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A slide of the talk in Logic and Engineering of Natural Language Semantics (LENLS) 12:

we analy...

A slide of the talk in Logic and Engineering of Natural Language Semantics (LENLS) 12:

we analyse Friedman-Sheared’s truth theory FS from the behavior of truth predicate by a proof theoretic semantics methodology.

- Truth as a

logical

connective

Shunsuke

Yatabe

Introduction

FS

Truth as a logical connective

PTS of Tr

A problem

HARMONY

McGee’s Theorem

Shunsuke Yatabe

Analysis

Interpreting γ

PTS for inductive

formulae

Center for Applied Philosophy and Ethics,

Graduate School of Letters,

Solution: The

HARMONY

Kyoto University

extended

PTS for coinductive

formulae

November 15, 2015

The failure of FC

Logic and Engineering of Natural Language Semantics 12

1 / 16 - ">Truth as a

The pourpose of this talk

logical

connective

Shunsuke

Yatabe

Introduction

•

FS

I do not insist that “truth conception should be treated not

PTS of Tr

as a predicate but as a logical connective”,

A problem

HARMONY

• I do want to analyse Friedman-Sheared’s truth theory FS

McGee’s Theorem

from the behaviour of truth predicate,

Analysis

Interpreting γ

• it seems to be possible to regard the truth predicate Tr as

PTS for inductive

formulae

a logical connective in truth theories with

Solution: The

HARMONY

ϕ

Tr(ϕ)

extended

NEC

Tr(ϕ)

ϕ

CONEC

PTS for coinductive

formulae

The failure of FC

• Since the condition of being a predicate is very flexible if

the theory is ω-inconsistent, such proof theoretic

semantics viewpoint is effective to analyse it.

2 / 16 - ">Truth as a

Friedman-Sheared’s Axiomatic truth theory FS

logical

connective

Shunsuke

FS = PA + the formal commutability of Tr

Yatabe

+ the introduction and the elimination rule of Tr

Introduction

FS

PTS of Tr

• Axioms and schemes of PA (mathematical induction for all

A problem

formulae including Tr)

HARMONY

McGee’s Theorem

• The formal commutability of the truth predicate:

Analysis

• logical connectives

Interpreting γ

• for amy atomic formula ψ, Tr( ψ ) ≡ ψ,

PTS for inductive

formulae

• (∀x ∈ Form)[Tr( ˙¬x) ≡ ¬Tr(x)],

Solution: The

• (∀x, y ∈ Form)[Tr(x ˙∧y) ≡ Tr(x) ∧ Tr(y)],

HARMONY

•

extended

(∀x, y ∈ Form)[Tr(x ˙∨y) ≡ ¬Tr(x) ∨ Tr(y)],

PTS for coinductive

• (∀x, y ∈ Form)[Tr(x ˙→y) ≡ ¬Tr(x) → Tr(y)],

formulae

The failure of FC

• 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.

3 / 16 - ">Truth as a

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

logical

connective

Shunsuke

Yatabe

Introduction

FS

PTS of Tr

FS’s two rules look like the introduction and the elimination

A problem

rule; Tr looks like a logical connective [Hj12]

HARMONY

McGee’s Theorem

ϕ

Analysis

Tr(ϕ)

NEC

Interpreting γ

Tr(ϕ)

ϕ

CONEC

PTS for inductive

formulae

Solution: The

HARMONY

extended

• A (naive) proof theoretic semantics: the meaning of a

PTS for coinductive

formulae

The failure of FC

logical connective is given by the introduction rule

and the elimination rule

4 / 16 - ">Truth as a

What is “HARMONY”?

logical

connective

Shunsuke

Yatabe

To say “Tr is a logical connective”, it must satisfy the

Introduction

HARMONY between the introduction rule and the

FS

PTS of Tr

elimination rule.

A problem

HARMONY

An example (TONK) The following connective without

McGee’s Theorem

Analysis

HARMONY trivialize the system.

Interpreting γ

PTS for inductive

formulae

A

tonk +

A tonk B tonk −

Solution: The

A tonk B

B

HARMONY

extended

PTS for coinductive

formulae

There are some criteria of “HARMONY”:

The failure of FC

• Concervative extension (Belnap, Dummett)

For any ϕ which does not include Tr, if FS proves ϕ, then

PA proves it.

