a d he i e. I ha e checked he la e i a i i he field i 2006 2007 b

feel afe e e ag a i .L ki g back, e ca b e e e c lica i i c ica i . I 1980 1981, I

a a de , de ake i ake , I ade i ake . P fe e h e i akea d he ega d de a bei g liable i ake a a . S a I ked h gh a

e ie f e I a ha e b il a e a i f e e h ake a f he . If ll ed e c e i l gic, a d a he a g e a i he i h El e Ba h, ihi d igh i ld ha e bee be e , gi e h a ch l g , fi fi i h he he

a da d c e l gic , h e i ellige ce, a d l he c i ici e acce edi d . Ye I a a de f ec e ic a d a de f l gic, I had e ed i e e i h e he c e l gic. I had l a i e e i he idea ha he

Lia a e e, a d I had he a e i i i a Fi le la e a ea ed ha e had

225



G del, ha he f al a g e be e l f ali ed. T a e ihil h a ha i e d a all ki d f c cl i f G del he e , hich

c cl i e e e i a cie ific i d. A cie i ca acce heG delia i fe e ce: , !I be e ha i ed ha eache did

hel e l e a f i ake a d h g ided e a d a be ede a di g f ha a i l ed. Al , h ee al ed l gic a a a al

c cl i a fi , a d he I a ked f i he fe ga e e hi c f K i kea icle, a d ha hel ed ge i g e cla i . The e a e he e e al i e f e i l g ,like ha I ed ake = i fal e a he Lia a he ha i elf, hich ai ee eb f h d he la e (b i fai if a e ha i e elf efe e ce a d

la h i ). B i h h ee al ed l gic I c ld ield ha la e c e i , i ce h eeal ed l gic all a be e e e i f e e ha e el la hi g he e ali

ig . B he i had bec e diffic l f c e e back h e h had ga . Vi i he i e i ie i he c hel ed a bi , b al ca ed a k a d

di c i b h e, i h he a ailabili f he ec f c ica i a d a blackb a d.

Whe I e i a a e he , a edi eac ed: hi (...) a eee i dica e ha a ha e died e l gic b g a f he i

ha e ead hea d ab i ed . B he a e l e he i e f he Lia

e d ced i hi b k ha he ejec i a d i le a ce a e a a ed. O eh ld a he bli h ch a a e ha all c e e e i i . (B f c e,hi i a lea f le a ce I ld a g e agai blica i f e e ...)

Wha e e he ci c a ce ha ca e e j ifica i f l gicia , I ill h ld hei i ha i i a he i ible ha e l gicia li e a g e . A i e

d i g he e di c i i a g a ed ha (a) he e a e e ha bec ei c i e b he G delia Lia , (b) he ce he G delia he e a e e f all

e ; hich a ha I c e ded; (c) hi i c i e c ca be l ed b h ee

al ed l gic; hich I b e e l c e ded. The e a eac i he l i fhe Lia e ce a e ha li e i i e f he ibili ie . Whe a eac i cha ge

f a ha he Ea h i fla i gi e a he i e e a i he a ei h f he b a ia i he hi i dica e a f f li e e a d i lie ha

e d e b ha i e e a i hile he e e l h ld be ei he a eha he a g e i alid h he e i i i alid. I a a .

Whe C lig a (1981), I e ia Phile a f C e alii ( bli hed)

c e diffe e f hi b k a ha ded i f a g ade a c e f l gic, hefe eac ed: (...) agai ha i e f i e fal e a e i a d

c ide a i hich e e all ki d f e e ial l gical hi g i a g a d

226



i leadi g a e . A l g a eb d d e g eall i he bjec f G del( ) a al i (e.g. he ac al c c i f he Lia e e ce, a d he like), he

ible gge i ab al e a i e e ai e a e a d ag e. Agai hi backg d I belie e ha I d e ice he acade ic ld (a d ei he ) b

gi i g he a ked fficial h i g (A il 6 1981). Agai hi i j a e dic a d b a ia i . I i al e i icall fal e i ce I had l ked he 1931 a icle a d

C lig a (1980) c ai ch a G delia c di g he fe did e e a khe he I had d e ( h gh C lig a (1981) d ed i ), a d i i a e ii ce i i l gicall e i ed e ch a c di g ( hich i h C lig a (1981)

d ed hi , a i hi b k). (F i hi : I did a al e , filled i he e ea e he e he e e e ec ed, g he g ade a a , b he a fe fec e ic a ed e i g ha a he a ical l gic a e i a h f

ec e ic . The dea f he a h de . fe S a had i e a le e heg ad a i c i ee ha a he a ical l gic a e i a h. Which e f c eca d b , i h ea , gi e he e a he a ical l gicia ha e bee aki g i ce1931. PM. Ec e ic i a eciali a i , a e hi k, b a ge e ali a i : ic ce he i e ec i f ec ic , a h a d a i ic , ha e ca d all i h

he e e al de e g a f c e.)

A efe ee f a j al a e : I i e f h e c f ed hi g hich c ai a i e

f i ge i , f al i ake a d igh bi . The efe ee gi e ea g e b he e a e ab l el fal e. F e a le: he ha hif fal e , a d , f a l g a i ake . The edi a a e l ag ee a d e : Iee ha he a e i he a da d f ig a d hi ica i e ec ed f

c e a k i hil hical l gic. (J e 15 1981)

I ld like c jec e ha all hi i he e a ea e le. I a eeha e le ake e b he e h ld ill de a d ha ac all i heai idea behi d ha i aid. Ce ai l i l gic a d cie ce e igh be c ide a e,

i ce he l gic f ce a e c ide a d acce ha he e le c ideigh bi a d i leadi g he , ell, ld a be i he

i i be ea ed like ha . O e ld like lead f a e a he e a d al cie ific a i de, e ha e el i a fa ade.

Th gh de i e ha e bec e e ag eeable e le d i g l gic, l giciaha e ca gh e . I e e e e al fac ha ade A i le flee f A he ,

ade Ockha e d f ea i i (a d die f he lag e a 47 ea f age), a d

ha all ed de l gicia li e l ge life , R ell 98, B e 85, G del 75. Ie e i e al fac l gic a a acade ic bjec ha ade l gicia eglec F egeBeg iff ch if f e ha 10 ea , ha ade he i i e e e B e , ha

227



ca ed he eglec Fi le (1926, 1967) h e 5 ea bef e G del, de el ed f ha a e a al i , a d h e d i h a ch de i i . The e

i e al fac ha e cha ged a d be a ici a ed b a e a i i g c ib e he de el e l gic a a bjec .

