This document discusses treating the truth predicate (Tr) as a logical connective in truth theories like Friedman-Sheared's theory (FS). It analyzes FS from the perspective of proof theoretic semantics, where Tr's introduction and elimination rules are like those of a connective. However, FS violates the "harmony" requirement for connectives, as it is not conservatively extending and proves the consistency of PA. The document then discusses interpreting paradoxical sentences like McGee's using coinduction and how guarded corecursion relates to the failure of Tr's formal commutability in FS.

1. Truth as a
logical
connective
Shunsuke
Yatabe
Introduction
FS
PTS of Tr
A problem
HARMONY
McGee’s Theorem
Analysis
Interpreting γ
PTS for inductive
formulae
Solution: The
HARMONY
extended
PTS for coinductive
formulae
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, 2015
Logic and Engineering of Natural Language Semantics 12
1 / 16

2. Truth as a
logical
connective
Shunsuke
Yatabe
Introduction
FS
PTS of Tr
A problem
HARMONY
McGee’s Theorem
Analysis
Interpreting γ
PTS for inductive
formulae
Solution: The
HARMONY
extended
PTS for coinductive
formulae
The failure of FC
The pourpose of this talk
• I do not insist that “truth conception should be treated not
as a predicate but as a logical connective”,
• I do want to analyse Friedman-Sheared’s truth theory FS
from 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 if
the theory is ω-inconsistent, such proof theoretic
semantics viewpoint is effective to analyse it.
2 / 16

3. Truth as a
logical
connective
Shunsuke
Yatabe
Introduction
FS
PTS of Tr
A problem
HARMONY
McGee’s Theorem
Analysis
Interpreting γ
PTS for inductive
formulae
Solution: The
HARMONY
extended
PTS for coinductive
formulae
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 all
formulae 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.
3 / 16

4. Truth as a
logical
connective
Shunsuke
Yatabe
Introduction
FS
PTS of Tr
A problem
HARMONY
McGee’s Theorem
Analysis
Interpreting γ
PTS for inductive
formulae
Solution: The
HARMONY
extended
PTS for coinductive
formulae
The failure of FC
A proof theoretic semantics (PTS) of Tr (naive version)
FS’s two rules look like the introduction and the elimination
rule; Tr looks like a logical connective [Hj12]
ϕ
Tr(ϕ)
NEC
Tr(ϕ)
ϕ CONEC
• A (naive) proof theoretic semantics: the meaning of a
logical connective is given by the introduction rule
and the elimination rule
4 / 16

5. Truth as a
logical
connective
Shunsuke
Yatabe
Introduction
FS
PTS of Tr
A problem
HARMONY
McGee’s Theorem
Analysis
Interpreting γ
PTS for inductive
formulae
Solution: The
HARMONY
extended
PTS for coinductive
formulae
The failure of FC
What is “HARMONY”?
To say “Tr is a logical connective”, it must satisfy the
HARMONY between the introduction rule and the
elimination rule.
An example (TONK) The following connective without
HARMONY trivialize the system.
A
A 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, then
PA proves it.
• Normalization of proofs (Dummett)
5 / 16

6. Truth as a
logical
connective
Shunsuke
Yatabe
Introduction
FS
PTS of Tr
A problem
HARMONY
McGee’s Theorem
Analysis
Interpreting γ
PTS for inductive
formulae
Solution: The
HARMONY
extended
PTS for coinductive
formulae
The failure of FC
The violation of “HARMONY”
However, we have the negative answers whether Tr satisfy
the 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

7. Truth as a
logical
connective
Shunsuke
Yatabe
Introduction
FS
PTS of Tr
A problem
HARMONY
McGee’s Theorem
Analysis
Interpreting γ
PTS for inductive
formulae
Solution: The
HARMONY
extended
PTS for coinductive
formulae
The failure of FC
ω-inconsistency?
Theorem[McGee] Any consistent truth theory T which include
PA 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 )) )
7 / 16

8. Truth as a
logical
connective
Shunsuke
Yatabe
Introduction
FS
PTS of Tr
A problem
HARMONY
McGee’s Theorem
Analysis
Interpreting γ
PTS for inductive
formulae
Solution: The
HARMONY
extended
PTS for coinductive
formulae
The failure of FC
ω-inconsistency? (conti.)
The intuitive meaning of γ is
γ ≡ ¬Tr( Tr( Tr( Tr( · · · ) ) ) )
Let us assume γ is true:
• Tr( Tr( · · ·
n
λ · · · ) ) is true for any n,
• By formal commutativity, ¬ Tr( Tr( · · ·
n
Tr( · · · )
¬λ
· · · ) )
is true,
• This is equivalent to ¬λ; contradiction!
8 / 16

