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Example:

I state that a coin will come up heads, then flip it. While the coin is flipping, does the statement 'the coin I just flipped will come up heads' have a truth value?

Based on my understanding, it's unknowable, so it doesn't have a truth value associated with it. I am not yet incorrect or correct.

But if I state that the coin comes up heads, flip it, and it comes up tails, the initial statement I made clearly doesn't match with reality, so its truth value is seemingly false.

From the perspective in the moment in time before the coin lands, to the moment after it lands, does the truth value retroactively change? Or would my initial perception (that the statement 'the coin I just flipped will come up heads' is one without a truth value) be fundamentally flawed, and the statement was always false, even though the information in question didn't exist yet?

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11 Answers 11

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Both, because propositions are expressed in language which is always both vague and ambiguous. Thus, a statement's truth value is dependent on the context of the discourse it finds itself.

Consider:

Today is Tuesday.

The proposition is true exactly one day every week, on Tuesday. But, we can make a similar proposition true by adding context:

'Today is Tuesday' is true only every Tuesday.

Now our proposition is true every day of the week. But, you might argue, you changed the proposition! True, but here's an alternative. Consider that you only, due to your religion, are allowed to utter 'Today is Tuesday' on Tuesday. Is the proposition always true? Sure, because at that point, you only utter it on Tuesdays, so the proposition because of a restriction of when it is uttered suddenly becomes always true when uttered.

But what if I read it on days other than Tuesday, you complain? Well, then it isn't always true. How exactly is that? Well according to some ways of thinking about truth, truth stems from having a judgement of truth, in this case of a proposition that purports to represent a truth condition about the state of affairs in the real world. So, now that you read it on days other than Tuesday, then the judgment you make about its truth will depend on the day you use it.

So am I saying that judgment is necessary to determine truth? Yes, very much so. This an important idea that goes back to at least Kant, but other philosophers have thought about it too. Consider Bretano's Theory of Judgment (SEP):

Judgements are one of the three basic kinds of intentional phenomena with which Brentano deals at great length in his Psychology. That is not to say, however, that Brentano’s theory of judgement is just concerned with psychological issues. Brentano also aims to show how an experience of judging, specifically the experience of judging correctly, can provide us with a basis for grasping concepts like existence, truth, and logical inference. Brentano’s investigation of the mental act of judgement promises therefore to advance logic, epistemology, and ultimately metaphysics.

So, if one is not an externalist in these matters, one simply can accept that the truth of a proposition relies on the judgment of the thinker. Since physical experience, rational thought, propositional content, one's truth-conditional semantics, propositional attitudes, and theories of truth all are relevant, it's easy to defend the notion that any truth-value of a proposition is highly-context dependent, an idea that particularly frustrates people who believe there is only way to think and believe.

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To answer this question profoundly, we first need to introduce terminology.

In formal logic (gray area field between math and computer science), statements are separated from interpretations. A statement in this context may be something like "the ball is red", "Archimedes is a man", "if the tide is high, then it will be low six hours later", or whatever other logical statement you can come up with. So far, so good.

Now, an interpretation is a function that assigns values to the names in such statements. An interpretation may assign a European soccer ball to "the ball" in the first statement. Which will make the statement false in most cases. It may also assign an American foot ball to "the ball", and the statement will usually be true. Or it might assign a specific dance event to "the ball", which renders the truth value of "the ball is red" ill-defined. Whenever you are reading a statement, you are applying some interpretation to the statement in order to make sense of it.

This interpretation is usually implicit. You read a text, and you assume you know what the different words are referring to. And you are doing that by considering some context. When you read "the ball is red" in a British sports magazine, you are likely using a different interpretation for "the ball" than if you were hearing it in the context of a baseball game. And it is exactly this implicit nature of interpretations that leads to so many confusions. Mind you, we are not even able to write down almost all relevant interpretations, simply because they assign real world objects to the terms in the statement. And you can't put a specific ball into writing without resorting to some other terms that will need to be interpreted themselves with a real world object. You can only write down the names of things, not the things themselves.

