Do these statements mean exactly the same thing: (a) You should not not buy that book. (b) You should buy that book.

Comments

(a) sounds a bit awkward and one might wonder whether it's ambiguous. Does it mean:

(a1) You should make it be the case that (it is not the case that (you do not buy that book)),

(a2) It is not the case that (you should make it be the case that (you do not buy that book)).

Using "S" to stand for "You should make it be the case that" and "N" for "It is not the case that", (a1) has the form: S(N(N(p))). But (a2) has the form: N(S(N(p))).

(a2) does not mean the same as (b), which has the form: S(p). But (a1) is arguably identical in meaning to (b). That's because "N(N(p))" means the same as "p".

A cautionary note: In general, most people would agree that "N(N(p))" means the same as "p", that is, that a statement means the same as its double negation. What would a world look like, you might wonder, in which those statements differ in their truth or falsity? We can't even coherently describe it. I say "most people," though, because some have wondered whether this equivalence holds regardless of what the statement "p" is about. For instance, some have wondered whether this equivalence is correct if the statement concerns the future: if it's not true that it's not going to rain tomorrow, does that mean that it is going to rain tomorrow? Well, that depends on whether you think it's already now determined that either it will rain or it will not. If you think the future is unsettled, or indeterminate, then you might not be willing to assert that either it will rain or it will not. And if you're unwilling to assert this disjunction, then you will be unwilling to say that if it's not the case that it's not going to rain tomorrow then it must be that it will rain tomorrow.

Another domain of discourse that makes some pause about the equivalence in question concerns mathematical statements about infinitely large collections. Those who do not believe such a statement is equivalent to its double negation are called intuitionists. For more on intuitionism, see Question 139, or search for "intuitionism" on this site.

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