Valid argument?

Question

An argument having ~(P -> P) as one of its premises is a necessarily valid argument?

Sorry I ask this question again, last time half the answers were yes and the other half no. Please give some reasons that I can understand with your answer.

Answer 1

Yes.

The way we determine validity is by looking at a truth table. If an argument is valid then when all of the premises are true the conclusion must be true. The way we determine if something is invalid is if, when the premises are all true the conclusion is false.

When you introduce a contradiction you introduce something which cannot be true. For all values ~(P -> P) is false. That means that we will never have a situation where all of the premises are true.

If we can never have all of our premises true at the same time then we can never have all of our premises true AND have a false conclusion. Since that scenario is the only way to tell if we have an invalid argument then we cannot, when we have a contradiction in the premises, have an invalid argument.

Answer 2

That is correct. Such an argument is necessarily valid.

It's like an argument that starts "if two plus two was five instead of four, then ...". There is nothing you can put after the 'then' that will be wrong.

So long as the premise is definitively impossible, no conclusion can be impossible.

Edit: To respond to Doctor Y, consider: "If two plus two were five, then two plus two would not be five." This is a valid argument because two plus two is never five. An "if X then Y" argument is valid if either the X is never true or the Y is always true. Since this has both of those, it must be valid. So an argument that concludes the opposite of one of its premises can still be valid. (Provided the premise cannot ever be true.)

Answer 3

Yes, any argument having a logically false premise is necessarily valid.

Recall the definition of validity: An argument is deductively valid  if and only if it is not possible for the premises to be true and the conclusion false.

An argument having a logically false premise satisfies this requirement. It is not possible for the premise, a logical falsity, to be true. Therefore, it is not possible for the premises to be true and the conclusion false -- again, because the premise cannot be true.

The argument is truth preserving because it will never take us from truths to a falsehood. It will not do so because the premise cannot be true, and hence there is no possibility of going from truths to a falsehood. Arguments with a premise that is logically false, while valid, are of course never sound.

Arguments of this sort are sometimes dismissed as not being arguments at all, precisely because their validity does not depend on the relation between the premises and conclusion. There are, however, systemic reasons for allowing these cases to constitute arguments and thus for recognizing them as valid deductive arguments. It is important to remember that such arguments are valid because they meet the requirement of truth preservation -- they will never take us from truths to a falsehood -- not because the premises support the conclusion in any intuitive way.

Similar special cases of validity involve arguments which have a logically true conclusion, or have premises which form an inconsistent set.

Answer 4

No.

For an argument to be valid, the conclusion must necessarily follow from the premises.  Assuming the premises are true, there can be no other outcome.

In this case, we don't know the conclusion.  For example, if your conclusion is...

(P -> P)

... then it couldn't possibly be valid because we assumed that this was NOT the case in a premise.

So no.  There is no way to tell if an argument is valid from just one or any number of premises.  You MUST have BOTH premises and conclusion to tell.