So, with that out of the way, we can finally answer your question: Statements get their truth value from some interpretation. As long as you use the same interpretation, the truth value cannot change. But whenever you apply a different interpretation, the truth value can change. And that is precisely what happens when people interpret time dependent statements. You interpret the statement "the coin will come up heads" differently before and after throwing the coin. However, when you apply an interpretation that assigns one specific throwing of the coin to this statement, the statement is either always true or always false, depending on the actual outcome of the throw. The result may not be known, as is the case when the throw is still in the future. In that case, observing the actual result tells you whether the statement was true/false all along. However, if you always interpret the statement "the coin will come up heads" to refer to the next throw of that specific coin, you never know its truth value when it is said.

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    @ScottRowe: Philosophy got out of the business of making predictions several hundred years ago, around the time that "natural philosophy" got renamed to "science." That does not make it useless, it simply means it is doing something else now.
    – Kevin
    Mar 16 at 23:42
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    "In math, statements are separated from interpretations." No they are not. Mathematicians say that 2 is an even number because they interpret the figure '2' are referring to the number 2. The difference with natural language is that (most of the time) the definitions for the terms involved in mathematical statements have a fixed interpretation. Mar 17 at 11:31
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    @ScottRowe "is it more accurate?" No. he is just switching to a technical sense of the word "interpretation" and that doesn't stop mathematicians interpreting the symbols that they use like everybody inevitably does. If there was somehow no interpretation, it just wouldn't be a language. Mar 17 at 17:05
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    I was just taught that precision is meaningless without accuracy.
    – Scott Rowe
    Mar 17 at 21:36
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    @ScottRowe And rightly so. You fully deserve the 1000 nitpicking points I award for your accuracy and precision. Happy? ;-) Mar 17 at 21:54
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Statements can have truth values that change with time. The statement, 'It is the 29th of February today' is periodically true, but is not true now.

A prediction, such as 'It will rain tomorrow', does not have a known truth value, since it refers to an unknown future state.

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  • The universe likes to keep us guessing.
    – Scott Rowe
    Mar 15 at 10:17
  • Many of the problems with changing truth values can be resolved by considering context and making implicit parts of the statement explicit. For example, the Feb. 29th statement implies, to quote Fatboy Slim, "Right here, Right now." Because on the other side of the date line it's Feb. 28th, tomorrow it's March 1st, etc. Same for statements like "it's raining". Mar 16 at 9:16
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    And if we want go to get a bit philosophical: The truth value of predictions may be unknown, but it may still exist! Namely if we believe that the future is already determined. Current science indicates that the future is categorically unknowable, but we don't know whether the science will change at some point! ;-) Mar 16 at 9:20
  • @Peter-ReinstateMonica Is the future already determined? Maybe someone should let everyone know?
    – Scott Rowe
    Mar 16 at 22:10
  • @Peter-ReinstateMonica I actually think that is the core idea the question is getting at. Despite many good answers here, I’m not sure anyone has really focused on that - that the question is actually asking if the future exists, before it has happened (from an observer’s temporal frame of reference). (Someone should add an answer expanding on that.) Mar 17 at 16:05
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The statement

The coin will come up head

is a sentence about a future event. After the coin came up, the statement is meaningless, because afterwards the coin will not come up any longer.

In 2-valued logic each statement is either true or false, even when we do not know the truth value. The problem is with future events. Your example is similar to Aristotle’s discussion about tomorrow’s see-battle. Aristotle concludes that future possible events do not necessarily have a truth value:

"Clearly, then, it is not necessary that of every affirmation and opposite negation one should be true and the other false. For what holds for things that are does not hold for things that are not but may possibly be or not be; with these it is as we have said." (Aristotle, On interpretation, Chap. 9)

For further discussion of the problem and different proposals for a solution see the Stanford encyclopedia.

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I think you are confusing your state of knowledge about reality (in this case, the future of the coin) with reality itself. They will only agree if you know all relevant information about reality and you reason coherently from the information you have.

The motion of the coin is completely deterministic, it is not unpredictable in principle. The only thing stopping us from predicting it's trajectory reliably is lack of information. There is arguably no such thing as randomness in the macroscopic world, just ignorance/uncertainty, which we tend to model as "random chance". When we view it as our state of knowledge, the ambiguity/paradox is resolved.

Whether the coin will come down heads or tails has a constant truth value, we just don't know what it is. When we say "the coin will come down heads" we are making a statement about a model of reality, but we don't usually need to explicitly state that.

