Upload
others
View
18
Download
5
Embed Size (px)
Citation preview
Logic and Philosophy of TimeThemes from Prior
Per Hasle, Patrick Blackburn, and Peter Øhrstrøm (Eds.)
Logic and Philos
ophy
of
Logic and Philosophy of TimeA.N. Prior (1914-69) in the course of the 1950s and 1960s founded a new and revolutionary paradigm in philosophy and logic. Its most central feature is the preoccupation with time and the development of the logic of time. However, this was inseparably interwoven with fundamental questions about human freedom, ethics, and existence. This remarkable integration of themes also embodies an original and in fact revolutionary conception of logic. The book series, Logic and Philosophy of Time, is dedicated to a deep investigation and also the further development of Prior’s paradigm.
The series includes:1 - Logic and Philosophy of Time: Themes from Prior
Series editorsPer Hasle, Patrick Blackburn & Peter Øhrstrøm
F
PH
P
♦ ! ♦
Pp ⊃ !Pp!(p ⊃ q) ⊃ ( ♦q ⊃ ♦p)p0 ∧ Fp0 ∧ ♦p0 p0
! ⊃∧ ⊃
p0 ∧ Fp0
p q
K
KH G P F
⊃ ⊃ ⊃⊃ ⊃ ⊃⊃⊃
⊢ p ⊢ Gp⊢ p ⊢ Hp
K
K
FPp ⊃ (Pp ∨ p ∨ Fp)
KPPp ⊃ Pp
K
FFp⊃Fp
< <
π × Φ Φ
(t, q)
π(t, q) 0 1
Fϕϕ
t Fϕ t′ t < t′ t′ ϕ
t, c q q π(t, q) = 1t, c ϕ t, c ϕt, c Fϕ t′∈c t < t′ t′, c ϕt, c Pϕ t′∈c t′ < t t′, c ϕ
t t
♦
t, c ♦ϕ c′ t∈c′ t, c′ ϕ
♦Fq!Fq Fq ! ♦
∀p : (p ∨ Pp∨ Fp) ⊃ FPp
pPFp FPp
nn n
n n
![δ[(p⊃0)⊃(q⊃r)]⊃δ[(r⊃p)⊃(q⊃p)]]!p⊃[δ(p⊃q)⊃δq]δ(0)⊃[δ(0⊃0)⊃δ(!p)]np⊃!(n⊃p)!n⊃p
nn
p⊃!(n⊃p)
n !n⊃p n!n
!n n
n
(1, 0)
δ ε ζ
δ
δ n n
p (p∨q)∼∼p (p∧∼∼p)
ab R
aRb bRa
p (p∨q)
Hp p
Gp p
p⊃PFp
Pp⊃!Pp p⊃PFp !PFp p Fp!(p⊃q)⊃(!p⊃!q) !Fp
l
p t T (t, p)p t
P FPp Fp
p Pp p
Fpp
H ¬P¬ G ¬F¬
!♦ ¬!¬
Pp → !Pp
pp
(p ⇒Diod q ∧ ♦p) → ♦q
q p pq
¬r ∧ ¬Fr ∧ ♦r r
r
⇒Diod
rt0
wt−1
¬r ∧ ¬Fr ∧ ♦r♦rPw w!Pw(∀t)(T (t, r) → T (t,¬Pw))
r ⇒Diod ¬Pw ¬Pw rt T (t, r) t t0
t
♦¬Pw¬!Pw
(p ∧Gp) → PGp!(p → HFp)
¬r ∧ ¬Fr ∧ ♦r♦r¬r ∧G¬rPG¬r!PG¬r!(r → HFr)♦HFr¬!PG¬r
! "g,w,t,t′
ϕ g w tt′
! ϕ"g,w,t,t′ !ϕ"g,w,t,t
! ϕ"g,w,t,t′ !ϕ"g,w,t′,t′
! "g,w,t,t′
′ ′
′ ′
ϕ ′ ′ g wt t′ t′′
! ϕ"g,w,t,t′,t′′ !ϕ"g,w,t,t,t′′
! ϕ"g,w,t,t′,t′′ !ϕ"g,w,t′,t′,t′′
! ϕ"g,w,t,t′,t′′ !ϕ"g,w,t,t′,t
! ϕ"g,w,t,t′,t′′ !ϕ"g,w,t′′,t′,t′′
ϕϕ w0
t0 !ϕ"g,w0 ,t0 ,t0 gψ ψ
w0 t0!ψ"g,w0 ,t0 ,t0 ,t0 g
w0 t0
t t0 < t t′
t′ < t0 d d w0 td w0 t′
t t′
t t′
AB A!→B
A !→B w
w A(A ∧ B) w (A ∧ ¬B)
h h′
h t h′
h′ t h h′
h h′
h′ t h′ t
❏❏❏❏❏
h′ h
t
h h′
! !!h′ t
tt
hO1
h1 O2 h2 O1 O2
h1 O1
h2 O2
h1h h1
O1 h2h h2
O1
S O2
h2h1 h !!
