Casper Storm Hansen

This book presents a new nominalistic philosophy of mathematics: semantic conventionalism. Its central thesis is that mathematics should be founded on the human ability to create language – and specifically, the ability to institute conventions for the truth conditions of sentences. This philosophical stance leads to an alternative way of practicing mathematics: instead of “building” objects out of sets, a mathematician should introduce new syntactical sentence types, together with their truth conditions, as he or she develops a theory. Semantic conventionalism is justified first through criticism of Cantorian set theory, intuitionism, logicism, and predicativism; then on its own terms; and finally, exemplified by a detailed reconstruction of arithmetic and real analysis. Also included is a simple solution to the liar paradox and the other paradoxes that have traditionally been recognized as semantic. And since it is argued that mathematics is semantics, this solution also applies to Russell’s paradox and the other mathematical paradoxes of self-reference. In addition to philosophers who care about the metaphysics and epistemology of mathematics or the paradoxes of self-reference, this book should appeal to mathematicians interested in alternative approaches.

Certain prior credence distributions concerning the future lead to inductivism, and others lead to inductive skepticism. I argue that it is difficult to consider the latter to be reasonable. I do not prove that they are not, but at the end of the paper, the tables are turned: in line with pre-philosophical intuitions, inductivism has retaken its place as the most reasonable default position, while the skeptic is called on to supply a novel argument for his. The reason is as follows. There are certain possibilities concerning the functioning of the world that, if assigned positive credence, support inductivism. Prima facie, one might think that the alternatives to those possibilities, if assigned similar or more credence, cancel out that support. However, I argue that it is plausible that reasonable credence distributions are such that the alternatives at most cancel themselves out, and thus leave the support for inductivism intact.

To ascertain the rational credences for the epistemic agents in the famous cases of self-locating belief, one should model the processes by which those agents acquire their evidence. This approach, taken by Darren Bradley (Phil. Review 121, 149–177) and Joseph Halpern (Ergo 2, 195–206), is immensely reasonable. Nevertheless, the work of those authors makes it seem as if this approach must lead to such conclusions as the Doomsday argument being correct, and that Sleeping Beauty should be a halfer. I argue that this is due to an implicit existential bias: it is assumed that the first step in those processes is the determination that the agent in question must necessarily exist. It is much more reasonable to model that determination as contingent and a result of other, earlier, steps in the process. This paper offers such alternative models. They imply an endorsement of what has mockingly been called “presumptuous” reasoning, and a massive shift of credences in favor of (1) the existence of a multiverse and (2) the Everettian interpretation of quantum mechanics.

Lasersohn has argued that the use of Russell's analysis of the definite determiner in Montague Grammar, which is responsible for giving the correct prediction in the case of the Temperature Paradox, is also responsible for giving the wrong prediction in the case of the Gupta Syllogism. In this paper I argue against Lasersohn, and show that the problem of the Gupta Syllogism can be solved by making a minor addition to Intensional Montague Grammar. This solution is one that Lasersohn discusses but rejects. I will show that his critique of it is ill-founded.

Two reverse supertasks—one new and one invented by Pérez Laraudogoitia —are discussed. Contra Kerkvliet and Pérez Laraudogoitia, it is argued that these supertasks cannot be used to conduct fair infinite lotteries, i.e., lotteries on the set of natural numbers with a uniform probability distribution. The new supertask involves an infinity of gods who collectively select a natural number by each removing one ball from a collection of initially infinitely many balls in a reverse omega-sequence of actions.

Radical glut and gap theorists deny—in opposite ways—that the liar sentence has exactly one of the two values true and not true. I describe a scenario where a signalman finds himself in a situation analogous to the liar paradox: if he lights a fire at a certain time, that is analogous to the liar being true, and if he does not, that is analogous to the liar not being true. It is obvious that he must make exactly one of those states of affairs come about. It is argued that there are no relevant differences between the liar and the signalman’s dilemma, implying that the glut and gap theorists are wrong about the former. A further point is that whether or not the liar is true/the signalman lights the fire, language/the signalman is misleading relative to the conditions under which the liar/the fire “ought” to be true/lit.

This paper offers a unification and systematization of the grounding approaches to truth, denotation, classes and abstraction. Its main innovation is a method for “kleenifying” bivalent semantics so as to ensure that the trivalent semantics used for various linguistic elements are perfectly analogous to the semantics used by Kripke, rather than relying on intuition to achieve similarity. The focus is on generalizing strong Kleene semantics, but one section is devoted to supervaluation, and the unification method also extends to weak Kleene semantics.