Philip Kremer

In the topological semantics, quantified intuitionistic logic, QH, is known to be strongly complete not only for the class of all topological spaces but also for some particular topological spaces — for example, for the irrational line, ${\Bbb P}$, and for the rational line, ${\Bbb Q}$, in each case with a constant countable domain for the quantifiers. Each of ${\Bbb P}$ and ${\Bbb Q}$ is a separable zero-dimensional dense-in-itself metrizable space. The main result of the current article generalizes these known results: QH is strongly complete for any zero-dimensional dense-in-itself metrizable space with a constant domain of cardinality ≤ the space’s weight; consequently, QH is strongly complete for any separable zero-dimensional dense-in-itself metrizable space with a constant countable domain. We also prove a result that follows from earlier work of Moerdijk: if we allow varying domains for the quantifiers, then QH is strongly complete for any dense-in-itself metrizable space with countable domains.

Shehtman introduced bimodal logics of the products of Kripke frames, thereby introducing frame products of unimodal logics. Van Benthem, Bezhanishvili, ten Cate and Sarenac generalize this idea to the bimodal logics of the products of topological spaces, thereby introducing topological products of unimodal logics. In particular, they show that the topological product of S4 and S4 is S4 \ S4, i.e., the fusion of S4 and S4: this logic is strictly weaker than the frame product S4 × S4. Indeed, van Benthem et al. show that S4 \ S4 is the bimodal logic of the particular product space , leaving open the question of whether S4 \ S4 is also complete for the product space . We answer this question in the negative

Dynamic topological logic provides a context for studying the confluence of the topological semantics for S4, topological dynamics, and temporal logic. The topological semantics for S4 is based on topological spaces rather than Kripke frames. In this semantics, □ is interpreted as topological interior. Thus S4 can be understood as the logic of topological spaces, and □ can be understood as a topological modality. Topological dynamics studies the asymptotic properties of continuous maps on topological spaces. Let a dynamic topological system be a topological space X together with a continuous function f. f can be thought of in temporal terms, moving the points of the topological space from one moment to the next. Dynamic topological logics are the logics of dynamic topological systems, just as S4 is the logic of topological spaces. Dynamic topological logics are defined for a trimodal language with an S4-ish topological modality □, and two temporal modalities, ○ and *, both interpreted using the continuous function f. In particular, ○ expresses f’s action on X from one moment to the next, and * expresses the asymptotic behaviour of f.

In the topological semantics for modal logic, S4 is well known to be complete for the rational line and for the real line: these are special cases of S4’s completeness for any dense-in-itself metric space. The construction used to prove completeness can be slightly amended to show that S4 is not only complete but strongly complete, for the rational line. But no similarly easy amendment is available for the real line. In an earlier paper, we proved a general theorem: S4 is strongly complete for any dense-in-itself metric space. Strong completeness for the real line is a special case. In the current paper, we give a proof of strong completeness tailored to the special case of the real line: the current proof is simpler and more accessible than the proof of the more general result and involves slightly different techniques. We proceed in two steps: first, we show that S4 is strongly complete for the space of finite and infinite binary sequences, equipped with a natural topology; and then we show that there is an interior map from the real line onto this space.

R. K. Meyer once gave precise form to the question of whether relevant implication can be defined in any modal system, and his answer was `no'. In the present paper, we extend $\mathbf{S4}$, first with propositional quantifiers, to the system $\mathbf{S4\pi}+$; and then with definite propositional descriptions, to the system $\mathbf{S4\pi}+^{lp}$. We show that relevant implication can in some sense be defined in the modal system $\mathbf{S4\pi}+^{lp}$, although it cannot be defined in $\mathbf{S4\pi}+$.

There is an intuition, notoriously difficult to formalise, that only some predicates express real properties. J. M. Dunn formalises this intuition with relevance logic, proposing a notion of relevant predication. For each first order formula Ax, Dunn specifies another formula that is intuitively interpreted as "Ax expresses a real property". Chapter I calls such an approach an object language approach, since the claim that Ax expresses a real property is rendered as a formula in the object language. On a metalanguage approach, on the other hand, the claim that Ax expresses a real property would be metalinguistic, mentioning but not using the formula Ax. ;Our Introduction begins by investigating Dunn's motivation for relevant predication, and argues that it implicitly presupposes some interpretation of identity. Indeed, part of the dissertation's work is to use relevant predication as a key to a coherent account of relevant identity. ;Chapter I re-motivates relevant predication. We consider P. Geach's distinction between real and "Cambridge" change, and suggest that there is an underlying distinction between real and Cambridge predicates. We further argue that no classical object language approach can formalise this distinction. So we turn to relevance logic, remotivating relevant predication, with Geach's distinction in mind. ;In light of this new motivation, we argue that relevant prediction relies on a logically weak notion of identity, according to which "x = y" means "x and y share all relevant properties". Identity is taken up in a more technical setting in Chapters IV and V. ;Subsequent chapters investigate technical issues that flow from Chapter I. Chapter II investigates metalinguistic grammatical characterisations of relevant predication. Chapter III concerns the semantics of first order relevance logic. Chapter IV investigates the identity in R and its relationship to relevant predication. In particular, Chapter IV argues against introducing the axiom, $\to$ Ay), in relevance logic, since the resulting systems are unstable, in a sense motivated by the notion of relevant identity underlying Dunn's notion of relevant predication. Finally Chapter V investigates the possibility of a coherent theory of identity in R's modal relatives ${\bf R}\sp\square$ and E.