Luca Incurvati

According to the iterative conception of set, sets can be arranged in a cumulative hierarchy divided into levels. But why should we think this to be the case? The standard answer in the philosophical literature is that sets are somehow constituted by their members. In the first part of the paper, I present a number of problems for this answer, paying special attention to the view that sets are metaphysically dependent upon their members. In the second part of the paper, I outline a different approach, which circumvents these problems by dispensing with the priority or dependence relation altogether. Along the way, I show how this approach enables the mathematical structuralist to defuse an objection recently raised against her view.

Maddy gave a semi-formal account of restrictiveness by defining a formal notion based on a class of interpretations and explaining how to handle false positives and false negatives. Recently, Hamkins pointed out some structural issues with Maddy's definition. We look at Maddy's formal definitions from the point of view of an abstract interpretation relation. We consider various candidates for this interpretation relation, including one that is close to Maddy's original notion, but fixes the issues raised by Hamkins. Our work brings to light additional structural issues that we also discuss.

ABSTRACTLinguistic evidence supports the claim that certain, weak rejections are less specific than assertions. On the basis of this evidence, it has been argued that rejected sentences cannot be premisses and conclusions in inferences. We give examples of inferences with weakly rejected sentences as premisses and conclusions. We then propose a logic of weak rejection which accounts for the relevant phenomena and is motivated by principles of coherence in dialogue. We give a semantics for which this logic is sound and complete, show that it axiomatizes the modal logic KD45 and prove that it still derives classical logic on its asserted fragment. Finally, we defend previous logics of strong rejection as being about the linguistically preferred interpretations of weak rejections.

Paul Horwich (1990) once suggested restricting the T-Schema to the maximally consistent set of its instances. But Vann McGee (1992) proved that there are multiple incompatible such sets, none of which, given minimal assumptions, is recursively axiomatizable. The analogous view for set theory---that Naïve Comprehension should be restricted according to consistency maxims---has recently been defended by Laurence Goldstein (2006; 2013). It can be traced back to W.V.O. Quine(1951), who held that Naïve Comprehension embodies the only really intuitive conception of set and should be restricted as little as possible. The view might even have been held by Ernst Zermelo (1908), who,according to Penelope Maddy (1988), subscribed to a ‘one step back from disaster’ rule of thumb: if a natural principle leads to contra-diction, the principle should be weakened just enough to block the contradiction. We prove a generalization of McGee’s Theorem, anduse it to show that the situation for set theory is the same as that for truth: there are multiple incompatible sets of instances of Naïve Comprehension, none of which, given minimal assumptions, is recursively axiomatizable. This shows that the view adumbrated by Goldstein, Quine and perhaps Zermelo is untenable.

Mathematical realists have long invoked the categoricity of axiomatizations of arithmetic and analysis to explain how we manage to fix the intended meaning of their respective vocabulary. Can this strategy be extended to set theory? Although traditional wisdom recommends a negative answer to this question, Vann McGee (1997) has offered a proof that purports to show otherwise. I argue that one of the two key assumptions on which the proof rests deprives McGee's result of the significance he and the realist want to attribute to it. I consider two strategies to deal with the problem --- one of which is outlined by McGee himself (2000) --- and argue that both of them fail. I end with some remarks on the prospects for mathematical realism in the light of my discussion.