I quote a philosopher Michael Antony:
Another common claim of the New Atheists is that you ‘can’t prove a negative’ – where what is typically meant is a negative existence claim of the form ‘X does not exist’. Rhetorically, this claim functions to legitimize the idea that evidence needn’t be provided for God’s nonexistence. After all, if evidence cannot be provided for a proposition it would be irrational to expect one to provide some, and so reasonable to believe that evidence isn’t needed. But the claim that you can’t prove a negative cannot help the atheist. That is because, on each of two possible ways of interpreting what it means to ‘prove’ something, it is generally false that you can’t prove a negative (and often true that you can’t prove a positive).
Consider first, proofs which deliver certainty, as in mathematics or logic. Such proofs are sometimes possible for negative existence claims, such as the claim that there is no greatest prime number. One can also prove with certainty that there are no Xs whenever the concept X can be shown to be incoherent (like the concepts round square, or 3pm on the sun). Of course, it is true that many negative existence claims cannot be proved with absolute certainty, but the same holds for positive existence claims, for example, from science or common sense, such as that there are electrons or tables and chairs. So there’s nothing special here about negative existence claims.
Turn next to proofs which aim to establish only the probable truth of their conclusions. These are the sorts of proofs which result from successful scientific and other empirical investigations. In this sense of ‘proof’, it is easy to prove the non-existence of many things: for example, that there is no pomegranate in my hand, or no snow-capped mountains in the Sahara Desert. And while it may be difficult or impossible to even in this weaker sense prove the non-existence of many things – goblins, sombreros in the Sombrero Galaxy – the same goes for many positive existence claims – that Aristotle sneezed on his 20th birthday; that there is a transcendent deity; that there is a sombrero somewhere in the Sombrero Galaxy. So, again, there is nothing unique about negative existence claims. The unfortunate saying that one can’t prove a negative should be dropped.
Another common claim of the New Atheists is that you ‘can’t prove a negative’ – where what is typically meant is a negative existence claim of the form ‘X does not exist’. Rhetorically, this claim functions to legitimize the idea that evidence needn’t be provided for God’s nonexistence. After all, if evidence cannot be provided for a proposition it would be irrational to expect one to provide some, and so reasonable to believe that evidence isn’t needed. But the claim that you can’t prove a negative cannot help the atheist. That is because, on each of two possible ways of interpreting what it means to ‘prove’ something, it is generally false that you can’t prove a negative (and often true that you can’t prove a positive).
Consider first, proofs which deliver certainty, as in mathematics or logic. Such proofs are sometimes possible for negative existence claims, such as the claim that there is no greatest prime number. One can also prove with certainty that there are no Xs whenever the concept X can be shown to be incoherent (like the concepts round square, or 3pm on the sun). Of course, it is true that many negative existence claims cannot be proved with absolute certainty, but the same holds for positive existence claims, for example, from science or common sense, such as that there are electrons or tables and chairs. So there’s nothing special here about negative existence claims.
Turn next to proofs which aim to establish only the probable truth of their conclusions. These are the sorts of proofs which result from successful scientific and other empirical investigations. In this sense of ‘proof’, it is easy to prove the non-existence of many things: for example, that there is no pomegranate in my hand, or no snow-capped mountains in the Sahara Desert. And while it may be difficult or impossible to even in this weaker sense prove the non-existence of many things – goblins, sombreros in the Sombrero Galaxy – the same goes for many positive existence claims – that Aristotle sneezed on his 20th birthday; that there is a transcendent deity; that there is a sombrero somewhere in the Sombrero Galaxy. So, again, there is nothing unique about negative existence claims. The unfortunate saying that one can’t prove a negative should be dropped.