• Normalization of proofs (Dummett)

5 / 16 - ">Truth as a

The violation of “HARMONY”

logical

connective

Shunsuke

Yatabe

Introduction

However, we have the negative answers whether Tr satisfy

FS

PTS of Tr

the HARMONY:

A problem

• FS is arithmetically sound:

HARMONY

McGee’s Theorem

• Any arithmetically sentence ϕ provable in FS is true in N,

Analysis

• FS proves the consistency of PA: adding NEC, CONEC is

Interpreting γ

PTS for inductive

not a conservative extension.

formulae

Solution: The

• Quite contrary: formal commutability + ω-consistency

HARMONY

implies a contradiction [M85],

extended

PTS for coinductive

•

formulae

this shows FS is ω-inconsistent,

The failure of FC

• 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 a

ω-inconsistency?

logical

connective

Shunsuke

Theorem[McGee] Any consistent truth theory T which include

Yatabe

PA and NEC and satisfies the following:

Introduction

FS

(1) (∀x, y)[x, y ∈→ (Tr(x ˙

→y) → (Tr(x) → Tr(y)))],

PTS of Tr

A problem

(2) Tr(⊥) → ⊥,

HARMONY

McGee’s Theorem

(3) (∀x)[x ∈→ Tr( ˙∀yx(y)) → (∀yTr(x(y)))]

Analysis

Interpreting γ

Then T is ω-inconsistent

PTS for inductive

formulae

Solution: The

proof We define the following paradoxical sentence

HARMONY

extended

PTS for coinductive

γ ≡ ¬∀

formulae

xTr( g(x, γ ))

The failure of FC

where g is a recursive function s.t.

g(0, ϕ ) =

Tr( ϕ )

g(x + 1, ϕ ) =

Tr( g(x, ϕ )) )

7 / 16 - ">Truth as a

ω-inconsistency? (conti.)

logical

connective

Shunsuke

Yatabe

Introduction

FS

The intuitive meaning of γ is

PTS of Tr

A problem

γ ≡ ¬Tr( Tr( Tr( Tr( · · · ) ) ) )

HARMONY

McGee’s Theorem

Analysis

Let us assume γ is true:

Interpreting γ

PTS for inductive

formulae

• Tr( Tr( · · · λ · · · ) ) is true for any n,

Solution: The

HARMONY

n

extended

• By formal commutativity, ¬ Tr( Tr( · · · Tr( · · · ) · · · ) )

PTS for coinductive

formulae

The failure of FC

n

¬λ

is true,

• This is equivalent to ¬λ; contradiction!

8 / 16 - ">Truth as a

logical

How can we interpret paradoxical sentences？

connective

Shunsuke

Yatabe

The provability of the consistency of PA and ω-inconsistency

shares the same reason:

Introduction

FS

• Tr identify (real) formulae with Godel codes.

PTS of Tr

A problem

• So we can apparently define recursive, infinitely

HARMONY

operation on formulae.

McGee’s Theorem

Analysis

• This makes us to define the following formulae:

Interpreting γ

PTS for inductive

McGee ¬Tr( Tr( Tr( Tr( · · · ) ) ) )

formulae

Solution: The

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

HARMONY

extended

PTS for coinductive

Yablo S

formulae

0 ≡ ¬Tr(S1) ¯

∧(¬Tr(S2) ¯∧(¬Tr(S3) ¯∧ · · · ))

The failure of FC

(we note that Si ≡ ¬Tr( Si+1 ) ¯∧Si+2)

• The original sentences are finite, but they generate (or

unfold) above infinite sentences (in that sense they are

“potentially infinite”).

• keyword: coinduction

9 / 16 - ">Truth as a

Induction

logical

connective

Shunsuke

Yatabe

Introduction

FS

PTS of Tr

To compare with the coinduction, first let us introduce a typical

A problem

example of the inductive definition.

HARMONY

McGee’s Theorem

• For any set A, the list ofA can be constructed as

Analysis

A<ω, η : (1 + (A × A<ω)) → A<ω by:

Interpreting γ

PTS for inductive

•

formulae

the first step: empty sequence

Solution: The

• the successor step: For all a0 ∈ A and sequence

HARMONY

extended

a1, · · · , an ∈ A<ω

PTS for coinductive

formulae

The failure of FC

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

10 / 16 - ">Truth as a

Thinking its meaning

logical

connective

Shunsuke

Gentzen (1934, p. 80)

Yatabe

The introductions represent, as it were, the ‘definitions’ of the

Introduction

symbols concerned, and the eliminations are no more, in the

FS

PTS of Tr

final analysis, than the consequences of these definitions.

A problem

HARMONY

The meaning of induction is given by the introduction rule,

McGee’s Theorem

Analysis

• the constructor η represents the introduction rule For

Interpreting γ

PTS for inductive

any a

formulae

0 ∈ A and sequence

a1, · · · , an ∈ A<ω,

Solution: The

HARMONY

η(a

extended

0, a1, · · · , an ) =

a0, a1, · · · , an ∈ A<ω

PTS for coinductive

formulae

A

B

The failure of FC

A ∧ B

• the elimination rule γ is determined in relation to the

introduction rule which satisfies the HARMONY,

A ∧ B

A

11 / 16 - ">Truth as a

Coinduction

logical

connective

Shunsuke

• infinite streams of A is defined as

Yatabe

A∞, γ : A∞ → (A × A∞) :

Introduction

• Their intuitive meaning is infinite streams of the form

FS

PTS of Tr

a0, a1, · · · ∈ A∞

A problem

HARMONY

γ( a

McGee’s Theorem

0, a1, · · · ) = (a0, a1, · · · ) ∈ ( A × A∞)

Analysis

Interpreting γ

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

PTS for inductive

formulae

stream.

