In this paper, we present a simple sequent calculus for the modal propositional logic S5. We prove that this sequent calculus is theoremwise equivalent to the Hilbert-style system S5, that it is contraction-free and cut-free, and finally that it is decidable. All results are proved in a purely syntactic way.

This paper traces the evolution of thinking on how mathematics relates to the world—from the ancients, through the beginnings of mathematized science in Galileo and Newton, to the rise of pure mathematics in the nineteenth century. The goal is to better understand the role of mathematics in contemporary science.

The paper argues that the view that the particular quantifier is ‘existentially loaded’ is a relatively new one historically and that it has become entrenched in modern philosophical logic for less than happy reasons.

The class of strong random reals can be defined via a natural conception of effective null set. We show that the same class is also characterized by a learning-theoretic criterion of ‘recognizability’.

It is shown that the standard definitions of truth-functionality, though useful for their purposes, ignore some aspects of the usual informal characterisations of truth-functionality. An alternative definition is given that results in a stronger notion that pays attention to those aspects.

One interpretation of the conditional If P then Q is as saying that the probability of Q given P is high. This is an interpretation suggested by Adams (1966) and pursued more recently by Edgington (1995). Of course, this probabilistic conditional is nonmonotonic, that is, if the probability of Q given P is high, and R implies P, it need not follow that the probability of Q given R is high. If we were confident of concluding Q from the fact that we knew P, and we have stronger information R, we can no longer be confident of Q. We show nonetheless that usually we would still be justified in concluding Q from R. In other words, probabilistic conditionals are mostly monotonic.

Several philosophers have argued that the logic of set theory should be intuitionistic on the grounds that the open-endedness of the set concept demands the adoption of a nonclassical semantics. This paper examines to what extent adopting such a semantics has revisionary consequences for the logic of our set-theoretic reasoning. It is shown that in the context of the axioms of standard set theory, an intuitionistic semantics sanctions a classical logic. A Kripke semantics in the context of a weaker axiomatization is then considered. It is argued that this semantics vindicates an intuitionistic logic only insofar as certain constraints are put on its interpretation. Wider morals are drawn about the restrictions that this places on the shape of arguments for an intuitionistic revision of the logic of set theory.

The two expressions ‘The cumulative hierarchy’ and ‘The iterative conception of sets’ are usually taken to be synonymous. However, the second is more general than the first, in that there are recursive procedures that generate some ill-founded sets in addition to well-founded sets. The interesting question is whether or not the arguments in favour of the more restrictive version – the cumulative hierarchy – were all along arguments for the more general version.

A decision procedure (PrSAT) for classical (Kolmogorov) probability calculus is presented. This decision procedure is based on an existing decision procedure for the theory of real closed fields, which has recently been implemented in Mathematica. A Mathematica implementation of PrSAT is also described, along with several applications to various non-trivial problems in the probability calculus.

We show that the set of ultimately true sentences in Hartry Field's Revenge-immune solution model to the semantic paradoxes is recursively isomorphic to the set of stably true sentences obtained in Hans Herzberger's revision sequence starting from the null hypothesis. We further remark that this shows that a substantial subsystem of second-order number theory is needed to establish the semantic values of sentences in Field's relative consistency proof of his theory over the ground model of the standard natural numbers: -CA0 (second-order number theory with a -comprehension axiom scheme) is insufficient. We briefly consider his claim to have produced a ‘revenge-immune’ solution to the semantic paradoxes by introducing this conditional. We remark that the notion of a ‘determinately true’ operator can be introduced in other settings.