Topics Logic

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

Accepted:
September 18, 2014

Comments

I'm inclined to say that no tautologies are ever logically equivalent, but only because no sentences are ever logically equivalent. I take it that any tautology is a sentence in some language, as opposed to the proposition expressed by that sentence. Indeed, the etymology of the term implies that a tautology is a sentence characterized by the repetition of words: Greek tauto ("the same") + logos ("word"). An example is the English sentence "All red things are red." Unlike sentences, propositions don't contain words, so tautologies can't be propositions, strictly speaking. I interpret "logically equivalent" to mean "matching in truth-value at every possible world." Two things match in truth-value at every possible world only if both things exist at every possible world. But no sentence -- no item of any language -- exists at every possible world, because the very language of the sentence might never have existed: all language is contingent. Therefore, no two sentences are ever logically equivalent: only the propositions (contingently) expressed by sentences ever are. If all tautologies are sentences, then no tautologies are logically equivalent. In short, logical relations obtain between propositions rather than between sentences. Or so it seems to me.

Stephen Maitzen raises some interesting philosophical issues, but, of course, his response is not the "textbook" answer to the question (but, then, isn't that what philosophy is all about? : Questioning "textbook" answers? :-)

The "textbook" answer would go something like this: By definition, a tautology is a "molecular" sentence (or proposition---textbooks differ on this) that, when evaluated by truth tables, comes out true no matter what truth values are assigned to its "atomic" constituents.

So, for example, "P or not-P" is a tautology, because, if P is true, then not-P is false, and their disjunction is true; and if P is false, then not-P is true, and their disjunction is still true.

Furthermore, by definition, two sentences (or propositions) are logically equivalent if and only if they have the same truth values (no matter what truth values their atomic constituents, if any, have).

So, because tautologies always have the same truth value (namely, true), they are always logically equivalent.

Moreover, two contradictions (sentences (or propositions) whose truth value is always "false") are also logically equivalent.

Finally, the sentence (or proposition) that asserts that two sentences (or propositions)---whether both true or both false---are logically equivalent is itself a tautology.

(For example, "(P or not-P) is logically equivalent to (not-(P & not-P))" is a tautology, because "P & not-P" is a contradiction, so its negation is a tautology, so that negation is logically equivalent to "P or not-P", which, by the way, can be proved using DeMorgan's Laws.)

I thank William Rapaport for his comment. I'll just point out that the claim

two sentences (or propositions) are logically equivalent if and only if they have the same truth values (no matter what truth values their atomic constituents, if any, have)

seems to imply the following odd consequence. Take two sentences lacking atomic sentential constituents: "Snow is white" and "Obama was born in Hawaii." Both sentences are true (sorry, birthers), but isn't it odd to hold that the two sentences are logically equivalent? Granted, they're materially equivalent, but that's just a technical way of saying that they in fact have the same truth-value. Something stronger seems required for genuine logical equivalence, which is why I prefer the definition I gave above. Fortunately, some standard textbooks do define it in that stronger way.

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".

Question Is it possible for two tautologies to not be logically equivalent? Accepted:September 18, 2014

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

Accepted:
September 18, 2014