02—Traditional Logichenz/cs3234/slides/slides_02.bw.pdf · Categorical Terms and their Meaning Propositions, Axioms, Lemmas, Proofs Manipulating Terms and Propositions Arguments

Categorical Terms and their MeaningPropositions, Axioms, Lemmas, Proofs

Manipulating Terms and PropositionsArguments and Syllogisms

02—Traditional Logic

CS 3234: Logic and Formal Systems

Martin Henz and Aquinas Hobor

August 19, 2010

Generated on Wednesday 18th August, 2010, 23:10

CS 3234: Logic and Formal Systems 02—Traditional Logic



1 Categorical Terms and their Meaning

2 Propositions, Axioms, Lemmas, Proofs

3 Manipulating Terms and Propositions

4 Arguments and Syllogisms




Origins and GoalsForm, not ContentCategorical TermsMeaning through models

1 Categorical Terms and their MeaningOrigins and GoalsForm, not ContentCategorical TermsMeaning through models








Traditional Logic

Origins

Greek philosopher Aristotle (384–322 BCE) wrote treatise PriorAnalytics; considered the earliest study in formal logic; widelyaccepted as the definite approach to deductive reasoning untilthe 19thcentury.

GoalExpress relationships between sets; allow reasoning about setmembership





Example 1

All humans are mortal.All Greeks are humans.Therefore, all Greeks are mortal.

Makes “sense”, right?

Why?





Example 2

All cats are predators.Some animals are cats.Therefore, all animals are predators.

Does not make sense!

Why not?





Example 3

All slack track systems are caterpillar systems.All Christie suspension systems are slack tracksystems.Therefore, all Christie suspension systems arecaterpillar systems.

Makes sense, even if you do not know anything aboutsuspension systems.

Form, not contentIn logic, we are interested in the form of valid arguments,irrespective of any particular domain of discourse.





Categorical Terms

Terms refer to setsTerm animals refers to the set of animals,term brave refers to the set of brave persons, etc

Terms

The set Terms contains all terms under consideration

Examples

animals ∈ Terms

brave ∈ Terms





Models

MeaningA modelM fixes what elements we are interested in, and whatwe mean by each term

Fix universe

For a particularM, the universe UM contains all elements thatwe are interested in.

Meaning of termsFor a particularM and a particular term t , the meaning of t inM, denoted tM, is a particular subset of UM.





Example 1A

For our examples, we haveTerm = {cats,humans,Greeks, . . .}.

First meaningMUM: the set of all living beings,catM the set of all cats,humansM the set of all humans,. . .





Example 1B

Consider the same Term = {cats,humans,Greeks, . . .}.

Second meaningM′

UM: A set of 100 playing cards, depicting living beings,catM: all cards that show cats,humansM: all cards that show humans,. . .





Example 2A

Consider the following set of terms:Term = {even,odd,belowfour}

First meaningM1

UM1 = N,evenM1 = {0,2,4, . . .},oddM1 = {1,3,5, . . .}, andbelowfourM1 = {0,1,2,3}.





Example 2B

Consider the same Term = {even,odd,belowfour}

Second meaningM2

UM2 = {a,b, c, . . . , z},evenM2 = {a,e, i ,o,u},oddM2 = {b, c,d , . . .}, andbelowfourM2 = ∅.




Categorical PropositionsSemantics of PropositionsAxioms, Lemmas and Proofs


2 Propositions, Axioms, Lemmas, ProofsCategorical PropositionsSemantics of PropositionsAxioms, Lemmas and Proofs







Categorical Propositions

All cats are predators

expresses a relationship between the terms cats (subject) andpredators (object).

Intended meaningEvery thing that is included in the class represented by cats isalso included in the class represented by predators.





Four Kinds of Categorical Propositions

Quantityuniversal particular

Qualityaffirmative All t1 are t2 Some t1 are t2negative No t1 are t2 Some t1 are not t2

ExampleSome cats are not brave is a particular, negativeproposition.





Meaning of Universal Affirmative Propositions

In a particular modelM, All Greeks are mortal meansthat GreeksM is a subset of mortalM





More formally...

(All subject are object)M =

{T if subjectM ⊆ objectM,F otherwise

Here T and F represent the logical truth values true and false,respectively.





Meaning of Universal Negative Propositions

In a particular modelM, No Greeks are cats means thatthe intersection of GreeksM and of catsM is empty.





More formally...

(No subject are object)M =

{T if subjectM ∩ objectM = ∅,F otherwise





Meaning of Particular Affirmative Propositions

In a particular modelM, Some humans are Greeks meansthat the intersection of humansM and of GreeksM is notempty.





More formally...

(Some subject are object)M =

{T if subjectM ∩ objectM 6= ∅,F otherwise





Meaning of Particular Negative Propositions

In a particular modelM, Some Greeks are notvegetarians means that the difference of GreeksM andvegetariansM is not empty.





More formally...

