by Corry Shores
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[Priest, Introduction to Non-Classical Logic, entry directory]
[The following is summary of Priest’s text, which is already written with maximum efficiency. Bracketed commentary and boldface are my own, unless otherwise noted. I do not have specialized training in this field, so please trust the original text over my summarization. I apologize for my typos and other unfortunate mistakes, because I have not finished proofreading, and I also have not finished learning all the basics of these logics.]
Summary of
Graham Priest
An Introduction to Non-Classical Logic: From If to Is
Part I:
Propositional Logic
3.
Normal Modal Logics
3.6b
Extensions of K^{t}
^{}
Brief summary:
(3.6b.1) We can apply constraints on the accessibility relation to obtain extensions of our modal tense logic K^{t}. (3.6b.2) These constraints on R condition the way ‘x is before y’ behaves. For example, the transitivity constraint makes beforeness transitive, and we can represent the beginninglessness or endlessness of time using the extendability constraint. (3.6b.3) Priest next notes some natural constraints for tense logic. {1} denseness (δ): if xRy then for some z, xRz and zRy, which places a moment between any two others; {2} forward convergence (ϕ): if xRy and xRz then (yRz or y = z or zRy); that is to say, when two moments come after some given moment, then they cannot belong to two distinct futures but must instead fall along the same timeline; and {3} backward convergence (β): if yRx and zRx then (yRz or y = z or zRy); in other words, two moments coming before some given moment must fall along a single succession. (3.6b.4) Priest next gives the tableau rules for constrained tense logics. (For convenience, I have here added ones in later sections to keep all the rules in one place.
Double Negation Development (¬¬D) |
¬¬A,i ↓ A,i |
Conjunction Development (∧D) |
A ∧ B,i ↓ A,i B,i |
Negated Conjunction Development (¬∧D) |
¬(A ∧ B),i ↓ ¬A ∨ ¬B,i |
Disjunction Development (∨D) |
A ∨ B,i ↙ ↘ A,i B,i |
Negated Disjunction Development (¬∨D) |
¬(A ∨ B),i ↓ ¬A,i ↓ ¬B,i |
Conditional Development (⊃D) |
A ⊃ B,i ↙ ↘ ¬A,i B,i |
Negated Conditional Development (¬⊃D) |
¬(A ⊃ B),i ↓ A,i ↓ ¬B,i |
(p.24, section 2.4.3 and 2.4.4)
Full Future Development ([F]D) |
[F]A,i irj ↓ A,j
(For all j) |
Partial Future Development (⟨F⟩D) |
⟨F⟩A,i ↓ irj A,j
(j must be new: it cannot occur anywhere above on the branch) |
Negated Full Future Development (¬[F]D) |
¬[F]A,i ↓ ⟨F⟩¬A,i |
Negated Partial Future Development (¬⟨F⟩D) |
¬⟨F⟩A,i ↓ [F]¬A,i |
Full Past Development ([P]D) |
[P]A,i jri ↓ A,j
(For all j) |
Partial Past Development (⟨P⟩D) |
⟨P⟩A,i ↓ jri A,j
(j must be new: it cannot occur anywhere above on the branch) |
Negated Full Past Development (¬[P]D) |
¬[P]A,i ↓ ⟨P⟩¬A,i |
Negated Partial Past Development (¬⟨P⟩D) |
¬⟨P⟩A,i ↓ ⟨P⟩¬A,i |
(p.50, section 3.6a.6, with my added names and other data at the bottoms)
α= World Equality (α=D) |
α(i) j=i ↓ α(j) . α(i) i=j ↓ α(j) .
where α(i) is a line of the tableau containing an ‘i’. α(j) is the same, with ‘j’ replacing ‘i’. Thus: if α(i) is A, i, α(j) is A, j if α(i) is kri, α(j) is krj if α(i) is i = k, α(j) is j = k |
(53; with my naming additions and copied text at the bottom)
ρ, Reflexivity (ρrD) |
ρ . ↓ iri |
σ, Symmetry (σrD) |
σ irj ↓ jri |
τ, Transitivity (ρrD) |
τ irj jrk ↓ irk |
η, Extendability (ηrD) |
η . ↓ irj .
