Akademik

Following Ramsey and the Italian mathematician G. Peano (1858–1932) it has been customary to distinguish logical paradoxes that depend upon a notion of reference or truth (semantic notions), such as those of the Liar family, Berry, Richards, etc., from the purely logical paradoxes in which no such notions are involved, such as Russell's paradox, or those of Cantor and Burali-Forti . Paradoxes of the first type seem to depend upon an element of self-reference, in which a sentence talks about itself, or in which a phrase refers to something defined by a set of phrases of which it is itself one. It is easy to feel that this element is responsible for the contradictions, although self-reference itself is often benign (the sentence ‘All English sentences should have a verb’ includes itself happily in the domain of sentences it is talking about), so the difficulty lies in forming a condition that excludes only pathological self-reference. Paradoxes of the second kind then need a different treatment. Whilst the distinction is convenient, in allowing set theory to proceed by circumventing the latter paradoxes by technical means, even when there is no solution to the semantic paradoxes, it may be a way of ignoring the similarities between the two families. There is still the possibility that while there is no agreed solution to the semantic paradoxes, our understanding of Russell's paradox may be imperfect as well.