# How is temporal logic not vacuously true?

This has been bothering me for a while and I really need your help. A proposition "It will rain tomorrow" isn't like any other proposition, it most resembles an implication of the form "If it is tomorrow, then it is raining". And when the antecedence is false, we get vacuous truths, it should also be that case that "If it is tomorrow, then it is not raining".

What are the assumptions of temporal logic to overcome this? Why does a statement such as F(p) (It will be the case that some time in the future p) hold, and F(-p) doesn't?

• What works about temporal logic have you studied? – Moritz Feb 9 '16 at 15:37

Instead of looking at temporal statements as playing out in a time-indexed classical logic, it is more realistic to look at temporal logic as a variety of (time-indexed) modal logic.

I disagree that your statement most naturally breaks down to "If it is tomorrow, it is raining." There is no world in which such statements have any purpose. What is missing is not logical segmentation of this sort. What is missing is the mood or modality in which you are putting forward the otherwise meaningless statement.

One simply cannot meaningfully say "Tomorrow, it will rain", where the will is an expression only of the future tense of 'is'. You cannot ever know this fact, so it is not a realistic approach to the use of language. Instead, here, 'will' is a modal verb, the future tense, not directly of 'is', but of 'must', the complement of 'can', indicating your prediction or belief, and not an indicative connector. You mean that for some refinement of 'can', it cannot fail to rain.

But in that translation, 'some refinement' is very important. "If the laws of physics are correct, it cannot fail to rain tomorrow" is very far from "If the wind does not shift unexpectedly, it cannot fail to rain tomorrow." or "If the way my knee aches right now is just the right way that faithfully represents a given barometric trend, and I have correctly assessed the degree of the pain, it cannot fail to rain tomorrow." Or even "(Without external basis) I believe that it will rain tomorrow."

There is an intermediate position between modal and ordinary logic that considers all modal propositions vacuously true. But only until enough context is supplied. One has enough context once the premises are supplemented enough to express the mode asserted at least well enough to give a notion of probability to the statement.

For some notion of 'can' -- in which you personally simply cannot every be wrong -- it is surely true that whatever you say, including 'It will rain tomorrow' is true. That world accords with the non-modal world where the statement is vacuously true. But if you live in that world you are megalomaniacally psychotic. Instead, all of us are supposed to guess by context the particular refinement of the meaning of 'can' involved in making sense of your statement.

That refinement is made up of a bunch of premises, the likelihood of each of which we can assess.

• Thanks! This is precisely what was confusing me. I was looking at it as time-indexed classical logic. – esnafga Feb 10 '16 at 11:09

In temporal logic, you consider words, infinite sequences of sets of atomic propositions. So, an atomic proposition could be 'rain' and the word {},{rain},{},{rain},... would mean that it isn't raining today, it is raining tomorrow, it isn't raining the day after tomorrow, etc. (if we take a day as the step).

A word can satisfy a formula. For example, the word above satisfies the formula "next: rain".

So, whereas in propositional logic and friends you consider just one set of atomic propositions, and check if it satisfies some formula, in temporal logic you consider an infinite sequence of sets of atomic propositions: this sequence represents all states in time. Because of this you can make claims about things in the future.

• I didn't quite understand that too well, could you perhaps try and rephrase before I try to ask a hopefully meaningful question? – esnafga Feb 9 '16 at 13:59
• @esnafga what is it precisely you don't understand? – user2953 Feb 9 '16 at 14:06
• Any statement about the future should be vacuously true because the antecedence is false. So both F(p) and F(-p) should be true right now because it is not yet tomorrow. So, how come this isn't the case in temporal logic? – esnafga Feb 9 '16 at 14:09
• @esnafga it is not always the case that F(p) and F(-p) will at some point be true. For example, if it is always raining, F(not rain) will never be true. Also, you cannot rewrite a statement in temporal logic to something in propositional logic, 'today it is the case that ...'. As I write in my answer, temporal logic talks about a sequence of sets of atomic propositions -- all moments are in that sequence, not just 'today'. So you can say that for some sequence/word it is the case that [tomorrow] it [will rain]. – user2953 Feb 9 '16 at 14:32
• I see, thank you. I misunderstood what F() meant. Could you perhaps say that not eventually, but in a certain, determined moment something will happen through temporal logic? – esnafga Feb 9 '16 at 15:27

The problem you are pointing out is not specific to temporal logic, it has to do with counterfactuals, conditionals where antecedents are contrary to fact. The material conditional of classical logic is inadequate for expressing various types of natural reasoning, but counterfactuals are perhaps the most obvious case. "If salt is placed in water it will not dissolve" comes out as true unless salt is actually placed in water. But in natural reasoning we would consider it false regardless. There are various ways to define the counterfactual conditional, but all of them are non-compositional, the truth value of the conditional is not determined by the truth values of its terms alone, "meaning" of the terms has to be taken into account. Statements about the future in temporal logic can then be treated as counterfactual, thus avoiding the problem.

There are other problems that arise when attempting to apply classical logic in temporal context, e.g. the problem of future contingents. In classical logic "it will rain tomorrow" is either true or false already today (bivalence or law of excluded middle), but then it would seem that logic alone predetermines the future. One way out of this oddity, suggested by Aristotle, is to assign no truth values to such statements at all.