It is applied to any integer, i, on a branch, provided that there is not already something of the form irj on the branch, and the j in question must then be new. |
δ, Denseness (δrD) |
δ irj ↓ irk krj
.
where k is new to the branch. |
ϕ, Forward Convergence (δrD) |
ϕ irj irk ↙ ↓ ↘ jrk j=k krj .
where i, j and k are distinct. |
β, Backward Convergence (βrD) |
β jri kri ↙ ↓ ↘ jrk j=k krj .
where i, j and k are distinct. |
(53; with my naming additions and copied text at the bottom)
(3.6b.5) In order to give the tableau rules for the ϕ and β constraints, we first need the rules for world equality. (See the table “α= World Equality (α=D)” above.) (3.6b.6) Priest next gives the rules for the ϕ and β constraints. (See them above.) (3.6b.7) Priest next gives an example tableau. (3.6b.8) We make counter-models in the following way. “For each number, i, that occurs on the branch, there is a world, w_{i}; w_{i}Rw_{j }iff irj occurs on the branch; for every propositional parameter, p, if p, i occurs on the branch, v_{wi}(p) = 1, if ¬p, i occurs on the branch, v_{wi}(p) = 0 (and if neither, v_{wi}(p) can be anything one wishes)” (p.27); however, “whenever there is a bunch of lines of the form i = j, j = k, . . . , we choose only one of the numbers, say i, and ignore the others” (54). (3.6b.9) “The tableaux for K^{t} and its various extensions are sound and complete with respect to their semantics” (55). (3.6b.10) The past cannot be altered, so there can only be one timeline of past events. But supposing that the future can be altered, then there are numerous timelines of sequences of future events. “Thus, one might suppose, time satisfies the condition β of backward convergence, but not the condition ϕ of forward convergence” (55-56). (3.6b.11) But just because there are two possible futures – ◊⟨F⟩p ∧ ◊⟨F⟩¬p) – does not mean there are two actual futures – ⟨F⟩p∧⟨F⟩¬p.
[Extensions of K^{t}]
[Modeling Time’s Structural Features Using Constraints on R]
[Denseness, Forward Convergence, and Backward Convergence]
[Tableau Rules]
[World Equality Rules]
[The Tableau Rules for ϕ and β]
[An Example Tableau]
[Counter-Models]
[The Soundness and Completeness of the Tableaux]
[Time as Backwardly Convergent but not Forwardly Convergent]
[The Non-Actuality of Divergent Futures]
Summary
[Extensions of K^{t}]
[We can apply constraints on the accessibility relation to obtain extensions of our modal tense logic K^{t}.]
[Recall from the prior section 3.6a that we are discussing the tense logic called K^{t}. In it, we model temporal succession using different worlds (for different moments), with the accessibility R relation having the intuitive sense of succession. So w_{1}Rw_{2} could be understood as saying something like: ‘w_{1} is earlier than w_{2}’. And “□A means something like ‘at all later times, A’, and ◊A as ‘at some later time, A’,” but “we will now write □ and ◊ as [F] and ⟨F⟩, respectively. (The F is for ‘future’)” (p.49, section 3.6a.2). Next recall from section 3.2.3 the constraints on the accessibility relation that generate variations of a modal logic:
ρ (rho), reflexivity: for all w, wRw.
σ (sigma), symmetry: for all w_{1}, w_{2}, if w_{1}Rw_{2}, then w_{2}Rw_{1}.
τ (tau), transitivity: for all w_{1}, w_{2}, w_{3}, if w_{1}Rw_{2} and w_{2}Rw_{3}, then w_{1}Rw_{3}.
η (eta), extendability: for all w_{1}, there is a w_{2} such that w_{1}Rw_{2}.
(p.36, section 3.2.3)
Priest says now that we can likewise apply constraints on the accessibility relation to obtain extensions of K^{t} , such as K^{t}_{ρ}, K^{t}_{ρσ} and so on.]
Extensions of K^{t} are obtained, as in the case of K, by adding conditions on the accessibility relation. In this way we obtain K^{t}_{ρ}, K^{t}_{ρσ}, etc.
(51)
[Modeling Time’s Structural Features Using Constraints on R]
[These constraints on R condition the way ‘x is before y’ behaves. For example, the transitivity constraint makes beforeness transitive, and we can represent the beginninglessness or endlessness of time using the extendability constraint.]
[These constraints on the R accessibility relation constrain the way that ‘x is before y’ behaves. Recall again the τ transitivity constraint:
τ (tau), transitivity: for all w_{1}, w_{2}, w_{3}, if w_{1}Rw_{2} and w_{2}Rw_{3}, then w_{1}Rw_{3}.
