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on this analysis, and although Oksanen quoted Russell's description of the paradox in detail, he did not show how it is explained in NFU after his resolution of the other related modal paradoxes; in fact, it is not at all clear how this might be done in NFU. 1PoM, p. 527. 2See, e.g., Grim 1991, pp. 92f. 3See, e.g., Grim 1991, p. 119 and Jubien 1988, p. 307. 4See Oksanen 1999. NFU is a modified version of Quine's system NF. It was first described in Jenson 1968 and recently has been extensively developed in Holmes 1999. Nordic Journal of Philosophical Logic, Vol. 5, No. 1, pp. 25--37. 2000 Taylor & Francis. 26 nino b. cocchiarella One reason why Russell's argument is dicult to reconstruct in NFU is that it is based on the logic of propositions, and implicitly in that regard on a theory of predication rather than a theory of membership. A mo

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In this article, we canvass a few of the interesting topics that philosophers can pursue as part of the simultaneous study of logic and metaphysics. To keep the discussion to a manageable length, we limit our survey to deductive, as opposed to inductive, logic. Though most of this article

ABSTRACT: Here is a crude list, possibly summarizing the role of paradoxes within the framework of mathematical logic: 1. directly motivating important theories (e.g. type theory, axiomatic set theory, combinatory logic); 2. suggesting methods of proving fundamental metamathematical results (fixed point theorems, incompleteness, undecidability, undefinability); 3. applying inductive definability and generalized recursion; 4. introducing new semantical methods (e. g. revision theory, semi-inductive definitions, which require non-trivial set theoretic results); 5. (partly) enhancing new axioms in set theory: the case of anti-foundation AFA and the mathematics of circular phenomena; 6. suggesting the investigation of non-classical logical systems, from contraction-free and many-valued logics to systems with generalized quantifiers; 7. suggesting frameworks with flexible typing for the foundations of Mathematics and Computer Science; 8. applying forms of self-referential truth and in Artificial Intelligence, Theoretical Linguistics, etc. Below we attempt to shed some light on the genesis of the issues 1–8 through the history of the paradoxes in the twentieth century, with a special emphasis on semantical aspects.