Logic

Is this argument valid?: A) The sky is blue. therefore B) 2+2=4 It may not seem that the premise is relevant to the conclusion. But an argument is supposed to be valid if its premises cannot be true without its conclusion being true. B is a necessary truth (we can imagine a world in which the sky is red, but a world in which 2+2=5 is just incoherent). B is always true, therefore B must be true in cases in which A is true. So this must be a valid argument. There's something horribly wrong with this thinking, but I can't figure it out.

Comments

The given argument is indeed valid on the classical definition of validity which you give (there is no possible way in which the premisses could be true and the conclusion false).

On the other hand, we are tempted at first blush to suppose that the conclusion of good argument ought to be connected by some relation of relevance to the premisses.

So either we have to reject the classical definition as unsatisfactory, or we have to revise our first thoughts about relevance as a requirement for being a good argument.

Different logicians jump different ways. Defenders of some variety of "relevance logic" insist on building in the relevance requirement for validity, and so need to revise the classical definition. But it turns out to be not at all easy to do this while preserving other (equally compelling) intuitions. And defenders of the relevance requirement fight among themselves as the best way of implementing it. Which is why the majority of logicians think it better to isolate the neat and clean notion of validity given by the classical definition. They then distinguish this key virtue of being guaranteed not to lead you from truth to falsehood from other virtues an argument might have (like persuasiveness).

(For a longer version of the same thoughts, see this excerpt from my Introduction to Formal Logic.)

It may not help very much, but the argument you describe is not (usually considered to be) logically valid. It's true that 2+2 could not have been other then 4, but almost no-one nowadays would suppose that it was logic that guaranteed that fact. So we might say that the argument is "mathematically valid", since there is no mathematical possibility of its conclusion being false while its premise is true. One other thing worth saying is that one shouldn't confuse the question what implies what with the question what you should infer from what. It does indeed seem silly to infer that 2+2=4 from the premise that the sky is blue (or pink, or green!). But it only follows that "the sky is blue" doesn't imply "2+2=4" if you assume that correct implication should always entail reasonable inference. But you shouldn't assume that.

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