Solution: The

HARMONY

extended

• Coinduction represents the intuition such that finitely

PTS for coinductive

formulae

The failure of FC

operations on the first element of the infinite sequence is

possible to compute (Productivity).

Actually CONEC only care about the productivity (or 1-step

computation)!

Tr(ϕ)

ϕ

CONEC

12 / 16 - ">Truth as a

Thinking its meaning

logical

connective

Shunsuke

Yatabe

Introduction

The meaning of coinduction is by the elimination rule.

FS

PTS of Tr

• de-constructor γ represents the elimination rule [S12], for

A problem

any infinite stream a

HARMONY

0, a1, · · ·

∈ A∞,

McGee’s Theorem

Analysis

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

Interpreting γ

PTS for inductive

formulae

γ is a function wich picks up the first element a

Solution: The

0.

HARMONY

extended

• the introduction rule is known to have to satisfy the

PTS for coinductive

formulae

condition （the guarded corecursion） which corresponds

The failure of FC

to the failure of the formal commutativity of Tr.

The final algebra which has the same structure to streams is

called weakly final coalgebra

13 / 16 - ">Truth as a

What is a guarded corecursive function?

logical

connective

Shunsuke

Yatabe

Introduction

FS

• Remember: a property of stream = infinite as a whole, but

PTS of Tr

1 step computation is possible

A problem

HARMONY

• Guarded corecursion guarantees the productivity of

McGee’s Theorem

functions over coinductive datatypes.

Analysis

Interpreting γ

• for any recursive function f,

PTS for inductive

formulae

map : ( A → B) → A∞ → B∞ is defined as

Solution: The

HARMONY

extended

map f x, x0, · · · = f (x)

( map f x0, x1, · · · )

PTS for coinductive

formulae

finite part

infinite part

The failure of FC

where

means the concatenation of two sequence.

• Here recursive call of map only appears inside ,

14 / 16 - ">Truth as a

The guarded corecursion and the failure of Formal

logical

connective

commutability

Shunsuke

Yatabe

Remember:

Introduction

ω

FS

-inconsistency = PA + the formal commutability of Tr

PTS of Tr

+NEC+CONEC

A problem

HARMONY

McGee’s Theorem

• The formal commutability violates the guardedness!

Analysis

Interpreting γ

Tr( γ → ¬Tr( Tr( · · · ) ) )

PTS for inductive

FS

formulae

Tr( γ ) → ¬Tr( Tr( Tr( · · · ) ) )

Solution: The

HARMONY

Both γ and Tr( Tr( Tr( · · · ) ) ) are coinductive objects:

extended

PTS for coinductive

• To calculate the value of Tr( γ → ¬Tr( Tr( · · · ) ) ), we

formulae

The failure of FC

should calculate the both the value of Tr( γ ) and

¬Tr( Tr( Tr( · · · ) ) ).

• But Tr( γ ) is also coinductive object, therefore the

calculation of the first value Tr( γ ) never terminates,

• therefore this violates the productivity.

• Therefore Tr with the formal commutability cannot be a

logical connective!!

15 / 16 - ">Truth as a

Reference

logical

connective

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

Shunsuke

Yatabe

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

Introduction

FS

PTS of Tr

Hannes Leitgeb. “Theories of truth which have no standard

A problem

models” Studia Logica, 68 (2001) 69-87.

HARMONY

McGee’s Theorem

Vann McGee. “How truthlike can a predicate be? A negative

Analysis

result” Journal of Philosophical Logic, 17 (1985): 399-410.

Interpreting γ

PTS for inductive

formulae

Halbach, Volker, 2011, Axiomatic Theories of Truth, Cambridge

Solution: The

University Press.

HARMONY

extended

PTS for coinductive

Ole Hjortland, 2012, HARMONY and the Context of Deducibility,

formulae

The failure of FC

in Insolubles and Consequences College Publications

Greg Restall “Arithmetic and Truth in Łukasiewicz’s Infinitely

Valued Logic” Logique et Analyse 36 (1993) 25-38.

Anton Setzer, 2012, Coalgebras as Types determined by their

Elimination Rules, in Epistemology versus Ontology.

Stephen Yablo. “Paradox Without Self-Reference” Analysis 53 16/16

(1993) 251―52.