by Corry Shores
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[Bracketed commentary and boldface are my own, unless otherwise noted (for example, there are some symbols that are boldface in the original). Please forgive my typos, as proofreading is incomplete. I am not trained in logic, so at times my summaries may be unhelpful or misleading. Please consult and trust the original text, which is absolutely wonderful.]
Logic: A Very Short Introduction
The Future and the Past: Is Time Real?
Very Brief Summary:
Tense logic allows us to examine the validity of inferences that refer to situations happening at different times. For a statement h referring to a state of affairs that could happen in one or in many temporally distinct situations, we can modify it to refer to a particular situation in the past (Ph) or in the future (Fh) or to all situations in the past (Hh) or in the future (Gh). We can construct a model of a chronological succession of situations and use the tense-modifiers to determine which temporally modified propositions are true and which are false. This helps for example in analyzing the validity of McTaggart’s argument that time is unreal. His reasoning is that an event happening only at one moment must be understood in two contradicting ways. It did not happen both in the past and the future, since it only happened one singular time. But since time flows, it once was true for the future and later was true for the past. Thus contrary to our prior claim, the event does happen both in the past and future. The tense-logic model shows the mistake in this reasoning. Never in one same temporally located situation does it happen both in the past and future, no matter how many additional tense modifiers we add to our formulations. Nonetheless, since this schema models the flow of time spatially, it is perhaps useless for this particular argument which is about the non-spatial flow of time.
We can use tense logic to analyze the validity of inferences that are based on statements referring to different moments in time. We first think of a one-dimensional series of situations arranged in their proper chronological sequence. We then think of statements of fact. They may or may not be true for one temporalized situation or another. Suppose a statement h is true only for the temporally situated moment s0. This statement refers to an instantaneous state of affairs, like the moment the first bullet entered Czar Nicolas’ heart. It will be false for all situations coming before and after that temporalized situation, since the event did not happen at those other moments. However, at a succeeding moment in the future, we can say truly that the event happened in the past. And likewise for a preceding moment in the past, we can say it will be happening in the future. We use the modifier P for past (“it was the case that”) and F for future (“it will be the case that”). So in moment s1, Ph is true, and for moment s-1, Fh is true. We can further designate temporal relations by compounding the modifiers. PPh would apply h to a situation coming before some other situation that is already in the past. FPh would apply h then to some situation coming after some other situation that is already in the past. Now, P and F refer to some determinate situation in the past or future. We can instead refer to all future situations with the modifier G (“it is always Going to be the case that”) and all past ones with the modifier H (“it Has always been the case that”). We can also make a model for this tense logic by arranging in sequence a number of s’s, placing s0 in the middle, and counting up and down the subscripts on both sides. This allows us to evaluate inferences based on tense modifiers. One example is McTaggart’s argument against the reality of time. If time is real, then the past and future are real, and thus they do not present logical contradictions. We then consider a sentence that is true just for the situation at one time-point. This means it did not happen in two temporally distinct time-points, and thus it did not happen both in the past and in the future: ¬(Ph&Fh). However, time flows, and so before it happened, it was in the future, and after it happened, it was in the past: Ph&Fh. The concepts of past and future present a contradiction, and thus time is unreal. One may object to the second formulation and say that it pretends that, for one situation that is located at one time point, the event can be both in the past and in the future. So to clarify the problem, we might then compound the modifiers and write ¬(PPh&FFh) to mean that the event did not happen at some determinate point coming before another in the past and at the same time happen at some determinate point coming after another in the future. Those following McTaggart’s reasoning can then say that still, because of the flow of time, PPh will be true and FFh was true, and thus, in contradiction with the prior, negated conjunction, PPh&FFh. But, by using the tense-logic model, we can display visually that the McTaggart argument is mistaken. There is never a singular temporalized situation where both terms in the past&future parings are true. Nonetheless, as this is a model that spatializes the flow of time, it might not be adequate for dealing with this argument about time’s non-spatial flow.
