"philosophers have often used first-order logic to analyze mathematical and scientific claims. however, we seem to grasp a notion of logical possibility prior to, and independent from, our grasp of mathematical objects like set-theoretic models. and powerful reasons to accept this notion as an additional logical primitive have emerged (gómez-torrente 2000; hanson 2006; boolos 1985,h. h. field 2008b,etchemendy 1990a). in this book, i'll make a case that philosophical analyses using (a natural generalization of) this notion of logical possibility can illuminate the philosophy of mathematics, metaphysics, and philosophy of language. much of this case will focus on pure mathematics and the philosophy of set theory. for example, i will show that formulating set theory in terms of logical possibility (along potentialist lines suggested by putnam and hellman in response to the burali-forti paradox) yields a new and more appealing justification for one of the standard zfc (zermelo-frankel with choice) axioms of set theory. this brings us closer to realizing the traditional hope of justifying mainstream mathematics from principles that seem clearly true. however, we will see that using a primitive logical possibility operator can also help us develop a modestly neo-carnapian philosophy of language. and philosophical analyses of scientific theories using the logical possibility operator can illuminatingly `factor' scientific claims into a logico-mathematical component and a remainder, in a way that reveals hidden heterogeneity in the role of mathematics in the sciences and clarifies debates over quinean and post-quinean indispensability arguments. 1.1 mathematics as a touchstone and the centrality of set theory "admittedly, the present state of affairs where we run up against the paradoxes is intolerable. just think, the definitions and deductive methods which everyone learns, teaches, and uses in mathematics, the paragon of truth and certitude, lead to absurdities! if mathematical thinking is defective, where are we to find truth and certitude? " mathematical proofs provide a touchstone of clarity and convincingness which serves as an inspiration to philosophy and other disciplines. while it is possible to doubt the results of mainstream mathematical arguments (philosophers are capable of doubting anything), there's something striking about just how convincing mathematical proofs often are. consider the standard argument that there are infinitely many primes. even philosophers who deny that there are numbers (and hence think the argument as usually stated is unsound) are strongly tempted to say that we know something like the premises and that these proofs provide some kind of valuable amplification of this knowledge. the premises we use in informal mathematical reasoning have a combination of prima facie obviousness, power and generality, which makes them exemplary tools for expanding our knowledge and resolving disputes in cases where people's initial hunches disagree. it's no surprise that leibniz wished philosophers could resolve their disputes like mathematicians by saying 'let us calculate' (or at least, 'let us each look for a proof'). in many ways, set theory lies at the heart of modern mathematics, and it does powerful mathematical (not just philosophical) work as a foundation for the whole. so, one might hope that the set-theoretic foundations for mathematics would share the clarity and convincingness we hope for from mathematical arguments. however, certain problems in the philosophical foundations of set theory raise worries. these concerns are more mathematical and specific to set theory than standard philosophical worries about, e.g., whether there are any abstract objects. and these concerns are more threatening to mathematical practice than philosophical doubts typically are, insofar as they raise doubts about whether the standard zfc (zermelo fraenkel with choice) axioms of set theory are even logically consistent. the development of set theory resolved a great many problems in analysis. it also provided a formal framework to allow interactions between various areas of mathematics - creating, as hilbert famously observed (hilbert 1926), a kind of mathematical paradise. however, contradiction, in the form of russell's paradox, threatened hilbert's paradise"--