synonymy - people.ucalgary.carzach/static/banff/moschovakis.pdf(1) some comments about synonymy...

Synonymy

Yiannis N. MoschovakisUCLA and ÌÐËÁ

Mathematical methods in philosophyBIRS, 2 February, 2007

Frege on synonymy

Frege in a letter to Husserl (1906):

“It seems to me that we must have an objective criterionfor recognizing a thought as the same thought, sincewithout such a criterion a logical analysis is not possible.”

I Does Frege want a precise definition or a decision procedurefor synonymy?

I The question is in the context of Fregean semantics, whichare concerned with the logical analysis of language

A ≈ B ⇐⇒ A is synonymous with B⇐⇒ A and B have the same meaning (sense)

Yiannis N. Moschovakis: Synonymy 1/21

Outline

(1) Some comments about synonymy (briefly)

(2) Meaning and synonymy in the formal system Lλr (K )

(3) The decision problem for synonymy

References (in www.math.ucla.edu/∼ynm)I Sense and Denotation as Algorithm and Value, Logic

Colloquium ’90, Lectures Notes in Logic v. 2 (1994), pp.210–249. (With a correction posted on the URL above.)

I A logical calculus of meaning and synonymy, Linguistics andPhilosophy v. 29 (2006), pp. 27–89.

I (with E. Kalyvianaki) Two aspects of situated meaning, toappear.


Intuitions about synonymy

A & B?≈ B & A

Abelard loves Eloise?≈ Eloise is loved by Abelard (Frege: yes)

2 + 3 = 6?≈ 3 + 2 = 6 (+ a primitive)

Busch is the president?≈ He is the president (context: He = Busch)

Do we have robust intuitions about synonymy?

I A theory of synonymy should primarily explain the differencebetween propositions with the same truth value—e.g.,

2 + 2 = 4 and there are infinitely many prime numbers

I Three additional reasons to study synonymy


(a) To give an account of faithful translation

Frege:

“the same sense has different expressions in differentlanguages, or even in the same language”

“the difference between a translation and the originalshould properly [preserve sense]”

I Faithful translation = synonymy in the union of two languages

I Bilingual speakers generally agree on translation accuracy. . . to within reason

I Better source for intuitions about meaning than entailment:“If Hamlet were bald and 2 + 2 = 4, then Hamlet was bald”

. . . can be easily translated, but is the inference justified?


(b) To understand the logic of propositional attitudes

Othello believed that Desdemona and Cassio were lovers

(probably in Italian—and so he did not believe the Englishsentence but its meaning)

I The belief carriers are meanings—but what kind of meanings?

In a situation where he = Busch:

I believe that Busch is the President

?≈ I believe that he is the President

I Synonymy theory should help clarify what sort of meaningsare the belief carriers


(c) To interpret self-referential sentence

liar ≡ this sentence is false

truthteller ≡ this sentence is true

Notwithstanding any other provisions of this agreement, . . .

Article V. The Congress, whenever two thirds of both houses shalldeem it necessary, shall propose amendments to this Constitution,

. . .

Clearly liar 6≈ truthteller but why?


Outline




The typed λ-calculus with recursion Lλr (K ) is a common extensionof the Formal Language of Recursion FLR (1994) and the Typedλ-calculus with acyclic recursion Lλar (K ) (2006), which extendsMontague’s Language of Intensional Logic


The typed λ-calculus with recursion Lλr (K ) - types

Basic types b ≡ e | t | s (entities, truth values, states)

Types: σ :≡ b | (σ1 → σ2)

Abbreviation: σ1 × σ2 → τ ≡ (σ1 → (σ2 → τ))

Every non-basic type is uniquely of the form

σ ≡ σ1 × · · · × σn → b

level(b) = 0level(σ1 × · · · × σn → b) = max{level(σ1), . . . , level(σn)}+ 1


Lλr (K ) - syntax

Pure Variables: vσ0 , vσ1 , . . . , for each type σ (v : σ)

