Truth as a logical connective?

Truth as alogical

connective?

ShunsukeYatabe

IntroductionFS

PTS of Tr

Motivations

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

October 26, 2016Kyoto Nonclassical Logic Workshop II

1 / 28

Truth as alogical

connective?

ShunsukeYatabe

IntroductionFS

PTS of Tr

Motivations

A problemHARMONY

McGee’s Theorem




The failure of FC

Abstract

• Question: Is it possible to think the truth predicate Tr ofFreidman-Sheared’s truth theory FS as a logicalconnective because FS has two rules NEC,CONEC?

φ

Tr(φ) NEC Tr(φ)φ CONEC

• Problem:• Tr(x) cannot be a logical connective because it violates

the HARMONY between NEC and CONEC,　• In particular, FS is ω-inconsistent!

• Analysis:• ω-inconsistency is due to the fact that infinite length

formulae are definable by finitely method.• Traditional HARMONY works for inductively defined

formulae effectively, but not for infinite formulae

• Solution: If we extend the concept of HARMONY forinfinite formulae, we have a positive answer.

2 / 28

Truth as alogical

connective?

ShunsukeYatabe

IntroductionFS

PTS of Tr

Motivations

A problemHARMONY

McGee’s Theorem




The failure of FC

T-scheme and the liar

• Tarskian T-schemata (unrestricted form): for any formulaφ,

φ ≡ Tr(⌈φ⌉)

where ⌈φ⌉ is a Godel code of φ.

• The fixed point lemma: For any P(x), there is a closedformula ψ such that ψ ≡ P(⌈ψ⌉).

• The liar paradoxΛ ≡ ¬Tr(⌈Λ⌉)

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Truth as alogical

connective?

ShunsukeYatabe

IntroductionFS

PTS of Tr

Motivations

A problemHARMONY

McGee’s Theorem




The failure of FC

Two solutions

• The solution 1: keep classical logic• abandan the full form of T-scheme（to exclude the liar

sentence）[Ha11]e.g.　 Friedman-Sheared’s FS

φ


• For any φ, T-sentence is provable• But the T-sentence of the liar is a theorem of FS.

Such restrictions prevents that the theory implies the liarsentence as a theorem.

• The solution 2: kepp the unrestricted form of T-schemata• and abandon classical logic

• paracomplete logic（[Fl08]）• paraconsistent logic（[B08]）• fuzzy logic（[?][R93]）, etc.

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Truth as alogical

connective?

ShunsukeYatabe

IntroductionFS

PTS of Tr

Motivations

A problemHARMONY

McGee’s Theorem




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(⌈ψ⌉) ≡ 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(⌈φ⌉) NEC Tr(⌈φ⌉)φ CONEC

where Form is a set of Godel code of formulae.5 / 28

Truth as alogical

connective?

ShunsukeYatabe

IntroductionFS

PTS of Tr

Motivations

A problemHARMONY

McGee’s Theorem




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


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

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Truth as alogical

connective?

ShunsukeYatabe

IntroductionFS

PTS of Tr

Motivations

A problemHARMONY

McGee’s Theorem




The failure of FC

A motivation: Proof theoretic semantics(PTS)A (naive) proof theoretic semantics:the meaning of a logical connective is given by theintroduction rule and the elimination rule• “the meaning of word is its use in the language”

(Wittgenstein)• Some PTS people pointed out that a truth predicate with

NEC, CONEC can be regarded as a logicalconnective[Hj12].

• To regard Tr as a logical connective involves the following:

• the essence of logical connectives is that the justificationof the consequence can be reduced to the justificationof the assumptions even though Tr is used in theargument(the the reducibility of the justifiability).

• So is the essence of Tr. In particular, we can reduce theargument in which we use Tr to that which does notcontain Tr by finite steps.

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Truth as alogical

connective?