(p.36, section 3.2.3)
Now in this temporal context, the τ constraint makes beforeness be transitive “(if x is before y, and y is before z, then x is before z)”. In fact, this is already one of our normal assumptions about time. Now recall the η constraint:
η (eta), extendability: for all w_{1}, there is a w_{2} such that w_{1}Rw_{2}.
(p.36, section 3.2.3)
This would make it such that there can be no last moment of time. Priest then says that we can note its reversal as η′, meaning “for all x, there is a y such that yRx”, which then makes it so that there is no first moment of time. But reflexivity would not seem to match any of our intuitions about time, because it would make it that every point in time comes after itself, and the symmetry constraint is also not plausible, as it would make it so that whenever one moment comes before a second one, then the second one also comes before the first one.]
Thought of in tense-logical terms, the conditions on R are constraints on the way in which the temporal relation ‘x is before y’ may | behave. Thus, the condition τ says that beforeness is transitive (if x is before y, and y is before z, then x is before z), which we normally suppose it to be. The condition η says that there is no last point in time, and its reversal, η′ (for all x, there is a y such that yRx) says that there is no first point in time. These are, perhaps, more contentious, but still very natural. The conditions ρ and σ have, by contrast, little plausibility. The first says that every point in time is later than itself; the second says that if x is before y then y is before x.
(52-53)
[Denseness, Forward Convergence, and Backward Convergence]
[Priest next notes some natural constraints for tense logic. {1} denseness (δ): if xRy then for some z, xRz and zRy, which places a moment between any two others; {2} forward convergence (ϕ): if xRy and xRz then (yRz or y = z or zRy); that is to say, when two moments come after some given moment, then they cannot belong to two distinct futures but must instead fall along the same timeline; and {3} backward convergence (β): if yRx and zRx then (yRz or y = z or zRy); in other words, two moments coming before some given moment must fall along a single succession.]
[(ditto).]
In the context of tense logic, some other constraints are very natural, however. Some notable ones are:
δ (delta), denseness: if xRy then for some z, xRz and zRy
ϕ (phi), forward convergence: if xRy and xRz then (yRz or y = z or zRy)
β (beta), backward convergence: if yRx and zRx then (yRz or y = z or zRy)
The first of these says that, for any two times, there is a time between them; the second says that time cannot branch forward, so that if y and z are both later than x, they cannot belong to distinct ‘futures’: if they are not the same, one must be before the other. Similarly, the third says that time cannot branch backwards. Note that ϕ and β are vacuously satisfied if y is z, or either is x. Hence, the conditions need apply only to distinct x, y and z.
(52)
[Tableau Rules]
[Priest next gives the tableau rules for constrained tense logics.]
[First recall from section 2.4.3 and section 2.4.4 the tableau rules for modal logic.
Double Negation Development (¬¬D) |
¬¬A,i ↓ A,i |
Conjunction Development (∧D) |
A ∧ B,i ↓ A,i B,i |
Negated Conjunction Development (¬∧D) |
¬(A ∧ B),i ↓ ¬A ∨ ¬B,i |
Disjunction Development (∨D) |
A ∨ B,i ↙ ↘ A,i B,i |
Negated Disjunction Development (¬∨D) |
¬(A ∨ B),i ↓ ¬A,i ↓ ¬B,i |
Conditional Development (⊃D) |
A ⊃ B,i ↙ ↘ ¬A,i B,i |
Negated Conditional Development (¬⊃D) |
¬(A ⊃ B),i ↓ A,i ↓ ¬B,i |
(p.24, section 2.4.3 and 2.4.4)
And recall from section 3.6a.6 the tableau rules for the tense operators:
Full Future Development ([F]D) |
[F]A,i irj ↓ A,j
(For all j) |
Partial Future Development (⟨F⟩D) |
⟨F⟩A,i ↓ irj A,j
(j must be new: it cannot occur anywhere above on the branch) |
Negated Full Future Development (¬[F]D) |
¬[F]A,i ↓ ⟨F⟩¬A,i |
Negated Partial Future Development (¬⟨F⟩D) |
¬⟨F⟩A,i ↓ [F]¬A,i |
Full Past Development ([P]D) |
[P]A,i jri ↓ A,j
(For all j) |
Partial Past Development (⟨P⟩D) |
⟨P⟩A,i ↓ jri A,j
(j must be new: it cannot occur anywhere above on the branch) |
Negated Full Past Development (¬[P]D) |
¬[P]A,i ↓ ⟨P⟩¬A,i |
Negated Partial Past Development (¬⟨P⟩D) |
¬⟨P⟩A,i ↓ ⟨P⟩¬A,i |
(p.50, section 3.6a.6, with my added names and other data at the bottoms)
Next recall from section 3.3.2 the tableau rules for the ρ, τ, σ constraints.
