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Truth as alogical

connective?

ShunsukeYatabe

IntroductionFS

PTS of Tr

Motivations

A problemHARMONY

McGees Theorem

AnalysisInterpreting

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

McGees Theorem

AnalysisInterpreting

PTS for inductiveformulae

Solution: TheHARMONYextendedPTS for coinductiveformulae

The failure of FC

Abstract

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

Tr() NECTr() CONEC

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

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

Analysis: -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

McGees Theorem

AnalysisInterpreting

PTS for inductiveformulae

Solution: TheHARMONYextendedPTS for coinductiveformulae

The failure of FC

T-scheme and the liar

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

Tr()

where is a Godel code of .

The fixed point lemma: For any P(x), there is a closedformula such that P().

The liar paradox Tr()

3 / 28

Truth as alogical

connective?

ShunsukeYatabe

IntroductionFS

PTS of Tr

Motivations

A problemHARMONY

McGees Theorem

AnalysisInterpreting

PTS for inductiveformulae

Solution: TheHARMONYextendedPTS for coinductiveformulae

The failure of FC

Two solutions

The solution 1: keep classical logic abandan the full form of T-schemeto exclude the liar

sentence[Ha11]e.g. Friedman-Sheareds FS

Tr() NECTr() CONEC

For any , 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.

4 / 28

Truth as alogical

connective?

ShunsukeYatabe

IntroductionFS

PTS of Tr

Motivations

A problemHARMONY

McGees Theorem

AnalysisInterpreting

PTS for inductiveformulae

Solution: TheHARMONYextendedPTS for coinductiveformulae

The failure of FC

Friedman-Sheareds 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 , Tr() , (x Form)[Tr(x) Tr(x)], (x, y Form)[Tr(xy) Tr(x) Tr(y)], (x, y Form)[Tr(xy) Tr(x) Tr(y)], (x, y Form)[Tr(xy) 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)

Tr() NECTr()

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

McGees Theorem

AnalysisInterpreting

PTS for inductiveformulae

Solution: TheHARMONYextendedPTS for coinductiveformulae

The failure of FC

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

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

Tr() NECTr() CONEC

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

6 / 28

Truth as alogical

connective?

ShunsukeYatabe

IntroductionFS

PTS of Tr

Motivations

A problemHARMONY

McGees Theorem

AnalysisInterpreting

PTS for inductiveformulae

Solution: TheHARMONYextendedPTS for coinductiveformulae

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.

7 / 28

Truth as alogical

connective?

ShunsukeYatabe

IntroductionFS

PTS of Tr

Motivations

A problemHARMONY

McGees Theorem

AnalysisInterpreting

PTS for inductiveformulae

Solution: TheHARMONYextendedPTS for coinductiveformulae

The failure of FC

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

sentencesby using truth predicate:e.g. Everything he said is true

Formally: Assume 0, 1, is a recursive enumeration ofsentences, and f is a recursive function s.t. f (n) = n.Then the following represents the infinite conjunction of0 1

xTr( f (x))

Deflationists insist that truths 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

McGees Theorem

AnalysisInterpreting

PTS for inductiveformulae

Solution: TheHARMONYextendedPTS for coinductiveformulae

The failure of FC

A problem: the violation of HARMONY

Tr(x) in FS cannot be a logical connectivebecause it violates the HARMONYbetween the

introduction rule and the elimination rule.

9 / 28

Truth as alogical

connective?

ShunsukeYatabe

IntroductionFS

PTS of Tr

Motivations

A problemHARMONY

McGees Theorem

AnalysisInterpreting

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 BB tonk

There are some criteria of HARMONY:

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

Normalization of proofs (Dummett)

10 / 28

Truth as alogical

connective?

ShunsukeYatabe

IntroductionFS

PTS of Tr

Motivations

A problemHARMONY

McGees Theorem

AnalysisInterpreting

PTS for inductiveformulae

Solution: TheHARMONYextendedPTS for coinductiveformulae

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 is

not a conservative extension!

In particular, McGees : FS is -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

McGees Theorem

AnalysisInterpreting

PTS for inductiveformulae

Solution: TheHARMONYextendedPTS for coinductiveformulae

The failure of FC

-inconsistency?

Theorem[McGee] Any consistent truth theory T which includePA and NEC and satisfies the following:(1) (x, y)[x, y (Tr(xy) (Tr(x) Tr(y)))],(2) Tr() ,(3) (x)[x Tr(yx(y)) (yTr(x(y)))]Then T is -inconsistent

proof We define the following paradoxical sentence

xTr(g(x, ))

where g is a recursive function s.t.

g(0, ) = Tr()g(x + 1, ) = Tr(g(x, )))

12 / 28

Truth as alogical

connective?

ShunsukeYatabe

IntroductionFS

PTS of Tr

Motivations

A problemHARMONY

McGees Theorem

AnalysisInterpreting

PTS for inductiveformulae

Solution: TheHARMONYextendedPTS for coinductiveformulae

The failure of FC

-inconsistency?