The bia a d di belief ha e ill e c e ill ake e feel a a b d i e hdde l fi d il he ad, hile hi a e ge a b hi head a d

ca ch he heel, hile he di c e ha i e e he a e ge h h e he il head. The i l li e a d a e e cc ied i h hi g a d di g.

The a e cie i h c ide a a g e b he a e b ea c a h checkhe e a a g e de ia e f acce ed i d a d he ch i he a da d

b . The li e b a h i , di belie e a d bia , b i agi a i a d i e . If

he a e illi g e e i a di c i , i i l ha he ake he i e e ha i e he if i f i i i ce 1931 a d .

The b ild e ca le i e e ba e la d , like he he f e a d heli i a i e he e , a d he ca e lai clea l ha he d . Had he ided

ha cla i , hi b k ld ha e bee ece a . Thi b k ide a cla i ha ill e c e el e he e ided f c e ha l gic ha bee ha il

ag a a I e e.

The e a e ge e ali a i , he e a e fac . A a cie i I e ec he fac . I l khe aigh i he e e a d he he d g a a he he a e j he e. She ha i i diffe e a d I a illi g e ac .

The e idea he Lia a d he G delia i 1981 a e e ha he ke i e. Whaal ha e ed back he i ha I al de el ed hi e ge e al idea f a

, b had ge ha e e a e e ha idea f he . Thei ellec al cli a e f he idea a a e l a ick a d i ld l e if f

a chi kee e i g e ick i d . I a l g d a d ece a ha

I c ld c ce a e ec e ic agai , b i a al a elief. B ha al eaha hi c ce f a a hel ed. O e he ea , i g he b ei h he e a d d af b k l gic a d he e h d l g f cie ce al g i h hee f e e h e a he , he i e i g e ha hi i igh be f

e ge e al al e. I ec ic , he bjec f blic ad i i a i a d he le fg e e b ea c acie i i a , a d e ill be a a e ha a b ea c ac

ai ai le . I ill c ib e he ge e al elfa e he lic ake a d b ea c a g a a e a d a e a gh i he e fi ea f hei ed ca i (if hege a ) ha le igh e i e e e ce i all f he aga i ie f life a dh a dece c .

The ec d ele e i hi h le i he c ea i f b S e he W lf a ,i i ed b he g a f Ma i Vel a , ha , f all ac ical

228



e , ake all he diffe e ce. He ce i i i h a diffe e a gle a d ef e hed jha he idea f 1980 1981 ha e bee i le e ed i . The l gicia

d c e i la , h gh a a c e e ce a e ge e a i ge a cha ce hi elf.

PM 1. I Feb a 2007, c ec i g he i g e i he Ja a 2007 i e , Ica add he f ll i g. I c ac ed he fe f 1980 1981, i f ed hi f he PDFa d f a e he eb, gge ed ha he f a e igh ake a diffe e ce ha he

igh be e de a d he a al i , added ha check f he a al i c fi ed iagai , i i ed hi agai , a d ffe ed gi e hi a a d f ha f a e d . He did eac . I al a ea ha he ha bee bli hi g a l d a ic l gica d ge i g h ge e ea ch g a f i . I 1980 1981 hi a he l hi g ha he aid

ha he liked i a e a d he e di c ed i back he i a al b i hahe had h gh ab ha a gle hi elf. I d k he he he gi e e

efe e ce . I i ef l f e add hi c e he i e e le igh hi kha I d gi e e efe e ce hi . F he ec d: he di i c i be ee a ic

a d d a ic c e f ec ic a d i 1980 i fel like a a al i e e a i fe a l i f he di i c i be ee i i a d i fe e ce. I i l a all

ele e i he h le f hi a al i .

PM 2. I al c ac ed he he ai e I di c ed hi a al i i h back he .

Thi e eac ed i h he a e e ha he did ag ee back he ha i ld beele f hi a d l k agai . Th 25 ea diffe e ce a d a fe i al ca ee

i ec e ic d ee c . A d he e l gicia each ch l gi abha a i al h gh i ... (h a i ali ca be ec c ed af e a d f

( ee i gl ) i a i al beha i ).

229



230



10. Notes on formalization

10.1 Introduction

A aid i he O e all I d c i , he e e i a e i ical del f ha ide el ed i hi b k, a d he ce i ca be f ali ed.

PM. I ca e e e d b he he a e e i a e ca efe he h lee , e igh al a ake a h a e a a e a le. Tha a h a e i

fallible i he e ec d e di e he ibili f ch elf efe e ce. (If eal ead had e hi g e fec he e ld eed l gic a d a he a ic ; i ead

e c c e hi g be e f a chi g he be a ec f he i e fec ia d .)

O igi all , C lig a (1981) e a bi f he i f al de el e . F he c e

di c i e e he c ce f a E a d he e ile eed f i . Thi cha e h e e ca be ef l h he backb e f a f ale i i . I ca be a g ide he e l k i he li e a e f he i f i e e .

10.2 General points

The b k d a ifie i e e a i he he ead gl , l k eda ic a d d e e iall c ib e a a g e . Jaakk Hi ikka e a k ha a a ifie

al a e i e a d ai h e e i ic l adhe ed , e e he he a ifie i a ed. The a f a i le f he a ifie f ee l gic i ha a c a a d f a he e i hile a a iable a d f a . Th ( œ D) ( œ A)

i dica e he i e ec i , hile ( œ D) ( œ A) i dica e ha bei g aD i a fficiec di i f bei g a A. Q a ifie he he e i da ge f c f i . Reale i e ce ld be i dica ed b œ , he e i he i e al e . Thi i ei hedefi ed l call he di c i igh c i f ba ic bjec a he l e le el( i di g Ca e e ). PM. The e e i a efe e ce, ha I l , a f ali a i

f a f ee l gic.

Wi h he e f e e ce a d œ a e e ce agai he e ge i ila i e likeh e f he e e ha e igh e i he he e eall ca efe a a

231



ali he he ch a ali i e l defi ed. B ca al be defi ed ai g f b l ha ca be i e e ed, a d he e ee ha i i e e e e

ch b l a he c llec i f all ch i g ( he e i i ed a ch).

10.3 Liar 1: I a fficie l ich e f al ed l gic he e d e e i a Lia .

Th he Lia ha h al e T e Fal e.

P f: See he b d f he e ha e ha" i : ∫ [ ] = 0 .

C lla : I a fficie l ich e f al ed l gic he e i eed f ecificf a i le f biddi g he f a i f he Lia .

PM. Thi e i e he di i c i be ee j f i g a e e ce a d de e i i ghe he i ca be a e ed.

