6 Jun 2014

Priest (11.4) In Contradiction, ‘The Leibniz Continuity Condition’, summary


by Corry Shores
Search Blog Here. Index-tags are found on the bottom of the left column.]

[Central Entry Directory]
[Logic & Semantics, Entry Directory]
[Graham Priest, entry directory]
[Priest’s In Contradiction, entry directory]

Graham Priest

In Contradiction:
A Study of the Transconsistent

Part III. Applications

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

11.4 The Leibniz Continuity Condition

Brief Summary:

One application of the dialetheic logic of change is Leibniz’ law of continuity. It says that what holds up to a limit holds at the limit too. Priest formulates this to mean that what holds at an intermediary moment holds as well at the surrounding moments it intermediates.


Priest previously formulated his dialectical conception of true contradiction in change in a formal semantical system using tense logic, which allows us to assign the truth value for instants of transition when it is both the case and not the case that something is in a particular state. The function that we use to assign truth values in that system was called v.  He ended that section by saying that “We will now look at a more sophisticated and important example of a condition that might naturally be placed on v.” (165)

He says now that this example incorporates a continuity principle into the semantical system. He begins with the historical background of this notion, going back to Leibniz’ Law of Continuity. [see excellent analysis here by Katz and Sherry. The Leibniz text Priest looks at is also examined here.] Quoting Leibniz

When the difference between two instances in a given series or that which is presupposed can be diminished until it becomes smaller than any given quantity whatever, the corresponding difference in what is sought or in their results must of necessity also be diminished or become less than any given quantity whatever. Or to put it more commonly, when two instances or data approach each other continuously, so that one at last passes over into the other, it is necessary for their consequences or results (or the unknown) to do so also.
[Priest 165, quoting Leibniz, (1687) ‘Letter of Mr. Leibniz on a General Principle Useful in Explaining the Laws of Nature through a Consideration of the Divine Wisdom; to Serve as a Reply to the Response of the Rev. Father Malebranche’. Published in English translation in Leibniz’ Philosophical Papers and Letters, ed. L. E. Loemker, Reidel, 1969, 351–4.]

[[Priest then gives a mathematical formulation for one interpretation of the principle. He seems to be working in a literal way with the first of Leibniz’ two explanations. If two values are brought infinitesimally close, then their difference will be 0 and they will be so close with so little difference between their values that they can be thought to be equal. In terms of the geometrical interpretation, this would be like two points brought so close together that no other points stand between them. In a sense, both points are inseparable and form a unit, because there is nothing extending between them to separate them. However, this unit can be thought of as a binary value, because there are still two points, two values. In terms of dialetheic logic, this allows for in the same instant (one infinitesimal interval) something to be in two states, or in the case of motion, two locations, all at once. In Russell’s at-at theory of motion, both moving objects and objects at rest are only ever at one location and never more. However, with the idea of the principle of continuity and the infinitesimal, we can think of there being moments when moving objects are between immediately neighboring locations and thus having one side facing the former and another side facing the latter.]] In the least Leibniz would be saying that:

(Priest 165)

But he is saying more as well. As his examples show, he intends it to apply to “limiting processes—not just arithmetic, but geometric, physical, temporal, and so on” (165).

In virtue of this, we might state the principle thus: given any limiting process, whatever holds up to the limit holds at the limit; or, as L’Huilier, who, like most eighteenth-century mathematicians, endorsed the principle, put it: if a variable quantity at all stages enjoys a certain property, its limit will enjoy the same property.

The continuity principle could not apply to all possible cases, because then we would say that “every real number is rational (since it is the limit of a sequence of rationals).” Also “it is not the case that every parabola is a closed and bounded figure, even though every ellipse is closed and bounded.” [165, see also Katz and Sherry, Leibniz, Leibniz again, and an explanation here for that example.] Priest assumes that Leibniz thought there was some limitation to the application of the principle, but it is not clear what that is. 

