Logic and Philosophy of Time

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

1

F

PH

P

♦ ! ♦

Pp ⊃ !Pp!(p ⊃ q) ⊃ ( ♦q ⊃ ♦p)p0 ∧ Fp0 ∧ ♦p0 p0

! ⊃∧ ⊃

p0 ∧ Fp0

♦p0P Fp0!P Fp0♦ P Fp0 !

!(p0 ⊃ P Fp0) ⊃ ( ♦ P Fp0 ⊃ ♦p0)♦p0

(Pp ∧ Pq) ⊃ (P (p ∧ q) ∨ P (p ∧ Pq) ∨ P (Pp ∧ q))

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 ! ♦

Fp t1c1 F p t2 c2

F G

Fpt1

t2

F = !F

HFp p

∀p : (p ∨ Pp∨ Fp) ⊃ FPp

pPFp FPp

∀t : (t < t)

P Q ∀i(Pi→Qi) iP i p i

q p

P Q∀i(Pi→Qi) i P ip i

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19P

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∀x(fx→gx)

fx∀x(fx→gx)

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x

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P Q

pp i P i

P i Qi

i P Q

i P i Qi

∀i(Pi→Qi)P Q

−∃i(Pi − Qi)

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1− x x = 1X x = 0

xy = 0 X YX Y

X Y

XY

X Y

X Y XY

XY

X X

1

a

n

a

a a n

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a

a

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![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

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n

(1, 0)

δ ε ζ

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n n nn n n

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n

n n⊃0

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Ut1p1p1 t1 p1 t1

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n

n

nW Q Wp p

Qp p

Wp = p ∧ ∀q [(q ⊃ !(p ⊃ q)]

Qp = ♦p ∧ ∀q [!(p ⊃ q) ∨ !(p ⊃ ¬q)]

a n

U

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[p⊃!(n⊃p)]

10

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n

Hp δ!(n⊃p)⊃δHp

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n

p (p∨q)∼∼p (p∧∼∼p)

ab R

aRb bRa

p (p∨q)

∨∨

∨ ⊃

aRb a b R ab aRb

R

a b

∨

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Pp p

Hp P p Gp F p

Hp p

Gp p

p⊃PFp

Pp⊃!Pp p⊃PFp !PFp p Fp!(p⊃q)⊃(!p⊃!q) !Fp

l

l

l

l

l l

lp q

p q

(p∧∼q∧Pq) ∨ P (p∧∼q∧Pq) ∨ F (p∧∼q∧Pq)

20

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

(p ⇒Diod q) (∀t)(T (t, p) → T (t, q))

!

(p ⇒Diod q) !(p → q)

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

w

ww

ww

v1 v2v1 v2v1 w v1

v2 v1v2

w

P ∧→ G f

P ∧ → G

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! 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′

∃ ∧

t t < t0 t′

t0 < t′ d d w0

t d w0 t′

tt′ t t′

′ ′

∃ ∧

′ ′

1 2 1 2

+

+

+

ϕ + g w tt′

+ ! + ϕ"g,w,t,t′ t′′ t′ < t′′

!ϕ"g,w,t,t′′

+

+ ∃ ∧

+

t t0 < t t′

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∀ → ∨

s s

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w

wi

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wi

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→

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1 2

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′

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tt

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h1 O2 h2 O1 O2

h1 O1

h2 O2

h1h h1

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h2h1 h !!

!

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!!✁✁✁✁✁✁

❆❆❆❆❆❆❆❆❆❆

h h2

h h1

O1 O2

h2h1 O2

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O2

h h′

t h′ tn

A !→B B An A B

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

th

18

18

∀x∀y∀P∀Q(x = y → (P (x) ↔ Q(x)))

t1

t2

t1

t1

t1

t2 t3

t1 t2

tn3

tn

tn

3

E3 E3(o, b, t1)E3(o, s, t2)

3

3

tn3

tn3

δt

t1t2

a1 a2

a1 a2

tn3

t1 φ t2 ¬φφ

R

R

R

R

R

12 ,

34 ,

78 ,

1516 , . . .

12

12

34

R

internalprocesses

action

persistence

impingements

reaction

resilience

external processes (activity)

stasis/stability

dissolution/destruction

(i)(ii)

x x

x

⇔

P

p | ¬ϕ | (ϕ ∧ ϕ) | [now]ϕ | !ϕ | $ ϕ | ϕ p ∈ P

!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

!ϕ → $ ϕ !ϕ → $ϕ[now]ϕ → ![now]ϕ

[now]ϕ→ $[now]ϕ

tt′

t tt′

t′ t t′

[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

⟨W,T,K⟩

Mϵ, (w, t) |= [i]ϕ⇔ Mτi , (w, t) |= [now]ϕ

t ϕt ϕ

!(T ′P ′→P )(T ′P ′→!P )

t t

t t

t

t t

tt

t t

t

t

tt

t

X

X

•••

⋄

⋄

⋄

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.