2: The Lia a i e f adhea i g he Defi i i f T h b ad i ga i a ce f he Al e a i e Defi i i f T h. I i ece a l k f f hee la a i f he c adic i ha ld cc if he Lia i i d ced.

P f: See he b d f he e ha h ha he Lia i a i a ce f he al e a i edefi i i f h , a d ha he defi i i a d i al e a i e a e all e cl i e.

C lla : (Thi i e e iall Occa a agai .) The le el f la g age f R ella d he defi abili f he c ce f h a e e i ed l e he Lia .Si ila l K i ke e la a i ha he Lia i g ded i i ele a . Si ce he

ble f he i le Lia ca be l ed b e i le l gical e h d , he cc e cef he Lia ca f a a g e f he i d c i f he c ce f R ell e al.

a d a i d c i f he e c ce ld ha e be ba ed hec ide a i , hich ld a he be f a l gical a e.

PM. S e f a e e (B ida ) ca ia b i i be h be Lia l a da be b ke a he e. O e igh al di ca d he h le e ha c ai ha

l . A ge e al ced e i c ea e a hie a ch f c ce i h a (g i g) c e fa e e ha a e acce ed a d a f i ge f a e e f ejec i .

3: Al h gh a a i f i g he ela i = [ ] = 0 ca bel g he ef al ed a e e e il ha he f eed i cl de i i e.

P f: Ei he hi i e b defi i i f h e e if l i gi e he i ca bee e ded.

C lla : The Lia i e cei ed a a ad ical i ce e belie e ha = , hichh e e ca be.

232



PM 1. Thi he e ee i c b a ea be e i ed beca e a e likeBe h (1959) clai ha he Lia e i . He gi e he e a le f a e e ce ha bc e efe i elf. I i diffic l la h a c e . A cie i h b e efac ld be i cli ed acce ha a e e ce a all e i a d efe i elf (f

e he i a i i e al he fal e ). He ce The e 3 i a e i e cehe e f he c le Lia . I al c e B ida l .

PM 2. A c e g a ha e el b i e ld c ea e a e dle l . Whea i he ec g i i f = [ ] = 0 ( i le Lia ) i ible (c cei abl i i )

he f he c le Lia e ha e (a) a c e a d e i c di i , (b) a i e edia ee a c llec i f e l a d check he he b h a d cc , (c) ead a d e a e he g a , i a d i e edia e e l . Whe he be f a iable a d

i e f l i la ge ha e he (a) a d (b) igh ffice. B he e a i e ag e a ea be ee ac ical ble a d he i ci le i l ed.

4: F a la g age a h ee al ed l gic i ece a .

P f: B The e 1 he Lia ha h al e. B he e i a c le Lia f hiche a e. He ce a hi d al e.

5: F la g age a he hf c i i eeded ha f.

P f: The d ai a d a ge diffe .

PM. Th e he Defi i i f T h he e i al e f h ee al ed l gic. Thehf c i a e defi ed a : Ø T e, Fal e : Ø 1, 0 a d : Ø T e,

Fal e , I de e i a e : Ø 1, 0 , 12 , i h he e e i el c le e e f

i i a d he e e ce . The e i l e e beddi g ha f ll i .Th gh he d ai a d a ge diffe e ake he libe ill c ide he be he

a e f c i a d e he a e b l = . B he e all f he cha ge fe ec i e he h he ical ea i g i che f al ed e h ee

al ed e . The b d f he e h he ical ea i g i i a f a ade a ei e e a i .

6: If h ee al ed l gic i al fficie f la g age he Ta ki c jec eha he e d e e i a ade a e defi i i f h f la g age i i c ec .

P f: The di c i i he e h ha (a) a DOT3 i ible, (b) ha i ecei ee a e ic i e e a i , (c) ha i ha ea able c ec i e a d c d

a e e .

F fficie c , e l ha e h ha i d e ge e a e e a ad e , i.e. haDOT3 i fficie .

1 : We ca di c i , a d e he DOT3 d .

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P f: Ob i . See he e f h he ical ea i g a d he d a ic e f c e ,al a licable f he d e a d fal e. While i ee ha e e lace DOT3 i hDOT ( he bec i g al ed agai ) he hi i eall e i ce DOT ic ai ed i DOT3.

PM. The Lia i a ble f al ed l gic b e f h ee al ed l gic.

2 : The a lica i f he DOT3 d e e l i c adic i .

P f: See he b d f he e . F all he e a e fficie a e e i al edl gic c i e l di c i ( e ial) e icali .

PM 1. Wi h Thi a e e i fal e e ele DeL g (1971:242) h ld ha hei ha i i fal e ld ake i e, beca e i a e ch; a d if e i ca be

e ele . Thi i a i a i , i ce ejec i f h ee al ed l gic e he ad G del a ach. Check he de i a i i he ai b d f he e he Lia f

h ee al ed l gic, a d fi d ha DeL g e he e.

PM 2. Si ce " i : ∫ [ ] = 0 , EFSQ h he ical ea i g.

3 : We ca defi e ch ha all ha ca be aid ab al bel g . Ihe d , Ta ki e a la g age i ece a a d he he f e a lie

( igidl ) la g age.

P f: While a f al f ld i l e he li i g f he f a i le i ifficie f hi b k gi e he i e i al defi i i ha i i l defi ed a ch

ha , if e hi g ca be aid ab he i ca be aid i . S ch a defi i i ce ai li acce able ( i h he i ha i e ai E gli h).

PM. La g age i e , ha Ca e e he e d e a l .

7: The DOT3 i fficie f la g age ().

P f: F le a 1 a d 3 f ll he e e i e c le e e . All a e e ca be

e e ed, a d al ed b h ee al ed l gic. Le a 2 gi e c i e c . The e a e he bl cki g c i e ia, f hich fficie c f ll .

: The DOT3 i ece a a d fficie f la g age ().

P f: C bi e he e 6 a d 7.

10.4 Predicates and set theory

Th ee al ed l gic i e i ed f edica e a d e he .

P f: See he b d f he e . Th gh e ca e e a c adic i f R ella ad b i g hi Ze el c i , i i eedle l e ic i e f bid R ell

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e e i e el b f a i le .

PM 1. T h he e i e ce f a e i i fficie h a ele e , b al hec i e c f he defi i i .

PM 2. S e he f e ca ill be ef l gi e Ca e e .

