Mathematics and philosophy have historically enjoyed a mutually beneficial and productive relationship, as a brief review of the work of mathematician–philosophers such as Descartes, Leibniz, Bolzano, Dedekind, Frege, Brouwer, Hilbert, Gödel, and Weyl easily confirms. In the last century, it was especially mathematical logic and research in the foundations of mathematics which, to a significant extent, have been driven by philosophical motivations and carried out by technically minded philosophers. Mathematical logic continues to play an important role in contemporary philosophy, and mathematically trained philosophers continue to contribute to the literature in logic. For instance, modal logics were first investigated by philosophers and now have important applications in computer science and mathematical linguistics.

This special issue presents a series of articles focusing on recent work in formal epistemology and formal philosophy. The articles in the latter category elaborate on the notion of context and content and their relationships. This work is not unrelated to recent developments in formal epistemology. Logical models of context, when connected with the representation of epistemic context, are clearly relevant for many issues considered by formal epistemologists. For example, the semantic framework Joe Halpern uses in his article for this issue has been applied elsewhere to solve problems in interactive epistemology (which can be seen as an active branch of contemporary formal epistemology).

In this article, we propose a belief revision approach for families of (non-classical) logics whose semantics are first-order axiomatisable. Given any such (non-classical) logic , the approach enables the definition of belief revision operators for , in terms of a belief revision operation satisfying the postulates for revision theory proposed by Alchourrón, Gärdenfors and Makinson (AGM revision, Alchourrón et al. (1985)). The approach is illustrated by considering the modal logic K, Belnap's four-valued logic, and Łukasiewicz's many-valued logic. In addition, we present a general methodology to translate algebraic logics into classical logic. For the examples provided, we analyse in what circumstances the properties of the AGM revision are preserved and discuss the advantages of the approach from both theoretical and practical viewpoints.

Public announcement logic is an extension of multiagent epistemic logic with dynamic operators to model the informational consequences of announcements to the entire group of agents. We propose an extension of public announcement logic with a dynamic modal operator that expresses what is true after any announcement: ⋄φ expresses that there is a truthful announcement ψ after which φ is true. This logic gives a perspective on Fitch's knowability issues: For which formulas φ, does it hold that φ → ⋄Kφ? We give various semantic results and show completeness for a Hilbert-style axiomatization of this logic. There is a natural generalization to a logic for arbitrary events.

Peter Gärdenfors proved a theorem purporting to show that it is impossible to adjoin to the AGM-postulates for belief-revision a principle of monotonicity for revisions. The principle of monotonicity in question is implied by the Ramsey test for conditionals. So Gärdenfors’ result has been interpreted as demonstrating that it is impossible to combine the Ramsey test for conditionals with the basic postulates for rational belief-revision. It is shown here that this interpretation of Gärdenfors’ result is unwarranted. A new diagnosis is offered of a methodological error made in the statement of the key principle of monotonicity. Crucial applications of this principle in Gärdenfors’ proof require one to regard as revisions what are really expansions. If monotonicity is stated only for genuine revisions, then Gärdenfors’ proof does not go through. Nor can it; for, when the monotonicity principle for revisions is correctly formulated, one can actually establish a contrary consistency result. This requires only a slight adjustment to the postulates of AGM-theory, in order to ensure that the three operations of expansion, contraction, and revision trichotomize the domain of theory-changes. It is further shown that being careful in this way about the proper domains of definition of the three operations of expansion, contraction, and revision also disposes of another, more direct, impossibility result, due to Arló-Costa, that targets the Ramsey test.

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.

As McKinsey and Tarski showed, the Stone representation theorem for Boolean algebras extends to algebras with operators to give topological semantics for (classical) propositional modal logic, in which the “necessity” operation is modeled by taking the interior of an arbitrary subset of a topological space. In this article, the topological interpretation is extended in a natural way to arbitrary theories of full first-order logic. The resulting system of S4 first-order modal logic is complete with respect to such topological semantics.

Just as set theory can be divorced from Ernst Zermelo's original axiomatization of it, counterpart theory can be divorced from the eight postulates that were originally stipulated by David Lewis (1968, p. 114) to constitute it. These were postulates governing some of the properties and relations holding among possible worlds and their inhabitants. In particular, counterpart theory can be divorced from Lewis's postulate P2, the stipulation that individuals are ‘world bound’—that none exists in more than one possible world

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.