Recursive variables: pσ0 , pσ1 , . . . , for each type σ (p : σ)

Constants: A set K , and for each c ∈ K , c : σ1 × · · · × σn → b

Terms – with assumed type restrictions and assigned types (A : σ)

A :≡ v | p | c(A1, . . . ,An) | B(C ) | λ(v)(B)| A0 where {p1 := A1, . . . , pn := An}

C : σ,B : (σ → τ) =⇒ B(C ) : τv : σ,B : τ =⇒ λ(v)(B) : (σ → τ)

A0 : σ =⇒ A0 where {p1 := A1, . . . , pn := An} : σ

Abbreviation: A(B,C ,D) ≡ A(B)(C )(D)Yiannis N. Moschovakis: Synonymy 9/21

Rendering natural language in Lλr (K )

Abelard loves Eloiserender−−−→ loves(Abelard,Eloise) : t̃

Busch is the presidentrender−−−→ eq(Busch,the(president)) : t̃

liarrender−−−→ p where {p := ¬p} : t

truthtellerrender−−−→ p where {p := p} : t

t̃ ≡ (s → t) (type of Carnap intensions)ẽ ≡ (s → e) (type of individual concepts)

Abelard,Eloise,Busch : ẽ

president : ẽ → t̃, eq : ẽ × ẽ → t̃¬ : t̃ → t̃, the : (ẽ → t̃) → ẽ


Rendering natural language in Lλr (K )

John stumbled and fellrender−−−→ λ(x)

(stumbled(x) & fell(x)

)(John)

(predication after coordination)

This is in Montague’s LIL, the language of Intensional logic(as it is interpreted in Lλr (K ))

John stumbled and he fellrender−−−→ stumbled(j) & fell(j) where {j := John}

(conjunction after co-indexing)

The logical form of this sentence cannot be captured faithfully inLIL — recursion models co-indexing preserving logical form


Lλr (K ) - denotational semantics

• We are given basic sets Te , Tt , Ts for the basic types

Tσ→τ = the set of all functions f : Tσ → TτPb = Tb ∪ {⊥} = the “flat poset” of Tb

Pσ→τ = the set of all functions f : Tσ → PτEach Pσ is a complete poset (with the pointwise ordering)• We are given a monotone c̄ : Pσ1 × · · · × Pσn → Pb

for each constant c : σ1 × · · · × σn → bI Pure variables of type σ vary over Tσ; recursive ones over PσI If A : σ and π is a type-respecting assignment to the variables,

then den(A)(π) ∈ Pσ.I Recursive terms are interpreted by the taking of

least-fixed-points den(liar) = den(truthteller) = ⊥Yiannis N. Moschovakis: Synonymy 12/21

Meaning in Lλr (K )

I In slogan form: The meaning of a term A is faithfully modeledby an algorithm int(A) which computes den(A)(π) for everyassignment π

I The referential intension int(A) is (compositionally)determined from A

I int(A) is an abstract (not necessarily implementable) recursivealgorithm which can be defined in Lλr (K )

I Referential synonymy: A ≈ B ⇐⇒ int(A) ∼ int(A)(where ∼ is a natural isomorphism relation between abstract,recursive algorithms)

I Claim: Meanings are faithfully modeled

I Claim: Synonymy is captured (defined)


Is this notion of meaning Fregean?

Evans (in a discussion of Dummett’s similar, computationalinterpretations of Frege’s sense):

“This leads [Dummett] to think generally that the senseof an expression is (not a way of thinking about its[denotation], but) a method or procedure for determiningits denotation. So someone who grasps the sense of asentence will be possessed of some method fordetermining the sentence’s truth value. . . ideal verificationism. . . there is scant evidence for attributing it to Frege”

Converse question: If you posses a method for determining thetruth value of a sentence A, do you then “grasp” the sense of A?

