1. 2010 2 21
1/1

2. Outline
T-
:
McGee [?]
T-
T-
[?] [?]
[?]
T-
T- =
ω- [?][?]
2/1

3. T-
T- (T-schemata
:
Tr( ϕ ) ≡ ϕ
ϕ ,
ϕ ϕ
3/1

4. “ ”( l)
Λ ≡ ¬Tr( Λ )
Λ
+ T-
(ZF, PA, etc.)
4/1

5. ( )
1:
T- e.g. (McGee
[?])
Tr( ϕ ) → ϕ
T-
2: T-
[?]
[?]
[?][?] etc.
5/1

6. 1:
: McGee Γ [?]
Γ Q
Γ
Γ ϕ Γ Tr( ϕ )
ϕ → Tr( ϕ )
Γ
K: Tr( ϕ → ψ ) → (Tr( ϕ ) → Tr( ϕ )),
D: Tr( ¬ϕ ) → ¬Tr( ϕ ),
Barcan: ∀x(Tr(sub( y , ϕ , nam(x))) → Tr( ∀yϕ )).
n nam(n)
numeral n
¯
6/1

7. 2:
L L ↔ ¬L
P
L ↔ ¬L L ↔ ¬L
[v : L] L → ¬L [w : ¬L] ¬L → L
[v : L] ¬L [w : ¬L] L
L ∨ ¬L ⊥ ⊥ −
⊥ ∨
LC
P
LEM: (Law of Excluded middle)
LC: (Law of contradiction)
CR: (contraction rule)
7/1

8. LEM: (Law of Excluded middle)
P ∨ ¬P
P
LC: (Law of contradiction)
⊥
P
P
CR: (contraction rule)
A ···
A ···
P
A→P
A, P
8/1

9. 2:
(transparent) [?]
Tr T-
T-
(transparent view of truth)
Quine (disquotational view of
truth)
9/1

10. (revenge)
10 / 1

11. negation complete A ¬A
A
¬A ∨ A
LEM
(indeterminacy)
3 Kripke
Hartry Field [?]
11 / 1

12. :
L ↔ ¬L L ↔ ¬L
[v : L] L → ¬L [w : ¬L] ¬L → L
[v : L] ¬L [w : ¬L] L
⊥ →+ ⊥ →+
¬L v ¬¬L w
⊥
LC
P
→ CR
indeterminacy
12 / 1

13. ⊥ ⊥
⊥
A→B A B
Schwichtenberg
JC Beal [?]
13 / 1

14. LC
C ↔ (C → P)
[v : C] C → (C → P)
[v : C] C→P
C ↔ (C → P) P
v→+
C ↔ (C → P) (C → P) → C C→P
C → (C → P) C ···
C→P C
P
C C ↔ (Tr( C ) → P)
P ⊥
→
14 / 1

15. →
→ Rule Modus Ponens (RMP)
(RMP) A, A → B B
Conditional Mopdus Ponens (CMP)
(CMP) A ∧ (A → B) → B
A B
A→B
→
by JC Beal
etc.
15 / 1

16. L
L → ⊥ (i.e. ¬L)
L → (L → ⊥)
A→B A → (A → B)
A B A
A→B
(Girard)
Comparative conception of truth
A B
(Hajek)
Fuzzy logics = logics of gradual properties
(Behounek)
Restall [?]
Hajek, Paris, Shefardson [?]
16 / 1

17. Łukasiewicz ∀Ł
:
(1) [0, 1]
(2) ϕ0 → ϕ1 = min{1, 1 − ϕ0 + ϕ1 }, ⊥ = 0,
¬A ≡ A → ⊥, etc.
(3) (∀x)ϕ(x) = inf{ ϕ(a) M : a ∈ |M|}.
∀Ł (i.e. ∀Ł ϕ CL ϕ
)
17 / 1

18. CR ∀Ł
A ∧ B = min{ A , B } (Additive)
∧ CMP
A ∧ (A → B) → B
∧ dual ∨ LEM
A ⊗ B = max{0, A + B − 1} (multipricative)
⊗ CMP
A ⊗ ( A → B) → B
& dual ⊕ LEM
2
CR
18 / 1

19. : ∀Ł PAŁTr2 [?]
PA PAŁTr2
T-
ϕ ϕ
ϕ ≡ Tr( ϕ )
Tr
PAŁTr2
Λ ≡ ¬Tr( Λ ) Λ = 0.5
19 / 1

20. : ω-
PAŁTr2
PAŁTr2 ω- [?] ϕ(x)
ϕ(0), ϕ(1), · · · , (∃x)¬ϕ(x)
d ϕ(d) d
PA ( PAŁ)
PAŁTr2
Adding a truth predicate “entails a drastic deviation from
the intended ontology of the theory” [?].
20 / 1

21. : ( )
ω-
Vann McGee [?] ω- Γ
Hartry Field [?] PAŁTr2 ω-
Halbach ω-
(Gupta & Belnap [?] (revision
sequence) [?]
ω-
[?]
21 / 1

22. ⇒
T- ⇒
⇒
⇒
graduality ⇒
Revenge problem
⇒
⇒ CMP
⇒
22 / 1

23. Reference
Jc Beal “Spandrels of Truth” Oxford University press (2008)
Jc Beal, Michael Glanzberg. “Where the paths meet: remarks on truth and paradox” In S. French, ed., Midwest
Studies: Truth, Notre Dame University Press (2008)
Hartry Field. “Saving Truth From Paradox” Oxford (2008)
Anil Gupta, Nuel Belnap “The revision theory of truth MIT Press (1993)
Volker Halbach “ Axiomatische Wahrheitstheorien” Wiley-VCH (1996)
´
Petr Hajek, Jeff B. Paris, John C. Shepherdson. “ The Liar Paradox and Fuzzy Logic” Journal of Symbolic Logic,
65(1) (2000) 339-346.
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.
Greg Restall “Arithmetic and Truth in Łukasiewicz’s Infinitely Valued Logic” Logique et Analyse 36 (1993) 25-38.
Shunsuke Yatabe “The revenge of the modest liar” in June 17, at Non-Classical Mathematics 2009
23 / 1