9. Truth as a
logical
connective
Shunsuke
Yatabe
Introduction
FS
PTS of Tr
A problem
HARMONY
McGee’s Theorem
Analysis
Interpreting γ
PTS for inductive
formulae
Solution: The
HARMONY
extended
PTS for coinductive
formulae
The failure of FC
How can we interpret paradoxical sentences？
The provability of the consistency of PA and ω-inconsistency
shares the same reason:
• Tr identify (real) formulae with Godel codes.
• So we can apparently define recursive, infinitely
operation 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)
• The original sentences are finite, but they generate (or
unfold) above infinite sentences (in that sense they are
“potentially infinite”).
• keyword: coinduction
9 / 16

10. Truth as a
logical
connective
Shunsuke
Yatabe
Introduction
FS
PTS of Tr
A problem
HARMONY
McGee’s Theorem
Analysis
Interpreting γ
PTS for inductive
formulae
Solution: The
HARMONY
extended
PTS for coinductive
formulae
The failure of FC
Induction
To compare with the coinduction, first let us introduce a typical
example 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

11. Truth as a
logical
connective
Shunsuke
Yatabe
Introduction
FS
PTS of Tr
A problem
HARMONY
McGee’s Theorem
Analysis
Interpreting γ
PTS for inductive
formulae
Solution: The
HARMONY
extended
PTS for coinductive
formulae
The failure of FC
Thinking its meaning
Gentzen (1934, p. 80)
The introductions represent, as it were, the ‘definitions’ of the
symbols concerned, and the eliminations are no more, in the
final analysis, than the consequences of these definitions.
The meaning of induction is given by the introduction rule,
• the constructor η represents the introduction rule For
any a0 ∈ A and sequence a1, · · · , an ∈ A<ω,
η(a0, a1, · · · , an ) = a0, a1, · · · , an ∈ A<ω
A B
A ∧ B
• the elimination rule γ is determined in relation to the
introduction rule which satisfies the HARMONY,
A ∧ B
A
11 / 16

12. Truth as a
logical
connective
Shunsuke
Yatabe
Introduction
FS
PTS of Tr
A problem
HARMONY
McGee’s Theorem
Analysis
Interpreting γ
PTS for inductive
formulae
Solution: The
HARMONY
extended
PTS for coinductive
formulae
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 the
stream.
• Coinduction represents the intuition such that finitely
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

13. Truth as a
logical
connective
Shunsuke
Yatabe
Introduction
FS
PTS of Tr
A problem
HARMONY
McGee’s Theorem
Analysis
Interpreting γ
PTS for inductive
formulae
Solution: The
HARMONY
extended
PTS for coinductive
formulae
The failure of FC
Thinking its meaning
The meaning of coinduction is by the elimination rule.
• de-constructor γ represents the elimination rule [S12], for
any 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 the
condition （the guarded corecursion） which corresponds
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

14. Truth as a
logical
connective
Shunsuke
Yatabe
Introduction
FS
PTS of Tr
A problem
HARMONY
McGee’s Theorem
Analysis
Interpreting γ
PTS for inductive
formulae
Solution: The
HARMONY
extended
PTS for coinductive
formulae
The failure of FC
What is a guarded corecursive function?
• Remember: a property of stream = infinite as a whole, but
1 step computation is possible
• Guarded corecursion guarantees the productivity of
functions 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

15. Truth as a
logical
connective
Shunsuke
Yatabe
Introduction
FS
PTS of Tr
A problem
HARMONY
McGee’s Theorem
Analysis
Interpreting γ
PTS for inductive
formulae
Solution: The
HARMONY
extended
PTS for coinductive
formulae
The failure of FC
The guarded corecursion and the failure of Formal
commutability
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 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

16. Truth as a
logical
connective
Shunsuke
Yatabe
Introduction
FS
PTS of Tr
A problem
HARMONY
McGee’s Theorem
Analysis
Interpreting γ
PTS for inductive
formulae
Solution: The
HARMONY
extended
PTS for coinductive
formulae
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 standard
models” Studia Logica, 68 (2001) 69-87.
Vann McGee. “How truthlike can a predicate be? A negative
result” Journal of Philosophical Logic, 17 (1985): 399-410.
Halbach, Volker, 2011, Axiomatic Theories of Truth, Cambridge
University Press.
Ole Hjortland, 2012, HARMONY and the Context of Deducibility,
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
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