Pietro Vigiani

We introduce a dynamic extension of the logic of relevant reasoners in classical worlds, where dynamic modalities model the update of agents’ beliefs. We argue that the system can be used to define a binary relative necessity operator, in the style of conditional logics, which captures Ramsey’s idea that conditional reasoning involves a form of belief revision. Recast as a relevant conditional logic, the system combines the best of existing classical and relevant conditional logics: it extends classical propositional logic, while adopting relevant implication to formalise the non-monotonic aspects of belief revision. The main technical result of the paper is a sound and complete axiomatisation of the logic and its extensions with respect to neighborhood Routley–Meyer models for dynamic relevant modal logic, enriched with possible worlds.

We introduce sans serif upper F upper D upper E Subscript c Superscript), an extension of FDE\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathsf {FDE}$$\end{document} with strict implication and a classicality constant, and we show that it formalizes the distinction between explicit and implicit belief. In the style of Levesque’s formalization of these two concepts, explicit beliefs are modelled as sets of formulas closed under our extension of FDE\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathsf {FDE}$$\end{document}, while implicit beliefs form, in a sense, the classical closure of explicit beliefs. We establish an embedding of Levesque’s logic of explicit and implicit belief into sans serif upper F upper D upper E Subscript c Superscript). This result shows that sans serif upper F upper D upper E Subscript c Superscript) is a viable generalization of Levesque’s logic lifting some of its limitations. Unlike a similar generalization introduced by Lakemeyer, sans serif upper F upper D upper E Subscript c Superscript) comes with an Australian-plan semantics and so it is an alternative potentially attractive to those who prefer the Australian plan over the American one.

The paper introduces a relevant containment logic according to which the analytic implication $$\varphi \twoheadrightarrow \psi$$ φ ↠ ψ is analyzed as the conjunction of two theses: $$\varphi$$ φ contextually entails $$\psi$$ ψ and $$\varphi$$ φ contextually contains $$\psi$$ ψ. By doing so, we are able to extend the ternary semantics of relevant logic to the analysis of topic containment, thus lifting some limitations of Richard Sylvan’s relevant containment logic. We offer a ternary account of topic containment, in that led by the consideration that topic inclusions are evaluated in situ, i.e. with respect to the discursive context fixed by an information state. The main technical result of the paper is a sound and complete axiomatisation of relevant containment logic. Finally, Sylvan’s logic turns out as a special case of our relevant containment logic.

We present a sound and complete axiomatisation of the epistemic logic \(\textsf {C.RC}\). In the logic, the propositional fragment is \(\textsf {C}\) lassical, while agents’ epistemic attitudes are closed under on–topic relevant consequence, as modeled by \(\textsf {R}\) elevant \(\textsf {C}\) ontainment logic. By doing so, \(\textsf {C.RC}\) complies with a principle of minimal mutilation of classical logic and lifts some limitations of existing frameworks, such as (i) logics of analytic implication, (ii) topic-sensitive analyses of epistemic modals, and (iii) the logic of relevant reasoners in classical worlds.

We present a neighbourhood-style semantic framework for modal epistemic logic modelling agents who process information using relevant logic. The distinguishing feature of the framework in comparison to relevant modal logic is that the environment the agent is situated in is assumed to be a classical possible world. This framework generates two-layered logics combining classical logic on the propositional level with relevant logic in the scope of modal operators. Our main technical result is a general soundness and completeness theorem.

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.

Combining relevant and classical modal logic is an approach to overcoming the logical omniscience problem and related issues that goes back at least to Levesque’s well known work in the 1980s. The present authors have recently introduced a variant of Levesque’s framework where explicit beliefs concerning conditional propositions can be formalized. However, our framework did not offer a formalization of implicit belief in addition to explicit belief. In this paper we provide such a formalization. Our main technical result is a modular completeness theorem.