!
S
O2
O1
!!✁✁✁✁✁✁
❆❆❆❆❆❆❆❆❆❆
h h2
h h1
h1 O1 h2O2
S h2
T ( ≼ d) M≼
M M≺ ,≽, ≻
≼
≼ m,m′
m′′
m′ ! m′′ m′′ ⊀ m′ m m ≺ m′ m ≺ m′′
m′ ≺ m m′′ ≺ m m′ ≼ m′′ m′′ ≺ m′
tth t ∩ h
th ≺ t′h th′ # t′h′ t t′
th ≺ t′h h
d (t, t′)n t t′
t′ = t′′ th ≺ t′h, t′′h′ d(t, t′) =
d(t, t′′)LT
Atom p, q, p1, . . .
,
LT
LT
M = (T , I) T IAtom × M {0, 1} I
{th, h} t/h
t/h # p I(p, th) = 1t/h # A ∃t′(t′h ≺ th t′/h # A)t/h # A ∃t′(th ≺ t′h t′/h # A)
LT !→ LTC
LC
LTC
A !→B t/h t
t/h # A !→Bt/h′ A h′
(A ∧ B) t/h′ t/h(A ∧ ¬B) t/h′′
t/h t/h′
h h′
t
h′ h h′′
h′ ∩ h ⊇ h′′ ∩ h
A t/h′ t/hA t/h′′ h′ h h′′
h h1 h2
/h
( o1 ∨ o2) !→ s
A t/h (A, t/h)A t/h
A t/h MO = (T , I)(A, t/h)
(A, t/h) ⊆ IM′
O = (T , I ′) (A, t/h) ⊆ I ′
M′O, t/h # A
f (A, t/h) ⊃ fM′
O = (T , I ′)I ′ ⊃ f M′
O, t/h $ A
(p, th)t h f A t/h
MO = (T , I) M′O = (T , I ′) f
A t/h M′O A t/h
" (A, t/h)A
A
(p ∨ ¬q) t/h{(p, th) 5→ 1} {(q, th) 5→ 0} (p ∨ ¬q) t/h
(A, t/h) tp t/h I(p, t′h) = 1 t′h ≺ th
{(p, t′h) → 1} p t/h t
(p ∧ q) t/h t/h{(p, th) 5→ 1, (q, t′h) 5→ 1}
t′h ≺ th
{(o1, h1) 5→ 1} O1
{(o2, h2) 5→ 1} O2
/h1 /h2
/h1/h2
′
(A, t/h′, h) t/h′ Ah
t∗ A t/h′
h h′ ′(A, t/h′, h) = t∗′(A, t/h′, h)
′′(A, t/h′, h)
h h′
h′
h′ h′ t/h′ A
′(A, t/h′, h) = d(h h′, ′(A, t/h′, h))
′(A, t/h′, h) ′(A, t/h′, h)A t/h′
h h′
′ A t/h′ t/hA t/h′′ ′(A, t/h′, h) ≤ ′(A, t/h′′, h)
′
′
pp!→B
th t′hh
p t′ t′
t′ t/h′
p th′ ′( p, t/h′, h)h′ h ′
( p, t/h′, h) t/h′
′′
A t/h MO
f ∈ I f At/h MO
A t/h MO
" (A, t/h) MO " (A, t/h) ∩ I ̸= ∅" (A, t/h) ∩ I A t/h
AA
p !→ Bt/h
{(t′h, p) 5→ 1 : t′h ≺ th} ′
