Truth as alogical

connective

ShunsukeYatabe

IntroductionFS

PTS of Tr

A problemHARMONY

McGee’s Theorem

AnalysisInterpreting γ

PTS for inductiveformulae

Solution: TheHARMONYextendedPTS for coinductiveformulae

The failure of FC

Truth as a logical connective

Shunsuke Yatabe

Center for Applied Philosophy and Ethics,Graduate School of Letters,

Kyoto University

November 15, 2015Logic and Engineering of Natural Language Semantics 12

1 / 16

Truth as alogical

connective

ShunsukeYatabe

IntroductionFS

PTS of Tr

A problemHARMONY

McGee’s Theorem

AnalysisInterpreting γ

PTS for inductiveformulae

Solution: TheHARMONYextendedPTS for coinductiveformulae

The failure of FC

The pourpose of this talk

• I do not insist that “truth conception should be treated notas a predicate but as a logical connective”,

• I do want to analyse Friedman-Sheared’s truth theory FSfrom the behaviour of truth predicate,• it seems to be possible to regard the truth predicate Tr as

a logical connective in truth theories with

ϕ

Tr(ϕ) NEC Tr(ϕ)ϕ CONEC

• Since the condition of being a predicate is very flexible ifthe theory is ω-inconsistent, such proof theoreticsemantics viewpoint is effective to analyse it.

2 / 16

Truth as alogical

connective

ShunsukeYatabe

IntroductionFS

PTS of Tr

A problemHARMONY

McGee’s Theorem

AnalysisInterpreting γ

PTS for inductiveformulae

Solution: TheHARMONYextendedPTS for coinductiveformulae

The failure of FC

Friedman-Sheared’s Axiomatic truth theory FS

FS = PA + the formal commutability of Tr+ the introduction and the elimination rule of Tr

• Axioms and schemes of PA (mathematical induction for allformulae including Tr)

• The formal commutability of the truth predicate:• logical connectives

• for amy atomic formula ψ, Tr(dψe) ≡ ψ,• (∀x ∈ Form)[Tr(¬̇x) ≡ ¬Tr(x)],• (∀x, y ∈ Form)[Tr(x∧̇y) ≡ Tr(x) ∧ Tr(y)],• (∀x, y ∈ Form)[Tr(x∨̇y) ≡ ¬Tr(x) ∨ Tr(y)],• (∀x, y ∈ Form)[Tr(x→̇y) ≡ ¬Tr(x) → Tr(y)],

• quantifiers• (∀x ∈ Form)[Tr(∀̇z x(z)) ≡ ∀zTr(x(z))],• (∀x ∈ Form)[Tr(∃̇z x(z)) ≡ ∃zTr(x(z))],

• The introduction rule and the elimination rule of Tr(x)ϕ

Tr(dϕe) NEC Tr(dϕe)ϕ CONEC

where Form is a set of Godel code of formulae.3 / 16

Truth as alogical

connective

ShunsukeYatabe

IntroductionFS

PTS of Tr

A problemHARMONY

McGee’s Theorem

AnalysisInterpreting γ

PTS for inductiveformulae

Solution: TheHARMONYextendedPTS for coinductiveformulae

The failure of FC

A proof theoretic semantics (PTS) of Tr (naive version)

FS’s two rules look like the introduction and the eliminationrule; Tr looks like a logical connective [Hj12]

ϕ

Tr(ϕ) NEC Tr(ϕ)ϕ CONEC

• A (naive) proof theoretic semantics: the meaning of alogical connective is given by the introduction ruleand the elimination rule

4 / 16

Truth as alogical

connective

ShunsukeYatabe

IntroductionFS

PTS of Tr

A problemHARMONY

McGee’s Theorem

AnalysisInterpreting γ

PTS for inductiveformulae

Solution: TheHARMONYextendedPTS for coinductiveformulae

The failure of FC

What is “HARMONY”?

To say “Tr is a logical connective”, it must satisfy theHARMONY between the introduction rule and theelimination rule.

An example (TONK) The following connective withoutHARMONY trivialize the system.

AA tonk B tonk+ A tonk B

B tonk−

There are some criteria of “HARMONY”:

• Concervative extension (Belnap, Dummett)For any ϕ which does not include Tr, if FS proves ϕ, thenPA proves it.

• Normalization of proofs (Dummett)

5 / 16

Truth as alogical

connective

ShunsukeYatabe

IntroductionFS

PTS of Tr

A problemHARMONY

McGee’s Theorem

AnalysisInterpreting γ

PTS for inductiveformulae

Solution: TheHARMONYextendedPTS for coinductiveformulae

The failure of FC

The violation of “HARMONY”

However, we have the negative answers whether Tr satisfythe HARMONY:• FS is arithmetically sound:

• Any arithmetically sentence ϕ provable in FS is true in N,• FS proves the consistency of PA: adding NEC, CONEC is

not a conservative extension.• Quite contrary: formal commutability + ω-consistency

implies a contradiction [M85],• this shows FS is ω-inconsistent,• NEC and CONEC make the domain of the models

drastically large!