(Some subject are not object)M =

{T if subjectM/objectM 6= ∅,F otherwise





Axioms

Axioms are propositions that are assumed to hold.

Axiom (HM)

The proposition All humans are mortal holds.

Axiom (GH)The proposition All Greeks are humans holds.





Graphical Notation

All humans are mortal

[HumansMortality]





Lemmas

Lemmas are affirmations that follow from all known facts.

Proof obligationA lemma must be followed by a proof that demonstrates how itfollows from known facts.





Trivial Example of Proof

LemmaThe proposition All humans are mortal holds.

Proof.

All humans are mortal

[HM]





Unusual Models

We can choose any model for our terms, also “unusual” ones.

Example

UM = {0,1}, humansM = {0}, mortalM = {1}

HereAll humans are mortal

does not hold.





Asserting Axioms

Purpose of axiomsBy asserting an axiom A, we are focusing our attention to onlythose modelsM for which AM = T .

ConsequenceThe lemmas that we prove while utilizing an axiom only hold inthe models in which the axiom holds.

ValidityA proposition is called valid, if it holds in all models.




ComplementConversionContrapositionObversionCombinations



3 Manipulating Terms and PropositionsComplementConversionContrapositionObversionCombinations






Complement

We allow ourselves to put non in front of a term.

Meaning of complementIn a modelM, the meaning of non t is the complement of themeaning of t

More formally

In a modelM, (non t)M = UM/tM





Double Complement

Axiom (NonNon)For any term t, the term non non t is considered equal to t.

· · · t · · ·

· · ·non non t · · ·[NNI]

· · ·non non t · · ·

· · · t · · ·[NNE]





Rule Schema

· · · t · · ·

· · ·non non t · · ·[NNI]

is a rule schema. An instance is:

Some t1 are t2

Some non non t1 are t2





Definitions

We allow ourselves to state definitions that may be convenient.Definitions are similar to axioms; they fix the properties of aparticular item for the purpose of a discussion.

Definition (ImmDef)The term immortal is considered equal to the term nonmortal.





Writing a Proof Graphically

LemmaThe proposition All humans are non immortal holds.

Proof.

All humans are mortal[HM]

All humans are non non mortal[NNI]

All humans are non immortal[ImmDef]





Writing a Text-based Proof

LemmaThe proposition All humans are non immortal holds.

Proof.

1 All humans are mortal HM2 All humans are non non

mortalNNI 1

3 All humans are non immortal ImmDef 2





Conversion switches subject and object

Definition (ConvDef)For all terms t1 and t2, we define

convert(All t1 are t2) = All t2 are t1convert(Some t1 are t2) = Some t2 are t1

convert(No t1 are t2) = No t2 are t1convert(Some t1 are not t2) = Some t2 are not t1





Which Conversions Hold?

IfAll Greeks are humans

holds in a model, then does

All humans are Greeks

hold?





Valid Conversions

Axiom (ConvE1)If, for some terms t1 and t2, the proposition

convert(Some t1 are t2)

holds, then the proposition

Some t1 are t2

also holds.





Valid Conversions

Axiom (ConvE2)

If, for some terms t1 and t2, the proposition

convert(No t1 are t2)

holds, then the proposition

No t1 are t2

also holds.





In Graphical Notation

In graphical notation, two rules correspond to the two cases.

convert(Some t1 are t2)

Some t1 are t2

[ConvE1]

convert(No t1 are t2)

No t1 are t2

[ConvE2]





Example

Axiom (AC)The proposition Some animals are cats holds.

LemmaThe proposition Some cats are animals holds.





Proof

Some animals are cats[AC]

convert(Some cats are animals)

[ConvDef]

Some cats are animals[ConvE1]





Example (text-based proof)

Proof.

1 Some animals are cats AC2 convert(Some cats are

animals)ConvDef 1

3 Some cats are animals ConvE1 2





Contraposition switches and complements

Definition (ContrDef)For all terms t1 and t2, we define

contrapose(All t1 are t2)= All non t2 are non t1

contrapose(Some t1 are t2)= Some non t2 are non t1

contrapose(No t1 are t2)= No non t2 are non t1

contrapose(Some t1 are not t2)= Some non t2 are not non t1





For which propositions is contraposition valid?

contrapose(All t1 are t2)

All t1 are t2

[ContrE1]

contrapose(Some t1 are not t2)

Some t1 are not t2

[ContrE2]





Obversion switches quality and complements object

Definition (ObvDef)For all terms t1 and t2, we define

obvert(All t1 are t2) = No t1 are non t2obvert(Some t1 are t2) = Some t1 are not non t2

obvert(No t1 are t2) = All t1 are non t2obvert(Some t1 are not t2) = Some t1 are non t2





Examples

Obversion switches quality and complements object

Example 1obvert(All Greeks are humans)= No Greeks are non humans

Example 2obvert(Some animals are cats)= Some animals are not non cats





Validity of Obversion

Obversion is valid for all kinds of propositions.

