1

1

φ φ

P P, Q Q, …

φ φ T2 φ

φ φ

T

T 3

T T

T 4

東京大学哲学研究室『論集』35号（2016年度）pp.77－90

2

5

T6

T

2

3

4

7

PA 8

3

9

LPA PA LPA+ PA

Tr φ LPA φ #φ

φ LPA LPA Tr( φ )←→φ

PA LPA LPA+

IND+ φ(0)& x(φ(x)→φ(x+1))→ xφ(x) φ LPA+

TB PA

LPA IND φ(0)& x(φ(x)→φ(x+1))→ xφ(x) φ

LPA TB− TB− TB

Tr( 0=1&5+7=12 )←→(Tr ( 0=1 )&Tr ( 5+7=12 )) TB−

~(T( 0=1 ) & T( ~0=1 ))

TB TB− TB10 TB

x(Sent(x&y) (Tr(x&y)←→Tr(x)&Tr(y)))

Sent(x)

x PA TB

TB TB− CT CT− CT LPA+

IND+ CT1–4 PA CT− PA

CT1–4 LPA IND

s t PA s t11 ~ &

x(t/v) x v t

PA

78 東京大学哲学研究室『論集』35号（2016年度）

2

5

T6

T

2

3

4

7

PA 8

3

9

LPA PA LPA+ PA

Tr φ LPA φ #φ

φ LPA LPA Tr( φ )←→φ

PA LPA LPA+

IND+ φ(0)& x(φ(x)→φ(x+1))→ xφ(x) φ LPA+

TB PA

LPA IND φ(0)& x(φ(x)→φ(x+1))→ xφ(x) φ

LPA TB− TB− TB

Tr( 0=1&5+7=12 )←→(Tr ( 0=1 )&Tr ( 5+7=12 )) TB−

~(T( 0=1 ) & T( ~0=1 ))

TB TB− TB10 TB

x(Sent(x&y) (Tr(x&y)←→Tr(x)&Tr(y)))

Sent(x)

x PA TB

TB TB− CT CT− CT LPA+

IND+ CT1–4 PA CT− PA

CT1–4 LPA IND

s t PA s t11 ~ &

x(t/v) x v t

PA

79保存性解釈に基づいた真理のデフレ主義批判について

4

CT1 s t(Tr( s=t )←→s =t )

CT2 x(Sent(~x)→(Tr(~x)←→~Tr(x)))

CT3 x y(Sent(x&y)→(Tr(x&y)←→Tr(x)&Tr(y))

CT4 v x(Sent( vx)→(Tr( vx)←→ tTr(x(t/v)))

CT1 CT2 CT3

CT412

CT CT−

CT CT− TB TB−

x(Sent(x&y) (T(x&y)←→T(x)&T(y)))

TB TB−

CT CT−

CT CT−

CT− IND LPA

CT IND+

CT− PA CT PA

T S T S S T

LS φ φ T T|–φ S S|–φ

CT− PA

LPA CT− PA13 CT− CT

Reflection Principle RP

CT

global reflection principle GRP 14

x(Sent(x)&ProvPA(x) Tr(x))

ProvPA(x) PA PA GRP PA

GRP 0=1 CT1

Sent( 0=1 ) PA ProvPA( 0=1 ) 0=1 PA

0 1 ~ProvPA( 0=1 ) PA

5

CT ~ProvPA( 0=1 ) LPA

~ProvPA( 0=1 ) PA CT

PA local

reflection principle LRP CT PA

ProvPA( 0=1 ) 0=1 15

ProvPA( φ ) φ

TB CT

TB CT Tr( Tr( 0=0 ) )

5+7=12

5+7=12 Tr( 0=1 )

Tr( 5+7=12 Tr( 0=1 ) )

16

CT

Friedman–Sheard FS 17 FS

FS1–4 NECCONEC PA IND+

Sent+(x) x LPA+ Sent(x)

Sent+(x)

FS1 s t(Tr( s=t )←→s =t )

FS2 x(Sent+ (~x)→(Tr(~x)←→~Tr(x)))