ShunsukeYatabe

IntroductionFS

PTS of Tr

Motivations

A problemHARMONY

McGee’s Theorem




The failure of FC

The source of the problem: DeflationismThe deflationism requires the following two mutuallyinconsistent conditions:• Deflationists allows to assert “infinite conjunctions of

sentences”by using truth predicate:

e.g. “ Everything he said is true”

Formally: Assume φ0, φ1, · · · is a recursive enumeration ofsentences, and f is a recursive function s.t. f (n) = ⌈φn⌉.Then the following represents the infinite conjunction ofφ0 ∧ φ1 ∧ · · ·

∀xTr( f (x))

• Deflationists insist that truth’s influence to the semanticsshould be limited:• Truth must not involve any particular change to the

ontology!• Connection to this talk: Logical connectives are typical

examples which do not involve ontological assertions8 / 28

Truth as alogical

connective?

ShunsukeYatabe

IntroductionFS

PTS of Tr

Motivations

A problemHARMONY

McGee’s Theorem




The failure of FC

A problem: the violation of HARMONY

Tr(x) in FS cannot be a logical connectivebecause it violates the “HARMONY”between the

introduction rule and the elimination rule.

9 / 28

Truth as alogical

connective?

ShunsukeYatabe

IntroductionFS

PTS of Tr

Motivations

A problemHARMONY

McGee’s Theorem




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)

10 / 28

Truth as alogical

connective?

ShunsukeYatabe

IntroductionFS

PTS of Tr

Motivations

A problemHARMONY

McGee’s Theorem




The failure of FC

The violation of “HARMONY”

However, Tr does NOT satisfy the criteria of HARMONY:

• FS proves the consistency of PA: adding NEC, CONEC isnot a conservative extension!

• In particular, McGee’s : FS is ω-inconsistent [M85],Actually NEC and CONEC make the number of provablesentences drastically large!

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

11 / 28

Truth as alogical

connective?

ShunsukeYatabe

IntroductionFS

PTS of Tr

Motivations

A problemHARMONY

McGee’s Theorem




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, ⌈γ⌉))

where g is a recursive function s.t.

g(0, ⌈φ⌉) = ⌈Tr(⌈φ⌉)⌉g(x + 1, ⌈φ⌉) = ⌈Tr(g(x, ⌈φ⌉))⌉)

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Truth as alogical

connective?

ShunsukeYatabe

IntroductionFS

PTS of Tr

Motivations

A problemHARMONY

McGee’s Theorem




The failure of FC

ω-inconsistency? (conti.)The intuitive meaning of γ (I don’t know how many Tr’s!)

γ ≡ ¬Tr(⌈Tr(⌈Tr(⌈Tr(⌈· · · ⌉)⌉)⌉)⌉)

Then we can prove γ: roughly

γ ≡ ¬Tr(⌈Tr(⌈Tr(⌈Tr(⌈· · · ⌉)⌉)⌉)⌉)γ → ¬Tr(⌈Tr(⌈Tr(⌈Tr(⌈· · · ⌉)⌉)⌉)⌉)

Tr(⌈γ → ¬Tr(⌈Tr(⌈Tr(⌈Tr(⌈· · · ⌉)⌉)⌉)⌉)⌉)Tr(⌈γ⌉) → Tr(⌈¬Tr(⌈Tr(⌈Tr(⌈Tr(⌈· · · ⌉)⌉)⌉)⌉)⌉)⌉)Tr(⌈γ⌉) → ¬Tr(⌈Tr(⌈Tr(⌈Tr(⌈Tr(⌈· · · ⌉)⌉)⌉)⌉)⌉)⌉)

Tr(⌈γ⌉) → γ

FCFC

And¬γ → ∀xTr(g(x, ⌈γ⌉))¬γ → Tr(g(0̄, ⌈γ⌉))¬γ → Tr(⌈γ⌉)

Therefore Tr(⌈γ⌉) → γ and ¬γ → Tr(⌈γ⌉) implies，FS ⊢ ¬¬γthough ¬Tr(g(n̄, ⌈γ⌉) is provable for any n.