What should we do if we want to say Tr is a logical connective?

6 / 16

Truth as alogical

connective

ShunsukeYatabe

IntroductionFS

PTS of Tr

A problemHARMONY

McGee’s Theorem

AnalysisInterpreting γ

PTS for inductiveformulae

Solution: TheHARMONYextendedPTS for coinductiveformulae

The failure of FC

ω-inconsistency?

Theorem[McGee] Any consistent truth theory T which includePA and NEC and satisfies the following:

(1) (∀x, y)[x, y ∈→ (Tr(x→̇y) → (Tr(x) → Tr(y)))],(2) Tr(⊥) → ⊥,

(3) (∀x)[x ∈→ Tr(∀̇yx(y)) → (∀yTr(x(y)))]Then T is ω-inconsistent

proof We define the following paradoxical sentence

γ ≡ ¬∀xTr(g(x, dγe))

where g is a recursive function s.t.

g(0, dϕe) = dTr(dϕe)eg(x + 1, dϕe) = dTr(g(x, dϕe))e)

7 / 16

Truth as alogical

connective

ShunsukeYatabe

IntroductionFS

PTS of Tr

A problemHARMONY

McGee’s Theorem

AnalysisInterpreting γ

PTS for inductiveformulae

Solution: TheHARMONYextendedPTS for coinductiveformulae

The failure of FC

ω-inconsistency? (conti.)

The intuitive meaning of γ is

γ ≡ ¬Tr(dTr(dTr(dTr(d· · · e)e)e)e)

Let us assume γ is true:

• Tr(dTr(d· · · d︸ ︷︷ ︸n

λe · · · e)e) is true for any n,

• By formal commutativity, ¬Tr(dTr(d· · · d︸ ︷︷ ︸n

Tr(d· · · e)︸ ︷︷ ︸¬λ

e · · · e)e)

is true,

• This is equivalent to ¬λ; contradiction!

8 / 16

Truth as alogical

connective

ShunsukeYatabe

IntroductionFS

PTS of Tr

A problemHARMONY

McGee’s Theorem

AnalysisInterpreting γ

PTS for inductiveformulae

Solution: TheHARMONYextendedPTS for coinductiveformulae

The failure of FC

How can we interpret paradoxical sentences？

The provability of the consistency of PA and ω-inconsistencyshares the same reason:• Tr identify (real) formulae with Godel codes.

• So we can apparently define recursive, infinitelyoperation on formulae.

• This makes us to define the following formulae:McGee ¬Tr(dTr(dTr(dTr(d· · · e)e)e)e)

(Non-standard length, e.g. infinite length!)

Yablo S0 ≡ ¬Tr(S1)∧̄(¬Tr(S2)∧̄(¬Tr(S3)∧̄ · · · ))(we note that Si ≡ ¬Tr(dSi+1e)∧̄Si+2)

• The original sentences are finite, but they generate (orunfold) above infinite sentences (in that sense they are“potentially infinite”).

• keyword: coinduction

9 / 16

Truth as alogical

connective

ShunsukeYatabe

IntroductionFS

PTS of Tr

A problemHARMONY

McGee’s Theorem

AnalysisInterpreting γ

PTS for inductiveformulae

Solution: TheHARMONYextendedPTS for coinductiveformulae

The failure of FC

Induction

To compare with the coinduction, first let us introduce a typicalexample of the inductive definition.• For any set A, the list ofA can be constructed as〈A<ω, η : (1 + (A × A<ω)) → A<ω〉 by:• the first step: empty sequence 〈〉• the successor step: For all a0 ∈ A and sequence〈a1, · · · , an〉 ∈ A<ω

η(a0, 〈a1, · · · , an〉) = 〈a0, a1, · · · , an〉 ∈ A<ω

10 / 16

Truth as alogical

connective

ShunsukeYatabe

IntroductionFS

PTS of Tr

A problemHARMONY

McGee’s Theorem

AnalysisInterpreting γ

PTS for inductiveformulae

Solution: TheHARMONYextendedPTS for coinductiveformulae

The failure of FC

Thinking its meaning

Gentzen (1934, p. 80)The introductions represent, as it were, the ‘definitions’ of thesymbols concerned, and the eliminations are no more, in thefinal analysis, than the consequences of these definitions.

The meaning of induction is given by the introduction rule,

• the constructor η represents the introduction rule Forany a0 ∈ A and sequence 〈a1, · · · , an〉 ∈ A<ω,

η(a0, 〈a1, · · · , an〉) = 〈a0, a1, · · · , an〉 ∈ A<ω

A BA ∧ B

• the elimination rule γ is determined in relation to theintroduction rule which satisfies the HARMONY,

A ∧ BA

11 / 16

Truth as alogical

connective

ShunsukeYatabe

IntroductionFS

PTS of Tr

A problemHARMONY

McGee’s Theorem

AnalysisInterpreting γ

PTS for inductiveformulae

Solution: TheHARMONYextendedPTS for coinductiveformulae

The failure of FC

Coinduction

• infinite streams of A is defined as〈A∞, γ : A∞ → (A × A∞)〉:• Their intuitive meaning is infinite streams of the form〈a0, a1, · · ·〉 ∈ A∞

γ(〈a0, a1, · · ·〉) = (a0, 〈a1, · · · 〉) ∈ (A × A∞)

γ is a function which pick up the first element a0 of thestream.