FS3 x y(Sent+ (x&y)→(Tr(x&y)←→Tr(x)&Tr(y))

FS4 v x(Sent+ ( vx)→(Tr( vx)←→ tTr(x(t/v)))

80 東京大学哲学研究室『論集』35号（2016年度）

4

CT1 s t(Tr( s=t )←→s =t )

CT2 x(Sent(~x)→(Tr(~x)←→~Tr(x)))

CT3 x y(Sent(x&y)→(Tr(x&y)←→Tr(x)&Tr(y))

CT4 v x(Sent( vx)→(Tr( vx)←→ tTr(x(t/v)))

CT1 CT2 CT3

CT412

CT CT−

CT CT− TB TB−

x(Sent(x&y) (T(x&y)←→T(x)&T(y)))

TB TB−

CT CT−

CT CT−

CT− IND LPA

CT IND+

CT− PA CT PA

T S T S S T

LS φ φ T T|–φ S S|–φ

CT− PA

LPA CT− PA13 CT− CT

Reflection Principle RP

CT

global reflection principle GRP 14

x(Sent(x)&ProvPA(x) Tr(x))

ProvPA(x) PA PA GRP PA

GRP 0=1 CT1

Sent( 0=1 ) PA ProvPA( 0=1 ) 0=1 PA

0 1 ~ProvPA( 0=1 ) PA

5

CT ~ProvPA( 0=1 ) LPA

~ProvPA( 0=1 ) PA CT

PA local

reflection principle LRP CT PA

ProvPA( 0=1 ) 0=1 15

ProvPA( φ ) φ

TB CT

TB CT Tr( Tr( 0=0 ) )

5+7=12

5+7=12 Tr( 0=1 )

Tr( 5+7=12 Tr( 0=1 ) )

16

CT

Friedman–Sheard FS 17 FS

FS1–4 NECCONEC PA IND+

Sent+(x) x LPA+ Sent(x)

Sent+(x)

FS1 s t(Tr( s=t )←→s =t )

FS2 x(Sent+ (~x)→(Tr(~x)←→~Tr(x)))

FS3 x y(Sent+ (x&y)→(Tr(x&y)←→Tr(x)&Tr(y))

FS4 v x(Sent+ ( vx)→(Tr( vx)←→ tTr(x(t/v)))

81保存性解釈に基づいた真理のデフレ主義批判について

6

NEC |–φ CONEC |– Tr( φ )

|–Tr( φ ) |–φ

FS1–4 LPA LPA+

18 NEC φ FS φ

φ CONEC φ

FS φ φ

φ φ

φ φ

NEC CONEC

FS CT FS NEC CONEC FS

FS1–4 LPA LPA+

19 FS

GRP+ 20

GRP+ x(Sent+(x)&ProvPA+(x) Tr(x))

GRP+ LPA+ PA+ PA IND+

GRP+ GRP PA+

FS ω 21 ω FS

PA NEC T

x(Sent+(~x)→(Tr(~x)→~Tr(x))) x y(Sent+(x&y)→(Tr(x&y)→Tr(x)&Tr(y)))

v x(Sent+ ( vx)→(Tr( vx)→ tTr(x(t/v)))) T ω 22

FS PA FS2–4 NEC

FS ω ω

FS 23

7

2 CT PA

24

25

CT

CT PA CT PA

CT

CT− CT PA CT−

IND+ IND

0

82 東京大学哲学研究室『論集』35号（2016年度）

6

NEC |–φ CONEC |– Tr( φ )

|–Tr( φ ) |–φ

FS1–4 LPA LPA+

18 NEC φ FS φ

φ CONEC φ

FS φ φ

φ φ

φ φ

NEC CONEC

FS CT FS NEC CONEC FS

FS1–4 LPA LPA+

19 FS

GRP+ 20

GRP+ x(Sent+(x)&ProvPA+(x) Tr(x))