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  • So you are siding with the idea that the truth value is unknown to us but is 'revealed' at some moment. The entire future already unchangingly is determined, we just haven't seen it all yet.
    – Scott Rowe
    Mar 15 at 18:55
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    @ScottRowe That would be the case in a deterministic (at the macroscopic scale) universe. I don't know if that is the case, but I think it is likely given our knowledge of physics (and particularly my understanding of it ;o). However, except in quantum physics, pretty much whenever we talk of randomness, we are talking about a simplified model of reality where we are lumping all of the details of things not of interest to us into "random chance". That would be the case even if the universe was not completely deterministic. Mar 16 at 15:32
  • Yeah, a universe where everything is determined AND we know everything that will happen would be... What it would be, and we would feel about that, what we would feel about that. Kind of removes the impetus to find out anything, but I guess we would have to do it anyway.
    – Scott Rowe
    Mar 16 at 15:54
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I can’t say anything of the following with confidence or definiteness, but I’ll try to give some perspective from what I myself have been learning lately.

  1. There are many different things in modern philosophy, modern scientific knowledge, and modern society called “logic”. A subset of those things can be considered “formal systems”. I do not know a perfect or definitive definition of what a “formal system” is, and maybe there is not yet one, since having a “perfect definition” might imply being able to do so in some “perfect system of meaning”; but since formal logic, as I see it, is motivated by a desire to uncover what a “perfect system of meaning” would be like, we would not be able to define “formal system” definitively until we had determined a definitive formal system to express that idea in. It would be interesting to imagine how a perfect formal system would define what a “perfect formal system” is, but obviously, there are so many frontiers in research in logic, so it’s (arguably) not a concrete question to pursue. Assuming we can still make progress in understanding, with our intuitive ability to discern what things are like, without offering perfect definitions of them, I can approximate that a formal system tends to be a set of rules which are “unambiguous”, like some of the rules of mathematics. I leave the question of what “unambiguous” means for a later time.
  2. In my opinion, it is way more modern to think of “logics” as formal systems, which means that there are as many different ones as humans can come up with. In other words, instead of asking if the nature of logic is such-and-such, the question is how a formal system could describe whatever phenomenon you have in mind well, and what kind. That is not to say that there are not some fundamental, universal laws of truth, but that whatever they are, we don’t know enough about it to ask a lot of questions about what it’s like, and we don’t need to consult with it to model the rules of systems we encounter in our daily lives. This is a suggested paradigm, not a truth claim I’m particularly attached to.
  3. The above approach makes your question easier. Once you have studied different systems of logic, it may be easier to imagine how symbolic notations can be used to represent patterns and phenomena we encounter. Developing such a notation is not necessarily easy, as a new notational system sometimes amounts to the birth of a new theory. Physics is full of examples of this. (Richard Feynman invented a particular kind of diagram for describing quantum phenomena in physics, which were later studied for their mathematics properties in their own right. Such diagrams have been named “string diagrams”.) The important thing seems to be that the rules of your symbolic, representational system mirror the structure or behavior of the “real world”-thing you are experiencing.
  4. Allowing yourself to freely create any symbolic-representational system invites one to ask if there is any real boundary between fields traditionally known as “logic” and “ontology”. Take modal logic for an example. Modal logic built on pre-existing logics by allowing a symbol to represent the conceptual idea of “possibly” and “necessarily”. From there, modal logic has seemingly rapidly expanded to encompass almost any imaginable way of semantically inflecting truths, and instituting a symbol to reflect it - I can make an operator that conveys “knowability”, as in, “this proposition is known”, or an operator that conveys “obviousness”, as in, “this proposition is obvious”. I think the idea that “pure logic” and “human conceptual meaning” are incompatible “layers” caused some confusion until it became clearer that they were not. In linguistics, notational systems have been developed to abstract away “conceptual meaning” from “logical structure” (for example, P(x) is a predicate that means “red” - but quantification of that predicate over an entity is apparently purely mathematical, encoded in the underlying logical system, not in the “conceptual” predicates). Modal logic is just one example that shows how the structure of concepts themselves can actually be represented by “formal” structures; the formal systems that come close to behaving like human conceptual meaning are just bigger and have more rules. That is how modern language-based AI works: they have learned patterns of human semantics and are able to model those rules in computer circuitry which ultimately reduces to very elementary mathematics (binary arithmetic, etc.)