′
g A t/h f ⊆ gA t/h A t/h
g − f
f h′ f ht a ∈ f
th′ a th
h A t/h′ fA t/h′ f h
A t/h
h t/h′
A A t/hA
t/h A t/h hA t/h′ h′
h A
t/h A t/h At/h′ h A t/h′
ht/h′ A (A, t/h′, h)
h A t/h′
( p, t/h′, h)I p
t ( p, t/h′, h)
′ ′
(A, t/h′, h) = d(h h′, (A, t/h′, h))
A t/h′ t/hA t/h′′ (A, t/h′, h) ≤ (A, t/h′′, h)
A t/h′
h′
′ A t/h h′
h h′ (A, t/h′, h) = ′(A, t/h′, h) At/h h A
t/h′ A t/h∅ (A, t/h′) h
A t/h′′
′ h′ h h′
h′
( p∧ q) t/h qh p t′′/h
t/h( p ∧ q) t/h′
{(p, t′′h′) 5→ 1, (q, t′h′) 5→ 1}
❚❚
❚❚
❚❚
❚❚
❚❚
❚h′ h
t
t′
t′′
h h1
! !!
!
!!!
q
p p
p q
t′′ ′(( p ∧ q), t/h′, h) = t′′ h( p∧ q) t/h′ {(q, t′h′) 5→ 1} h
( p∧ q) t/h(( p ∧ q), t/h′, h) = t′ (( p ∧ q), t/h′, h) ̸=
′(( p ∧ q), t/h′, h)
′
t/h t/h′
h h′
t/h′ t/h
internalprocesses
action
persistence
impingements
reaction
resilience
external processes (activity)
stasis/stability
dissolution/destruction
!ϕ ϕ$ϕ ϕ
t tt
ϕ→ !ϕϕ→ $ϕ
!ϕ→ ϕϕ→ ! ϕ
ϕ→ $ ϕϕ→ !$ ϕ
t0
t0 ϕ ¬!ϕt t ¬ϕt t ¬ ϕ
t0 ϕ ¬! ϕ
M0 = ⟨W,T, t0,K, V ⟩
WTK ⊆ W ×W WV : P → ℘(W × T )
K
K(w,w)K(w, v) ⇒ K[v] ⊆ K[w] K[w] = {v | K(w, v)}
M, (w, t) |= p ⇔ (w, t) ∈ V (p)M, (w, t) |= ¬ϕ⇔ M, (w, t) ̸|= ϕM, (w, t) |= ϕ ∧ ψ ⇔ M, (w, t) |= ϕ and M, (w, t) |= ψM, (w, t) |= [now]ϕ⇔ M, (w, t) |= ϕM, (w, t) |= !ϕ⇔ ∀t′(t′ ∈ T ⇒ M, (w, t′) |= ϕ)M, (w, t) |= $ϕ⇔ ∀t′(t′ ∈ T ⇒ M, (w, t′) |= ϕ)M, (w, t) |= ϕ⇔ ∀v(K(w, v) ⇒ M, (v, t) |= ϕ)
[now]ϕ ϕ
M, (w, t) |= !ϕ⇔ M, (w, t) |= $ϕ
ϕ→ ϕϕ→ ϕ
[now]ϕ,!ϕ,$ϕ
[now]ϕ↔ ϕ |=([now]ϕ↔ ϕ) |=[now]ϕ↔ ϕ |=
[now]
!ϕ↔ $ϕ |=!ϕ↔ $ ϕ |=
! ϕ↔ $ ϕ |=
!φ→ φ
!φ→ ! φ!φ↔ ! φ!φ↔ $ φ