Argument-places play an important role in our dealing with relations. However, that does not mean that argument-places should be taken as primitive entities. It is possible to give an account of ‘real’ relations in which argument-places play no role. But if argument-places are not basic, then what can we say about their identity? Can they, for example, be reconstructed in set theory with appropriate urelements? In this article, we show that for some relations, argument-places cannot be modeled in a neutral way in V[A], the cumulative hierarchy with basic ingredients of the relation as urelements. We argue that a natural way to conceive of argument-places is to identify them with abstract, structureless points of a derivative structure exemplified by positional frames. In case the relation has symmetry, these points may be indiscernible.

We propose an approach to assigning propositional content to deliberate acts of arbitrary type, as opposed to just speech acts. This approach, which is based on the idea that the content of an act is the decision maker's expectation concerning the change that would take place if the act were to be performed, is shown to be related to the concept of expected utility that has played a central role in various accounts of rationality.

This note explains the circumstances under which a type 〈1〉 quantifier can be decomposed into a type 〈1, 1〉 quantifier and a set, by fixing the first argument of the former to the latter. The motivation comes from the semantics of Noun Phrases (also called Determiner Phrases) in natural languages, but in this article, I focus on the logical facts. However, my examples are taken among quantifiers appearing in natural languages, and at the end, I sketch two more principled linguistic applications.

We develop a formal apparatus to be used as a tool in analyzing common kinds of context dependence in natural language, and their interaction with temporal and spatial modalities. It is based on context-operators, which act on wffs. The interplay between the various modalities and the context-operators is one of the main targets of the analysis. Statements made by different people at different times in different places, using the same personal temporal and spatial indexicals, can be represented in the system, and can be combined by sentential connectives and be subject to quantification. The use of spatial modality and the suggested treatment of adverbial phrases are new as far as we know. So is a certain variant of temporal modality. In the nontechnical part, consisting of Sections 1 and 2, we discuss the role that formalisms can, in principle, play in the analysis of linguistic usage; this is followed by a philosophical discussion of various kinds of context dependence. The semitechnical part, Section 3, introduces the system's components, the context, and the modal operators, and explains their use via natural language examples. In Section 4 the formal language and its semantics are defined, in full detail. The temporal and spatial sublanguages constitute separate sorts, which interact through the modal operators and the context-operators. A sound deductive system is given and a completeness result is stated, without proof.

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.

1.1 In standard modal logics, the worlds are 2-valued in the following sense: there are 2 values (true and false) that a sentence may take at a world. Technically, however, there is no reason why this has to be the case. The worlds could be many-valued. This paper presents one simple approach to a major family of many-valued modal logics, together with an illustration of why this family is philosophically interesting.

We propose a model for belief which is free of presuppositions. Current models for belief suffer from two difficulties. One is the well known problem of logical omniscience which tends to follow from most models. But a more important one is the fact that most models do not even attempt to answer the question what it means for someone to believe something, and just what it is that is believed. We provide a flexible model which allows us to give meaning to beliefs in general contexts, including the context of animal belief (where action is usually our only clue to a belief), and of human belief which is expressed in language.

A new formal theory DT of truth extending PA is introduced, whose language is that of PA together with one new unary predicate symbol T (x), for truth applied to Gödel numbers of suitable sentences in the extended language. Falsity of x, F(x), is defined as truth of the negation of x; then, the formula D(x) expressing that x is the number of a determinate meaningful sentence is defined as the disjunction of T(x) and F(x). The axioms of DT are those of PA extended by (I) full induction, (II) strong compositionality axioms for D, and (III) the recursive defining axioms for T relative to D. By (II) is meant that a sentence satisfies D if and only if all its parts satisfy D; this holds in a slightly modified form for conditional sentences. The main result is that DT has a standard model. As an improvement over earlier systems developed by the author, DT meets a number of leading criteria for formal theories of truth that have been proposed in the recent literature and comes closer to realizing the informal view that the domain of the truth predicate consists exactly of the determinate meaningful sentences.

For nonmonotonic logics, Cumulativity is an important logical rule. We show here that Cumulativity fans out into an infinity of different conditions, if the domain is not closed under finite unions.

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’.