Therefore, I guess there are two approaches to the question: what is the nature of “knowing” or “truth”, itself, in our world (which is definitely useful and interesting); or, what I find more inviting, how could I model what I observe as the patterns of some system, in symbols? There are tons of logics developed for those kinds of purposes, like temporal logic, dynamic logic, imperative logics, erotetic logics, dialethic logics, and others that I don’t know of.

One framework I can pass on a recommendation for is David Spivak’s Temporal Type Theory. While I believe intended for the modeling of temporal systems, since it presumably includes a concept of “truth”, it could maybe be used to model “the truth of a proposition at a certain time.” (In my opinion, David Spivak is a genius who has changed the world by more than anyone thrusting category theory into real-world domains outside of mathematics, as a “universal modeling language”, in the field of applied category theory. In fact, I worship him.)

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Do statements always have the same truth value, and that truth value is just revealed, or can they have different truth values at different times?

It is trivial that if a statement is true (or false), it is true (or false) of the real world, and that each statement has to be interpreted in context. The statement "Donald is a criminal" may be true in a number of contexts and false in others, it all depends on which Donald the statement is referring to.

It would be absurd to assert that a statement has no truth value because it is sometimes true and sometimes false. If that was the case, we would have to say that the statement "the statement has no truth value" has itself no truth value because it would have different truth values according to the different truth values of the statement. Thus, asserting "the statement has no truth value" would then be both asserting that the statement is true and that it has no truth value, which is absurd.

So, a statement is only a statement in context. What this means is essentially that we have to interpret what the statement is used to mean according to the context of its utterance. We all do that all the time and there is no problem.

The implicit fallacy in the question is to pretend that it is the same statement irrespective of context.

I state that a coin will come up heads, then flip it. While the coin is flipping, does the statement 'the coin i just flipped will come up heads' have a truth value?

Yes, it has.

You don't know which truth value it is yet, but you will if you wait long enough.

Whether you are here or not to verify the result, it will be empirically true or empirically false that the coin you just flipped has come up heads, and it is already true or false now that it will or will not.

But, if I state that the coin comes up heads, flip it, and it comes up tails, the initial statement I made clearly doesn't match with reality, so it's truth value is seemingly false.

The prediction that the coin will come up heads implicitly concerns the next time that the coin lands. Statements have to be interpreted in context.

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  • So the real question: how is an unknown and probably unknowable future truth value different in a practical way from a truth value that is as yet undefined? For us as humans I mean? Can't we abbreviate "we don't know it yet and cannot" with 'undetermined'? We usually make the distinction with the thought that eventually we will be able to know. Physicists (and Philosophers?) seem to hate that "unknowable in principle" idea and hide it under the rug of: "it is there but we can't see it", which I don't get. Knowledge is for us, not for stones.
    – Scott Rowe
    Mar 16 at 12:18
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    ” It would be absurd to assert that the statement has no truth value because this would be equivalent to asserting that the statement "the statement has no truth value" is both true and has no truth value.” Could you expand on that? I’m not seeing the implication.
    – PW_246
    Mar 16 at 17:06
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    I differ with "it is already true or false now that it will or will not." The future categorically does not exist until it becomes the present. Maybe that's just my ignorance of reality, but it seems that way to me. Do we really know that it is absolutely determined that it will or will not rain here next Wednesday? If so, I think I'll just stop trying to do anything, because it either will or will not happen anyway. Why try to push the river?
    – Scott Rowe
    Mar 16 at 22:04
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    Millions of people believe that. And almost every last one of them, sometimes all of them, are wrong. I don't want belief, I want knowledge.
    – Scott Rowe
    Mar 17 at 12:52
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    @ScottRowe "I don't want belief, I want knowledge." Sure, but I was answering the question of whether statements always have a truth value . . . Further, we don't use knowledge, we use beliefs, because this is all we can get. And it works. Humanity has already survived more than 300,000 years. Plus, all animals do the same. Come back to earth. Mar 17 at 17:14
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Any claim to know about a future event when one doesn’t is surely invalid whether or not it turns out to be correct. Here the exact phrasing matters; ‘the coin will come up heads’ is invalid if the speaker doesn’t know, but ‘I predict that the coin will come up heads’ is valid because the speaker is free to make predictions. Whether the prediction turns out to agree with the coin is a separate matter.

