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I'm studying logical proposition. And I know that $P$ is a proposition if it's either true (T) or false (F), there are no other possibilities than being (T) or (F) and it can't be true and false. I know for exemple that "tomorrow it will be sunny" is not a proposition, because we can't know if it's true or false.

  1. What about "yesterday it was sunny" ? I guess that we could check on the newspaper and say at the moment where we say it if it's true or false. So in somehow, I would say that it is a proposition, but in the other hand, the fact that the truth can change everyday, maybe it's not. What do you think ?

  2. Since "tomorrow it will be sunny" is not a proposition, can I consider the proposition : "if tomorrow it will be sunny, then I'll go to the swimming pool" ? The question maybe strange, but I know that to consider the implication $\implies $, we consider the implication of two propositions. Since "tomorrow it will be sunny" is not a proposition, can I say that "if tomorrow it will be sunny, then I'll go to the swimming pool" is not a proposition ?

joshua
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    This seems to me not a useful definition of a proposition. Philosophically it can be argued that we never know anything about the real world with absolute certainty, so if absolute certainty is required then “it was sunny yesterday” is not a proposition. On the other hand a modal logic may say something like “eventually P will be true,” which is something you can never know to be false by observation. Where did you get this definition? – David K Feb 25 '24 at 12:59
  • In propositional calculus facts are not time-dependent. The proper examples are: "2 is even", "3 is less than 2" and so on. – Mauro ALLEGRANZA Feb 25 '24 at 13:22
  • @DavidK: I modified my definition. Is there less ambiguity like that ? – joshua Feb 25 '24 at 13:58
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    No, the problem isn't ambiguity. It's just not a good definition for use in mathematical logic. A better definition is given in the answer that begins, "As far as mathematical logic is concerned, a proposition is simply whatever you can assign a truth value to." Did someone tell you that "tomorrow it will be sunny" is not a proposition? If so, who? If not, how do you "know" that it's not a proposition? – David K Feb 25 '24 at 20:03
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    When dealing with propositions the truth values of which may vary over time (as here), it is advisable to explicitly introduce quantifiers over a variable for time. – Dan Christensen Feb 25 '24 at 21:44
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    "I know for exemple that "tomorrow it will be sunny" is not a proposition, because we can't know if it's true or false." We can if we wait until tomorrow. Of course, whether it has truth value in advance seems more of a philosophical question than a mathematical question. – rus9384 Feb 25 '24 at 22:12
  • See Formal definition of proposition and Sentence vs proposition. – Mauro ALLEGRANZA Feb 26 '24 at 09:34

2 Answers2

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As far as mathematical logic is concerned, a proposition is simply whatever you can assign a truth value to. But it doesn’t matter how you do it, what the sentence means and whether your assignment of truth values would be convincing to anyone or possible to justify in the real world. Propositional logic defines certain rules how to construct propositions from other propositions, and how to draw conclusions from propositions once you know their structure, but it doesn’t dictate what basic propositions you start with, and whether they can involve the past or the future.

You can simply consider “yesterday it was sunny” or “tomorrow it will be sunny” as atomic propositions that cannot be broken up any further. Logic doesn’t care how you interpret such propositions or evaluate their truth values. Maybe you remember yesterday’s weather, maybe you were only told what it was; maybe you have a perfect forecasting ability that allows you to see tomorrow’s weather, maybe you don’t. You might as well declare either true or false by fiat. Logic doesn’t care: it is only concerned with what conclusions you can draw once the basic facts are established.

For an example, imagine you have a machine that you can ask yes-no questions about tomorrow’s weather and it will answer with perfect accuracy. Propositional logic tells you that if you get YES answers for questions “will tomorrow be sunny or rainy?” (S ∨ R) and “will tomorrow not be rainy?” (¬R), you don’t need to additionally ask if tomorrow will be sunny (S), because you can draw that conclusion yourself from the answers you got already. And if you do and the machine replies NO (¬S), then you know must be broken, because it described an inconsistent state of affairs. But whether such a machine can actually exist falls outside the remit of logic.

user3840170
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  • What about the OP's point #1, that the truth value of the proposition is relative? A statement about yesterday, today, or tomorrow depends on when you state it. A question that refers to "my borther" depends on who states it. – Barmar Feb 25 '24 at 22:38
  • This is true in first-order logic. But there are other logics which incorporate the past and the future as quasi-modal operators, and in those logics, a sentence such as "yesterday it was raining" is not atomic. – Kevin Feb 26 '24 at 00:19
  • @Barmar You either find a way to allow such statements, you conveniently ignore the possibility of saying them, or you create a restricted grammar of statements--although if you're going to go to that trouble, maybe just do mathematical formulas from the start. – David K Feb 26 '24 at 05:01
  • @Kevin The point of the answer isn’t that you can’t express the passage of time formally in first-order logic (though indeed you can’t); the point is that the choice of how to use a logic’s expressive power is arbitrary, and assignment of truth values is an extra-logical concern. Even if the asker had a logic that captures the structure of time, the problem of assigning a truth value to “tomorrow it will be raining” remains. (Also, this is not even first-order logic, but mere propositional calculus. There are only propositional variables here, no terms, no domain of discourse.) – user3840170 Feb 26 '24 at 07:21
  • @user3840170 I find this answer quite off mark (but am especially puzzled by the upvoting): by that token you could simply reduce every argument, any argument, to a single proposition "P" and be done, but that is totally pointless and it is certainly nothing to do with Logic... -- Indeed the main issue rather is: what is an "atomic proposition". – Julio Di Egidio Feb 26 '24 at 14:41
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I know for exemple that "tomorrow it will be sunny" is not a proposition, because we can't know if it's true or false.