Priest will now focus on changes in physical states of affairs. If we think of these changes as events, we could formulate it as: “any event that is occurring at a continuous set of times is occurring at any limit of those times”. If we do away with the “limit jargon” we could formulate it thus:

anything going on arbitrarily close to a certain time is going on at that time too. Let us call this, in honour of Leibniz, the Leibniz Continuity Condition, LCC for short.

Leibniz’ justification for the principle is this. If at the limit the principle did not hold, then for some reason, that would mean the behavior at the limit would be capricious. [This seems to assume that it would not hold because of a law of non-continuity which would say that whatever holds up to the limit does not hold at the limit, in which case it would be law governed.] But since God, the designer of the world, is not capricious, that means the world cannot behave in this capricious way. Later Priest will reexplain this justification in a non-anthropomorphic way.

Priest now looks for a way to establish the LCC. We cannot verify it empirically, since no clock is precise enough [to measure the infinitely brief]. In mathematics, the value of a function at some argument is logically independent of the value at others, so capricious behavior is not mathematically impossible. But in nature, neighboring moments are not independent.

How might one establish the LCC? Clearly there is no possibility of verifying the principle by experiment. No measuring instrument, particularly no clock, is accurate to more than a finite number of decimal places. There is therefore no way in which we might hope to observe the situation at a certain time to the exclusion of states at arbitrarily close times. Neither is there any question of proving the principle by pure mathematics. There is nothing mathematically impossible in such capricious behaviour. This is because the mathematical representation of states of affairs is quite atomistic. The value at some argument of a mathematical function in extension is logically independent of its value at all others. But it is precisely here that nature may plausibly be thought to differ from such a representation. For succeeding states of affairs in nature are not atomistic: there are connections. This would be denied by a Humean. For her, if the principle held it could only be by a global accident. I therefore see no possibility of convincing a Humean of the plausibility of the principle. But of course, for a Humean, every sequence of events is a global accident; hence there is no possibility of convincing her of anything. Let us therefore leave this scepticism aside. For the non-sceptic there are nexuses that serve to make the state of affairs at a certain time dependent on those at other times.

Priest will now discuss why a change that violates LCC would be unintelligible. Unless we say that there is a moment when a changing thing is in its former and following states at the same time, then we would have to say the change happened during no time at all. If we say that something is in one state up to a particular time, with not transitional state, then when does the transition happen? We cannot say the change happened before the limit, because it has not yet happened. We cannot say it happened at the limit, because then it already happened. But since there is no time in between or no transitional status where both states were together at the same time, then the change must have happened without happening in time.

I suspect that a change which violates the LCC is capricious in the sense that it is incompatible with the existence of some of these nexuses. How does this work exactly? There is, I think, a good deal to be said about this, and the following is at least part of it. A change that violated the LCC would be unintelligible because of the following sorts of considerations. Let us suppose that a state of affairs, s, holds before, and all the way up to, a limit time, t, but fails at t. Then, clearly, a change has occurred. But when did this change occur? It cannot occur before t, since at any time before t there are later times at which s held; but it cannot occur at t (or at any subsequent time), because at this time the change is all over: s is already terminated! We can reason similarly if the state holds after, and at all times down to, a prior limit time, t, but not at t. When did the change occur? It cannot happen after t—that is too late: at any time after t there are prior times at which s already holds; but it cannot happen at t (or at any prior time), because at that time the change has not yet started: the old state is still in place. It therefore seems, in either case, that something, namely a change, has occurred, but that it took place at no time. But this is very strange. We may countenance things that happen very quickly, but if something happens it must take some time, if only an instant. (For just this reason, theories of action at a distance, which require something to happen in no time, namely the transmission of an effect, have always been felt philosophically puzzling.) A possible response to this train of | thought is simply to deny that there is any such thing as change itself. A change occurs when one state is replaced by another, and that’s that. This response just endorses the cinematic account of change, which we met in section 11.2. As we noted there, it, too, is highly counter-intuitive.