10.5 Intuitionism

10 B e i i i i ic l gic i l ade a el e e e ed b a( ecific) c bi a i f de i i i i ic e h d .

P f: See he b d f he e . If e a e he dich ie f (1) h a d fal eh d,(2) ece i a d c i ge c , (3) f a d ab e ce f f, (4) e e a d e e,

he B e ble ca be ade a el e e e ed. If e d e he edich ie he e i i e e a i al ble a i e, i a ic la i h he d

.

C lla : He i g a i a i a i f i i i i ic l gic ca l be e lde d a a ded c i el i c le e a i a i a i f al ed i i al

l gic. (P f: i ca e e e . If i i added he e e ge hecla ical l gic.)

10.6 Proof theory and the Gödeliar

11 The e e i a e a icall c ec e f hich he f edica eb e he h edica e, ch ha he G delia c lla e he Lia . A d h ee

al ed l gic i ece a deal i h ha .

P f: The del i he ld. Na e. See he ec i he Lia f h ee al ed l gic.

PM. Thi i l e i d ha a he a icia e d f ge ab he ld. The

d ee i a a del . I i al e i d ha he G delia i a c i f he Liaa d j a e ele .

12: If e ha e a fficie l de el ed e ch ha (i) i c ai al ed i i al l gic, (ii) i ca di c i elf a d i a e i , (iii) i c ai he

a f a i le" i : ( ¢ ) ( ¢ ( ¢ )), he he G delia ed ce he Liaagai , i h all i c e e ce , a d he The e f G del a l be EFSQ.

P f: See he b d f he e .

C lla : If he e d e a i f c di i (ii) a d (iii) he i a e ei da e e likeñ .

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C lla : U ef l e f e hi ica i be h ee al ed.

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11. Reading notes

11.1 Introduction

The f ll i g e a e le e a he e i i ab e a d a be f e he ea lie i e ec i e.

11.2 Prerequisites in mathematics

Thi b k di c e l gic a d i fe e ce a d d e de el he he f f c ia d ela i . S de i g hi b k ill ge e all ha e had e a he a ic bef e a d ill e d de a d i e he e. E a le f ch e a e he f c i (bijec i ) a d . If he e a ea e deficie c i he backg d

he he i e e ill ide a le e la a i . S e l gic b k al de el he

(ba ic) he f f c i a d ela i b he idea he e i ha e a c ce a e l gic a d i fe e ce ha h e i e a be be ea ed el e he e, a ld al

h ld f ge e a d i a i a ic a d a al ical ge e . I igh be a g ed haea i g i h de i g (J h i a a ce f Ca l a d Ca l i a a ce f Magg ,h ...) igh be a ecial e f edica e l gic, de e i g a e i i a b k

Eele e a L gic a ell, b he 250 age f hi b k al ead a ea ake ffficie i ha de i g ca be e be deal i h el e he e.

11.3 Logical paradoxes in voting theory

Thi a h ha bee i g DeL g (1971) A file f a he a ical l gic i h g eadeligh . P fe DeL g al e a b k A he e i i g (1991) eeC lig a (2008) a d a ki g Jeffe ia Telede c ac (1997). The i g

a ad e e i e i i g i ce i g he i a he a ea i h c f i , eeC lig a (2001, 2007, 2011), V i g he f de c ac a d C lig a (2007i)

he Pe e eighi g che e. L gic a d de c ac a e ic ha de e e a e i .

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11.4 Cantor’s theorem on the power set

11.4.1 Introduction

Se A a d B ha e he a e i e he he e i a bijec i e e f c i be ee he . Ab e e adhe e b defa l Ca The e ha a e i al a

alle ha i e e ( e f i b e ), ee age 131, 196 a d 231 235. U J l 2007 hi a h h gh ha hi he e h ld i ge e al, b he I ead Wallace(2003) a d e C lig a (2007j), c cl di g ha he he e h ld f fi i e ea d he de e able e f a al be , b i ge e al. Thi lack f ge e ali

ld all eak ab a e f all e . A adde d i 2011 i ha I

al ejec he diag al a g e f he de e able e f a al be : hiche he ibili ha a d igh ha e he a e i e.

Thi b k i l gic a d i fe e ce a d h kee e di a ce f be hea d i e f he i fi i e. Hi icall , l gic de el ed a allel ge e a d he ie

f he i fi i e (Ze a ad e ). A i le ll gi i h All , S e a dN e hel ed di c he i fi i e. R ell e a ad clea l deal i h i fi i ee a d G del he e e a i h e ic i h i i fi i e be i g e abli helf efe e ce. Ye , de el l gic a d i fe e ce e , i a ea ed ha hi b k

c ld ki he ick bi f be he , E clidea ge e , he de el ef li i , a d Ca de el e f he a i fi i e. Th gh i i cl e i ible

di c l gic i h e i i g he bjec a e ha l gic i a lied , e illke a dece di a ce f h e bjec he el e . Al he e di c ed

a he a ical i d c i , a d h ba icall elied he a al be , e did l i c le e i d c i i ge e al, a d i h a e ha i he i ha

ded c i e i fe e ce bec e i d c i e if e all h he e f ha ki d.

I al be b e ed ha hi a h i e e Ca The e . We aejec he f gi e b Wallace (2003) b e ha he e a e he f . A a gi al

check he i e e h ha hi f i he l e gi e a e i e ha ee a e b hi a l ea ha i i a la f. Th f i ee

i al gi e a h di c i f aid f f Ca The e a d i c eejec i , a d f he efe C lig a (2007j) a d f e di c i .

11.4.2 Cantor’s theorem and his proof

Ca The e h ld ha he e i bijec i be ee a e a d i e e . Ffi i e e hi i ea h (b a he a ical i d c i ). The ble i f( ag el defi ed) i fi i e e A ch a he a al eal be . The f (i

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Wallace (2003:275)) i a f ll . Le : A Ø 2 A be he h he ical bijec i be ee Aa d i e e . LeF = œ A – [ ] . Clea l F i a b e f A a d h he e i aj = f - 1[F] ha [j ] = F. The e i a i e he hej œ F i elf. We fi d ha j œ

F ñ j – [j ] ñ j – F hich i a c adic i . E g , he e i ch .Thi c le e

he c e l e i i g f f Ca he e . The b e e di c i i hha hi f ca be acce ed.

