We present a logic of evidence that reduces agents’ epistemic idealisations by combining classical propositional logic with substructural modal logic for formulas in the scope of epistemic modalities. To this aim, we provide a neighborhood semantics of evidence, which provides a modal extension of Fine’s semantics for relevant propositional logic. Possible worlds semantics for classical propositional logic is then obtained by defining the set of possible worlds as a special subset of information states in Fine’s semantics. Finally, we prove that (...) evidence is a hyperintensional and non-prime notion in our logic, and provide a sound and complete axiomatisation of our evidence logic. (shrink)

Relevant propositional dynamic logics have been sporadically discussed in the broader context of modal relevant logics, but have not come up for sustained investigation until recently. In this paper, we develop a philosophical motivation for these systems, and present some new results suggested by the proposed motivation. Among these, we’ll show how to adapt some recent work to show that the extensions of relevant logics by the extensional truth constants \ are complete with respect to a natural class of ternary (...) relation models to show a similar result for the constant-free versions of the logic. In addition, we prove that the logics in question satisfy the variable sharing property, vindicating the claim that they really are relevant logics. (shrink)

In this paper, we extend the canonicity methodology in Ghilardi & Meloni (1997) to arbitrary lattice expansions, and syntactically describe canonical inequalities for lattice expansions consisting of -meet preserving operations, -multiplicative operations, adjoint pairs, and constants. This approach gives us a uniform account of canonicity for substructural and lattice-based logics. Our method not only covers existing results, but also systematically accounts for many canonical inequalities containing nonsmooth additive and multiplicative uniform operations. Furthermore, we compare our technique with the approach in (...) Dunn et al. (2005) and Gehrke et al. (2005). (shrink)

In relevant logics, necessary truths need not imply each other. In justification logic, necessary truths need not all be justified by the same reason. There is an affinity to these two approaches that suggests their pairing will provide good logics for tracking reasons in a fine-grained way. In this paper, I will show how to extend relevant logics with some of the basic operators of justification logic in order to track justifications or reasons. I will define and study three kinds (...) of frames for these logics. For the first kind of frame, I show soundness and highlight a difficulty in proving completeness. This motivates two alternative kinds of frames, with respect to which completeness results are obtained. Axioms to strengthen the justification logic portions of these logics are considered. I close by developing an analogy between the dot operator of justification logic and theory fusion in relevant logics. (shrink)

Halldén completeness closely resembles the relevance property. To prove Halldén completeness in terms of Kripke-style semantics, the van Benthem–Humberstone theorem is often used. In relevant modal logics, the Halldén completeness of Meyer–Fuhrmann logics has been obtained using the van Benthem–Humberstone theorem. However, there remain a number of Halldén-incomplete relevant modal logics. This paper discusses the Halldén completeness of a wider class of relevant modal logics, namely, those with some Sahlqvist axioms.

Kripke-completeness of every classical modal logic with Sahlqvist formulas is one of the basic general results on completeness of classical modal logics. This paper shows a Sahlqvist theorem for modal logic over the relevant logic Bin terms of Routley- Meyer semantics. It is shown that usual Sahlqvist theorem for classical modal logics can be obtained as a special case of our theorem.

Here, I combine the semantics of Mares and Goldblatt [20] and Seki [29, 30] to develop a semantics for quantified modal relevant logics extending ${\bf B}$. The combination requires demonstrating that the Mares–Goldblatt approach is apt for quantified extensions of ${\bf B}$ and other relevant logics, but no significant bridging principles are needed. The result is a single semantic approach for quantified modal relevant logics. Within this framework, I discuss the requirements a quantified modal relevant logic must satisfy to be (...) “sufficiently classical” in its modal fragment, where frame conditions are given that work for positive fragments of logics. The roles of the Barcan formula and its converse are also investigated. (shrink)

In this paper, we investigate neighbourhood semantics for modal extensions of relevant logics. In particular, we combine the neighbourhood interpretation of the relevant implication (and related connectives) with a neighbourhood interpretation of modal operators. We prove completeness for a range of systems and investigate the relations between neighbourhood models and relational models, setting out a range of augmentation conditions for the various relations and operations.

Sedlár and Vigiani [18] have developed an approach to propositional epistemic logics wherein (i) an agent’s beliefs are closed under relevant implication and (ii) the agent is located in a classical possible world (i.e., the non-modal fragment is classical). Here I construct first-order extensions of these logics using the non-Tarskian interpretation of the quantifiers introduced by Mares and Goldblatt [12], and later extended to quantified modal relevant logics by Ferenz [6]. Modular soundness and completeness are proved for constant domain semantics, (...) using non-general frames with Mares–Goldblatt truth conditions. I further detail the relation between the demand that classical possible worlds have Tarskian truth conditions and incompleteness results in quantified relevant logics. (shrink)

This paper briefly overviews some of the results and research directions. In the area of substructural logics from the last couple of decades. Substructural logics are understood here to include relevance logics, linear logic, variants of Lambek calculi and some other logics that are motivated by the idea of omitting some structural rules or making other structural changes in LK, the original sequent calculus for classical logic.

Following a result by De Rijke for modal logic, it is shown that the basic weak entailment model-theoretic language with absurdity is the maximal model-theoretic language having the finite occurrence property, preservation under relevant directed bisimulations and the finite depth property. This can be seen as a generalized preservation theorem characterizing propositional weak entailment formulas among formulas of other model-theoretic languages.