7 Jun 2014

Priest (11.5) In Contradiction, ‘The LCC and Contradiction’, summary

by Corry Shores
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[The following is summary. My own comments are in brackets, but please consult the original text, as I am not a logician. All boldface and underlining are my own.]

Graham Priest

In Contradiction:
A Study of the Transconsistent

Part III. Applications

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

11.5 The LCC and Contradiction

Brief Summary:

By applying Liebniz’ Law of Continuity in states of change, we can say that during such transitions something is both in a state and not in that state.


[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.

The Δ sort of change involves something being both in a state and in its succeeding state at the same time.] Priest will now apply Leibniz’ Law of Continuity (LCC, for Leibniz’ Continuity Condition) to Δ type changes. [If something changes continuously from one state to another, there had to be a moment when both states held together.]

So much for the LCC itself. Let us now return to the question of the existence of type Δ changes, and apply the LCC. For the LCC implies that any change from a continuous state of p to a continuous state of ¬p is a type Δ change. More generally, suppose that φ and ψ are any distinct literals (propositional parameters or their negations). Suppose that prior to time t system s is in state s0: φ is true. Posterior to time t, s is in state s1: ψ is true. Since s0 occurs arbitrarily close to t (and continuously), it occurs at t by the LCC. But s1 occurs arbitrarily close to t (and continuously). Hence it too occurs at t. Thus, at t there is a nexus state at which both φ and ψ are realised. In particular, if φ is p and ψ is ¬p, p ∧ ¬p is realised at t. The LCC therefore implies that contradictions are realised at the nodal points of certain sorts of change.
(Priest 169)

Priest will now formulate this in tense logic semantics [see especially section 11.3. Recall that W is the set of moments. Propositions describe the states at certain moments. We can add the following modifiers to designate temporal locations.

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 …”
Galton’s Stanford Encyclopedia article on tense logic.)


We can reproduce this argument in the tense logical semantics. Suppose that W is an appropriate stretch of time and that | g is the nodal point of a change from φ to ψ; i.e., Hφ ∧ Gψ holds at g. Then, assuming only that g is suitably distant from the ends of a < chain, it follows that FF(q ∨ ¬q) and PP(q ∨ ¬q) hold at g, whence, by (1) of the previous section [{PP(q ∨ ¬q), Hp} ⊨ p ], φ ∧ ψ holds at g. The dialetheia produced at a type Δ change need not be instantaneous (for all I have said so far, though this is a plausible additional constraint). For example, the interpretation with real time where vx(p) is {1} if x < 0, {1, 0} if 0 ≤ x ≤ 1, and {0} if x > 1, is quite compatible with the LCC. Still, there must be at least an instantaneous dialetheia.
(Priest 169-170)

[So we think of moment g that is a transitional place between a prior period when φ held and following period when ψ will hold. This means that there is some specific prior moment or moments when (q ∨ ¬q) and some specific future moment when this holds as well too. (Again I do not know why it is important for it to be this particular formulation). But this means that whatever held in the prior period and whatever held in the forthcoming must both be held at g, in accordance with Priest’s formulation of LCC. Thus at g both φ and ψ hold, making it a type Δ change. Regarding the idea that the change need not be instantaneous, he seems to be saying that we can think of a range of moments when both φ and ψ together hold, but this is still compatible with LCC, since what holds just prior to that period also holds within it.]

Priest will now show that LCC can be applied not just to discrete changes like the above but also to continuous ones. He has us consider a body moving according to a particular equation. In it, x is the position and t is the time: x = kt (k≠0). So this formula will have a value at a particular time. It will have a different value before and after that time. But by LCC, that means it both will and will not have these different values.

It is not only for discrete changes that the LCC can be applied to show that contradictions arise. The LCC entails that contradictions arise in continuous change too. For example, consider a body that moves in accordance with the equation x = kt (k ≠ 0), where x is its position and t is the time, both with respect to some suitable coordinate system. Consider a point t0. At t0, x = kt0. But for all points after (and before) t0, x ≠ kt0. Hence, by the LCC, at t0, x ≠ kt0. Thus at t0, x = kt0 and x ≠ kt0. And since t0 was arbitrary, we see that motion produces a continuous state of contradiction. What this might possibly mean I will return to in a moment; we can at least see it as vindicating dialecticians, such as Hegel, who claimed that change would be impossible without contradiction. As he put it, [the following is quotation]
. . . contradiction is the root of all movement and vitality; and it is only in so far as something contains a contradiction within it that it moves, has an urge and activity.
(Priest 170, quoting Hegel [1982] p.439 of the English translation)

People sometimes complain that they cannot conceive how it is for something to be in a state of true self-contradiction. But as we can see, it happens in moments of transition, which we commonly experience.

The thesis that contradictions arise at the nodal points of certain transitions can also be used to free the mind of a certain mental cramp that often arises when people consider dialetheism. A commonly heard complaint is as follows (said with an air of puzzlement): ‘I just cannot see what it would be like for a contradiction to be true, what it would be like, for example, for something to be a cup and not a cup, or for a person to be in a room and not in a room.’ The answer to this (objection?) should now be obvious: something is a cup and not a cup the instant it breaks into pieces. Someone is in and out of the room the instant they leave. Contradictions occur at the nodal points of certain transitions and, as such, are perfectly familiar.

Regarding such moments of change when contrary states coincide, we can even think of it as a state in itself, a state of change. [In this respect is might be similar to Leibniz’ status transitus.] However, this might lead to infinite regress, for we then need to account for how we got to the state of change. Priest says there is not this problem, because in the nexus state getting into the state of change, the changed state both holds and does not hold [see footnote 17 below]:

We have seen that a certain kind of change from a holding to β holding, produces a nexus state where α ∧ β holds. We may, however, go a step further. We may take the nexus state produced to be the state of change itself. The state described by α ∧ ¬α just is the state described by a changing into the state described by ¬α. Thus, there is such a thing as a state of change, and it does take time, if only an instant. Notice how this relates to the discussion of the LCC in section 11.4. Not only is there a state of change that takes time, but it commences while the prior state obtains and terminates only after the posterior state has begun.17
[Foootnote 17: If we suppose there to be states of change, does this not start an infinite regress? For what of the change between, e.g. the prior state, described by a, and the state of change, described by α ∧ ¬α? | There is no infinite regress. The nexus state between these two states is described by a α ∧ (α ∧ ¬α), i.e. α ∧ ¬α, which is the original nexus state. Thus, to be changing into a state of change is already to be in that state of change, as one might expect.]

Motion is in a continual state of contradiction, because each instant it is both entering and leaving its location.

The notion that a contradictory state is a state of change also starts to make sense of the fact that motion is a continuous state of contradiction. For the contradictory state of the body at t0 in the above example, x = kt0 ∧ x ≠ kt0, is then indicative of the fact that the body is not only occupying the spot kt0, but, since its occupation is instantaneous, is at the same time both entering and leaving the spot. All this suggests that the thesis that certain kinds of contradictory state are states of change should be investigated further. To this I turn in the next chapter.



Most citations from:

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

Or otherwise as noted from:

Hegel, G. W. F. (1812) Wissenschaft der Logik, published in English translation by A. V. Miller as Hegel’s Science of Logic, Allen & Unwin, 1969.




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