Anthony F. Peressini

The Quine/Putnam indispensability approach to the confirmation of mathematical theories in recent times has been the subject of significant criticism. In this paper I explore an alternative to the Quine/Putnam indispensability approach. I begin with a van Fraassen-like distinction between accepting the adequacy of a mathematical theory and believing in the truth of a mathematical theory. Finally, I consider the problem of moving from the adequacy of a mathematical theory to its truth. I argue that the prospects for justifying this move are qualitatively worse in mathematics than they are in science.

In the intriguing final chapter of his book Warrant and Proper Function, Alvin Plantinga argues that naturalism, conjoined with a neo-Darwinian picture, is self-defeating. This argument has drawn its fair share of critical response. Plantinga in turn has recently responded in his as yet unpublished manuscript ‘Naturalism Defeated’. This first volley of debate has helped bring into focus several points of contention between Plantinga and his critics, but to date, the logical structure of the argument has not been an element of the discussion. In this paper I argue that the logical structure of Plantinga’s argument has been misconstrued – that it does not show that naturalism is self-defeating. I argue further that properly reconstructed, it fails to make the case for the (ordinary) defeat of naturalism as well.

In this paper I examine the nature of Russell's ramified type theory resolution of paradoxes. In particular, I consider the effect of construing the types in Church's cumulative sense, that is, the range of a variable of a given type includes the range of every variable of directly lower type. Contrary to what seems to be generally assumed, I show that the decision to make the levels cumulative and allow this to be reflected in the semantics is not neutral with respect to the solution of the paradoxes. I introduce a distinction between syntactical and semantical cumulativeness. It turns out that noncumulative type theories (in either sense) are equally capable of dealing with the paradoxes. Furthermore, whether cumulativeness is appropriate appears to be context dependent.