Transparent intensional logic, -r ule and Compositionality

Transparent intensional logic, -rule and Compositionality

Marie DužíVSB-Technical University Ostravahttp://www.cs.vsb.cz/duzi

http://www.cs.vsb.cz/duzi

Rules of substition 2

Attitude Logic(s) A reliable test on Compositionality Attitudes:

Notional Propositional

We are dealing with a fine difference between the meanings of sentences like

(P1) Charles believes that the Pope is in danger(P2) Charles believes of the Pope that he is in

danger Some authors even claim that (P1) is ambiguous, that

it can be also read as (P2).


Attitude logics and belief sentencesIn our opinion it is not so. We can, for instance,

reasonably say (it may be true) thatCharles believes of the Pope that he is not the

Pope,whereas the sentence

Charles believes that the Pope is not the Popecannot be true, unless our Charles is completely irrational. The sentences like (P1) and (P2) have different meanings, and their difference consists in using ‘the Pope’ in the de dicto supposition (P1) vs. the de re supposition (P2).

The two sentences are neither equivalent, nor is any of them entailed by the other.


Belief sentences in doxastic logics In the usual notation of doxastic logics the distinction

is characterised as the contrast between BCharles D[p] (de dicto) (x) (x = p BCharles D[x] (de re) But there are worrisome questions (Hintikka, Sandu

1989):Where does the existential quantifier come from in the de

re case? There is no trace of it in the original sentence. How can the two similar sentences be as different in their

logical form as they are? Hintikka, Sandu propose in their (1996) a remedy by

means of the Independence Friendly (IF) first-order logic:


Belief sentences in doxastic logics “Independence Friendly (IF) first-order logic deals with

a frequent and important feature of natural language semantics. Without the notion of independence, we cannot fully understand the logic of such concepts as belief, knowledge, questions and answers, or the de dicto vs. de re contrast.”

Hintikka, Sandu (1989): Informational Independence as a Semantical Phenomenon. In J.E. Fenstad et el (eds.), Logic, Methodology and Philosophy of Science, Elsevier, Amsterodam 1989, pp. 571-589.

Hintikka, Sandu (1996): A revolution in Logic? Nordic Journal of Philosophical Logic, Vol.1, No.2, pp. 169-183.


Belief sentences and IF semantics Hinttika, Sandu solve the de dicto case as above, and propose the

de re solution with the independence indicator ‘/’: BCharles D[p / BCharles]

This is certainly a more plausible analysis, closer to the syntactic form of the original sentence, and the independence indicator indicates the essence of the matter:

There are two independent questions: ”Who is the pope” and ”What does Charles think of that person”. Of course, Charles has to have a relation of an ”epistemic intimity”

to a certain individual, but he does not have to connect this person with the office of the Pope (only the ascriber must do so).

(Chisholm,R.(1976): Knowledge and Belief: ‘De dicto’ and ‘de re’. Philosophical Studies 29 (1976), 1-20. )


Belief sentences and Intensional logics

BCharles D(p) (de dicto)x BCharles D(x)(p) (de re)

But: x BCharles D(x)(p) BCharles D(p) !(applying the rule of -reduction).

What then is the difference between de dicto and de re?

Why is it “forbidden” here to perform the fundamental rule of -calculi?


Solomon Feferman (1995): Logic of DefinednessIntroduces the axioms (λp) for Partial Lambda Calculus

as follows (t↓ means - the term t is defined):i. λx.t ↓ ii. (λx.t(x))y t(y). The axiom (ii) corresponds to the trivial β-reduction,

but the limitation on instantiation in PLC restricts its application to:

s↓ (λx.t(x))s t(s). (but why this restriction?, proof?)

Our system (TIL) introduces a generally valid β-reduction for the Partial Higher-Order Hyper-intensional Lambda Calculus.

04/22/23 TIL & beta-rule 9

Transparent Intensional LogicFormally:

The language of TIL constructions can be viewed as a hyper-intensional -calculus operating over partial functions.

“hyper-intensional”: -terms are not interpreted as set-theoretical mappings (”modern functions”) but as algorithmically structured procedures (which produce as an output the (partial) mapping).

Procedures, known as TIL constructions, are objects sui generis: they can be not only used but also mentioned within a theory.

04/22/23 TIL & beta-rule 10

Suppositio (substitution) A lot of misunderstanding and many

paradoxes arise from confusing different ways in which a meaningful expression can be used.

We are going to show that these different ways consist in using and mentioning entities (by means of an expression)

In which way can an entity be used or mentioned?

