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I know that an argument's validity has nothing to do with the truth of its premises, but does it have something to do with the validity of its premises? I'm thinking specifically of where a premise is itself a syllogism.

For example:

P1. (A → B) ∧ ¬A → ¬B
P2. (A → B)
P3. ¬A
C: ¬B

Would this count as a valid argument?

Michael
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2 Answers2

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A premise is not valid or invalid, it is either true or false. Validity only applies to deductions.

Maybe the confusion comes from the fact that you're conflating the logical implication "->" and the deduction rule. Logical implication is a logical operator that says that either its antecedent is false or its consequence is true, but it does not say that B is deducible from A. For example if "p:=tigers are mammals" is true and "q:=it is raining" is true, "p->q" is true even though q cannot be deduced from p.

In your example, the premise is not a syllogism, but a logical statement that can be true or false depending on what you mean by A and B. From this sentence and the other premises you can deduce the conclusion. The argument is valid. Whether the premise is true or not will depend on what you mean by A and B, but the premise is neither invalid or valid: it's not a deduction, but a statement.

Quentin Ruyant
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Officially, it can be 'valid' but not 'sound'. As all of the Lewis Carroll logic exercises involving flying pigs and boiling oceans attest, logic has nothing to do with reality per se. Validity, the primary property logic pursues in an argument, preserves truth, but some source outside logic needs to fill in the truth-values of the source premises.

Many authors introduce the separate notion of a 'sound' argument which is one that is both valid and 'grounded in reality'. Of course, at that point they have departed from logic in its strictest sense because logic cannot tell you how to verify your axioms.

And, as to the example, yes, deducing anything from a contradiction is always a valid argument. If your premises contradict one another, logic can't help you, and you need to address your sources of information, instead.