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I studied logic and found no category for statements that are not impossible but never true (e.g. "The earth is flat.") to differ them from statements that could be true ("I'm wearing blue shoes.") or that were or are occassionally true ("I drink beer."). Why not? Wouldn't it be hepful to classify not only what is necessarily true and false but also what is or was always or never true or false? I don't think it's the same as a contradiction (e.g. "The small house is larger than the big house.") since a contradiction is false in relation to itself and per definition false whereas a statement that was never true is not impossible and could have been true of another world ("The stock market gained 500 % today").

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    Isn't it simply a false proposition? – stevebot Dec 11 '13 at 21:06
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Tense Logic makes this distinction. In tense logic, there are all of the typical logic connectives plus 4 modal operators {P, F, H, and G} defined as follows:

  • Px = "x was true at some time"
  • Fx = "x will be true at some time"
  • Hx = "x has always been true"
  • Gx = "x will always be true"

To use the example you've given of "The earth is flat," we could express it like so:

Lx = x is flat   and    e = Earth

  ~P(Le) ^ ~F(Le)   // It's not the case that the Earth was flat at some time,
                    // and it's not the case that the Earth will be flat at some time.
  or

  H(~Le) ^ G(~Le)   // It has always been the case that the Earth is not flat,
                    // and it will always be the case that the Earth is not flat.

You may also want to explore Modal Logic (from which Tense Logic is an extension). In Modal Logic there are Necessary Truths (things that must be true in all possible worlds), Possible Truths (things that may or may not be true in a given possible world), and Contingent Truths (things that happen to be true in our world but are not necessary). Your examples could all be considered false contingent truths.

  • How do you know ~F(Le) is true? – Lukas Dec 12 '13 at 8:22
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The way "The earth is flat" differs from "I'm wearing blue shoes" is that the latter sentence includes an indexical: "I." An indexical is a term that can change its reference based on the context in which it's uttered. "You" is another example, as are "this" and "nearby." Sentences containing indexical terms take on different truth-values in a more obvious way than sentences like "The earth is flat" do. But even a sentence like that can vary in truth-value. Suppose I like going to a bar called "the earth." Maybe for some reason I want to say that it's flat, not rectangular like some other bars. My point is that other than that one contains indexicals, there's no difference between the sentences in terms of the relationship between their structures and truth. That is unlike a logical contradiction, which has a structure that makes it false. A false proposition, on the other hand, as stevebot notes, is simply false, because we say that different utterances of sentences (or instances of them in different contexts) sometimes express different propositions, which are true, or false, or possibly something intermediate.

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I think you got the notion of contradiction kind of wrong. "There exists a horse and there does not exist a horse" is a contradiction, but not in relation to itself and not per definition. They are never true.

Your example, "The earth is flat", is not never true. It is just not true here, but possibly true.

To clarify your question you should define the scope of "never". Either it means our world, then the simple actual true/false suffices. Or you understand it modally, then something that is necessary false is never true.

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Adding to what @Lukas said, I think the only statement I can think of that never is true is "This statement is false", and that's known as the liar paradox.

  • What about: "There exists a squared circle", on my book its never true and necessarily false. – Lukas Dec 12 '13 at 8:20
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How about impossible? However...

Borges, the Argentinian writer in short story The Library of Babel, wrote that any proposition in some system of thought may be true. This is his library of thought, or of worlds.

In model theory which is the mathematical investigation of truth & proofs in formal systems there are what are called undecidable propositions - that is propositions which are true but cannot be formally proved so. One way to look at this is that in some worlds it can be proved true, and in other worlds it will be proved false; so generally, that is formally it isn't provable. This means, in this architecture of truth & proof, a provably false propositon is one that is false in all models - that is in no model can it be true.

False propositions, like the flat-earth theory can also have significant truth value when looked at the right way. For example, we can say that the Earth looks flat wherever you are on the Earth, and given the actual physical landscape this is actually an abstract thought of a kind. Notably, it is this definition which is used in the mathematical discipline of manifold theory whose most famous application is in Einsteins theory of Gravity. So, although, some people dismissively say that flat space was proved wrong by Einstein, a more subtle understanding is that it became a local segment of the theory.

In Hegelian discourse, thought is a process and not a proposition. A given proposition gives rises to its opposite and are then sublated in a synthesis. No proposition is wholly true, nor wholly false, but must seen in a context of every increasing sophistication. So Truth is not simply a binary affair of true & false, that is the very categories of thought - true & false are false - or properly partly true & partly false.

In Jain Logic, some propositions, are false and indescribable, or true and indescribable. One might suppose for example, Negative Theology, which describes God by what he is not, because by being an Infinite Being any description is naturally not wholly true; one might say He is true but indescribable. This is part of a system of Jain thought called Anekantavada which means objective knowledge can only be obtained by Kevalis who see objects in all aspects & all manifestations.

One might suppose a proposition that exhibited itself as a contradiction must be false. For example, '0 is 1', or 'grass is not grass'. But this can be argued against, simply by seeking some context where it is true; for example the first statement is true in what is called the trivial ring, and the second is not true of a painting of grass. One might argue then, one should include the context explicitly. But this then leads immediately, at least in thought, to an infinite regression. But of course this does not happen in practise - and one could argue that might be one way of understanding or arguing that consciousness is one.

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