(Goodsell, Journal of Philosophical Logic, 51(1), 127-150 2022) establishes the noncontingency of sentences of first-order arithmetic, in a plausible higher-order modal logic. Here, the same result is derived using significantly weaker assumptions. Most notably, the assumption of rigid comprehension—that every property is coextensive with a modally rigid one—is weakened to the assumption that the Boolean algebra of properties under necessitation is countably complete. The results are generalized to extensions of the language of arithmetic, and are applied to answer a question (...) posed by Bacon and Dorr (2024). (shrink)

I propose a revision of Cantor’s account of set size that understands comparisons of set size fundamentally in terms of surjections rather than injections. This revised account is equivalent to Cantor's account if the Axiom of Choice is true, but its consequences differ from those of Cantor’s if the Axiom of Choice is false. I argue that the revised account is an intuitive generalization of Cantor’s account, blocks paradoxes—most notably, that a set can be partitioned into a set that is (...) bigger than it—that can arise from Cantor’s account if the Axiom of Choice is false, illuminates the debate over whether the Axiom of Choice is true, is a mathematically fruitful alternative to Cantor’s account, and sheds philosophical light on one of the oldest unsolved problems in set theory. (shrink)

An increasing amount of contemporary philosophy of mathematics posits, and theorizes in terms of special kinds of mathematical modality. The goal of this paper is to bring recent work on higher-order metaphysics to bear on the investigation of these modalities. The main focus of the paper will be views that posit mathematical contingency or indeterminacy about statements that concern the ‘width’ of the set theoretic universe, such as Cantor’s continuum hypothesis. Within a higher-order framework I show that contingency about the (...) width of the set-theoretic universe refutes two orthodoxies concerning the structure of modal reality: the view that the broadest necessity has a logic of S5, and the ‘Leibniz biconditionals’ stating that what is possible, in the broadest sense of _possible_, is what is true in some possible world. Nonetheless, I suggest that the underlying picture of modal set-theory is coherent and has attractions. (shrink)

I argue that it is metaphysically necessary that: (i) every fundamental entity is conscious, and (ii) every fundamental property is a phenomenal property.

How should we account for the extraordinary regularity in the world? Humeans and Non-Humeans sharply disagree. According to Non-Humeans, the world behaves in an extraordinarily regular way because of certain necessary connections in nature. However, Humeans have thought that Non-Humean views are metaphysically objectionable. In particular, there are two general metaphysical principles that Humeans have found attractive that are incompatible with all existing versions of Non-Humeanism. My goal in this paper is to develop a novel version of Non-Humeanism that is (...) consistent with (and even entails) both of these general metaphysical principles. By endorsing such a view, one can have the explanatory benefits of Non-Humeanism while at the same time avoiding two of the major metaphysical objections towards Non-Humeanism. (shrink)