Priest will now incorporate LCC into the semantics of tense logic. [On the real number line, there are intervals with limits that lie outside the interval. Recall Russell’s description of intervals whose limits lie outside them: “a limit may or may not belong to the class u of which it is a limit, but it always belongs to some series in which u is contained, and if it is a term of u, it is still a limit of the class consisting of all terms of u except itself.” (p. 279 of Bertrand Russell. Principles of Mathematics. London/New York: Routledge, 2010 [1st published 1903].)]

we notice that in the real line (which is the paradigm representation of time), with the usual ordering and topology, the (open) continuous intervals are just sets of the form {x | r < x < s} for real numbers r and s; and r and s are the only limits of the interval that are not already in it.

[To understand the following formulation, first note that a ‘parameter’ is “A variable to which arbitrary values may be assigned for a specific purpose” (p. 159 of Greenstein Dictionary of Logical Terms and Symbols). A propositional parameter would be a variable which would take one proposition or another. “basic propositions will be represented in PL [propositional logic] by simple capitals letters (called “ sentence letters,” “propositional constants,” or “propositional parameters”): A, B, C, …, P, Q, R, …” (p.31-32 of Smith Logic: The Laws of Truth). Priest emphasizes that his formulation works with propositional parameters, because he means that the propositions involved do not have a tense, so they refer only to their own moment and not to one coming before or after. We can say that ‘I am alive’ applies to all moments we are alive and as well to the transitional moment into our death, during which we as well are not alive. But ‘there is a later moment of my life’ does not apply at that final transition. The formulation then seems to say that if a proposition holds (or does not hold) at some moment, then it holds (or does not hold) during its immediate predecessor and successor as well. In this formulation, it seems to imply that the moments are in immediate succession, and the brackets seem to mean that the formulation can be read as having all trues or all falses.]

For every propositional parameter, p, and every x, yW, if 1[0] ∈ vz(p) for every z such that x < z < y, then 1[0] ∈ vx(p) and ∈ vy(p).

under no circumstances should it be extended to tensed formulas. For suppose the LCC did apply to tensed formulas, and consider the moments of someone’s life. Being alive is certainly a continuous state of affairs, and so we can apply the LCC to conclude that this set contains all its limit points. In particular, it has a last moment, assuming, of course, that it does not go on for ever. Call this z. At any point prior to z, ‘There will be a (later) time of life’ is true. If we could apply the LCC to tensed sentences we could apply it to this one to conclude that it is true at z, which, manifestly, it is not. The point, of course, is that, though ‘There will be a later time of life’ may be true at time t, it does not describe a state of affairs that holds at time t in the pertinent sense.

[To understand the next formulation, we should establish a few things. The first is that if the is no tense modifier, we can perhaps assume the proposition refers to this current moment or the moment it is spoken.

(6') T'(It is sunny and warm today)
a certain redundancy is present in (6'); not only does the statement operator T signify present tense but the interior statement in is the present tense as well. Thus the T operator is superfluous and can be dispensed with.
(McArthur 3)


'It did rain in Boston but isn't now' being Pp & ~p
(McArthur 4)

The next thing to note is what happens when we combine two P tense markers.

Below are statements in two common perfect tenses (past and future) with symbolizations.
(8) It had rained in Boston = PPp
(9) It will have rained in Boston = FPp.
(McArthur 4)

Note that the past perfect implies two temporal positions in the past, with one preceding the other.

Past Perfect
[had + past participle]
Examples: You had studied English before you moved to New York.
Past perfect. englishpage
(Text and image above from englishpage.com)

So in the following proposition, it seems we are referring to q or not-q holding a moment before another past moment, p always having held in the past, and p holding now in the present.]

it is clear that the LCC will validate the following inference:
{PP(q ∨ ¬q), Hp} ⊨ p

[This seems to mean that if q or not-q hold a moment before a past moment, and if p always held in the past, then right now p holds. Putting Priest aside for a moment, if we say that p holds all instants in the past (understood as infinitesimal intervals of time), including the infinitesimal interval leading into the present, then p holds for the present interval too. But as we saw before, Priest uses three moments to define LCC:

For every propositional parameter, p, and every x, yW, if 1[0] ∈ vz(p) for every z such that x < z < y, then 1[0] ∈ vx(p) and ∈ vy(p).