04/22/23 Use / Mention 11

Using / Mentioning EntitiesExpression

used mentioned to express its meaning:

procedure (‘TIL construction’)

de dicto / de re mentioned used to produce a function:

mentioned used to point at …


TIL constructions Abstract procedures, structured from the algorithmic

point of view. structured meanings: Instructions specifying how to arrive

at less-structured entities. Being abstract, they are reachable only via a verbal

definition. The ‘language of constructions’: a modified version of the

typed -calculus, where Montague-like -terms denote, not the functions constructed, but the constructions themselves.

Henk Barendregt (1997): -terms denote functions, yet “... in this interpretation the notion of a function is taken to be (hyper-)intensional, i.e., as an algorithm.”

Operate on input objects (of any type, even constructions) and yield as output objects of any type: they realize functions (mappings)

04/22/23 Constructions 13

Kinds of constructions1. Atomic: do not contain as a used constituent any

other construction but themselves (supply objects …) Variables x, y, p, c, … v-constructing Trivialisation of X: 0X

2. Compound. Composition [X X1…Xn]: the instruction to apply a

(partial) function f (constructed by X) to an argument A (constructed by X1,…,Xn) to obtain the value (if any) of f at A.

-Closure [x1…xn X]: the instruction to abstract over variables in order to obtain a function.

Double execution 2X: the instruction to use a higher-order construction X twice over as a constituent.


TIL Ramified Hierarchy of Types

The formal ontology of TIL is bi-dimensional.

One dimension is made up of constructions.

The other dimension encompasses non-constructions, i.e., partial functions mapping (the Cartesian product of) types to types.


TIL Ramified Hierarchy of Types

1st-order: non-constructionsBase: , , , , partial functions ((())), (()), …, (01…n)2nd-order: Base: *1 constructions of 1st-order entities, partial functions involving such constructions: (01…n), i = *1

3rd-order: Base: *2, constructions of 2nd-order entities, partial functions involving such constructions: (01…n), i = *2, or *1

And so on, ad infinitum


-intensions; examples Functions of type () Usually both modal and temporal

parameters: (()) Abbreviation:

Propositions /

(individual) offices / Magnitudes /

Empirical functions (attributes)/()

Attitudes / (n)


Definition Used* vs. Mentioned*Let C be a construction and D a sub-construction of C. Then an

occurrence of D is used* as a constituent of C iff: If D is identical to C (i.e., 0C = 0D) then the occurrence of D is

used* as a constituent of C. If C is identical to [X1 X2…Xm] and D is identical to one of the

constructions X1, X2,…,Xm, then the occurrence of D is used* as a constituent of C.

If C is identical to [x1…xmX] and D is identical to X, then the occurrence of D is used* as a constituent of C.

If C is identical to 1X or 2X and D is identical to X, then the occurrence of D is used* as a constituent of C.

If C is identical to 2X and X v-constructs a construction Y and D is identical to Y, then the occurrence of D is used* as a constituent of C.

If an occurrence of D is used* as a constituent of an occurrence of C’ and this occurrence of C’ is used* as a constituent of C, then the occurrence of D is used * as a constituent of C.

If an occurrence of a sub-construction D of C is not used* as a constituent of C, then the occurrence of D is mentioned* in C.


Definition Used* vs. Mentioned*

Let C be a construction and D a sub-construction of C.

Then an occurrence of D is mentioned* in C iff it is not necessary to execute D in order to execute C;

Otherwise D is used* as a constituent of C.

Makes a fine individuation possible; finer than just an equivalence.


Two kinds of using a construction:de dicto vs. de re supposition.

Roughly: C = [… D … ], D () 1. D occurs in C with de dicto supposition iff D is

not composed with a construction A ; the respective function / () is just mentioned

2. D occurs in C with de re supposition iff D is composed with a construction A , and D does not occur as a constituent of a de dicto occurrence D’ (de dicto context is dominant); the respective function / () is used as a pointer

to its actual, current value /


Contextssuppositio substitution

The President of USA knows that John Kerry wanted to become the President of USA.

The President of USA is (=) the husband of Laura Bush.

Hence what ?Did John Kerry want to become the husband of

Laura Bush?

04/22/23 21

Contextssuppositio substitution

C1 wt [0= [wt [0Preswt 0USA]]wt [0Husbandwt 0Bush]]

extensional context: of using* de re C2 wt [0Wwt 0K [wt [0Bwt 0K wt [0Pres 0USA]]] ]

intensional context: of using* de dicto

C3 wt [0Knowwt [wt [0Preswt 0USA]]wt 0[wt [0Wwt 0K wt [0Bwt 0K wt [0Preswt 0USA]]]] ]

hyper-intensional context: of mentioning*


Using / Mentioning Constructions

Dividing six by three gives two and dividing six by zero is improper.

Types: 0, 2, 3, 6 / , Div / (), Improper / (1)the class of v-improper constructions for all v

[[[0Div 06 03] = 02] [0Improper 0[0Div 06 00]]]

used* constituents mentioned*



There is a number such that dividing any number by it is improper.