ρ, Reflexivity (ρrD) |
ρ . ↓ iri |
σ, Symmetry (σrD) |
σ irj ↓ jri |
τ, Transitivity (ρrD) |
τ irj jrk ↓ irk |
We did not yet cover the rule for η from section 3.4. It is:
η, Extendability (ηrD) |
η . ↓ irj .
It is applied to any integer, i, on a branch, provided that there is not already something of the form irj on the branch, and the j in question must then be new. |
(see p.42, section 3.4; with my naming additions)
In section 3.4.5, Priest explains how in open tableaux, this rule causes it them to be infinite. Now recall from section 3.6b.3 above the δ denseness constraint: if xRy then for some z, xRz and zRy. It places a moment between any two others. As you can imagine there are infinitely many such moments within any finite interval no matter how small. Its rule is given below, and Priest notes that open branches will be infinite.
δ, Denseness (δrD) |
δ irj ↓ irk krj
.
where k is new to the branch. |
(As you can see, this tableau rule for density places a world-moment, k, between any pair, i-j.) He also gives an example. We should first try to interpret these formulations. Recall from section 3.6a.2 and section 3.6a.3 that [P]A means something like “at all past times, A”. But how do we read, [P][P]A ? Is it: “At all past times of all past times, A”? With what is said in Priest’s Logic: A Very Short Introduction, ch.8 that might be read as, “It always was the case that it always was the case that, A” (see pages 56 and 59). Priest will illustrate, showing that:
[P] [P] A ⊬_{Kt}[P] A
[P] [P] A ⊢_{Ktδ}[P] A
So let us try first to make sense of that. I am guessing that the idea is that no matter what moment in time you chose in the past, not only does A hold in that moment, but there will always be another past moment coming after it (but not including the present moment) when it held. So no matter how recent the past moment is to the present, there will be another one between it and the present (where A holds). This would work if time is dense, but not if there is a finite limited series of past events leading up to the present. But I am guessing. Please read the quotation.]
The tableau rules for ρ, τ, σ and η are as usual. That for η′ is an obvious modification of that for η. The rule for δ is:
δ, Denseness (δrD)
δ
irj
↓
irk
krj
.
where k is new to the branch.
(52; with my naming additions)
where k is new to the branch. (If a branch fails to close, this rule makes it infinite, as does the rule for η.) In K^{t}_{δ}, we have [F] [F] A ⊢ [F] A and its mirror image, [P] [P] A ⊢ [P] A (neither of which is valid in K^{t}, as may easily be checked). Here, for example, is a tableau for the latter:
⊢_{Ktδ} [P][P]A ⊢ [P]A
1.
.
2.
.
3.
.
4.
.
5.
.
6.
.
7.
.
8.
.
.
[P][P]A,0
↓
¬[P]A,0
↓
⟨P⟩¬A,0
↓
1r0
↓
¬A,1
↓
1r2, 2r0
↓
[P]A,2
↓
A,1
×
P
.
P
.
2¬[P]
.
3⟨P⟩
.
3⟨P⟩
.
4δr
.
1,6b[P]
.
7,6a[P]
(8×5)
valid
(52-53, enumeration and step accounting are my own and are probably mistaken)
The line 1r2, 2r0 is generated by the rule for δ.
(52-53)
[World Equality Rules]
[In order to give the tableau rules for the ϕ and β constraints, we first need the rules for world equality.]
[Recall from section 3.6b.3 above the ϕ and β constraints:
forward convergence (ϕ): if xRy and xRz then (yRz or y = z or zRy). It makes it so that when two moments come after some given moment, then they cannot belong to two distinct futures but must instead rather fall along the same time-line.
backward convergence (β): if yRx and zRx then (yRz or y = z or zRy). It makes it that two moments coming before some given moment must fall along a single succession.
