Topics Logic

Question Frequently, one finds the following statment: "You cannot prove a negative." My question is, in this context, what is meant by the word "negative?" I understand how the word is used in mathematics and I "think" I know the meaning when used in logic. I just cannot seem to get a handle on how it is used here. Moreover, does it, perhaps, refer to a total position in the debate over the existence of God? Any comments you would make would be greatly appreciate. I enjoy your application very much and, moreso, since I am so old. Thanks. JH

Accepted:
September 6, 2012

Comments

This is a pretty confusing expression. What's usually meant, I think, is that a negative general proposition -- a proposition asserting that a certain kind never occurs -- requires much more by way of justification from its defender than from its opponent. Take the proposition "there are no black swans," for example. To prove it, you would have to comb through the whole universe, presumably all the way backward and forward in time, to demonstrate conclusively that nothing contained therein is a black swan. To disprove the proposition, by contrast, all you need do is produce a single black swan. Given this asymmetry, it thus makes sense to saddle the opponent, rather than the proponent, of a negative general proposition with the burden of proof.

What's confusing here is that the same sort of asymmetry is present with affirmative general propositions as well. Thus the proposition "all elks like mushrooms" requires much more by way of justification from its defender than from its opponent. To prove it, you would have to comb through the whole universe, presumably all the way backward and forward in time, to investigate all elks in regard to their fondness for mushrooms. To disprove the proposition, by contrast, all you need do is produce a single elk who doesn't like mushrooms. Given this asymmetry, it makes sense once again to saddle the opponent, rather than the proponent, of such an affirmative general proposition with the burden of proof.

In its most general formulation, the point your queried statement tries to express is then not one about negative versus affirmative propositions but one about propositions governed by different logical quantifiers. Any proposition -- negative or affirmative -- that is universally quantified requires much more by way of justification from its defender than from its opponent, who need merely produce a single counterexample, that is, prove an existentially quantified proposition.

To see this, we might state our two sample propositions in logical language. Both of them are governed by a universal quantifier ("for all x, ..."):

(1) For all x, if x is a swan then x is not black.

(2) For all x, if x is an elk then x likes mushrooms.

the negations of these two propositions, both governed by existential quantifiers ("there is an x such that...") are, respectively,

(1-) There is an x, such that x is a swan and x is black.

(2-) There is an x, such that x is an elk and x does not like mushrooms.

Clearly, much more is required of the defenders of (1) and (2) than of the defenders of (1-) and (2-). This is an interesting point, but not one well expressed in the common statement you cite.