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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?

user2766
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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 02 '12 at 15:16
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    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 02 '12 at 18:02

3 Answers3

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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.

Niel de Beaudrap
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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 02 '12 at 14:01
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    @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 02 '12 at 17:43
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Any argument with a tautology as the conclusion is valid, no matter what the premises are. Validity is a technical term in formal logic meaning that the conclusion cannot fail to be true if the premises are true. Since a tautology is always true it follows for such an argument that the conclusion can not fail to be true if the premises are true.

However, in the larger picture such an argument is useless, since the premises have no necessary relationship to the conclusion.

Understanding this kind of result is important if you want to grasp the ways in which formal logic can often be counter-intuitive.

Chris Sunami
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Thank you for your question I've just understood a great deal about logic, from where I come from it is very dear.

  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

It seems like this could go endlessly. you could then say.

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

And so on and so on. This might be correct in some logic systems...But

From a previous question I asked about the liars paradox, which is what this reminds me of, M. Cort Ammon replied that a certain M.Gödel concerned himself with systems which could "admit arithmetic". These systems being "strong" logical systems.

And showed that in these systems :

For any formal effectively generated theory T including basic arithmetical truths and also certain truths about formal provability, if T includes a statement of its own consistency then T is inconsistent.

so for example:

"1.(1) is not true" Is an inconsistent statement

Now your argument implies that we can easily swap "a proposition" by "(1)" without changing any of the meaning since "a proposition" can be any proposition. That gives us

  1. (1) is either a tautology or not a tautology

which is inconsistent in a system that admits arithmetic.

Alexandre Babeanu
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