Priest will now formulate the tableau rules for these constraints. But we need now to first add a pair of rules for =:
α= World Equality (α=D) |
α(i) j=i ↓ α(j) . α(i) i=j ↓ α(j) .
where α(i) is a line of the tableau containing an ‘i’. α(j) is the same, with ‘j’ replacing ‘i’. Thus: if α(i) is A, i, α(j) is A, j if α(i) is kri, α(j) is krj if α(i) is i = k, α(j) is j = k |
(53; with my naming additions and copied text at the bottom)
]
To formulate tableau rules for ϕ and β, we have to complicate things a little. Lines concerning the accessibility relation are now allowed to be of the form i = j as well as irj. There is a new rule (or to be precise, pair of rules) for =:
α(i) i = j ↓ α(j) α(i) j = i ↓ α(j) α(i) is a line of the tableau containing an ‘i’.
α(j) is the same, with ‘j’ replacing ‘i’. Thus: if
α(i) is A, i, α(j) is A, j
if α(i) is kri, α(j) is krj
if α(i) is i = k, α(j) is j = k
In fact, in the first case, we never need to (or will) apply the rules to lines where A is anything other than a propositional parameter or the negation of one (though the rule works whatever A is). And obviously, we do not need to apply the rule if it would produce a line that is already there (counting i = j as the same as j = i).
(53)
[The Tableau Rules for ϕ and β]
[Priest next gives the rules for the ϕ and β constraints.]
[(ditto)]
The rules for ϕ and β are now, respectively:
ϕ, Forward Convergence (δrD)
ϕ
irj
irk
↙ ↓ ↘
jrk j=k krj
.
where i, j and k are distinct.
β, Backward Convergence (βrD)
β
jri
kri
↙ ↓ ↘
jrk j=k krj
.
here i, j and k are distinct.
(53; with my naming additions and copied text at the bottom)
here i, j and k are distinct.
(53)
[An Example Tableau]
[Priest next gives an example tableau.]
[(ditto)]
In K^{t}_{ϕ}we have ⟨F⟩p ∧ ⟨F⟩q ⊢ ⟨F⟩(p ∧ q) ∨ ⟨F⟩(p ∧ ⟨F⟩q) ∨ ⟨F⟩(⟨F⟩ p ∧ q), which is not valid in K^{t}, as may easily be checked.^{7} (And the same for the mirror image of this in K^{t}_{β}.) Here is the tableau:
⟨F⟩p∧⟨F⟩q ⊢ _{Ktϕ}⟨F⟩(p∧q) ∨ ⟨F⟩(p∧⟨F⟩q) ∨ ⟨F⟩(⟨F⟩p∧q) | ||
1. . 2. . 3. . 4. . 5. . 6. . 7. . 8. . 9. . 10. . 11. . 12. . 13. . . 14. . 15. . 16. . . | ⟨F⟩p∧⟨F⟩q0
↓ ¬(⟨F⟩(p∧q)∨⟨F⟩(p∧⟨F⟩q)∨⟨F⟩(⟨F⟩p∧q))0 ↓ ¬⟨F⟩(p∧q)0 ↓ ¬⟨F⟩(p∧⟨F⟩q)0 ↓ ¬⟨F⟩(⟨F⟩p∧q)1 ↓ ⟨F⟩p0 ↓ ⟨F⟩q0 ↓ 0r1 ↓ p1 ↓ 0r2 ↓ q,2 ↙ ↓ ↘ 1r2 1=2 2r1 ↓ ↓ ↓ ¬(p∧⟨F⟩q)1 ¬(p∧q)1 ¬(⟨F⟩p∧q)2 ↓ ↓ ↓ ↙ ↘ ↙ ↘ ↙ ↘ ¬p1 ¬⟨F⟩q1 ¬p1 ¬q1 ¬⟨F⟩p2 ¬q2 × ↓ × ↓ ↓ × [F]¬q1 ¬q2 [F]¬p2 ↓ ↓ ¬q2 ¬p1 × × | P . P . 2¬∨ . 2¬∨ . 2¬∨ . 1∧ . 1∧
. 6⟨F⟩ . 6⟨F⟩ . 7⟨F⟩ . 7⟨F⟩ . 8,10ϕ . 4/8,3/8, 5/10,¬⟨F⟩ . 13¬∧ (13×9,11) 14¬⟨F⟩, 12/14α= 15/12[F] (16×11,9) valid |
(54, commas omitted to save space, enumeration and step accounting are my own and are probably mistaken)
The last formula on the right fork of the middle branch is obtained by the = rule.
