Is it possible to positively prove a negative?

Why on earth not? Elementary mathematics is full of proofs of negative claims. There are familiar school-room proofs that no negative number has a real square root, that no fraction is the square root of two, that there is no largest prime number.

People often say this and it can be baffling to logicians! Perhaps your use of "positively" hints at what you're getting at though. Let's assume by "prove a negative" you mean something like: establish that something of a particular kind does not exist. For instance, your "negative" statement might be: Martians do not exist. And perhaps by "positively prove" you mean: establish by pointing to a particular thing that does exist. Then an instance of your claim might be that it's not possible to establish that Martians do not exist by displaying any particular non-Martian. And that's right: just because this particular object is not a Martian it doesn't follow that there are no Martians. In general, from the fact that a particular object is not an F we cannot logically infer that there are no Fs. So, if that's what you mean, logicians will agree that it's not possible "to positively prove a negative." However, that does not mean that one cannot logically prove statements of the form: there are no Fs. Those statements can be the conclusions of logically correct arguments.

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