Self-Reference and Negativity as Conditions of Paradoxes

Evgeny Borisov identifies two conditions of set-theoretical paradoxes: self-reference and deducibility of a contradiction from it. Based on the general scheme of paradoxes suggested by W. Lawvere the article shows that self-reference as well as negativity is not necessarily present in their statements. Hence neither self-reference, nor deducibility of a contradiction from it can serve as paradoxes conditions even in the limited case of set-theoretical paradoxes. The notion of negativity is even more obscure but at least in some senses it is absent in the formulations of paradoxes. The article examines indefinitely extensible concepts (Dummett) and unnoticed model shift in non-wellfounded set theory (Barwise, Moss) as candidates for paradoxes conditions. However, even in these cases the principal interest lies in particular concepts – ordinals, truth, denotation etc. – which we found as misconceived. Whereas universal schemes of paradoxes are useful for setting out the general direction of our research but not more.