There he seemed to be saying that whatever proposition holds for an (immediately) intermediary moment holds for those (immediately) surrounding moments neighboring it. So perhaps that is why it is not enough for Priest to just say Hp ⊨ p. So let’s think of the present moment as t3. We know that p holds for all prior moments t2, t1, etc. In our evaluation so far, we do not yet know if p holds for present moment t3. But we do know that at least two moments in the past q or not-q held, while at the same time p held. And this this one precedes another past moment that is the intermediary moment, t2, during which p held. Since p holds in an intermediary moment t2, standing between two other moments, then what was true of t2 must hold for them as well, and thus p must hold for the present moment. In the following explanation, Priest seems to have us suppose that PP(q ∨ ¬q) and p hold at (present) moment g. To clarify what the PP means, let’s look at this formulation from Galton’s Stanford Encyclopedia article on tense logic.

“If it will be the case that p, it will be — in between — that it will be”

Here the FFp refers to a moment nearer to the present than Fp.]

For suppose the premises hold at g. Then there is some x < g such that p and P(q ∨ ¬q) hold at x. Hence there is some y < x. By the LCC, p holds at y and g. The inference holds if we replace p by its negation. The future-symmetric analogues of these inferences also hold. However, all of these inferences may break down if the LCC does not hold.

[Let me offer an experimental explanation (meaning likely wrong). Let’s assume as he says that PP(q ∨ ¬q) holds at moment g. This means literally ‘It has  at some time been the case that it has at some time been the case that q or not-q.’ So ‘It has at some time been the case [intermediary moment x]’ ‘that it has at some time been the case [most prior moment y]’ ‘that q or not-q’. This means that at one moment prior, P(q ∨ ¬q), because at that intermediary moment there was a prior where q or not-q held. I am assuming then that at most prior moment (q ∨ ¬q) held, but we cannot say at middle moment x what held other than p, which always had held. So we seem to have this sequence:

moment y, at which p and (q ∨ ¬q) held;
moment x, at which p and P(q ∨ ¬q) held; and
moment g, at which Hp and PP(q ∨ ¬q) holds.

Now recall Priest’s formulation for LCC:

For every propositional parameter, p, and every x, yW, if 1[0] ∈ vz(p) for every z such that x < z < y, then 1[0] ∈ vx(p) and ∈ vy(p).

So since p holds in this case in the middle moment x, it must also hold at neighboring moments y and g. But without this condition, we might not have any basis to conclude p. The role of (q ∨ ¬q) is not clear to me.]

[Now Priest will say that it is not interesting if LCC holds for an order of moments that is linear and discrete, because it would mean that what is said of one moment holds for all (through transitivity). He says it does become interesting when the sequence is continuous or dense. Perhaps what he means is that the density makes us lose the total transitivity. If between x and z there is y where p holds, then it holds for x and z. But between x and y is another moment d, which maybe cancels the application to z (since it is now two steps away); and, between x and d there is e, and so on. Perhaps what he is saying is that the transitivity breaks down or is called into question when we cannot assign an immediate neighbor, and thus we cannot say that a proposition holds for all possible intermediary values.]

In general, the effects of the LCC are not very interesting if the order is both linear and discrete. For, given any three consecutive points x, y, and z, the LCC ensures that whatever propositional parameters or their negations hold at y hold at x and z. If this does not render the evaluations at all indices identical, it does so near enough to make the situation rather uninteresting. The LCC assumes real interest mainly when the ordering is continuous, or at least dense.


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

Or otherwise as indicated from:

Greenstein, Carol Horn. Dictionary of Logical Terms and Symbols. New York / Cincinnati / Toronto / London / Melbourne: Van Nostrand Reinhold, 1978.

Smith, Nicholas J.J. Logic: The Laws of Truth. Princeton / Oxford: Princeton University Press, 2012.

No comments:

Post a Comment