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.