11.4.3 Rejection of this proof

Si ila like i h R ell e a ad ( ee ab e) e igh h ld ha ab eF i badldefi ed i ce i i elf c adic de he h he i . A badl defi ed e hi g

a j be a ei d e e i a d eed e e e a e e . A e hi li e fea i g i i e a all c i e c c di i , gi i g F = œ A – [ ] œ

F . N e c cl de ha j – F i ce i ca a i f he c di i f e be hi , i.e.e ge j œ F ñ (j – [j ] j œ F) ñ (j – F j œ F) ñ .P i icall eaki g, he

F defi ed i 11.4.2 diffe le icall f heF defi ed he e, i h he fi e e i bei g e ical a d he e e e c i e . I ill be ef l e e e he eF

f he e defi i i a d eF' f he e e i i 11.4.2. The la e b l i af he le ical de c i i b d e ea i gf ll efe a e . U i g hi , e ca al

eF* =F ‹ j a d e ca e e c i e l haj œ F*. S he f ab e ca beee a i g a c f ed i e fF a d F*.

I f ll :

1. ha he f f Ca The e (i.e. a ed ab e) i ba ed a badldefi ed a d i he e l a ad ical c c , a d ha hi f e a a e ce a

d c c i ed.

2. ha he he e i ill e f ( ag el defi ed) i fi i e e ( ha i , hia h i a a e f he f ). We ld be e eak ab CaI e i Ca S ed The e , i.e. li k he li e a e ha

ai ai ha he e . I i i e a c jec e i ce Ca igh ha ed e ch a c jec e ( i h f) if he ld ha e k ab ab e

ef a i .

3. ha i bec e fea ible eak agai ab he e f all e . Thi ha head a age ha e d eed di i g i h (i) e e cla e , a d/ e haa ell (ii) all e a .

4. ha he a fi i e ha a e defi ed b i g Ca The e e a a e i h i .

5. ha he di i c i be ee he a al be a d he eal be e( l ) he ecific diag al a g e ( ha diffe f he ge e al f). SeeC lig a (2007j) f hi a ec . H e e , a adde d he e i 2011 i ha e

239



ca clea l ejec ha diag al a g e .

Ad 5) C ide he deci al be be ee 0 a d 1 i bi a e a i . A fi egi e 0, 1 . A ec d e gi e 0.0, 0.1, 1.0 . A hi d e gi e 0.00, 0.01, 0.10, 0.11,1.00 . A f h e gi e 0.000, 0.001, 0.010, 0.011, 0.100, 0.101, 0.111, 1.000 . E ce e a.F each he e i a fi i e li . Le i gØ ¶ gi e all be be ee 0 a d 1. (I i

i al al defi e ha each be e di g i a 0 ha i fi i e e , ha i ii e; a d he each add a li f e al e . I hi a e a able f c i Ø

ca be c ea ed.) F each a diag al i defi ed. H e e , f he li i i a i he ei e defi i i f a diag al. He ce Ca diag al a g e ca bea lied. I e e e hi g ha i e l defi ed. PM. Pe ha he e ia he a g e h gh ha√ i de e able.

Si ce e a e f c ed l gic a d be he i ffice hedi c i he e.

11.5 Paradoxes by division by zero

(PM. Thi ec i a i e f he fi edi i f 2007 a d ha bee de el edi C (2011). The e e e e ai a ef l a .)

Dijk e h i (1990) gge ha he A cie G eek did de el algeb a (a d

b e e l a al ical ge e ) i ce he ed hei al habe de e be ha a + a = b al ead had he ea i g 1 + 1 = 2, a d he ce i ld be le ea hi

he idea ea a a a iable. We ld c ide i a ge e e.g. 15 a aa iable a gi g e ¶ +¶ . Thi e la a i i e i el c i ci g i ce he

G eek did e a e like Pla A i le a d h igh ha e ed a a e de e a a iable (like Va iab le ). B a i clea l a e f he b acle

e c e. We e a h had ai ill 1200 AD bef e he e ca e f I dia iaA abia ge he i h he A abic digi ( he e b h e a d ci he a e j i lde i ed f he A abic if = e ). A abic e al a e ea ie k i h ha

a e al , e.g. di ide MCM b VII, e hi ad a ce ca e i h he c hahe e ca ed a l f a ad e . We e a h l ed ble b f biddi g

di i i b e b he ble e i ed i calc l , he e he diffe e ial ieelie i fi i e i al ha agicall a e b h e bef e di i i b e af e i .

Ka l Weie a (1815 1897) i c edi ed i h f la i g he ic c ce f he li i deal i h he diffe e ial ie . Ye e e he e he li i f e.g. / f Ø 0 i

be defi ed f he al e = 0 he h i al a i b i defi ed f ac all= 0 b l f ge i g cl e i , hich i a ad ical i ce= 0 ld be he

al e e a e i e e ed i . C lig a (2007k) gge di i g i h he a i e

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di i i e l f he ac i e di i i ce . I he ac i e de f di idi g b ea fi i lif algeb aicall de he a i ha ∫ 0 hile b e e l hee l i al decla ed alid f = 0 (i.e. e e di g he d ai a d j e i g =

0). Ma he a icia ill e d ega d / a al ead defi ed f he a i e e l

i h i lifica i , ha he ac i e i ld be e a d f e a le de eda // . O he , i.e. e le h a e fe i al a he a icia , ill e d ake /

a a ac i e ce a d he igh de e // f he a i e e l . All i all, ild a e ch, i ce e igh c i e i e / a d all b h

i e e a i de e di g c e . I ha a ch a ad e a e e lai ed bc f i f a ach a d b he i ha a he a icia ca be le b e a f

ha e le e d d . A he c cl i i ha calc l igh e algeb a f hediffe e ial ie i ead f efe i g i fi i e i al li i .

T ake hi ic , le / be a i i ed c e l b a he a icia a d // be hef ll i g ce g a : a e∫ 0, i lif he e e i / , decla e he

e l alid al f he d ai ã 0 . Thi defi i i l h ld f a iable , ha// = 1 b f be , e.g. 4 // 0 ge e a e 4 / 0 hich i defi ed. The de i a i edeal i h f la , a d j be , a d ld be he g a '[ ] = „ / „ ªD // D , he e D = 0 hich e b h haD // D e e d he d ai D = 0 a dha he i c i eD = 0 ac all e ic he e l ha i . I a a , hi

defi i i i hi g e i ce i e el c difie ha e le ha e bee d i g i ceLeib i a d Ne . I a he e ec , he a ach i a bi diffe e i ce hedi c i f i fi i e i al , i.e. he a i ie a i hi g e , i a ided. Al ,

he i e e a i gi e b Weie a a d c dified i he i f a li i i ejec edi ce ha defi i i f he li i e cl de he al eD = 0 hich ac all i eci el heal e f i e e a he i he e he li i i ake .