• Coinduction represents the intuition such that finitelyoperations on the first element of the infinite sequence ispossible to compute (Productivity).

Actually CONEC only care about the productivity (or 1-stepcomputation)!

Tr(ϕ)ϕ CONEC

12 / 16

Truth as alogical

connective

ShunsukeYatabe

IntroductionFS

PTS of Tr

A problemHARMONY

McGee’s Theorem

AnalysisInterpreting γ

PTS for inductiveformulae

Solution: TheHARMONYextendedPTS for coinductiveformulae

The failure of FC

Thinking its meaning

The meaning of coinduction is by the elimination rule.

• de-constructor γ represents the elimination rule [S12], forany infinite stream 〈a0, a1, · · · 〉 ∈ A∞,

γ(〈a0, a1, · · · 〉) = (a0, 〈a1, · · · 〉) ∈ (A × A∞)

γ is a function wich picks up the first element a0.

• the introduction rule is known to have to satisfy thecondition（the guarded corecursion） which correspondsto the failure of the formal commutativity of Tr.

The final algebra which has the same structure to streams iscalled weakly final coalgebra

13 / 16

Truth as alogical

connective

ShunsukeYatabe

IntroductionFS

PTS of Tr

A problemHARMONY

McGee’s Theorem

AnalysisInterpreting γ

PTS for inductiveformulae

Solution: TheHARMONYextendedPTS for coinductiveformulae

The failure of FC

What is a guarded corecursive function?

• Remember: a property of stream = infinite as a whole, but1 step computation is possible

• Guarded corecursion guarantees the productivity offunctions over coinductive datatypes.• for any recursive function f ,

map : (A → B) → A∞ → B∞ is defined as

map f 〈x, x0, · · · 〉 = 〈 f (x)〉︸︷︷︸finite part

_ (]map f〈x0, x1, · · · 〉)︸ ︷︷ ︸infinite part

where _ means the concatenation of two sequence.• Here recursive call of map only appears inside ],

14 / 16

Truth as alogical

connective

ShunsukeYatabe

IntroductionFS

PTS of Tr

A problemHARMONY

McGee’s Theorem

AnalysisInterpreting γ

PTS for inductiveformulae

Solution: TheHARMONYextendedPTS for coinductiveformulae

The failure of FC

The guarded corecursion and the failure of Formalcommutability

Remember:

ω-inconsistency = PA + the formal commutability of Tr+NEC+CONEC

• The formal commutability violates the guardedness!

Tr(dγ → ¬Tr(dTr(d· · · e)e)e)Tr(dγe) → ¬Tr(dTr(dTr(d· · · e)e)e) FS

Both γ and Tr(dTr(dTr(d· · · e)e)e) are coinductive objects:• To calculate the value of Tr(dγ → ¬Tr(dTr(d· · · e)e)e), we

should calculate the both the value of Tr(dγe) and¬Tr(dTr(dTr(d· · · e)e)e).

• But Tr(dγe) is also coinductive object, therefore thecalculation of the first value Tr(dγe) never terminates,

• therefore this violates the productivity.• Therefore Tr with the formal commutability cannot be a

logical connective!!15 / 16

Truth as alogical

connective

ShunsukeYatabe

IntroductionFS

PTS of Tr

A problemHARMONY

McGee’s Theorem

AnalysisInterpreting γ

PTS for inductiveformulae

Solution: TheHARMONYextendedPTS for coinductiveformulae

The failure of FC

Reference

Jc Beal. Spandrels of Truth. Oxford University press (2008)

Hartry Field. “Saving Truth From Paradox” Oxford (2008)

Hannes Leitgeb. “Theories of truth which have no standardmodels” Studia Logica, 68 (2001) 69-87.

Vann McGee. “How truthlike can a predicate be? A negativeresult” Journal of Philosophical Logic, 17 (1985): 399-410.

Halbach, Volker, 2011, Axiomatic Theories of Truth, CambridgeUniversity Press.

Ole Hjortland, 2012, HARMONY and the Context of Deducibility,in Insolubles and Consequences College Publications

Greg Restall “Arithmetic and Truth in Łukasiewicz’s InfinitelyValued Logic” Logique et Analyse 36 (1993) 25-38.

Anton Setzer, 2012, Coalgebras as Types determined by theirElimination Rules, in Epistemology versus Ontology.

Stephen Yablo. “Paradox Without Self-Reference” Analysis 53(1993) 251―52.

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