(54)
[Counter-Models]
[We make counter-models in the following way. “For each number, i, that occurs on the branch, there is a world, w_{i}; w_{i}Rw_{j }iff irj occurs on the branch; for every propositional parameter, p, if p, i occurs on the branch, v_{wi}(p) = 1, if ¬p, i occurs on the branch, v_{wi}(p) = 0 (and if neither, v_{wi}(p) can be anything one wishes)” (p.27); however, “whenever there is a bunch of lines of the form i = j, j = k, . . . , we choose only one of the numbers, say i, and ignore the others” (54).]
[Recall from section 3.6a.8 how we make a counter-model in K^{t} for open branches: “Counter-models are read off from tableaux just as they are for K” (p.51, section 3.6a.8) and from section 2.4.7 for K:
Counter-models can be read off from an open branch of a tableau in a natural way. For each number, i, that occurs on the branch, there is a world, w_{i}; w_{i}Rw_{j }iff irj occurs on the branch; for every propositional parameter, p, if p, i occurs on the branch, v_{wi}(p) = 1, if ¬p, i occurs on the branch, v_{wi}(p) = 0 (and if neither, v_{wi}(p) can be anything one wishes).
(p.27, section 2.4.7)
Priest says we do it the same way, except “whenever there is a bunch of lines of the form i = j, j = k, . . . , we choose only one of the numbers, say i, and ignore the others” (54), and he gives an example.]
Reading off a counter-model from a tableau is the same as for K^{t}, except that whenever there is a bunch of lines of the form i = j, j = k, . . . , we choose only one of the numbers, say i, and ignore the others. (It does | not matter which we choose, because of the = rule.) For example, in K^{t}_{ϕ}, we have ⟨F⟩ p ⊭ [F](p∧q). Here is the tableau:
⟨F⟩p ⊭ _{Ktϕ}[F](p∧q) | ||
1. . 2. . 3. . 4. . 5. . 6. . 7. . 8. . 9. . 10. . 11. . . | ⟨F⟩p
↓ ¬[F](p∧q),0 ↓ ⟨F⟩¬(p∧q),0 ↓ 0r1 ↓ p,1 ↓ 0r2 ↓ ¬(p∧q),2 ↙ ↓ ↘ 1r2 1=2 2r1 ⫶ ↙ ↘ ⫶ ¬p,2 ¬q,2 ↓ ↓ p,2 p,2 × ↓ ¬q,1 | P . P . 2¬[F] . 1⟨F⟩ . 1⟨F⟩ . 3⟨F⟩ . 3⟨F⟩
. 4,6ϕ . 7¬∧ . 5,8α= (10×9) 8,9α= (open) invalid |
(55, enumeration and step accounting are my own and are probably mistaken)
The last two lines on the open branch shown are obtained by applying the = rule. All other applications of the rule produce lines that are already present. In reading off the counter-model from the completed open branch, since 1 = 2 occurs on the line, we can simply ignore all lines marked 2 to obtain:
xxx
xxxw_{0}xxx→xxxw_{1}
xxxw_{0}xxx→xxxp
xxxw_{0}xxx→xxx¬q
xxx
It is easy to check that the counter-model works.
(54-55)
[The Soundness and Completeness of the Tableaux]
[“The tableaux for K^{t} and its various extensions are sound and complete with respect to their semantics” (55).]
[(ditto)]
The tableaux for K^{t} and its various extensions are sound and complete with respect to their semantics. This is shown in 3.7.
(55)
[Time as Backwardly Convergent but not Forwardly Convergent]
[The past cannot be altered, so there can only be one timeline of past events. But supposing that the future can be altered, then there are numerous timelines of sequences of future events. “Thus, one might suppose, time satisfies the condition β of backward convergence, but not the condition ϕ of forward convergence” (55-56).]
