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> I don’t think this is a standard definition. Well, I suppose it depends on your definition of standard. That's how I have been taught logic. I also believe i

• denotational 3y ago
  > Well, I suppose it depends on your definition of standard. Of course :) I believe my distinction between validity and truth is the one generally used in the literature (I have listed four examples above), and the one that would be understood by most working mathematicians and analytic philosophers who care about mathematical logic. We both agree that there is a clear distinction between formulae that are true in some model (specified, or inferred from context) and formulae that are true in all models; the latter are not particularly interesting to most mathematicians once one has agreed on the logic (e.g. classical, constructive, etc.) in which one operates, hence I think it’s reasonable to use “true” to refer to the former, as indeed many authors do. > That's called satisfiability. Many logicians say that a formula is true in a model (sometimes true in a structure) if it’s satisfied in that model under all assignments. Can you find me a reference in the literature where “true” is used to mean “true in all models” consistently?
  • loicd 3y ago
    > We both agree that there is a clear distinction between formulae that are true in some model (specified, or inferred from context) and formulae that are true in all models; [...] Sure. But I feel we are deviating from the subject. We have obviously been educated differently so it is pointless to argue about that, but there is a language issue. You insist on comparing what I mean by "true" (alone) with "true in a model". However, that's an apple to orange comparison. We should be comparing what I mean by "true" (alone) with what you mean by "true" (alone), and by that, you mean: "true in the standard model". (I don't think your references validate that use, although I don't have access to all of them at the moment.) The obvious problems with that are: - I don't think there is such a thing as a standard model in set theory (actually you cannot prove that a model of set theory exists). - When most mathematicians say something like "X is true", what they mean is "X can be proved from the axioms of set theory", which in logical terms means "X is valid". Are you really arguing against that? - And of course (back to the original point), you get that confusing idea that "undecidable" means "true but unprovable" (I had never heard of the incompleteness theorem being presented that way before.). I argue "undecidable" is "neither provable nor disprovable". EDIT: "X is valid" should read "X is valid in set theory".
    • denotational 3y ago
      I think there are two points of disagreement here: 1. The definition of “truth”; and 2. My claim that most mathematicians who aren’t logicians use “true” in the sense of being true in some model, not in all of them. Re. 1, I accept that this is just naming, but I will insist that your usage of “true” is nonstandard, and should carry a disclaimer as such :) I think this is evidenced by the fact that “true but unprovable” is a very widely used characterisation of the Gödel sentence for a logic, and it is completely wrong given your definition of “true”, and further evidenced by the references I provided that use “valid” for true in all models. I think point 2 is a lot more fuzzy because we’re trying to talk about what is happening inside other people’s minds, in particular their view of what a mathematical result actually is; I’m happy to concede that I may be guessing wrong regarding what most mathematicians are actually thinking. As per 1, my position is that there is no such thing as “true alone”, at least not in mathematical logic as it is conventionally presented English speaking world (and apparently in the Spanish speaking world, if one source is enough to generalise); truth only exists in the context of a model. Accordingly, when we talk about the truth of a formula, we have a model (or possibly a class of models) in mind; I claim that when a mathematician says the Gödel sentence is true, they mean it is true in the standard model of the naturals, which is why the “true but unprovable” characterisation is used. I don’t know if my references support this, since it’s harder to find than just looking for the definition of “validity” in the index, but if you Google “truth of the Gödel sentence” you’ll find a lot of people using “true” to mean true in the standard model (of the naturals). I suspect that most mathematicians are Platonists (this may be my bias creeping in) and they believe the objects they work with are real; I certainly believe the naturals are real, and I believe that non-standard models of the naturals are not the (real) naturals, even though I’m perfectly happy to “play” with non-standard models as an intellectual exercise. I claim that most mathematicians have other (real) objects in mind when dealing with things other than the naturals, and those form the meta-mathematical model for the notion of truth. > When most mathematicians say something like "X is true", what they mean is "X can be proved from the axioms of set theory", which in logical terms means "X is valid". Are you really arguing against that? That doesn’t mean “X is valid”; if something follows from the axioms of set theory then it holds in all models of set theory (yes, assuming a sound deductive calculus, but that’s a given), but that’s not enough to consider it valid, it would need to hold in all models compatible with the language. So yes, I’m arguing against that, I think what you have written is factually wrong, unless it’s a typo? > you get that confusing idea that "undecidable" means "true but unprovable" I don’t think I ever said this, I agree that this is confusing, and in fact it’s just wrong as stated (under my usage of “truth”), since of course there are undefinable sentences that are false (but irrefutable), such as the negation of the Gödel sentence.