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Given this definition:

A deduction is valid if and only if its conclusion is true whenever all of its hypotheses are true.

Can an argument be valid if it has a tautology as a conclusion?

An example I can think of is:

  1. A proposition is either a tautology or not a tautology
  2. (1) is a proposition
  3. (1) is either a tautology or not a tautology

That seems strange/wrong to me, even though it appears to meet the definition. Are there other conditions for an argument to be valid?

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One would think that most tautologies are so because they can in principle be a conclusion of valid arguments given any premises. Am I missing something? –  Paul Ross Dec 2 '12 at 15:16
I like the line of argument. It appears bizarre, because through our own natural language capabilities we see staightaway that 1 is a tautology. Whereas the conclusion asserts that it may not be. I would say that you need a definition of what a tautology is, and using this, show that 1 is a tautology. So that the line of argument you show is wasteful: it asserts more than is the case. –  Mozibur Ullah Dec 2 '12 at 18:02
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1 Answer

A tautology is a conclusion which requires no hypotheses. Then, in particular, it doesn't have any hypotheses which are false. Therefore, as we say in the business, all of its hypotheses (vacuously) are true. Then the tautology is a true conclusion.

You should perhaps think of logical validity not as "truth-preserving", but more accurately as "not increasing falsity". (This depends on being rather staunchly Boolean in one's view of logic, so that you would for instance regard 0 = 1 to be no larger a falsehood than 0 = 0.0001; but it is certainly not an extremely controversial view.) Then, if you have no premisses, you have no falsehood, and anything you can derive from no premisses can therefore contain no falsehood, i.e. is necessarily true.

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I haven't come across this definition of a tautology. Working in a formal system, are its axioms to be taken literally, and not hypothetically? –  Mozibur Ullah Dec 2 '12 at 14:01
@Mozibur Ullah: I would say that a tautology does not involve any (non-logocal) axioms; and because both by training and personal inclination I would subsume any logical axioms into the rules of inference, I tend to describe tautologies as involving no axioms whatsoever. –  Niel de Beaudrap Dec 2 '12 at 17:43
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