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My logic/philosophy exercises include the following statement:

"Every tautology is knowable a priori but not every tautology is necessarily true."

I'm bamboozled. How can a tautology not be necessarily true? If it's a tautology, then it's true in every possible world, surely?

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In logic, a tautology is defined as a logical truth of the propositional calculus. If your preferred semantics of logical truth is 'true in all possible worlds' then yes, a tautology is true in all possible worlds and hence necessarily true. Other semantics for logical truth include model theory, category theory and various kinds of substitutivity. In some of these, one does not say that a logical truth is 'necessarily' true: indeed one may not believe in necessary truths at all.

It is possible that your exercise is referring to Kripkean examples of contingent apriori statements, such as the recently discussed question about the standard metre. Maybe in a general sense the statement "the standard metre is one metre long" could be described as tautologous, but this is not the strict sense of tautology used in logic.

  • Thanks I think this answer is closest to the mark. The statement was actually on a previous exam paper so I think it is unlikely to be a typo. – Brendan Hill Nov 1 '15 at 6:27
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I think your exercise has a typo, or the point was to catch the error in the exercise. Quoting:

In logic, a tautology (from the Greek word ταυτολογία) is a formula that is true in every possible interpretation.
...
A formula of propositional logic is a tautology if the formula itself is always true regardless of which valuation is used for the propositional variables.

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I suppose a tautology may be meaningless.

All four-horned unicorns have four horns.

seems to be tautological, but since the set of four-horned unicorns is empty, you can also say

All four-horned unicorns have three horns.

And yes, all four-horned unicorns (zero out of zero) have three, or four, or five hundred and twenty seven, horns. And so the tautology in case isn't either true or false.

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It is possible that the exercise has a typo. However, there are logics that construe its claim as correct. Tautologies are logically true, by logic alone, it is a linguistic notion. Necessity, on the other hand, is a modal notion, and a priori is an epistemological notion. There is no conceptual reason why the three can not come apart. This said, contingent tautologies are quaint, and logicians rarely entertain them, even Kripke explicitly stipulates that he only considers analytic truths, which includes tautologies, that are necessary. However, Kaplan's sentence "I am here now" can be construed as a contingent tautology, it is tautologically true in any utterance, but it could always have been otherwise.

In Demostratives Kaplan suggests that the distinction between logical and necessary truths reflects the distinction between character and content:

"The bearers of logical truth and of contingency are different entities. It is the character (or, the sentence, if you prefer) that is logically true, producing a true content in every context. But it is the content (the proposition, if you will) that is contingent or necessary".

An alternative path to contingent tautologies is to allow logic itself to vary across possible worlds. In worlds where the classical logic obtains the law of excluded middle will be a tautology, but not in intuitionistic worlds. Thus, the law of excluded middle will not be a necessary tautology. Zalta discusses more examples in Logical and Analytic Truths That Are Not Necessary.

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