Time is something we deal with in our everyday behavior and thinking, and we often make intuitively valid inferences about time, like the following ones:
It is raining.
It will have been raining.
It will be true that it has always been raining.
It is raining.
But as soon as we begin to reflect more on time, we get confused. One problem is that time itself seems to change, but it is the seemingly constant form that measures how much everything else changes (56). This issue lies “at the heart of several conundrums concerning time,” and one of which was posed by John McTaggart Ellis McTaggart. “Like many philosophers, McTaggart was tempted by the view that time is unreal – that, in the ultimate order of things, time is an illusion” (56d).
We now need to fashion some symbols for our analysis of temporal relations in logic. We begin with the past tense designation, “it was the case that,” which we will notate in boldface as P (for past). So when we mean “The sun was shining” or “it was the case that the sun is shining” we write “P the sun is shining”. If we further abbreviate “The sun is shining” as s, then we can write for these sentences: Ps. We do something very similar for future time. For “The sun will be shining” or “It will be the case that the sun is shining,” we will write Fs, since F (future) will mean “it will be the case that”.
Priest writes, “P and F are operators, like ◻ and ◇, that affix to whole sentences to make whole sentences. Moreover, like ◻ and ◇, they are not truth functions” (56). To illustrate, he shows how by adding the future operator to two true sentences, in one case it stays true but in the other it becomes false.
‘It is 4 p.m.’ and ‘It is 4 p.m. on August 2nd, 1999’ are both true (at the instant I write); ‘It will be 4 p.m.’ is also true (at the present instant) – it is 4 p.m. once every day – though ‘It will be 4 p.m. August 2nd, 1999’ is not.
In logic, P and F are called tense operators. They can be iterated, meaning that they can be compounded. So, using our above example, we could have FPs, meaning, “The sun will have been shining” or put more formally, “It will be the case that it was the case that the sun is shining.” Or we could have PPs, meaning, “The sun had been shining” or “It was the case that it was the case that the sun is shining” (56). Priest also notes that we can iterate the modal operators. [I am not exactly sure how yet. What does it mean that it is possible that it is possible that the sun is shining, or necessary that it is necessary?] We might however have difficulty rendering iterated tense operators into standard English tenses. FPFs for example would have to be “It will be the case that it was the case that the sun will be shining”. Priest notes however that “The iterations, though, make perfectly good grammatical sense. We can call iterations of P and F, like FP, PP, FFP, compound tenses” (56).
One part of McTaggart’s argument is that without past and future, you would not have time, since they are of the essence of time. “Yet pastness and futurity, he argued, are inherently contradictory; so | nothing in reality can correspond to them” (56-57). One reason that past and future are contradictory is that they are incompatible. [The reasoning is a bit tricky. Perhaps the basic insight is that we can regard temporal relations from the perspective of a singular determinate position in the flow of time, or we can see time as one continuous flow that binds together all moments of time. In the first case, past and future mutually exclude one another. In the second case, they intertwine. If we orient ourselves within the flow of time by placing ourselves at one particular moment or another in the succession, we would see that a singular instantaneous event cannot happen both in the past and in the future. However, if we stand outside the flow of time and see all moments lined up in a series, we can see that each moment is past or future, relative to other moments. But no such moment is a point of origin for comparison under this view, so each moment is both past and future, because we have not oriented that moment by giving it a privileged position for comparison.]
If some instantaneous event is past, it is not future, and vice versa. Let e be some instantaneous event. It can be anything you like, but let us suppose that it is the passing of the first bullet through the heart of Czar Nicholas in the Russian Revolution. Let h be the sentence ‘e is occurring’. Then we have:
But e, like all events, is past and future. Because time flows, all events have the property of being future (before they happen) and the property of being past (after they happen):
So we have a contradiction.