Types: Div / (), Improper / (1), ,/(()), x, y .Exists x for all y [0Improper 0[0Div x y]].But x, y occur in the hyper-intensional context of mention*;

they are not free for evaluation or substitution. How to quantify? To this end we use functions Sub and Tr:Sub / (1111)the mapping which takes a construction C1,

variable x, and a construction C2 to the resulting construction C3, where C3 is the result of substituting C1 for x in C2.

Tr / (1)the mapping which takes a number and returns its trivialisation



(*) [0y [0x [0Improper [0Sub [0Tr y] 0y’ [0Sub [0Tr x] 0x’ 0[0Div x’ y’ ]]]]]].

Let a valuation v assign 0 to y and 6 to x. Then the sub-construction [0Sub [0Tr y] 0y’ [0Sub [0Tr x] 0x’ 0[0Div x’ y’ ]]]

v-constructs the construction [0Div 06 00], which belongs to the class Improper. This is true for any valuation v’ that differs from v at most by assigning another number to x.

The construction (*) constructs True.


De dicto / de re supposition The temperature in Amsterdam equals the temperature in

Prague. The temperature in Amsterdam is increasing.

--------------------------------------------------------- The temperature in Prague is increasing.

Types: Temp(erature in …)/(), Amster(dam), Prague/, Increas(ing)/().

wt [wt [0Tempwt 0Amster]wt = (de re) wt [0Tempwt 0Prague]wt]

wt [0Increaswt [wt [0Tempwt 0Amst]] the magnitude is (de dicto) mentioned.

27

Rules of Substitution (logic of partial functions !)

“Homogeneous” substitution: no problemLebniz’s law Used* de re extensional context de re Used* de dicto intensional context de dicto Mentioned* construction hyperintensional context

Used* constructions – constituents: De re (extensional) context: [Cx] = [C’y]

co-incidental constructions substitutable De dicto (intensional context): C = C’

equivalent constructions substitutable Mentioned* (hyper-intensional) context: 0C = 0C’

Only identical constructions substitutable

28

Rules of Substitution (logic of partial functions !)

Heterogeneous substitutions. Construction of a lower-order into a higher-

order context (which is dominant): We must not carelessly draw a construction

D occurring in a lower-order context into a higher-order context.

Why not? The substitution would not be correct even if there is no collision of variables, due to partiality


De re rules

The president of CR is (is not) an economist. de reThe president of CR exists.

The president of CR is eligible. de dictoThe president of CR may not exist.

In the de re case there is an existential presupposition, unlike the de dicto case.


Charles believes of the president of CR that he is an economist.

Types: Ch/, B/(), Pr(esident of …)/(), CR/, Ec/()

Synthesis (h , a free variablethe meaning of “he”):He is an economist: wt [0Ecwt h] v (anaphora)The President of CR: wt [0Prwt

0CR]

a) The President of CR is believed by Charles to be an economist – the passive variantwt [h [0Bwt 0Ch wt [0Ecwt h]] wt [0Prwt

0CR]wt ] Now, can we perform -reduction ???

Yes, but only the trivial one: wt [0Prwt

0CR]wt | [0Prwt 0CR]

Collision of variables? Let us rename them:



-reduction “by name” :wt [h [0Bwt 0Ch w’t’ [0Ecw’t’ h]] [0Prwt

0CR] ] | ??? wt [0Bwt 0Ch w’t’ [0Ecw’t’ [0Prwt

0CR]]] No collision of variables,

But. [h [0Bwt 0Ch w’t’ [0Ecw’t’ h]] [0Prwt 0CR] ]

[0Existwt wt [0Prwt

0CR]] = [0x [x = [0Prwt

0CR]] Unlike the latter.

Therefore, don’t perform -reduction (!?!)



b) The direct analysis of the active form, using Tr and Sub.

-reduction “by value”:Now we have to substitute for h the construction

of the individual (if any) that actually plays the role of the president:

) wt [0Belivewt 0Charles 2[0Sub [0Tr wt [0Prwt 0CR]wt] 0h (extens.) 0[wt [0Ecwt h]]]] (intens.)