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Truth as alogical

connective?

ShunsukeYatabe

IntroductionFS

PTS of Tr

Motivations

A problemHARMONY

McGee’s Theorem




The failure of FC

ω-inconsistency? (conti.)

This is ω-inconsistency:

• Since γ is provable, allTr(g(0̄, ⌈γ⌉)),Tr(g(1̄, ⌈γ⌉)),Tr(g(2̄, ⌈γ⌉)), · · · are provable.Therefore, for any natural number n, Tr(g(n̄, ⌈γ⌉)) isprovable.

• However, γ, i.e. ¬∀xTr(g(x, ⌈γ⌉)) is provable.

Model theoretically speaking, there is a non-standard naturalnumber d such that ¬Tr(g(d, ⌈γ⌉)) in any model of FS.Intuitively it is of the form ¬Tr(⌈Tr(⌈Tr(⌈Tr(⌈· · · ⌉)⌉)⌉)⌉)︸︷︷︸

length d

.

14 / 28

Truth as alogical

connective?

ShunsukeYatabe

IntroductionFS

PTS of Tr

Motivations

A problemHARMONY

McGee’s Theorem




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(⌈Tr(⌈Tr(⌈Tr(⌈· · · ⌉)⌉)⌉)⌉)

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

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

• These infinite sentences are definable because of thedeflationism.

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Truth as alogical

connective?

ShunsukeYatabe

IntroductionFS

PTS of Tr

Motivations

A problemHARMONY

McGee’s Theorem




The failure of FC

How to define a logical connective to infinite formulae？

• We should extend the concept of “being a logicalconnective” for infinite formulae naturally.

• If it is enough natural, then we can expect that Tr does notimply the ontrogical involvement, i.e. ω-inconsistency• because Tr can be regarded as a logical connective, i.e.

with no ontrogical commitment, then.• Such natural extansion way is known as coinduction.

16 / 28

Truth as alogical

connective?

ShunsukeYatabe

IntroductionFS

PTS of Tr

Motivations

A problemHARMONY

McGee’s Theorem




The failure of FC

Remember: 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

17 / 28

Truth as alogical

connective?

ShunsukeYatabe

IntroductionFS

PTS of Tr

Motivations

A problemHARMONY

McGee’s Theorem




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

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

18 / 28

Truth as alogical

connective?

ShunsukeYatabe

IntroductionFS

PTS of Tr

Motivations

A problemHARMONY

McGee’s Theorem




The failure of FC

The elimination rule and recursive computationPrawitz (1965, p. 33)“ inversion principle”　Let α be an application of an elimination rule that has β asconsequence. Then, deductions that satisfy the sufficientcondition [...] for deriving the major premiss of α, whencombined with deductions of the minor premisses of α (if any),already “contain” a deduction of β; the deduction of β is thusobtainable directly from the given deductions without theaddition of α.

....A

....β

A ∧ β ∧+[v : A]....β

βα ⇒

....A....β

The proof by the elimination rule should have thecorresponding recursive function:• Input: the proof tree of β with α lefthand-side• Output: the proof tree of β without α righthand-side

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Truth as alogical

connective?

ShunsukeYatabe

IntroductionFS

PTS of Tr

Motivations

A problemHARMONY

McGee’s Theorem




The failure of FC

Thinking its meaning (revised)

• the constructor η represents the introduction rule For anya0 ∈ A and sequence ⟨a1, · · · , an⟩ ∈ A<ω,

η(a0, ⟨a1, · · · , an⟩) = ⟨a0, a1, · · · , an⟩ ∈ A<ω

• the elimination rule γ is represented as a recursivefunction over the datatype of formulae inductively definedby the introduction rule.

The recursive function can be identified with the recursiveprocedure of the cut elimination or the normalization of proofs.