Axiom (ObvE)If, for some proposition p

obvert(p)

holds, then the proposition p also holds.

obvert(p)

p[ObvE]





Example

Axiom (SHV)The proposition Some humans are vegans holds.

Lemma (NNVeg)The proposition Some humans are not non vegans holds.





Proof

Some humans are vegans[SHV]

Some humans are non non vegans[NNI]

obvert(Some humans are not non vegans)[ObvDef]

Some humans are not non vegans[ObvE]





Proof (text-based)

Proof.

1 Some humans are vegans SHV2 Some humans are non non

vegansNNI 1

3 obvert(Some humans are notnon vegans)

ObvDef 2

4 Some humans are not nonvegans

ObvE 3





Another Lemma

Lemma (SomeNon)

For all terms t1 and t2, if the proposition Some non t1 are nont2 holds, then the proposition Some non t2 are not t1 alsoholds.

A lemma of the form “If p1 then p2” is valid, if in every model inwhich the proposition p1 holds, the proposition p2 also holds.





Proof

Lemma (SomeNon)

For all terms t1 and t2, if the proposition Some non t1 are nont2 holds, then the proposition Some non t2 are not t1 alsoholds.

Proof.

1 Some non t1 are non t2 premise2 convert(Some non t2 are non t1) ConvDef 13 Some non t2 are non t1 ConvE1 24 obvert(Some non t2 are not t1) ObvDef 35 Some non t2 are not t1 ObvE 4





“iff” means “if and only if”

Lemma (AllNonNon)For any terms t1 and t2, the proposition All non t1 are nont2 holds iff the proposition All t2 are t1 holds.

All non t1 are non t2

All t2 are t1

All t2 are t1

All non t1 are non t2




ArgumentsSyllogismsBarbaraFun With Barbara




4 Arguments and SyllogismsArgumentsSyllogismsBarbaraFun With Barbara





Argument

An argument has the form

If premises then conclusion

Sometimes also

premises therefore conclusion

Example:

Lemma (SomeNon)For all terms t1 and t2, if the proposition Some non t1 are nont2 holds, then the proposition Some non t2 are not t1 alsoholds.





Syllogisms

A syllogism is an argument with two premises, in which threedifferent terms occur, and in which every term occurs twice, butnever twice in the same proposition.

ExampleAll cats are predators.Some animals are cats.Therefore, all animals are predators.





Barbara

Axiom (B)For all terms minor, middle, and major, if All middle are majorholds, and All minor are middle holds, then All minor aremajor also holds.

All middle are major All minor are middle

All minor are major[B]





Why is Barbara valid?

If the first premise holds, then areas 1 and 4 are empty, and ifthe second premise holds, then areas 2 and 3 are empty. Theconclusion simply states that areas 1 and 2 are empty, whichclearly follows from the premises, regardless what otherproperties the model under consideration has.





Example

LemmaThe proposition All Greeks are mortal holds.

Proof.

1 All Greeks are humans GH2 All humans are mortal HM3 All Greeks are mortal B 1,2





Officers as Poultry?

PremisesNo ducks waltz.No officers ever decline to waltz.All my poultry are ducks.

ConclusionNo officers are my poultry.





Formulation in Term Logic

Lemma (No-Officers-Are-My-Poutry)If

No ducks are things-that-waltz holds,No officers are non things-that-waltz holds,andAll my-poutry are ducks holds,

then No officers are my-poultry also holds.





Proof

1 No officers are nonthings-that-waltz

premise

2 obvert(All officers arethings-that-waltz)

ObvDef 1

3 All officers arethings-that-waltz)

ObvE 2

4 No ducks arethings-that-waltz)

premise

5 convert(No things-that-waltzare ducks)

ConvDef 4

6 No things-that-waltz areducks

ConvE2 5





Proof (continued)

7 No things-that-waltz are nonnon ducks

NNI 6

8 obvert(All things-that-waltzare non ducks)

ObvDef 7

9 All things-that-waltz arenon ducks

ObvE 8

10 All my-poultry are ducks premise11 All my-poultry are non non

ducksNNI 10

12 All non non my-poultry arenon non ducks

NNI 11





Proof (continued)

13 contrapose(All non ducks arenon my-poultry)

ContrDef 12

14 All non ducks are nonmy-poultry

ContrE1 13

15 All things-that-waltz arenon my-poultry

B 9,14

16 All officers are nonmy-poultry

B 3,15

17 obvert(No officers aremy-poultry)

ObvDef 16

18 No officers are my-poultry ObvE 17





Admin

Assignment 1: out on module homepage; due 26/8,11:00amCoq Homework 1: out on module homepage; due 27/8,9:30pmMonday, Wednesday: Office hoursTuesday: Tutorials (clarification of assignment)Wednesday: Labs (Coq Homework 1; start earlier!)


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02—Traditional Logichenz/cs3234/slides/slides_02.bw.pdf · Categorical Terms and their Meaning Propositions, Axioms, Lemmas, Proofs Manipulating Terms and Propositions Arguments