The e ble i h h hi e defi i i f„ / „ ake e e. Le c ea e calc l i h de e di g i fi i e i al li i di i i b e . Le

F[ ] be he face de = [ ] ill , f k F a d k hich i bede e i ed ( e hi de ). The he cha ge i face iDF = F[ + D ] F[ ] a dclea l DF = 0 he D = 0. A a i a i he face cha ge i f e a leD ( + D /2) i h D = [ +D ] [ ]. The e f ch a a i a i ill be a f c i fD agai . We ca i e DF i e f = [ ] ( be f d) a d a ge e al e ee[D ], he e he la e ca al be i e ae[D ] = D [D ] he e [D ] i a ela i e

e ai de . We di i g i h D ∫ 0 a d D = 0, a d bel e e i (*) i dica e ai lici defi i i f [D ] a d (**) a e lici defi i i :

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D = D + e@D D if D ∫ 0 H*L@D D ª D F ê D - if D ∫ 0 H**L

DF = 0 = D + e@D D if D = 0, f a ; elec = H*L

@D D ª 0 = - if D = 0, f = H**LWe l i l b e a d he e di ide b e i fi i e i al . N , i lifDF /D algeb aicall f D ∫ 0 a d de e i e he he e i gD = 0 gi e a defi ed

c e. Whe he la e i he ca e, akea ha c e = DF // D , he e D = 0 .The elec i f = i ba ed a e e f c i i , i he e e f li i bi he e e f a e f la , i ha (*) a d (**) ha e he a e f i e ec i e f he

al e f D . We he fi d = = [ ] hich ca be de ed a F' [ ] a ell.

The dee e ea ( ick ) h hi c c i k i ha (*) e ade hee i ha he c e fe[D ] // D ld be b (**) ide a defi i i he

he e i ee a a f la. The e lici defi i i (**) i lici e i (*) gi ee ac l ha e eed f b h a g d e e i f he e a d b e e l he

de i a i e aD = 0. The dee e ea ( agic ) h hi k i ha e ha edefi ed F[ ] a he face ( i eg al), b h i h a a i a i a d a e fa i a i ha i acc a e fD = 0, ha he he e i e he e k

ha F[ ] gi e he face de he= = [ ] = F' [ ] ha e f d. If e a i e

ha e ha e d e he e fi d he g a F' [ ] = „ F / „ ª DF // D , he e D = 0 .The defi i i (*) (**) gi e he a i ale f e e di g he d ai i hD = 0,

a el .

F e a le, he de i a i e f F[ ] = x2 gi e „ x2 / „ = (H x + D xL2 x2) // D , he D :=0 = 2 + D & D := 0 = 2 . Thi c ai a ee i g di i i b e hile ac all

he e i ch di i i beca e i l a i e he ced e (*) a d (**). Whilehe Weie a a ach e edica e l gic ide if he li i al e , hi al e a i e

a ach e he l gic f f la a i la i .Hi icall , he i d c i f he 0 i E e a d AD 1200 ga e a

ble ha ce h e e e ge i g l ed, he l i ha e ca di ide be a c dified i e, a d il i he ch l f E e ld ee i h bad

g ade , e e e i h e a d i fa if he ld i agai h e ac a c le .T agicall , a bi la e he hi ical i eli e, di i i b e ee ed be i af he diffe e ial ie . Ra he ha ec ide i g ha di i i ac all ea ,Leib i , Ne a d Weie a decided k a d hi , c ea i g he c ce fi fi i e i al he li i . I hi a he ac all c lica ed he i e a d c ea ed

a ad e f hei . L gical cla i a d d e ca be e ed b di i g i hi g be ee he (f al) ac f di i i a d he ( e ical) e l f di i i .

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11.6 Non-standard analysis

Thi b k e lai ha G del he e h ld he e acce eak a i a d

ha he e a a e he e ad i i e e e g h. I al de el he i ha hil hical a g e he c e e ce f he he e igh ill h ld b

f he ea ha he he e , i ce he ee i l ch g a ia d h ld e a a e i h he he e . A a he a icia e hi a h : I a

i ed ha a he a icia ade e e f i , ha e ha e e hea h [i.e. G del], i h hi a he a icia ha , ade e e f i . O e

h ld al a be a e f a he a icia h ici he he a alki g abhil h . (U all i i be i l ig e e e hi g he a i ha de.)

Thi ill lea e he i ha he he e h ld de h e eake a i . O ede el e ha e G del he e (i.e. likel de h e eake a i ) i

a da d a al i . Thi be e i ed i ce a he a icia e d fi d i ag ea i e i . The a e a he a icia : O e f he bea if l hi g e ca d i hG del i e i i d ce i fi i e i al . I fi d i i ible a a hi g ab

hi i ce I ha e died a da d a al i a d d ha e he i e d i . I be ed ha hi ld l e he ble f di i i b e , a di c ed

ab e, i ce he e ill ld be i fi i e i al ( ha he ld be e ). Thial lead be he a d a be c ide ed be l ge a i e f l gic a di fe e ce i elf.

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11.7 Gödel’s theorems

11.7.1 Gödel-Rosser

G del (1931:616), ed hi b k 211 a d 213, e lai hi e l i e fdecidabili , hich h ld de he a i f e a ical c ec e , i.e. he e

f a del i e e a i ( hi b k 208 209). F all ee (i.e. i e e ed) hihe e l h abili . I a R e 1936 h e abli hed decidabili

e , ee DeL g (1971:180). The c cl i f hi b k a e affec ed, h gh.Fi f all, i i e a e e a ical c ec e a del i ce i ld beki d f a ge de el a f al del a d be able e i . Sec dl , hi b k

9.2 3 h ha he e de ea able a i bec e i c i e a de e e hi g, ha R e fi di g i ele a .

11.7.2 Proof-consequentness

S e l gicia e e a e ail ha f c e e e" : ( ¢ ) ( ¢ ( ¢ ))ld al ead be k de he a e a d e e be alid f Pea A i h e ic, =

PA. Thi a a be e, b d e ee ele a i i elf. Wha h ld fPA d e ece a il e e d he di i c i be ee elf efe e ce f e e ce

a d elf efe e ce f e . The i e i i a e ial he ba ic i ha addi gea able e ie a e ca e a ge e l , i h e e all he c lla e fhe G delia he Lia . The a e f c e e e e ai ef l i ce hi

e e e ha i c e e i hi e ec .