[Someone might note that the second formulation, Ph&Fh, is confused about the structure of the series of moments. We might observe that, yes, there is one situation at one moment where Ph is true, and there is a prior situation where Fh is true, but there is never a situation where both is true. There was a moment in the past before it happened when it was still to happen in the future, FPh, then after this past event happened, it was in the past, PPh. On this basis, Priest will then make the formulation ¬(PPh&FFh), but I am not sure I follow how we get there. Why is this not ¬(PPh&FPh)? Perhaps it is ¬(PPh&FFh) to speak more generally, rather than to refer only to past historical events. But again, I do not know how yet. PPh&FFh seems to be referring to a point in the past, before which an event is said to happen, and a point in the future, after which an event is said to happen. Thus we are interposing a duration of time between the past and future moments. But maybe that is not what it means.]
This argument isn't likely to persuade anyone for very long. An event can’t be past and future at the same time. The instant the bullet passed through the Czar's heart was past and future at different times. It started off as future; became present for a painful instant; and then was past. But now – and this is the cunning part of McTaggart's argument – what are we saying here? We are applying compound tenses to h. We are saying that it was the case that the event was future, PFh; then it was the case that it was past, PPh. Now, many compound tenses, like simple tenses, are incompatible. For example, if any event will be future, it is not the case that it was past:
But, just as with the simple tenses, the flow of time suffices to ensure that all events have all compound tenses too. In the past, Fh; so in the distant past FFh. In the future, Ph; so in the distant future, PPh:
And we are back with a contradiction.
[It is still not entirely clear to me how to conceptualize the way this problem regresses, since like before, I do not understand how we go from PFFh in the example to FFFh in the conjunction. But the basic idea seems to be that the objector keeps designating a point before and a point after the event which each independently have their own different temporal relation to the event, when really we should be orienting our our statements about the event to one singular moment when it happens.]
Those who have kept their wits about them will reply, just as before, that h has its compound tenses at different times. It was the case that FFh; then, later on, it was the case that PPh. But what are we saying here? We are applying more complex compound tenses to h: PFFh and PPPh; and we can run exactly the same argument again with these. These compound tenses are not all consistent with each other, but the flow of time ensures that h possesses all of them. We may make the same reply again, but it, too, is open to the same counter-reply. Whenever we try to get out of the contradiction with one set of tenses, we do so only by describing things in terms of other tenses that are equally contradictory; so we never escape contradiction. That is McTaggart’s argument.
We will now see why we have this problem by examining “the validity of inferences concerning tenses” (58). We begin with a schema that does not use temporal modal operators like P and F. Rather, we begin with a series of temporally various situations s that follow one another along the one-dimensional extension of time. We orient the series around a middle point s0, with left-most being past-most, like so:
[We might have a statement, like “Czar Nicholas is shot in the heart”. We might call it a. Then, a is true in a situation s if it happened at that situation’s designated time-point. Suppose it really happens at s0. Then a is true for s0. This also means that Pa is true for all situations happening after s0.]
As usual, each s provides a truth value, T or F, for every sentence without tense operators. What about sentences with tense operators? Well, Pa is T in any situation, s, just if a is true in some situation to the left of s; and Fa is true in s just if a is true in some situation to the right of s.
[In our prior example where a meant “Czar Nicholas is shot in the heart”, the statement a was true only for one situation s. Now we are to think of a proposition being true in all situations in the past or in the future. I do not know how to exemplify this. Perhaps we might make a statement like, “the cosmos exists in one form or another”. And supposing that the cosmos has no beginning or end, then this statement will hold for all s’s coming before and after some given one.]
While we are doing all this, we can add two new tense operators, G and H. G can be read ‘It is always Going to be the case that’, and Ga is true in any situation, s, just if a is true in all situations to the right of s. H can be read as ‘It Has always been the case that’, and Ha is true in any situation, s, just if a is true in all situations to the left of s. (G and H correspond to F and P, respectively, in just the way that ◻ corresponds to ◇.)
