## Is it possible for two tautologies to not be logically equivalent?

The term "tautology" has no established technical usage. Indeed, most logicians would avoid it nowadays, at least in technical writing. But when the term is used informally, it usually means: sentence (or formula) that is valid in virtue of its *sentential* (as opposed to predicate, or modal) structure. I.e., the term tends to be restricted to sentential (or propositional) logic.

It is clear that Rapaport is assuming the sort of usage just mentioned: "a tautology is a 'molecular' sentence...that, *when evaluated by truth tables*, comes out true no matter what truth values are assigned to its 'atomic' constituents". Hence, on this definition, "Every man is a man" would not be a "tautology". Which is fine. It's logically valid, but not because of *sentential* structure.

It is all but trivial to prove, as Rapaport does, that all tautologies are logically equivalent. In fact, however, it is easy to see that Rapaport's proof does not depend upon the restriction to sentential logic. One can prove (as he of course knows) by exactly the same argument that all "valid" sentences are logically equivalent. (All logic texts prove this. It is, for example, general law (12) on p. 64 of Warren Goldfarb's *Deductive Logic*, which is the text we use at Brown.) The argument goes through so long as "valid" means "true in every interpretation" and "equivalent" means "have the same truth-value in every interpretation". In particular, it doesn't matter what is meant by an interpretation (at least as long as the interpretations are classical).

A couple other remarks on the discussion.

First, when Rapaport says that "two [interpreted] sentences...are logically equivalent if and only if they have the same truth values (no matter what truth values their atomic constituents, if any, have)", he is talking not about logical equivalence *in general* but about "truth-functional" equivalence, since only sentential constituents can have truth-values. And "Snow is white" and "Obama was born in Hawaii" are not, of course, *truth-functionally* equivalent. The sole atomic constituent of each sentence is that sentence itself. (Constituency is so defined as to make it a reflexive relation, trivially.) And an interpretation can perfectly well assign these two sentences different truth-values. So this is just a case of P not being equivalent to Q.

Second, as Rapaport notes, there is some controversy whether the basic notion here should apply to sentences (or formulas) or to propositions. But if it applies to syntactic items (as I would prefer), then the sentences and formulas have to be regarded as "interpreted", i.e., as having fixed meanings, or at least the "logical constants" have to be so regarded. Otherwise, indeed, no sentence will be "always true". But if the meanings are held fixed, then, for many purposes, the difference makes no difference, since the notions are inter-definable: A sentence is (logically) true if the proposition it expresses is (logically) true; a proposition is (logically) true if there is a (logically) true sentence that expresses it. The remaining philosophical issue is which notion is more fundamental.

Finally: For lots of interesting material on the word "tautology" and the history of its use in logic, see Burton Dreben and Juliet Floyd, "Tautology: How Not To Use a Word".

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