11.7.3 Method of proof

Page 212 a d 218 e he e h d f f b i g a d e ac i g h he e b hef ca al be a la ed b e lici l a i g e ie , ee al C lig a

(2007 ). The e ie a e e hi g ab a d he d h h i k i .Thi a e ha g ea ec i ce i all f a ide a ie f i e al e h d i

e hile a he a e i e i all di c he c e e ce . A bjec iigh be ha hi lack a e i e ce f ha he e i a lea e i e al e h d

ge e a e h e a d e ie . Bel h e e gi e e all e a le ( a b )a d hi h h ha e i ca be deal i h e ge e all . While e lack a

b , a ac ical l i ld be a k e e i i a black b a d ac a ha

b , a d e ld all be able ell he ha e ld chea (f he e ef , ha i ).

244



11.7.3.1 For page 212

P e 1 i ha if i e a d he c e e ce ld be ha i e , he hee i a e gh ee i elf ha . The e d e ha e be eal a ,

i a l e i e a b a i a b h he i , e ha e fall i

he b c e e ce , a d he ick b h a e e a d a fe hee b . We a f ge ab ha b a d he ha e he e i elf:

1. " : (( ¢ ) ( ¢ )) ( ¢ ( ))

P e 2 i f c e e e , i.e. he e e a he hi e i alilli g a ha he he aid .

2. " : ( ¢ ) ( ¢ ( ¢ ))

O e ill ickl e ha e ca b i e= ( ¢ ) i (1) a d ha d e (1) a d (2) ge e a e:

(1) & (2) " : ¢ ( ( ¢ ))

hich i half f he defi i i f h c c i (i.e. he e ld decla e haf i i c ce f h a e i ). We a e ge i g cl e he Lia a ad .

Whe e ake = i (1) & (2) he e ge ¢ ( ( ¢ )) ¢ ( ), a d he ce (¢ ). QED. Th e ha e e d ced he e l f age 210.

The f ll i g d e he a e i alle e . I he ge e al c di i ( ¢ ) ( ¢ ( ¢

)) e b i e = (li e 2) a d ( ¢ ) = (li e 3). Li e 4 e i e 1 a dhe li e 5 b i e = a d = . The e a l d e li e 5 a d 3.

1 HS ¢ H g Í Ÿ g LL2 HHS ¢ g L HS ¢ HS ¢ g LLL3 HHS ¢ g L HS ¢ Ÿ g LL4 HHHS ¢ pL HS ¢ qLL HS ¢ H p qLLL5 HHHS ¢ g L HS ¢ Ÿ g LL HS ¢ H g Ÿ g LLL6 HS ¢ H g Ÿ g LLErgo ————— 7 HS ¢ Ÿ g L

N e ha ¢ ea ha ef e , ha , if he e i c i e , i able, a d i ce i a , i i e. He ce he e ef e a e a e e . Gi e

he e i ale ce be ee a d , e al ha e ¢ , ea i g ha he e e i i c i e c ( hile e ld e d a e i c i e c ). N e al ha he

i e i decidable i.e. decidable. Thi ld c flic i h a i g e a icalc ec e ( ee 208 209) i ce ha ld ca e be decidable hile i idecidable he e.

245



11.7.3.2 For page 218

P e 3 i a bi e c le . If e a d if f e he c bi a i f ca e a i c i e c , he i a e gh c cl de ha .

3. " , : (( ¢ ) ( ¢ (( ) ¬ ))) ( ¢ ( ¬ ))

The ele a ce f hi a ach ill bec e clea e he i i h ha hi lead (¢ ). Thi di c i ill al h ha e 3 i i e ea able.

Whe he e acce Te i Da (TND) he i ca acce TND i cehi ld ca e a c adic i , a d c e el . Thi ha a c e e ce f i g

c i e c . C i e c i e i ale he f he f ( e , h ee i e ). We ca h he la e b i g he ke i ha $ .. a d ...

he e i he a e a $ ... a d ... he e .

HS ¢ TND L ñ IS ¢ " p H p Í Ÿ pLM HS ¢ ŸTNDL ñ IS ¢ $ p H p Ï Ÿ pLM ñ $ p HHS ¢ pL Ï HS ¢ Ÿ pLL ñ

HŸ HS ¢ Ÿ TND LL ñ " p HHŸ HS ¢ pLL Í HŸ HS ¢ Ÿ pLLL ñ

The e ge he f ll i g f. I ill be ef l b i e A = ( ¢ TND) a d B = ( ¢

TND):

1) ¢ (((S ¢ TND) (S ¢ TND)) ) defi i i f i h = TND

2) ¢

(( A B) ) li e 1 i h h e A a d B3) ¢ TND a i f

4) ¢ ( ¢ TND) f c e e

5) ¢ A li e 4 i h A = (S ¢ TND)

6) (( ¢ ) ( ¢ (( ) ¬ ))) ( ¢ ( ¬ )) property 4 for some p and q

7) ¢ ( A B) f 2, 5, a d 6 f = A a d = B

) ¢ (( ¢ TND) ( ¢ TND)) a la e back

) ¢ (S ¢ TND) d e 4 a d 8

10) ¢ a la i g agai

Which gi e a i c i e c i h he ea lie ¢ .

A c e a g e e 3 igh be ha e e c ld all ch a e ai decidable. The i h e e i ha e ca i agi e eak e bha e 3 i ea able a d e ld a acce i f ea able

e . S e de i g deci i e j a Lia i a G delia i ea able he he ble ca al be l ed b h ee al ed l gic.

246





g al a cie ld h ld a ai , ee di c i de ic l gic, he i a ii diffe e f cie ce, i ce he e he bjec i e ha bee gi e b defi i i , i.e. cie cea i e a di c e i g he h, a d i i he e ha he de c i e ele e f he ela e a he a icia a d .

H b eak he a gleh ld f he age ld Lia a ad de a he a ic ?Thi b k f ll he a ach f h i g e e ie ha a e ea able b

he el e a d ha , he added he G delia e ca e i c lla e. Of c e,he a he a icia a g ill a ha i i ea i cl de e a i , ch a¢

( ) ¢ ( ( ¢ )). Thi a he a icia a g ill l c ide hea b

c e a d ill ha e g i i he a i . If he i cl i ca e ai c i e c he he a ell c cl de ha hi i a beggi g f he e i ha

e f h e addi i be e e e . ¢ ( ) ld be ejec ed i ce i idi ec l e i ale a i e al f f c i e c ¢ , a d ¢ ( ( ¢ )) ld be ejec ed i ce i di ec l i d ce a f he Lia a ad i. S ch eac i cai deed be b e ed. H e e , f he e i ical cie i i i a ge ejec ef l

e ie j ai ai a he e , a d i i a ge ha ld be all ed k i elf ha i c ce f f ea . O e ld a he l e he Lia a ad

ha f bid i c ea i . O e a eg b eak he deadl ck i a a a f h eb i e ie a d i d ce he eake e i ed ab e. The e e ie

agai he a e ki d f c i ici f he a he a icia a g h i i la edagai a c ide a i he ch ice f a i . Ne e hele , ejec i bec eha de , a d b i g he eake i ca bec e clea e ha ea able e ie a e

ejec ed a d ha hi ca ie he high ice f i g hi a f a he a ici ele a . E e all , e hi g ld ha e gi e. O e i e i ded f he Chi e e

a i g The i a i a bea able. I c ld l ge c i e. I la ed 300 ea .