GRP+ LPA+ PA+ PA IND+

GRP+ GRP PA+

FS ω 21 ω FS

PA NEC T

x(Sent+(~x)→(Tr(~x)→~Tr(x))) x y(Sent+(x&y)→(Tr(x&y)→Tr(x)&Tr(y)))

v x(Sent+ ( vx)→(Tr( vx)→ tTr(x(t/v)))) T ω 22

FS PA FS2–4 NEC

FS ω ω

FS 23

7

2 CT PA

24

25

CT

CT PA CT PA

CT

CT− CT PA CT−

IND+ IND

0

83保存性解釈に基づいた真理のデフレ主義批判について

8

CT CT−

CT

CT

CT CT26

GRP CT CT

S S

B S27

GRP28 PA PA

GRP

GRP PA PA

GRP PA

29 CT

PA

RP RP

CT CT

CT1–4 IND+

CT

30

GRP

PA CT1–4 IND+ CT GRP

CT1–4 IND+

9

IND+

IND+

CT1–4 IND+ CT CT

CT IND+ IND+

RP

CT CT RP

CT− CT

2 FS CT

FS ω

ω FS

GRP

PA GRP

GRP+

84 東京大学哲学研究室『論集』35号（2016年度）

8

CT CT−

CT

CT

CT CT26

GRP CT CT

S S

B S27

GRP28 PA PA

GRP

GRP PA PA

GRP PA

29 CT

PA

RP RP

CT CT

CT1–4 IND+

CT

30

GRP

PA CT1–4 IND+ CT GRP

CT1–4 IND+

9

IND+

IND+

CT1–4 IND+ CT CT

CT IND+ IND+

RP

CT CT RP

CT− CT

2 FS CT

FS ω

ω FS

GRP

PA GRP

GRP+

85保存性解釈に基づいた真理のデフレ主義批判について

10

PA+

PA+ PA

PA+ 3

PA+ PA GRP

PA+ GRP+

GRP+

CT GRP GRP+

FS FS

FS ω

FS

ω

CT

11

CT

FS

FS GRP+ FS

1

2 Horwich 1998, 6 φ φ φ φ

3 L L L T

L L L L

4 Horwich 1998 , 40–43) McGee 1992, 236 5 Horwich (1998, 42) 6 McGee 1998 7 Halbach 2011 Horsten 2011 8 PA PA

PA

PA 9 T Tarski 1956, 187–188 T

3 10 Tarski 1956 3 11 Halbach 2011 12 PA LPA

86 東京大学哲学研究室『論集』35号（2016年度）

10

PA+

PA+ PA

PA+ 3

PA+ PA GRP

PA+ GRP+

GRP+

CT GRP GRP+

FS FS

FS ω

FS

ω

CT

11

CT

FS

FS GRP+ FS

1

2 Horwich 1998, 6 φ φ φ φ

3 L L L T

L L L L

4 Horwich 1998 , 40–43) McGee 1992, 236 5 Horwich (1998, 42) 6 McGee 1998 7 Halbach 2011 Horsten 2011 8 PA PA

PA

PA 9 T Tarski 1956, 187–188 T

3 10 Tarski 1956 3 11 Halbach 2011 12 PA LPA

87保存性解釈に基づいた真理のデフレ主義批判について

12

13 Halbach 2011 14 Halbach 2011, 104 15 PA|– φ PA|– ProvPA( φ )→φ LRP

PA φ PA 16 –

Kripke 1975 17 Friedman and Sheard 1987 FS Halbach 2011

NEC Necessitation |–φ / |–□φ

FS NEC

18 FS

FS1 Tr(s)

FS1 CT1 s(Tr( Tr(s) ) ←→Tr(s )) FS2

FS 19 Friedman and Sheard 1987 20 GRP+ Halbach 2011, 162 21 FS Friedman and Sheard 1987 Halbach 2011, 172

T T|– xφ(x) n T|– ~φ(n

22 McGee 1990, 1.5 FS ω

Leitgeb 2001 23 Horsten 2011, 112 24 Ketland 1999, 79 25 Shapiro 1998, 499–500 26

TB

[TB ] [TB

]

