A quick note: someone whipped out this old bit of folk-logic that “you can’t prove a negative” at me earlier today. This statement shows up an awful lot in all sorts of debates, but despite its folklore position as some sort of rule of elementary logic, no logician ever has actually proposed it.

And that’s because it is pretty clearly not true, after all, it contains within itself its own handy refutation: “you can’t prove a negative” is itself a negative statement, so if you can’t prove a negative, you can’t prove that “you can’t prove a negative.”

But of course there are plenty of examples of negative statements that people can prove. One of the real elementary laws of logic is that any proposition *P* is identical to the negation of its negation, that is, to *not-not-P*. So if you can prove a positive statement, then you also prove a negative statement which is equivalent to it. (If Descartes could prove that he existed with *cogito, **ergo sum*, he could also thus prove that he wasn’t nonexistent.)

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Prove is such a nasty word within the realm of logic.

I’m sorry, what? A nasty word? Prove is a well-defined term in logic, as is proof, and one of the central concepts.

How is that nasty?

I meant that it leads to unnecessary conflict and emotional responses. I think very little can ever be proven beyond any kind of doubt. But I also think many people misuse it within inquiry and discussion. For example, a theist may claim “I can prove God exists” when really they mean to say “I can attempt to make a case that shows belief in God is more reasonable than not.”

Sure, fine, I agree with you, but none of that is “within logic.”

The actual intended statement is that “you can’t prove a *universal* negative”, e.g. There is no such thing as a black swan (back when folks didn’t know such swans actually do exist in Australia).

To some extent this is a justified statement, if we restrict ourselves to empirical observations of the known universe. But if we are talking about logical universes (universes of discourse, e.g.) then of course it is often possible to ‘universally’ prove the non-existence of some proposed element.

Also, if we are talking about logically defined entities, then we can also prove that there are no circles with points on the circumference that are further away from the centre than the radial distance, or that there are no such things as triangles with other than three sides, or three sided polygons that are not also triangles.

Often, the objection is to an atheist saying “God doesn’t exist.” To preempt this objection, some atheists simply amend that to “God *probably* doesn’t exist.”

Yet, for certain logical definitions of the word ‘god’, it is possible to prove that *that* ‘god’ doesn’t exist. A good example would be the tri-omni (all good, all knowing, all powerful) god that so many people believe exists. The problem of evil logically disproves this god’s existence (Epicurus’ trilemma as an example disproof).

Unfortunately, people who repeat the statement omitting the important ‘universal’ qualifier often don’t understand enough logic yet to see where they went wrong, in which case, the points you make in this post are very helpful.

Yes, absolutely, I agree: there are lots of universal negatives (mostly involving empirical questions) that one can’t really prove. But I still don’t think that even “you can’t prove a universal negative” is a valid objection to an attempt to prove such a thing.

But the problem with proving that there is no such thing as a black swan is

notthat it is a universal negative. Rather, it is that every proof of this is necessarily inductive: so far every swan we see isn’t black, therefore no swans are black. Inductive proof will always be problematic in this way (although it is still essential to human life and rational thought.)Trying to prove an existential negative, or universal/existential positive, inductively suffers the same flaws. So even in these instances, I would argue, it is incorrect to respond to the claim that “there is no black swan” or “there is no god” with “you can’t prove a universal negative” because (a) as you note there *are* universal negatives you can prove, and (b) the problem with the proof is not that it is a negative or universal claim, but that the method of inductive proof is inherently flawed (outside of mathematical induction, obviously.)

I agree with your analysis there. I don’t use the objection myself, I was just trying to state what that objection is usually intended to mean (induction vs. proving universal negatives). By understanding the intention more completely, I find that it’s easier to unpack a person’s usage of that objection more thoroughly, and so show them the ins and outs of it, and why it stands or falls in any particular case. Basically, in order to help set them straight. 😉