The likel c e i ha (af e h e 300 ea ) he a he a icia a g illad h ee al ed l gic a d ceed a bef e, i ce li le ill ha e cha ged i

a i de, l e hea b. Ne e hele , f he ld, a d de e i i e l gical a ad e , i ld be a be eficial cha ge. A d f he a he a ical c ii igh bec e a ic f c ide a i he he i i i e a i e bec e a

a he a icia a g. Ma he a ic i a fac l f he i d a d a a f life ald ha e e ali . A i le, A chi ede, Ne , Leib i did e

dece a h hile ge i g l i i . M de a h did ge l . Y g i da i i g a a he a ic ld be e be i e i elec i g hei le del . A hei a i ld be ha a De a e f Ma he a ic a he h ld beli ked ai l he Ph ic De a e b be ce al all he de a e

ch a Ec ic , P ch l g e ce e a i cl di g a highe ega d f e gi ee i g.C lig a (2007i) gi e a he e a le he e fe f a h a d h ic g

248



a a i.e. he e he ad he Pe e a e le f i g eigh , a d dhi e e i a e le e he g e e i he E ea U i , hile hei

a i ca be ai ai ed i ce he eglec a e i ical a d a i icala g e , a d b e e l c f e fac i h al . I e a f hi he i d,

he a he a icia i la e hi he elf f he ld b ad i g hea ba i de, b i he he a f he i d i a ea ha d d hi c e e l . Ra he

ha ai ai he fic i ha he a he a ical fac l f i d ca al bec e hee hi elf he elf, i ld be be e i e e a he a icia i he eal

ld ha he a e be e a a e f he i fall a d ce ai ie l ki g he e. Theld l ge be a he a icia b e e cie i i g a he a ic . O e

ac ical le c ld be ha a acade ic i e e a e a eal ble ce i ea he e eali i defi ed i e f e e i e al da a. A he le ld be

ha a he a icia h ca deal i h he ld h ld be he c e f heiDe a e b a he he f i ge. A he i e i ha e ea che f he didac ic f

a he a ic h ld hide i hei e ea ch j al b be ac all i l ed i heeachi g f a he a ic . I he e e , he a ie ha e ake e he a l a d

ab e he g ea ac ical hi f a he a ic l e he ld al g i hei l acie .We ld a he ee ha e ac icall i ded cie i ec he ld f all

ha i he f e. M e hi iE (2009).

11.8 Scientific attitude revisited

The e i de f Ja a 2007 Ma ch 2011 c fi ha ha bee aid i he leaf a cie ific a i de , 9.3.14 ab e. Yea a ed, i h he a ailable e f

he i e e a d , i hich l gicia a aged eglec he a al i i hi b k. C i g Se e be ill h he a f a he acade ic ea he e e

de ill be i d c i a ed i h e e agai . Ma he a ic ld be a de c a ic a d libe a i g ac i i i ce he e i a h i ha e f ce a he e b

l he a i al ec g i i b f. U f a el , hi ill e i e a e ai ai a e i d e i igh a d e f ef a i . Ge e all , i ihe cie ific a i de ha f e ch a e i d. L gicia a d a he a icia fail

ha a i de. Thi ha c e e ce f ha he ell cie , l l gic a dc e e b al i g he a d i f de c ac , a d ibl hea ea ( h gh died b hi a h ). I i diffic l a h ick he a ie i .Ye , a ld i h a e la i f 6.9 billi l g i g a e

e la i f 9 billi l ld ha e he l e d e hi . O e ca b c cl de ha cie eed a f da e al cha ge i he a h i deal i h l gica d a he a ic .

249



250



Conclusion

Thi b k ha di c ed he bjec f l gic f he b . The ba ic ele e i

l gic a e i h c e i : al ed e , i i al l gic, edica e l gic, ehe , i fe e ce, a lica i ali a d he like, a i a i a i a d he c i e ia

f ded c i e e ch a c i e c . Ye he age ld a ad f he Lia i a b ef c e i ha ca e diffe e l i a ache ch a he he f e , he

di aba d e f h i fa f f he a d decidabili , h ee al edl gic. The fi a ache c ea e hei ble . The l ac ifice kef f elf efe e ce b al a he i d i h le el i la g age a e e like

he G delia ha a e c i f he Lia ha e feel ha ac all hi g eall ha bee l ed. O l h ee al ed l gic all f a c i e de el e ha iag eeable h e le de a d la g age a d ea i g. Ye h ee al ed l gic i

f fa i h he l gic c i beca e f ed c le i ie . The e i ai a i al ele e i ha i ce h ee al ed l gic ce ai l i i le ha hec le i ie f he he a ache .

L ki g a he e i i g li e a e l gic, e fi d ha i i g l i ade a e a dhighl i leadi g he e ke i e . I a ic la , G del he e a ea be

a he a ical c c ha ha e a diffe e ea i g ha gge ed b i a h a dhe d hi c cl i hich h e e a e ge e all ad ed i he

li e a e. B di c i g he e e a d h i g he e he clai e e a a ed,e ha e i fac e a ed ha e i dica ed i he begi i g: a el h i g h l gic

a d i fe e ce ca be clea a d d.

O di c i ha al highligh ed e a ea he e e e ea ch i l gic ld bef i f l. F al a he a icia ld a ake he a g e fficie l f al

ha he he el e ca acce i . Si ce he bjec a e i be iddled i ha ad e a d i de a di g , e cla ifica i i a g d gge i i deed.

Sec dl , e ha e i dica ed ha e ce i igh ha e a c c e ( le ), b he e a be he a a d e f i f l a i a d ced e . A he ii he e a ical i e e a i f h ee al ed l gic. We ha e i dica ed a ce fd a ic a d h he ical a e i b e igh a ee e ha . A fi al

i i he i le e a i he c e . The g a ided he e a d iblh e he achi e a e a all e a d be e deci i b i

be ilde i g ld, i h he h ge challe ge f he ld la i g h a d hei al f ci ili a i a e k i , e eed e a d be e . The e a e, i he

d , ill ac ical ble a ell.

251



252



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A Logic of Exceptions