Actually, a proposition is by definition true or false, whether we know or don't know which it is. Of some propositions we can even prove that we cannot indeed know, and those we call (provably) undecidable. (Provability is relative to a specific formal system, but I won't get here into any more technical details.)

That said, IMO, "tomorrow it will be sunny" is a proposition: for the simplest formalization, just take "(to be) tomorrow" as an atomic proposition, denote it by $T$, then denote "(to be) sunny" by $S$, and you have the proposition $T \to S$ (which reads "to be tomorrow implies to be sunny": to express that "when it is tomorrow, then it will be sunny", i.e., in common English, that "tomorrow it will be sunny").

Now, assuming that we cannot in fact, here and now, know whether that proposition is or is not true, we should indeed conclude that the proposition is undecidable (we'd have to formalize what I have just said to technically conclude such a thing, but I hope it is easy to see that it would follow).

But notice that the assumption is not really a little assumption: we'd have to assume that predicting the future is impossible, and, while this is in most cases reasonable, it is something that not even contemporary physics would say for certain...

On the other hand, for a more "mundane" approach, and maybe more useful in this case, a fuzzy propositional logic could be used: where we could attach a level of confidence to the truth value of that proposition, as e.g. supported by a weather forecast.

Julio Di Egidio
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    The implication $T \to S$ ("If it is tomorrow, then it is sunny") isn't actually equivalent to "Tomorrow, it will be sunny". Right now, $T$ is false, so $T \to S$ is vacuously true, regardless of whether it will be sunny tomorrow. – Misha Lavrov Feb 25 '24 at 16:41
  • I actually read it "to be tomorrow implies to be sunny", which to my/our purposes I'd say is faithful (enough). But the point would be (IMO) that this analysis works if we go from informal to formal, i.e. with a precise semantics, it becomes a problem only if going in the opposite direction, i.e. from arbitrary symbols... – Julio Di Egidio Feb 25 '24 at 16:48
  • One could even say that $T$ is always false, since tomorrow is by definition the day that does not contain now. It is never tomorrow, it is always today. – user3840170 Feb 25 '24 at 16:51
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    No, (IMO) you are simply misreading / not correctly formalizing: "to be tomorrow implies etc." is not the same as "it is tomorrow and etc.". -- Anyway, I'll at least amend my question in that sense for clarity. – Julio Di Egidio Feb 25 '24 at 16:58
  • What is the difference between "to be tomorrow" and "it is tomorrow"? Are there situations in which "to be tomorrow" is true and "it is tomorrow" is false, or vice versa? – Misha Lavrov Feb 25 '24 at 17:58
  • @MishaLavrov "to be tomorrow" is just not plain English, of course: it is a way (the one I'd suggest, understanding- and explanation-wise) to give a reading to the formal translation, where the point of the infinitive is that propositions do not have any intrinsic notion of time, so we formalize them in the infinitive... Does it work? You be the judge, but here is an example: "Socrates is mortal" --> "(to be) Socrates is/implies (to be) mortal"... – Julio Di Egidio Feb 25 '24 at 18:40
  • Actually, I think the problem is not the notion of time; the problem is that we can't look at the two halves independently. The second half "is mortal" doesn't make sense unless we're talking about a specific entity, just like the second half "will be sunny" doesn't make sense unless we're talking about a specific day. We could define $M(x)$ to be "$x$ is mortal", $S(x)$ to be "$x$ is Socrates", and then say $\forall x, (S(x) \to M(x))$. Similarly, if we have $T(x)$ and $U(x)$ for "$x$ is tomorrow" and "$x$ is sunny", we can say $\forall x, (T(x)\to U(x))$. – Misha Lavrov Feb 25 '24 at 18:51
  • @MishaLavrov But then it is not anymore propositional logic... <<"is mortal" doesn't make sense unless we're talking about a specific entity>> which is valid in itself, still "is mortal" is not the same as "to be mortal" (which is not even the same as "be mortal"), and rigour here matters! That said, I'd have hard times explaining it any better in this space. So, while I will think more about it and whether I can (and should) explain all that in my answer... (should I?), if you'd like to keep discussing, I'd propose we continue in a chat. – Julio Di Egidio Feb 25 '24 at 19:03
  • P.S. The main issue really is: what is an "atomic proposition": and what the copula means/does in natural language. Which is the despair of logicians: there are letters from Wittgenstein to Russell, approximately at the time of the Tractatus, specifically discussing that issue. But transforming every answer into a scholarly article I find just too much -- and not even very healthy as to (also) learning how to reason by oneself. – Julio Di Egidio Feb 26 '24 at 14:46