[Recall our inferences from the beginning of the chapter. 1) It is raining. Therefore, it will have been raining. 2) It will be true that it has always been raining. Therefore, it is raining.] Using these new tense operators, we can formulate the two inferences from the beginning of the chapter like this:
[We look initially at the first one. r means “it is raining”. We assume it is true in some situation s0. I do not follow so well, but it seems we then add P, which then orients us at some point in the future, where r is now true in the past. Then we add F, which takes us back to the present, where it is true that in the future Pr is true. But there must be a much better way to restate the following:]
The first inference is valid, since if r is true in some situation, so, then in any situation to the right of so, say s1, Pr is true (since so is to its left). But then FPr is true in so, since s, is to its right. We can depict things like this:
[The second inference again was: It will be true that it has always been raining. Therefore, it is raining. The H modifier to r will orient us somewhere in the future, and it says that r is true in all prior moments. Since so is one such prior moment, the inference is true.]
The second inference is valid, since if FHr is true in so, then in some situation to the right of so, say s2, Hr is true. But then in all situations to the left of s2, and so in particular so, r is true.
Priest then notes that certain combinations of tenses are impossible. [If an event happens just within one time-point, then it cannot happen in more than one time-point. Thus it cannot be both past and future at any point in time.]
if h is a sentence that is true in just one situation, then Ph&Fh is false in every s. Both conjuncts are false in so, the first conjunct is false to the left of so; the second conjunct is false to the right. Similarly, e.g., PPh&FFh is false in every s.
We return now to McTaggart’s argument. [We were dealing with an event holding exclusively for a determinate time-point. The contradiction we found was between two sets of seemingly equally valid claims. The first claim is that the event cannot happen both in the past and in the future, ¬(Ph&Fh), and yet, since time flows, the event would have to have been in the future before it happened and in the past after it happened, Ph&Fh. The next step was an objection to the second claim. It implies that the event is future and past at the same time. But really, we need to be clear that it was past at one time and future at another. For this, we add another tense marker to make it clear that the McTaggart argument is oriented around two temporally separate moments. And thus it is not saying that in one same moment the event is in both the past and the future. The one taking the McTaggart view will then say that because of the flow of time, any more specific designations will still be both past and future, for the same reason as before. (In fact I am not exactly sure how things work with these compounded tenses in the McTaggart debate, since, as I mentioned before, the literal translation of the formulations is hard for me to situate in the way it is presented here.)]
Now, how does all this bear on McTaggart's argument? The upshot of McTaggart's argument, recall, was that, given that h has every possible tense, it is never possible to avoid contradiction. Resolving contradictions in one level of complexity for compound tenses only creates them in another. The account of the tense operators that I have just given, shows this to be false. Suppose that h is true in just so. Then any statement with a compound tense concerning h is true somewhere. For example, consider FPPFh. This is true in s2, as the following diagram shows:
Clearly, we can do the same for every compound tense composed of F and P, zigzagging left or right, as required. And all this is perfectly consistent. The infinitude of different situations allows us to assign h all its compound tenses in appropriate places without violating the various incompatibilities between them, e.g., by having Fh and Ph true in the same situation. McTaggart's argument, therefore, fails.
Priest then notes that we have disproved McTaggart’s argument of the reality of time by building a model of time and of the relations between temporally various moments of time. But models might distort what they model, or leave out important features. In this case, we have modeled time using space. But perhaps the flow of time is not analogous to the flow of space.
Now, it is exactly the flow of time that produces the supposed contradiction that McTaggart was pointing to. No wonder this does not show up in the model! Exactly what, then, is missing from the model? And once that is taken into account, does the contradiction reappear?
[The following is quotation.]
Main Ideas of the Chapter
● Every situation comes with an associated collection of earlier and later situations.
● Fa is true in a situation if a is true in some later situation.
● Pa is true in a situation if a is true in some earlier situation.
● Ga is true in a situation if a is true in every later situation.
● Ha is true in a situation if a is true in every earlier situation.
(quoted from Priest, 62, boldface his, with F, P, G, and H in extra bold)
Priest, Graham. Logic: A Very Short Introduction. Oxford: Oxford University, 2000.