M0 = ⟨W,T, t0,K, V ⟩
WT t0 ∈ TK ⊆ W ×W WV : P → ℘(W × T )
M0 t0K
K(w,w)K(w, v) ⇒ K[v] ⊆ K[w] K[w] = {v | K(w, v)}
M0, (w, t) |= p ⇔ (w, t) ∈ V (p)M0, (w, t) |= ¬ϕ⇔ M0, (w, t) ̸|= ϕM0, (w, t) |= ϕ ∧ ψ ⇔ M0, (w, t) |= ϕ and M0, (w, t) |= ψM0, (w, t) |= [now]ϕ⇔ M0, (w, t0) |= ϕM0, (w, t) |= ϕ⇔ ∀v(K(w, v) ⇒ M0, (v, t) |= ϕ)M0, (w, t) |= !ϕ⇔ ∀t′(t′ ∈ T ⇒ M0, (w, t′) |= ϕ)M0, (w, t) |= $ϕ⇔ ∀t′(t′ ∈ T ⇒ Mt′ , (w, t) |= ϕ)
[now]ϕϕ
M0, (w, t) |= [now]ϕ⇔ M0, (w, t) |= ![now]ϕ
[now]ϕ
M0, (w, t) |= [now]ϕ ̸⇔ M0, (w, t) |= $[now]ϕ
ϕ t0 t1[now]ϕ t0 $[now]ϕ
ϕ→ ϕϕ→ ϕ
[now]ϕ
[now][now]([now]ϕ↔ ϕ)
[now]ϕ,!ϕ,$ϕ
[now]ϕ↔ ![now]ϕ[now]ϕ ̸↔ $[now]ϕ
ϵ
P I
p | ¬ϕ | (ϕ ∧ ϕ) | [i]ϕ | !ϕ | $ ϕ | ϕ p ∈ P
[now]ϕϕ
[i]ϕ ϕ i
now
i
Mϵ = ⟨W,T, τ, ϵ,K, V ⟩
WTτ : I → TϵK ⊆ W ×W WV : P → ℘(W × T )
Mϵ ϵ Kτ I
τi ∈ T
Mϵ, (w, t) |= p ⇔ (w, t) ∈ V (p)
Mϵ, (w, t) |= ¬ϕ⇔ Mϵ, (w, t) ̸|= ϕ
Mϵ, (w, t) |= ϕ ∧ ψ ⇔ Mϵ, (w, t) |= ϕ and Mϵ, (w, t) |= ψ
Mϵ, (w, t) |= [i]ϕ⇔ Mϵ, (w, τi) |= ϕ
Mϵ, (w, t) |= !ϕ⇔ ∀t′(t′ ∈ T ⇒ Mϵ, (w, t′) |= ϕ)
Mϵ, (w, t) |= $ϕ⇔ ∀t′(t′ ∈ T ⇒ Mϵ, (w, t′) |= ϕ)
Mϵ, (w, t) |= ϕ⇔ ∀v(Kϵ(w, v) ⇒ Mϵ(v, t) |= ϕ)
ti
[i]ϕϕ τi
ϕ→ ϕϕ→ ϕ
[i]ϕ,!ϕ,$ϕ
[i]ϕ↔ ![i]ϕ[i]ϕ↔ $[i]ϕ
This book is the first volume in the series, Logic and Philosophy of Time. Its various contributions take their inspiration from A.N. Prior’s paradigm for the study of time, hence the subtitle “Themes from Prior”. The volume contains important research on historical as well as modern systematic challenges related to Prior’s work and thought.