 Classical logic  gets round this by widening the interpretation of disjunction: it interprets \(P \vee Q\) as \(\neg(\neg P\wedge \neg Q)\), or in other words, “it is contradictory that both \(P\) and \(Q\) be false”. In turn, this leads to the idealistic interpretation of existence, in which \(\exists xP(x)\) means \(\neg \forall x\neg P(x)\) (“it is contradictory that \(P(x)\) be false for every \(x\)”). It is on these interpretations of disjunction and existence that mathematicians have built the grand, and apparently impregnable, edifice of classical mathematics which serves a foundation for the physical, the social, and (increasingly) the biological sciences. However, the wider interpretations come at a cost: for example, when we pass from our initial, natural interpretation of \(P \vee Q\) to the unrestricted use of the idealistic one, \(\neg(\neg P\wedge \neg Q)\), the resulting mathematics cannot generally be interpreted within computational models such as recursive function theory.