Priest (11.3) In Contradiction, ‘Dialectical Tense Logic’, summary

by Corry Shores
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Graham Priest

In Contradiction:
A Study of the Transconsistent

Part III. Applications

Ch.11. The Metaphysics of Change I:
The Instant of Change

11.3 Dialectical Tense Logic

Brief Summary:

Priest needs a semantical system for expressing his dialetheic logic of change. Here he formulates one using tense logic, which allows for specifications to be made about past, present, and future.

Summary

Priest’s “dialectical idea that contradictions may be realised in a process of change” needs a suitable logical system. Here he will formulate a tense logic for propositional languages [see Galton’s Stanford Encyclopedia article for more on tense logic].

In such a language, we will have formulas [propositions I suppose, or their parts perhaps too]. We will call the set of all such formulas L. [Tense logic is a sort of modal logic. In modal logics, we allow for operators to qualify the statement, for example, by adding ‘it is possible that’ or ‘usually’. In the case of tense logic, there are four modalities:

P     “It has at some time been the case that …”
F     “It will at some time be the case that …”
H     “It has always been the case that …”
G     “It will always be the case that …”

We can obtain H and G from P and F. So first consider a proposition that can take F, for example, ‘It rains in Boston’. Then Fp would be ‘It will at some time be the case that it rains in Boston’. F¬p would then be ‘It will at some time be the case that it does not rain in Boston,’ and since there will be at least one time it does not rain, we can render it as ‘It will not always rain in Boston.’ This would be different than ¬Fp, which is literally ‘It is not the case that it will at some time rain in Boston.’ This means that there will be no time that it rains in Boston and thus we can render it “It will never rain in Boston.” If we say that ¬F¬p, then this would be ‘It is not the case that there is not a time in Boston when it will rain.’ So if there is not time that it does not rain, then it rains at all times and thus we can render it “It will always rain in Boston.” Instead of ¬F¬p, we can just write Gp, with G meaning “It will always be the case that”. And all this holds for the past P, and ¬P¬p, which means “It is not the case that there was not a time in Boston when it rained” and thus “It always rained in Boston”, can be given as Hp, with H meaning “It has always been the case that.” (based on pp.4-5 of McArthur, Tense Logic.) What seems to be important here is that for H and G, the proposition holds for all times, such that if it were about raining in Boston, then it means it rains at all times.]

Let L be the set of formulas of the extensional language of section 5.2 augmented by the two monadic tense operators P and F, thought of as meaning ‘it was the case that’ and ‘it will be the case that’, respectively. The operators H and G (‘it was always the case that’ and ‘it will always be the case that’) are thought of as defined in the usual ways as ¬P¬ and ¬F¬ respectively. (163)

[Normally in logic we have two truth values, true and false or 0 and 1. But dialetheic logic has a third option, being both 0 and 1.]

As before, let π be the set of truth values
{{0}, {1}, {0, 1}}.
(Priest 163)

[It seems that in this language Priest will want to make statements like, ‘it is true that one instant comes before this other one.’ To allow for such interpretations, he will need to formulate a ‘model’ which gives the terms of the language and also the rules for assigning functional interpretations of those terms. Here he will have a set of instants in a set called W. The before relation among those terms (the instants) is < and the after is >. Then v is a function which assigns a truth value to a formulation in this language.]

An interpretation for the language is a triple < W, <, v >, where W is a set of temporal instants, < is a relation on W and, for any x ∈ W, v(x) (v_x) is a map from propositional parameters to π. < is thought of as a relation of temporal precedence and, despite the notation, need not be an order, though this is a very natural further condition to put on it. For the present, we will impose no requirements on <. The converse relation of < will be written as >.
(163)

[Recall that P means ‘it was the case that’. The following formulation seems to mean that if we say at moment x that ‘it was the case that something happened’, then this is true if there is a moment coming before x when that something happened. The following might be read as, ‘true is one of the values for a statement modified by P (and finding its reference point at x) when there is a prior moment y when that statement was true.]

The additional clauses required for the tense operators are:
(Pa) 1 ∈ v_x(Pα) iff for some y < x, 1 ∈ v_y(α)
(163)

[Likewise, it would not be true if the statement were false for all prior moments.]

(Pb) 0 ∈ v_x(Pα) iff for all y < x, 0 ∈ v_y(α)
(163)

[Now recall that F meant ‘It will at some time be the case that’. This is true if there is a moment following the one of the statement where it is true, and it is false if there are no following moments when it is true.]

(Fa) 1 ∈ v_x(Fα) iff for some y > x, 1 ∈ v_y(α)
(Fb) 0 ∈ v_x(Fα) iff for all y > x, 0 ∈ v_y(α)
(163)

[Recall that H means ‘It has always been the case that’ and G means ‘It will always be the case that’. We just use the same reasoning to find their values. For example, for a statement modified by H at instant x, that statement would need to hold for all prior moments.]

(Ha) 1 ∈ v_x(Hα) iff for all y < x, 1 ∈ v_y(α)
(Hb) 0 ∈ v_x(Hα) iff for some y < x, 0 ∈ v_y(α)

(Ga) 1 ∈ v_x(Gα) iff for all y > x, 1 ∈ v_y(α)
(Gb) 0 ∈ v_x(Gα) iff for some y > x, 0 ∈ v_y(α)
(Priest 163)

[Now recall from section 11.2 the four types of changes we discussed.

Before a time t₀, a system s is in a state s₀, described by α. After t₀ it is in a state s₁, described by α. What state is it in at t₀? A priori, there are four possible answers:
(A) s is in s₀ and s₀ only.
(B) s is in s₁ and s₁ only.
(Γ) s is in neither s₀ nor s₁.
(Δ) s is in both s₀ and s₁.
(160)

The delta sort of change involves something being both in a state and in its succeeding state at the same time.]

This semantical system will allow us to model Δ sorts of changes. So let’s suppose that person b is in a room before time t₀. Then he leaves the room exactly at that time. Let’s also suppose that p stands for ‘b is in the room.’ [Now, this statement would be true if the moment in question comes before the time he leaves; it would be false if it happens after; but it would be both true and false if it happened exactly at that moment. So]

v_x(p) = {1} if x < t₀

v_x(p) = {0} if x > t₀

v_x(p) = {0, 1} if x = t₀
At x = t₀, 1 ∈ v_x(p ∧ ¬p), showing the contradiction realised in this type Δ change.
(164)

Priest then adds to the system to allow for entailment [see p.164 for details, and additional discussion pp.164-165].

[Priest now seems to make a formulation saying that, for example: moment x comes before moment y, but it is true b is in the room at x but false at y, then there must have been an intermediary moment during which b was both in and not in the room in that instant.]

If x < y and v_x(p) ≠ v_y(p), then there is a z such that x ≤ z ≤ y and v_z(p) = {0, 1}
(165)

[The next formulation seems to be saying that if something holds (now), but at one time in the past did not hold, then either it both holds and does not (now) or at one time in the past it held and did not. So if you are in the room but previously you were not, then either right now you are in a transitional state when you are both in and not in, or there was a transitional moment in the past when you were both in and not in.]

This condition is sufficient to verify what we might call ‘Zeno’s principle’:

{p ∧ P¬p} ⊨ (p ∧ ¬p) ∨ P(p ∧ ¬p)
(165)

Most citations from:

Priest, Graham. In Contradiction: A Study of the Transconsistent. Oxford/New York: Oxford, 2006 [first published 1987].

Or otherwise as noted from:

McArthur, Robert P. Tense Logic.  Dordrecht / Boston: D. Reidel, 1976.