2-phase -reduction: how does it work?

wt [0Belwt 0Ch 2[0Sub [0Tr wt [0Prwt 0CR]wt] 0h 0[wt [0Ecwt h]]]]

1. Let wt [0Prwt 0CR]wt be v-improper (the president does not exist).

Then [0Tr wt [0Prwt 0CR]wt] is v-improper and The function Sub does not have an argument to operate on: [0Sub [0Tr wt [0Prwt 0CR]wt] 0h 0[wt [0Ecwt h]]]

v-improper. (And so is the Double execution.) The so-constructed proposition does not have a truth-value, as

it should be (the existential presupposition)


Substitution by value (-reduction)

wt [0Belwt 0Ch 2[0Sub [0Tr wt [0Prwt 0CR]wt] 0h 0[wt [0Ecwt h]]]]

2. Let wt [0Prwt 0CR]wt be v-proper (the president exist). Then the construction [0Prwt 0CR] v-constructs particular individual

Y (For instance V. Klaus.) Then [0Tr wt [0Prwt 0CR]wt] v-constructs 0Y, and Sub inserts it for

the variable h. the result is the construction: [wt [0Ecwt 0Y]] that is

executed (Double execution) in order to construct the proposition that is believed by Charles.

36

Substitution by value (-reduction)

Type checking: 2[Sub [0Tr [0Prwt 0CR]] 0h 0[wt [0Ecwt h]]]

(*1 ) (*1*1*1*1) *1 *1 *1

*1 ( ) 1. step

2. step (if the 1st did not fail):

1[wt [0Ecwt 0Y]]

wt [0Belwt 0Ch 20[wt [0Ecwt 0Y]]


-reduction, another example(*) [y [0Deg z [0: z y]] 0x ] ( = square root) (Deg/(())-a degenerated function)(*n) -reduced “syntactically-by-name”: [0Deg z [0: z 0x]] [[0Exist x] 0] ??? NO(*v) -reduced “by value”:

2[0Sub [0Tr 0x] 0y 0[0Deg z [0: z y]]]

for: value of (*) of (*n) of (*v)

x > 0 False False False

x = 0 True True True

x < 0 Undefined True Undefined


Valid rule of -reduction (2-phase)

Let C(y) be a construction with a free variable y, y , and let D . Then

[[y C(y)] D] 2[0Sub [0Tr D] 0y 0C(y)]

is a valid rule (proof, see above).


Rules of inference:Types: y β, x , D / (β), [Dx] β, C(y) α, y C(y) (αβ), [[y C(y)] [Dx]] α.

Compositionality:[0Improper 0[Dx]] | [0Improper 0[[y C(y)] [Dx]]][0Improper 0[Dx]] | [0Improper 02[0Sub [0Tr [Dx]] 0y 0C(y)]] [0Proper 0[Dx]] | 2[0Sub [0Tr [Dx]] 0y 0C(y)] = [[y C(y)] [Dx]] =

C(y/[Dx]) Special case: Existential presupposition de reExist / (( (β)) )the property of a (β)-function of being defined at a

-argument, [Exist x] ( (β))

[[0Exist x] D] | [0Improper 0[[y C(y)] [Dx]]][[0Exist x] D] | [0Improper 02[0Sub [0Tr [Dx]] 0y 0C(y)]]

But not: C(y/[Dx]) | [[0Exist x] D] …


The two “de re principles”: a) existential presupposition

Example: [y [0Deg z [0: z y]] [0x]] | [[0Exist x] 0] 2[0Sub [0Tr [0x]] 0y 0[0Deg z [0: z y]]] | [[0Exist x] 0]

Indeed: The square root does not exist for x < 0; for x < 0 the left-hand side is (v-)improper. If the left-hand side is true or false, then the square root exists and x 0.

However, the result of the “syntactical” β-reduction does not meet these rules:

[0Deg z [0: z 0x]] and not (for x < 0) [[0Exist x] 0 ].


The two “de re principles”: b) inter-substitutivity of co-incidentals

[Dx] = [D’ ]

[[y C(y)] [Dx]] = [[y C(y)] [D’ ]] =2[0Sub [0Tr [D’ ]] 0y 0C(y)]

Example:The US President is the husband of Laura.The US President is a Republican. Hence: The husband of Laura is a Republican.But not: John Kerry wanted to become the

husband of Laura.


Substitutions in general

Types: c n, 2c , A , y

a) “by name” (homogeneous substitution):2[0Sub 00A 0c 0C(c)] = C(c/0A)2[0Sub 0A 0y 0C(y)] = C(y/A)

b) “by value” (generally valid, even for heterogeneous substitution): 2[0Sub [0Tr A] 0y 0C(y)] = [y [C(y)] A]

C(y/A)


Conclusions The top-down, fine-grained approach of TIL makes it

possible to adequately model structured meanings, and thus: to formulate meaning-driven (non ad hoc) rules of

substitution taking into account the Use/Mention distinction at all levels;

to adhere to Compositionality and anti-contextualism (even in the cases of anaphora, de re attitudes with anaphoric reference, hyper-intensional attitudes, …);

to take into account partiality; to meet the two de re extensional principles (existential

presupposition, inter-substitutivity of co-referentials).

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Transparent intensional logic, -r ule and Compositionality