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The Wikipedia article on the corresponding conditional contains the following sentence:

An argument is valid if and only if its corresponding conditional is a logical truth.

Some sources use "tautology" in place of "logical truth":

An argument is valid if and only if its corresponding conditional is a tautology.

This got me thinking about what "tautology" and "logical truth" actually mean, because a tautological corresponding conditional does not seem to be a tautology in the same way that P v ~P is a tautology. P v ~P seems to be a tautology by virtue of the definition of "v". It will always be tautology, regardless of what sentence P represents. However, whether or not a corrsponding conditional is a tautology depends on the truth values of the premises and conclusion of the argument that the conditional represents. So, in what sense is a tautological corresponding conditional a tautology and does this differ to the sense in which P v ~P is a tautology?

My confusion might be stemming from my very hazy understanding of the concepts of "logical truth", "tautology" and "necessary truth":

  • "Tautology" seems to be a term of propositional logic which describes a sentence that is true on every possible valuation/truth-value assignment.
  • "Logical truth" seems to be a term of first-order logic, but when used within the context of propositional logic, it is synonymous with tautology (I'm not sure why, as I haven't studied FOL yet).
  • "Necessary truth" seems to be something that is fundamentally true. All tautologies are necessary truths, but not all necessary truths are tautologies, e.g. the statement "1 = 1" is a necessary truth, but, in propositional logic, it can only be expressed using a single sentence letter, which cannot be a tautology on its own.

I also came across this page, which draws a distinction between 1) tautologies which are true by virtue of the logical terms they contain (e.g. "every", "some" and "is") and are synonymous with logical truths, and 2) truth-functional tautologies, which are true by virtue of the connectives they contain (so, something like P v ~P?). However, the paragraph is missing citations and I can't find any other sources that distinguish between tautologies/logical truths and truth-functional tautologies.

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We say in general that a formula is valid when it is true in every interpretation.

A tautology is a valid formula of propositional logic [sense 2) above: "truth-functional tautologies, which are true only by virtue of the logical connectives"].

Thus, P ∨ ¬P is a tautology (valid propositional formula), while ∀x (x=x) is a valid formula of predicate logic.

If we agree on this, the informal concept of "logical truth" is formalized with valid formula [sense 1) above: "true by virtue of the logical terms they contain, e.g. the logical connectives, 'every', 'some' and 'is' "].

We can extend the use of "valid" to arguments: a formal argument is valid when there is no interpretation where the premises are true and the conclusion is false.

In most logical systems we can prove the so called Deduction (meta-)theorem that links valid arguments with valid formulas.

In this case, we have that A ⊢ B implies ⊢ A→B that means that if the argument deriving conclusion B from premise A is valid, then the formula A→B is valid.

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  • Some authors don't restrict "tautology" to the propositional calculus. They use the word for any logic to mean any proposition that is true in all models. Commented Dec 14, 2022 at 11:34
  • @DavidGudeman Indeed, alas. One of many, many examples in logic where different authors use the same word for different things, or different words for the same thing. You would think logicians would be able to sort themselves out, but there is considerable lack of consensus on terminology. Nice answer from Mauro.
    – Bumble
    Commented Dec 14, 2022 at 11:56
  • @DavidGudeman - indeed... IMO the best way is to use "valid" for the most general case (included e.g. modal logic, etc.) and restrict "tautology" to the truth-functional case. But it is my opinion. In addition, the modern use of "tautology" has been introduced by Wittgenstein (1922): "Tractatus 4-46 Among the possible groups of truth-conditions there are two extreme cases. In one of these cases the proposition is true for all the truth-possibilities of the elementary propositions. We say that the truth-conditions are tautological [and] we call the proposition a tautology". Commented Dec 14, 2022 at 11:57
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    I agree with preferring the restricted use of tautology. Some authors, such as Enderton, even speak of 'tautological consequence' as something distinct from logical consequence, which wouldn't make sense if tautologies and validities were synonymous.
    – Bumble
    Commented Dec 14, 2022 at 19:32
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    @DavidGudeman At the end of the day it is only a matter of terminology. One might say that it serves a pedagogical purpose in that most textbooks introduce propositional logic first and then proceed on to first order, and they may wish to observe a distinction. Some authors make a deal out of the fact that we can effectively determine whether a formula is a tautology, but not whether a formula is true in every interpretation. Perhaps one might simply say that we already have a good term 'validity' or 'valid sentence' for the general case, so why make 'tautology' nothing more than a synonym.
    – Bumble
    Commented Dec 15, 2022 at 13:26

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