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Truth as alogical

connective?

ShunsukeYatabe

IntroductionFS

PTS of Tr

Motivations

A problemHARMONY

McGee’s Theorem




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 NEC, CONEC only care about the productivity (or1-step computation)!

φ


21 / 28

Truth as alogical

connective?

ShunsukeYatabe

IntroductionFS

PTS of Tr

Motivations

A problemHARMONY

McGee’s Theorem




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

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Truth as alogical

connective?

ShunsukeYatabe

IntroductionFS

PTS of Tr

Motivations

A problemHARMONY

McGee’s Theorem




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 ♯,

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Truth as alogical

connective?

ShunsukeYatabe

IntroductionFS

PTS of Tr

Motivations

A problemHARMONY

McGee’s Theorem




The failure of FC

The introduction rule and guarded corecursive function

Analogy to the inductive case: the proof by the introductionrule should have the corresponding guarded corecursivefunction:

• Input: the (coinductive) proof tree with the use of theintroduction rule

• Output: the proof tree without the use of theintroduction rule

24 / 28

Truth as alogical

connective?

ShunsukeYatabe

IntroductionFS

PTS of Tr

Motivations

A problemHARMONY

McGee’s Theorem




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(⌈γ → ¬Tr(⌈Tr(⌈· · · ⌉)⌉)⌉)Tr(⌈γ⌉) → ¬Tr(⌈Tr(⌈Tr(⌈· · · ⌉)⌉)⌉) FS

Both γ and Tr(⌈Tr(⌈Tr(⌈· · · ⌉)⌉)⌉) are coinductive objects:• To calculate the value of Tr(⌈γ → ¬Tr(⌈Tr(⌈· · · ⌉)⌉)⌉), we

should calculate the both the value of Tr(⌈γ⌉) and¬Tr(⌈Tr(⌈Tr(⌈· · · ⌉)⌉)⌉).

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

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

logical connective!!25 / 28

Truth as alogical

connective?

ShunsukeYatabe

IntroductionFS

PTS of Tr

Motivations

A problemHARMONY

McGee’s Theorem




The failure of FC

The solution

If we abandon the FC, we can regard Tr as a logicalconnective in the extended sense.

• the truth theory Gi over Heyting arithmetic within theintuitionistic logic with two rules NEC,CONEC, andvarious formal commutativity conditions except, inparticular, Tr(A → B) → Tr(A) → Tr(B), satisifies thefollowing theorem [LgR12].

theorem(Leigh-Rathjen) Gi is ω-consistent.

Gi proves that it is consistent that Tr can be regarded as alogical connective and it is ω-consistent.

• This seems to suggest that their relationship between tobe a logical connective and to be ω-consistent.

26 / 28

Truth as alogical

connective?

ShunsukeYatabe

IntroductionFS

PTS of Tr

Motivations

A problemHARMONY

McGee’s Theorem




The failure of FC

Truth predicate as a de-constructor of a coinductivedatatype of formulae

In Gi, we can say the following:

• the meaning of truth predicate is given by the eliminationrule (CONEC),

• the introduction rule is given by the guarded corecursion,

If we extend the concept of HARMONY for coinductive objects,then we can say “Tr in Gi is a logical connective”.

• The nature of truth predicate here is to make itpossible to define coinductive formula of infinitelength.

• The truth conception here guarantees the reducibility ofthe argument,

• the essence is that the truth predicate never prevent thereducibility from the assumption to the consequence.

27 / 28

Truth as alogical

connective?

ShunsukeYatabe

IntroductionFS

PTS of Tr

Motivations

A problemHARMONY

McGee’s Theorem




The failure of FC

ReferenceJc 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

Graham E. Leigh and Michael Rathjen. The Friedman-Sheardprogramme in intuitionistic logic Journal of Symbolic Logic 77(3)pp. 777-806. (2012)

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) 25152.

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Science

Truth as a logical connective?