[~Tr(x) ~Tr(~x)]

13

x

T

[ ]

CT1–4

RP

PA

0+1=1+0 1 2 2 1

x y (x+y=y+x)

27 Ketland 2005, 78 28 Ketland 2005 Shapiro 1998

29 φ ψ φ ψ

Bridge Principle MacFarlane unpublished

88 東京大学哲学研究室『論集』35号（2016年度）

12

13 Halbach 2011 14 Halbach 2011, 104 15 PA|– φ PA|– ProvPA( φ )→φ LRP

PA φ PA 16 –

Kripke 1975 17 Friedman and Sheard 1987 FS Halbach 2011

NEC Necessitation |–φ / |–□φ

FS NEC

18 FS

FS1 Tr(s)

FS1 CT1 s(Tr( Tr(s) ) ←→Tr(s )) FS2

FS 19 Friedman and Sheard 1987 20 GRP+ Halbach 2011, 162 21 FS Friedman and Sheard 1987 Halbach 2011, 172

T T|– xφ(x) n T|– ~φ(n

22 McGee 1990, 1.5 FS ω

Leitgeb 2001 23 Horsten 2011, 112 24 Ketland 1999, 79 25 Shapiro 1998, 499–500 26

TB

[TB ] [TB

]

[~Tr(x) ~Tr(~x)]

13

x

T

[ ]

CT1–4

RP

PA

0+1=1+0 1 2 2 1

x y (x+y=y+x)

27 Ketland 2005, 78 28 Ketland 2005 Shapiro 1998

29 φ ψ φ ψ

Bridge Principle MacFarlane unpublished

89保存性解釈に基づいた真理のデフレ主義批判について

14

RP

Tennant 2002 Ketland 2010, 428–429

RP

30 Field 1999, 538

Field, Hartry 1994. “Deflationist Views of Meaning and Content,” Mind, 103, 411, 249–284.

Field, Hartry 1999. “Deflating the Conservativeness Argument,” Journal of Philosophy, 96, 533–540.

Friedman, Harvey and Sheard, Michael 1987. “An Axiomatic Approach to Self-Referential Truth,”

Annals of Pure and Applied Logic, 33, 1–21.

Halbach, Volker 1994. ‘A System of Complete and Consistent Truth,’ Notre Dame Journal of Formal

Logic, 35, 311–327.

2011. Axiomatic Theories of Truth, Cambridge University Press.

Horsten, Leon 2011. The Tarskian Turn: Deflationism and Axiomatic Truth, MIT Press.

Horwich, Peter 1990. Truth, Basil Blackwell, Oxford; 2nd edition, 1998, Clarendon Press.

Ketland, Jeffrey 1999. “Deflationism and Tarski’s Paradise,” Mind, 108, 429, 69–94.

2005. “Deflationism and Gödel Phenomena,” Mind, 114, 453, 75–88.

2010. “Truth, Conservativeness, and Provability: Reply to Cieslinski,” Mind, 19, 423–436.

Kripke, Saul 1975. “Outline of a Theory of Truth,” Journal of Philosophy, 72, 690-716.

Leitgeb, Hannes 2001. “Theories of Truth which have no Standard Models,” Studia Logica, 68, 69–87.

MacFarlane, John unpublished. “In What Sense (If Any) is Logic Normative for Thought?,”.

McGee, V. 1990. Truth, Vagueness and Paradox, Hackett Publishing.

1992. “Maximal consistent set of Tarski’s Schema (T),” Journal of Philosophical Logic, 21, 690–716.

Quine, Willard V. O. 1970. Philosophy of Logic, 2nd edition, Harvard University Press.

Reinhardt, William 1986. “Some Remarks on Extending and Interpreting Theories with a Partial Predicate

for Truth,’ Journal of Philosophical Logic, 15 (2), 219-251.

Shapiro, Stewart 1998. “Proof and Truth: Through Thick and Thin,” Journal of Philosophy, 95, 493–521.

Tarski, Alfred 1956. Logic, Semantics, Metamathematics, Clarendon Press.

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