(Sounds more like Davidson rather than Frege)Yiannis N. Moschovakis: Synonymy 14/21

Reduction, Canonical Forms and the Synonymy Theorem

I A reduction relation A ⇒ B is defined on terms of Lλr (K )I Each term A is reducible to a unique (up to congruence)

irreducible recursive term, its canonical form

A ⇒ cf(A) ≡ A0 where {p1 := A1, . . . , pn := An}

I int(A) = (den(A0), den(A1), . . . , den(An))

I The parts A0, . . . ,An of A are irreducible, explicit terms

I cf(A) models the logical form of A

I Synonymy Theorem. A ≈ B if and only if

B ⇒ cf(B) ≡ B0 where {p1 := B1, . . . , pm := Bm}

so that n = m and for i ≤ n, den(Ai ) = den(Bi )


Some of the examples

A & B ≈ B & A

Abelard loves Eloise ≈ Eloise is loved by Abelard

loves(a, e) where {a := Abelard,e := Eloise}

≈ is loved by (e, a) where {e := Eloise, a := Abelard}

2 + 3 = 6 ≈ 3 + 2 = 6

eq(s, r) where { s := t + t ′ , t := 2,t ′ := 3, r := 6}

eq(s, r) where { s := t ′ + t ,t ′ := 3, t := 2, r := 6}Busch is the president 6≈ He is the president

(at state a, where He = Busch)

eq(b, p)(ā) where { b := Busch ,p := the(President)}

6≈ eq(b, p)(ā) where { b := He , p := the(President)}Yiannis N. Moschovakis: Synonymy 16/21

Outline





Is referential synonymy decidable?

Synonymy Theorem. A ≈ B if and only if

A ⇒cf(A) ≡ A0 where {p1 := A1, . . . , pn := An}B ⇒cf(B) ≡ B0 where {p1 := B1, . . . , pn := Bn}

so that for i = 0, . . . , n and all π, den(Ai )(π) = den(Bi )(π).

I Synonymy is reduced to denotational equality forexplicit, irreducible terms

I Denotational equality for arbitrary terms is undecidable(there are constants, with fixed but arbitrary interpretations)

I The explicit, irreducible terms are very special— but by no means trivial!


The synonymy problem for Lλr (K ) (with finite K )

I The decision problem for Lλr (K )-synonymy is open

Theorem If the set of constants K is finite, then synonymy isdecidable for terms of adjusted level ≤ 2

These terms include the renditions in Lλr (K ) of most (perhaps all)sentences of natural language—those involving nouns, adjectives,verbs, adverbs, logical and modal operators, descriptions, etc. Theyalso include all sentences about mathematical structures which areincluded among the basic types

Proof is by reducing this claim to the Main Theorem in the 1994paper

(There was a gap in the 1994 proof; a correct version is posted inwww.math.ucla.edu/∼ynm)


Explicit, irreducible identities that must be known

I Los Angeles = LA (Athens = ÁèÞíá)

I x & y = y & x

I between(x , y , z) = between(x , z , y)

I love(x , y) = be loved(y , x)

A dictionary is needed—but what kind and how large?

ev2(λ(u1, u2)r(u1, u2,~a), b, z) = ev1(λ(v)r(v , z ,~a), b)

Evaluation functions: both sides are equal to r(b, z ,~a)

The dictionary line which determines this is (essentially)

λ(s)x(s, z) = λ(s)y(s) =⇒ ev2(x , b, z) = ev1(y , b)


The form of the decision algorithm

I A finite list of true dictionary lines is constructed, whichcodifies the relationships between the constants

I Given two explicit, irreducible terms A,B of adjusted level≤ 2, we construct (effectively) a finite set L(A,B) of lines

I A = B if and only if every line in L(A,B) is congruent to onein the dictionary

I It is a lookup algorithm, justified by a finite basis theorem


synonymy - people.ucalgary.carzach/static/banff/moschovakis.pdf(1) some comments about synonymy...

Documents