[Now things get philosophically interesting. Priest next discusses how these constraints can help clarify certain debates about the structure of time. Recall for instance Diodorus’ “Master Argument” from Goldschmidt’s Le système stoïcien et l'idée de temps section 2.1.4.3.49.1 and section 2.1.4.1.35.2. Part of the reasoning in it is that the past cannot be altered, so it is necessary. So there can only be one past. That furthermore means that anything that in fact did happen in the past would have to both be on the same timeline. Now recall yet again from section 3.6b.3 above the ϕ and β constraints:
forward convergence (ϕ): if xRy and xRz then (yRz or y = z or zRy). It makes it so that when two moments come after some given moment, then they cannot belong to two distinct futures but must instead rather fall along the same time-line.
backward convergence (β): if yRx and zRx then (yRz or y = z or zRy). It makes it that two moments coming before some given moment must fall along a single succession.
As we can see, the past is most probably backwardly convergent. But what about the future? Many think that there can be many different possible futures, depending on the decisions that are made. We see this asymmetrical temporal structure in Nolt’s “Picture of Time” diagram from his Logics section 13.2.1 (after which is quotation of Nolt’s writing):
Ordinarily we understand time as a linearly ordered sequence of moments. We have a position in time, the present moment. All other moments lie in either the past or the future. The present constantly advances toward the future, and this advance gives time a direction. The past is a continuum of moments stretching behind us, perhaps to infinity. It is unalterable. Whatever has been is now necessarily so. The future, however, is not frozen into unalterability but alive with possibilities. Starting with the present, events could take various alternative courses. There is, in other words, more than one possible future. Though only one of these courses of events will in fact be realized (we may not, of course, know which one), still the others are genuinely possible, in a way that alternative pasts are not genuinely possible. These intuitions suggest a model on which time is like a tree with a single trunk (the past) that at a certain point (the present) begins to split and split again into ramifying branches (various possible futures). As time moves forward, the lower branches (formerly live possibilities, lost through passage of time ) disappear. Only one path through the tree represents the actual course of time, that is, the actual world. More and more of the path is revealed as time moves on and lower branches vanish. If time were finite, eventually all the branches representing merely possible futures would disappear and only this single path from trunk to branch tip would remain: the entire history of the actual world from the beginning to the end of time. But we might also think of time as infinite-at least toward the future and perhaps also backward into the past. If time is infinite toward the past, then the tree's trunk extends endlessly downward, never touching ground; and if time is infinite toward the future, then its branches stretch endlessly upward, never touching the sky. In either case, we might picture at least a part of the tree like the diagram in Figure 13.1.
[...]
This is a picture we often use in decision making. Suppose I am considering whether to go to the mountains for a hike or just stay at home and relax this weekend. These are (we assume) real possibilities, though undoubtedly not the only ones. Corresponding to each is at least one possible world – that is, at least one course of events that the world might take from the beginning of time through and beyond the moment of my decision. Suppose I decide to hike and I carry out that intention. Then the world (or one of the worlds) in which I hike is the actual world, and the worlds in which I stay at home that weekend are possible but nonactual. In these nonactual worlds, everything up to the moment of my decision occurs exactly as it does in the actual world, though events depart from their actual course more or less dramatically thereafter.
(Nolt p.365; 368 section 13.2.1)
So as Priest writes: “Thus, one might suppose, time satisfies the condition β of backward convergence, but not the condition ϕ of forward convergence” (55-56).]
Some of the interesting philosophical issues related to tense logics concern the structure of time itself. For example, it is natural to suppose that the future is open in a way that the past is not. Let p describe some future event that it is within my power to make true, and within my power to make false. (So p might be ‘I will father a third child’ — well, with a little help from at least one other person!) Then there would seem to be different futures, in one of which p is true, and in the other of which it is not. The same is not the case for a q about the past (e.g., ‘I fathered at least two children’). I am not now able to render this either true or false at will. (It is | just true, and nothing I do can change this.) Thus, one might suppose, time satisfies the condition β of backward convergence, but not the condition ϕ of forward convergence.
(55-56)
[The Non-Actuality of Divergent Futures]
[But just because there are two possible futures – ◊⟨F⟩p ∧ ◊⟨F⟩¬p) – does not mean there are two actual futures – ⟨F⟩p∧⟨F⟩¬p.]
