Indirect proof

We showed in Section 3.7(a) that the law of double negation, ¬¬ A ⊃ A, is not derivable in intuitionistic logic. The proof of underivability was done with formal detail in Section 4.4, example (d). The difference between A and ¬¬ A was explained in Section 3.7(c): The former is a direct proposition, the latter expresses the impossibility of something negative, the best example being direct existence against the impossibility of non-existence. The former can be established by showing an object with a required property, the latter by showing that it is impossible that no object has the property.

One reason for the natural tendency to accept the law of double negation, or the related law of excluded middle, is as follows. If there is only a finite number of alternatives, the question of A or ¬ A can be decided by going through all of these. Say, if we claim, for natural numbers less than 100, that there are three and only three successive odd numbers that are all prime, we can go through all possible cases and find that 3, 5, and 7 are precisely those three numbers. More generally, the constructive interpretation of A ∨ ¬ A is that it expresses the decidability of A.