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  • I'm going to begin every sentence with, "I predict that..." from now on. I think...
    – Scott Rowe
    Mar 16 at 12:15
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    And you’ll be right every time
    – Frog
    Mar 16 at 20:52
  • "Would you rather be right or be happy?" Can we as humans do anything other than make predictions? Can we really know?
    – Scott Rowe
    Mar 16 at 22:06
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    In formal philosophy we know almost nothing as I’m sure you’re aware. Informally though, we might say that we know something once a reasonable standard of certainty has been achieved (P >> 0.5), so in that context, saying ‘the coin will land heads up’ is valid if that level of confidence exists but invalid if we are just guessing.
    – Frog
    Mar 18 at 4:40
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Tl; dr: Absolute statements about the future have no definite truth value. All we can do is make probabilistic statements.

The future is categorically unknown; quantum physics is one of the two great fundamental shifts in our understanding of nature brought about in the 20th century (the other one being that time and space differ in different reference systems). It effectively killed Laplace's demon.

Quantum physics posit that on the microscopic level, future events are not only unknown but unknowable. Each one of a plethora of possible micro-events (e.g. the spontaneous splitting of an unstable nucleus; the transition of an electron to a lower orbit, emitting a photon) is assigned a probability to happen in a given interval in space and time. It may be more likely to happen at time T1 and location L1 than at some other T2/L2 location, but we really do not and cannot now where it will actually take place.

This principle unknowability caused intense debates at the intersection of physics and philosophy. Famously, Einstein, one of the main proponents, could not accept the "Copenhagen interpretation" which essentially embraces the unknowability, saying "Der Alte würfelt nicht.". I think most physicists today would disagree, even though no proper consensus has been found about what it all means then.

For macroscopic aggregates of particles, these probabilities usually equal out and coalesce to a predictable future, which is why engineering works: Houses do not collapse, bullets hit their targets.

But the fundamental microscopic uncertainty underlying the macroscopic state prevents an entirely known future. Even worse, on the microscopic scale, several possibilities can overlay into a state called "superposition": Even the present time contains more than one possible state; reality proper is indeterminate.

It should be clear that any attempt to assign absolute truth values to future events is laughable. The future is unknowable. For statistical reasons, many macroscopic events are very likely; but the uncertainty creeps in through random effects of microscopic events in non-linear systems (like the coin toss) and through "error accretion" over long time spans.

Does a prediction then change its truth value? I would indeed think that a three-valued logic is necessary to properly reason about predictions, and that yes, the truth value of most predictions changes after the event they predicted happened or not. The exceptions are predictions about — usually microscopic — events which are still part of a superposition, like Schrodinger's cat; the prediction that the cat will be dead at a time T1 in the unopened box has the value "unknown" (or, more precisely, a certain probability) before T1; it retains this "indeterminate" value even after T1 until the box is opened.

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Truth values are strongly associated with logical contexts, the object of study of Epistemic Contextualism.

To put it in simple words, Aristotle's three laws of reason (e.g. Law of Identity) depend on an epistemic context. If you tell a story about Einstein existing to the moon, the statement "Einstein exists on the moon" is true: but ONLY in a certain epistemic context. And it is true as long as the story is told, imagined, used. Outside such context, in any new context, truth is relative (the statement can be false or true, depending on the new context).

With more rigor, epistemic contexts depend on several elements: language, semantics, environment, amount of subjectivity and objectivity, etc. In addition, epistemic contexts allow certain levels of dynamics (that is, breaking the law of identity in the context): e.g.

...

- But men arrived to the moon after Einstein died!

- So, Einstein existing on the moon is necessarily false; it can't be true anymore.

...

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  • "Einstein exists on the moon" in context with a fictional story and "Einstein exists on the moon" in context with non-fictional story are two different statements, despite appearing similar. Truth is absolute, not relative. The statements can change, and you may conflate two different statements to be the same, or may interpret the same statement multiple ways, but changing the interpretation changes the statement, not the truth.
    – David S
    Mar 22 at 22:41
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I state that a coin will come up heads, then flip it. While the coin is flipping, does the statement 'the coin I just flipped will come up heads' have a truth value?

This sounds like a restatement of Schrodinger's cat.

While the coin is flipping it is simultaneously Heads and Tails until it lands like Schrodinger's cat is simultaneously Dead and Alive until the box is opened. So the prediction about the final state of the coin is also simultaneously True and False until the coin lands.

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