Can an argument be valid even when its premise refutes the conclusion? (Trying to disprove my professor)

Question

I want to ask about something that I saw in philosophy class today.

Let's say that this is a valid argument, and let's call it argument A:

• Premise 1: P1
• Premise 2: P2
• Conclusion: C

And there's another argument, let's call it argument B, that goes like this:

• Premise 1: P1
• Premise 2: P2
• Premise 3: C', whereas C' is the logical opposite of C
• Conclusion: C

My question is: Is argument B valid, just because argument A is valid?

My professor says that argument B is valid, because by ignoring Premise 3, argument B will become exactly the same as argument A, and hence, argument B must also be valid.

But I find this very fishy, because by definition of validity, if all the premises of argument B are true (and hence C' is true), it is impossible for C, the conclusion, to be true!

What do you think? Are there scholarly sources on philosophical logic than can help me build a case against my professor/or to realize that I am wrong?

Answer 1

Yes, argument B is valid.

However, if you write argument B in any context, including in a philosophy, logic, or math exam, you're opening yourself to very simple criticism: while argument B is valid, if you stop at its conclusion, then you're missing an obvious follow-up conclusion.

The obvious follow-up would result in the following argument, of which argument B is a subargument (or lemma):

• Lemma:

• Premise: P1
• Premise: P2
• Premise: C'
• Intermediary conclusion: C
• Conclusion: contradiction

• Conclusion: not(P1) or not(P2) or not(C').

In other words, argument B is valid, but stopping at its conclusion is missing an obvious and much more enlightening conclusion.

You are right to find this fishy.

In any real situation, is someone is making an argument, but deliberately ignoring a consequence of the conclusion of their argument which is both obvious, and obviously more useful than the conclusion they settled on, then they're being deliberately misleading.

Answer 2

There is a difference between soundness and validity of an argument. An argument is valid if the conclusion can be (formally) derived from the premises. For the argument to be sound, there is the additional condition that the premises must be true. In this sense, argument B is valid but not sound, as the premises P1, P2 and C‘ cannot all be true.

It may seem counterintuitive that an argument with contradictory premises can be formally valid. But when you think about it, common sense is not a good guide when dealing with contradictions anyway.

You may be interested paraconsistent logic, that is logical systems that try to deal with contradictions without using the „principle of explosion“, i. e. the principle that from a contradiction, you can validly derive whatever you want.

Answer 3

Your Argument B is not valid even if your argument A is.

As you say, if we assume that C' is true and C' is contradictory to C, then C is false, and so we cannot ever derive C logically from any set of premises which includes C'.

Probably a majority of academic specialists will say otherwise, as your professor himself does, and as many answers on this forum will also do, but they only say what they say because this is what they have learned at school, but the same specialists cannot properly explain human logic, as demonstrated by academic papers on the Paradox of the Liar.

The idea that you could derive logically something which is necessarily false is simply absurd, and this is all we need to know to decide that this idea is wrong. Yet, this is an idea taught in probably all universities around the world.

This is the same idea which has been repeated here:

From contradictory premise everything follows.

This is clearly absurd, but this is what most academics now believe, and what most pundits on this forum will repeat.

You will have to make up your own mind to decide whether this makes sense.