[Priest next notes a complication with the idea from the above section 3.6b.10. We are supposing that there are two possible futures. We would write that as ◊⟨F⟩p ∧ ◊⟨F⟩¬p. In one future Priest will have three kids, and in another he will never have three kids. He says that nonetheless that formulation is compatible with future convergence, but it would require that we fashion a language that handles both modal and tense operators. (It is not immediately clear how intuitively speaking they are compatible. I will guess the following. Maybe the idea is that future convergence does not apply to possible futures but it does apply for certain futures. I really do not know. But also interesting here is that Priest says that “it is not clear that ⟨F⟩p∧⟨F⟩¬p,” that is, “that there are two actual futures” (56). This is especially interesting to consider. I want to wander down the following path, but feel free not to join me. In section 2.5 we asked, what exactly do possible worlds and their semantics represent, philosophically speaking? In section 2.6 we considered modal realism, which says that possible worlds are real worlds that exist at different times and/or places. In section 2.7 we saw modal actualism, which says that possible worlds are abstract entities, like numbers. And in section 2.8, we learned that according to Meinongism, possible worlds are thought to be non-existent objects. What I have in mind is Deleuze’s notions of undecidability, the affirmation of chance outcomes, and incompossible worlds. Let me propose “modal virtualism”. According to this view, there is the actual world, there are possible worlds, and there are virtual worlds. A possible world is a non-existent object, although it can become actual if it is actualized. A virtual world is one that is not actual, but is no less real than the actual world. A possible world raises to the status of a virtual world, that is, it becomes real but not actual, when it enters the actual timeline of events. But, although it is on the actual timeline, it is not actual. One way we can see something like this idea is with Deleuze’s use of Stoic prohairesis. The world is causally headed in one direction, but we can introduce alternate tendencies of development to bend the trajectory of time. We bifurcate the timeline by giving the world another path it can go down, and we convert the possible alternatives to real and virtual alternate futures by affirming how chance will decide. So in that moment that we affirm this chance, we make the alternate possibilities real, as they are virtual (that is, they are on the actual timeline), even though they are not yet actualized. The point here is that in reality there can be two futures at particular critical present actual moments, because in those moments we literally find ourselves tending really and actually down two different future paths. Again, that does not mean both futures will be actualized. Only one will. But then, consider when in the future one of them is in fact actualized and the other does not obtain. There will still be a time in the past when it was a real future. As such, in the future when the alternative is actualized instead, we can think of the non-actualized one as coexisting with the actual, still in a virtual state. Consider moments when you made a critical decision that drastically changed the course of your life and fundamentally changed who you are as a person. There may have been a moment before your decision when you affirmed that the path you were going along is not determined, that the momentum of your life can actually change. So before you acted out your choice, you may first have affirmed that the future you were going down was not the only real future you had. You saw also that you have an alternate real future. And with the initial effort and impulse you send through you body to change your behavior (a friend of mine for example describes throwing his pack of cigarettes out the window), you have sent into the world tendencies of development taking the present also down another path. In other words, you combine simultaneously the tendencies of the world’s given momentum with these alternate tendencies that open the door to the world going down a divergent path. So the world in that present moment is actually going down two divergent virtual future paths. And, you did not determine the path, because we do not really have that much determinative power of the course of events in the cosmos. Rather, you are affirming that in the future there is a difference to the path you are on, and you affirm that difference, and thereby make the future undecidable. That does not mean you have falsified the future paths or even subtracted their truth values, making them truth-gaps. Rather, you have affirmed that both are real, even though neither are actualized yet, and thus they are undecidable. Thus in those cases ⟨F⟩p∧⟨F⟩¬p would be evaluated as true (my friend was both a smoker in his future and a non-smoker, that instant he first affirmed the virtuality of this alternative and just began to send new impulses to motion into his muscles, in simultaneous combination with the ones already there from his smoking habits), and it would be understood from the perspective of modal virtualism as being that both futures are real, because the present is actually tending in both directions of virtual futures, even though only one of the two will eventually be actualized.)]
This is less than clear, though. Granted, there are two possible futures concerning p; it does not follow that there are two actual futures. Certainly, ◊⟨F⟩p ∧ ◊⟨F⟩¬p; but it is not clear that ⟨F⟩p∧⟨F⟩¬p. The first of these is quite compatible with future convergence. To establish this, however, requires a semantics for a language with both tense and modal operators. I leave details of this as a non-trivial exercise.
(56)
From:
Priest, Graham. 2008 [2001]. An Introduction to Non-Classical Logic: From If to Is, 2nd edn. Cambridge: Cambridge University.
Or if otherwise noted:
Nolt, John. 1997. Logics. Belmont, CA.
Priest, Graham. 2000. Logic: A Very Short Introduction. Oxford: Oxford University.
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