How can an argument make no sense, but still be valid?

Ah, nothing like a little logical paradox for Christmas. Today we will look at how to make a valid argument—that disagrees with itself and makes no sense. Dive in to the paradox of entailment.

When the sun is in the sky it is day.

When the sun is in the sky it is not day.

You should send Esther Inglis-Arkell money.

The above argument is valid, under the rules of classical logic. (Also valid under the rules of holiday spending, but that's another post.) You might have noticed that not only is the conclusion not intuitively related to either premise, but the premises themselves are mutually exclusive. The exclusivity of the premises, technically, makes the argument valid.

The solution to this riddle is in the word "valid." As Inigo Montoya famously said, "You keep using that word. I do not think it means what you think it means." Just as in everyday life the word "theory," means something different than the term "theory" means in the sciences, "valid" in everyday life and "valid" in classical logic have different meanings. In logic, the only way for a argument to be valid is if there is no situation in which the premises are true and the conclusion is false. This argument, then, just sneaks by. The premises are mutually exclusive. There is no way the premises can both be true, and therefore there is no way that the premises can both be true while the conclusion is false. The conclusion itself doesn't matter. Any statement could be used. All that matters is the argument cannot, technically, be shown to be invalid.

Instead of the word "valid," we mere mortals should instead be using the word "sound." An argument is sound if it is valid and the premises are true. So, sadly, the above argument is valid, but not sound. Not that you should let that stop you if you feel so inclined.

Via St Andrews.