A second-order relevance logic with modality

In this paper a system, RPF, of second-order relevance logic with S5 necessity is presented which contains a defined, notion of identity for propositions. A complete semantics is provided. It is shown that RPF allows for more than one necessary proposition. RPF contains primitive syntactic counterparts of the following semantic notions: (1) the reflexive, symmetrical, transitive binary alternativeness relation for S5 necessity, (2) the ternary Routley-Meyer alternativeness relation for implication, and (3) the Routley-Meyer notion of a prime intensional theory, as well as defined syntactic counterparts, of the semantic notions of a possible world and the Routley-Meyer ^* operator.     

Semantics for relevance logic with identity

Models are constructed for a variety of systems of quantified relevance logic with identity. Models are given for systems with different principles governing the transitivity of identity and substitution, and the relative merits of these principles are discussed. The models in this paper are all extensions of the semantics of Fine's Semantics for Quantified Relevance Logic (Journal of Philosophical Logic 17 (1988)).I am indebted to J. M. Dunn for reading earlier versions of this paper. I am also grateful to Nuel D. Belnap, Jr., Robert K. Meyer, Timothy Day, and Adriano Palma for discussing the topic with me and to Kit Fine for helpful correspondence. Moreover, I am grateful to an anonymous referee for many useful suggestions. A slightly different version presented in a seminar in the philosophy department of the Research School for Social Sciences at The Australian National University. While working on this paper I was funded by postdoctoral fellowship 456-89-0128 from the Social Sciences and Humanities Research Council of Canada.     

Combining classical logic, paraconsistency and relevance

We present a logic with has both a simple semantics and a cut-free Gentzen-type system on one hand, and which combines relevance logics, da Costa's paraconsistent logics, and classical logic on the other. We further show that the logic has many other nice properties, and that its language is ideal from the semantic point of view.     

Notes on conditional logic

This paper consists of some lecture notes in which conditional logic is treated as an extension of modal logic. Completeness and filtration theorems are provided for some basis systems. These notes were originally drafted for a course given at the University of Auckland in 1987 (Philosophy 29.308). The work reported in Section 5 is due to Michael Strevens, who took the course and presented his conception of a filtration in a term paper [5]. The author wishes to thank Strevens for many stimulating discussions and for permission to include his results here     

Quantum logic as a dynamic logic

We address the old question whether a logical understanding of Quantum Mechanics requires abandoning some of the principles of classical logic. Against Putnam and others (Among whom we may count or not E. W. Beth, depending on how we interpret some of his statements), our answer is a clear “no”. Philosophically, our argument is based on combining a formal semantic approach, in the spirit of E. W. Beth’s proposal of applying Tarski’s semantical methods to the analysis of physical theories, with an empirical–experimental approach to Logic, as advocated by both Beth and Putnam, but understood by us in the view of the operational- realistic tradition of Jauch and Piron, i.e. as an investigation of “the logic of yes–no experiments” (or “questions”). Technically, we use the recently-developed setting of Quantum Dynamic Logic (Baltag and Smets 2005, 2008) to make explicit the operational meaning of quantum-mechanical concepts in our formal semantics. Based on our recent results (Baltag and Smets 2005), we show that the correct interpretation of quantum-logical connectives is dynamical, rather than purely propositional. We conclude that there is no contradiction between classical logic and (our dynamic reinterpretation of) quantum logic. Moreover, we argue that the Dynamic-Logical perspective leads to a better and deeper understanding of the “non-classicality” of quantum behavior than any perspective based on static Propositional Logic.     

The logic of historical necessity as founded on two-dimensional modal tense logic

We consider a version of so called T × W logic for historical necessity in the sense of R.H. Thomason (1984), which is somewhat special in three respects: (i) it is explicitly based on two-dimensional modal logic in the sense of Segerberg (1973); (ii) for reasons of applicability to interesting fields of philosophical logic, it conceives of time as being discrete and finite in the sense of having a beginning and an end; and (iii) it utilizes the technique of systematic frame constants in order to handle the problem of irreflexivity in tense logics, well known since Gabbay (1981). Axiomatizations are given for two infinite hierarchies of two-dimensional modal tense logics, one without and one with the characteristic operators for historical necessity and possibility. Strong and weak completeness results are obtained for both hierarchies as well as a result to the effect that two approaches to their semantics are equivalent, much in the spirit of Di Maio and Zanardo (1996) and von Kutschera (1997).