Sobocinski in his paper on Leśniewski's solution to Russell's paradox (1949b) argued that Leśniewski has succeeded in explaining it away. The general strategy of this alleged explanation is presented. The key element of this attempt is the distinction between the collective (mereological) and the distributive (set-theoretic) understanding of the set. The mereological part of the solution, although correct, is likely to fall short of providing foundations of mathematics. I argue that the remaining part of the solution which suggests a specific (...) reading of the distributive interpretation is unacceptable. It follows from it that every individual is an element of every individual. Finally, another Leśniewskian-style approach which uses so-called higher-order epsilon connectives is used and its weakness is indicated. (shrink)

SummaryLeśniewski presented his logical systems in a way which conformed to his nominalism, so the question arises whether Leśniewski's logic can be given a natural formal semantics which, unlike current versions, avoids commitment to abstract entities. Building on hints in Wittgenstein's Tractatus, I develop the idea of a way of meaning which is the basis for what I call combinatorial semantics. I then consider whether this commits us to abstract objects or an intensional metalogic.

The new system of axioms we propose is based on the foundational theory of De Giorgi et al. Scuola Normale Superiore di Pisa, Preprints di Matematica 26: 1 (1996) slightly modified. In that paper (which is dedicated to a new axiomatic framework for mathematics, informatics and logic) the authors use two kinds of primitive notions: relations and qualities. Since their system is based on the distribution paradigm, they start from distinction. We propose to shift the perspective and to start from (...) unity and then from within unity to pass to distinction; to this end we apply ideas of Lesniewski, Nijhoff International Philosophy Series 44 (1992). We introduce only one kind of entity as a primitive notion, namely relations, and treat qualities as articulations of relations. The new concept of "primary relation" permits the introduction of a dynamic, non-standard form of identity, which we hope will find application in various fields where self-referential structures are required. (shrink)