Showing papers on "Normal modal logic published in 2021"

Being in a position to know

Abstract: The concept of being in a position to know is an increasingly popular member of the epistemologist’s toolkit. Some have used it as a basis for an account of propositional justification. Others, following Timothy Williamson, have used it as a vehicle for articulating interesting luminosity and anti-luminosity theses. It is tempting to think that while knowledge itself does not obey any closure principles, being in a position to know does. For example, if one knows both p and ‘If p then q’, but one dies or gets distracted before being able to perform a modus ponens on these items of knowledge and for that reason one does not know q, one is still plausibly in a position to know q. It is also tempting to suppose that, while one does not know all logical truths, one is nevertheless in a position to know every logical truth. Putting these temptations together, we get the view that being in a position to know has a normal modal logic. A recent literature has begun to investigate whether it is a good idea to give in to these twin temptations—in particular the first one. That literature assumes very naturally that one is in a position to know everything one knows and that one is not in a position to know things that one cannot know. It has succeeded in showing that, given the modest closure condition that knowledge is closed under conjunction elimination (or ‘distributes over conjunction’), being a position to know cannot satisfy the so-called K axiom (closure of being in a position to know under modus ponens) of normal modal logics. In this paper, we explore the question of the normality of the logic of being in a position to know in a more far-reaching and systematic way. Assuming that being in a position to know entails the possibility of knowing and that knowing entails being in a position to know, we can demonstrate radical failures of normality without assuming any closure principles at all for knowledge. (However, as we will indicate, we get further problems if we assume that knowledge is closed under conjunction introduction.) Moreover, the failure of normality cannot be laid at the door of the K axiom for knowledge, since the standard principle NEC of necessitation also fails for being in a position to know. After laying out and explaining our results, we briefly survey the coherent options that remain.

Logic-Sensitivity of Aristotelian Diagrams in Non-Normal Modal Logics

Abstract: Aristotelian diagrams, such as the square of opposition, are well-known in the context of normal modal logics (i.e., systems of modal logic which can be given a relational semantics in terms of Kripke models). This paper studies Aristotelian diagrams for non-normal systems of modal logic (based on neighborhood semantics, a topologically inspired generalization of relational semantics). In particular, we investigate the phenomenon of logic-sensitivity of Aristotelian diagrams. We distinguish between four different types of logic-sensitivity, viz. with respect to (i) Aristotelian families, (ii) logical equivalence of formulas, (iii) contingency of formulas, and (iv) Boolean subfamilies of a given Aristotelian family. We provide concrete examples of Aristotelian diagrams that illustrate these four types of logic-sensitivity in the realm of normal modal logic. Next, we discuss more subtle examples of Aristotelian diagrams, which are not sensitive with respect to normal modal logics, but which nevertheless turn out to be highly logic-sensitive once we turn to non-normal systems of modal logic.

Modal and Intuitionistic Variants of Extended Belnap–Dunn Logic with Classical Negation

Abstract: In this study, we introduce Gentzen-type sequent calculi BDm and BDi for a modal extension and an intuitionistic modification, respectively, of De and Omori’s extended Belnap–Dunn logic BD+ with classical negation. We prove theorems for syntactically and semantically embedding BDm and BDi into Gentzen-type sequent calculi S4 and LJ for normal modal logic and intuitionistic logic, respectively. The cut-elimination, decidability, and completeness theorems for BDm and BDi are obtained using these embedding theorems. Moreover, we prove the Glivenko theorem for embedding BD+ into BDi and the McKinsey–Tarski theorem for embedding BDi into BDm.

Non-normal modal logics and conditional logics: Semantic analysis and proof theory

Abstract: We introduce proper display calculi for basic monotonic modal logic, the conditional logic CK and a number of their axiomatic extensions. These calculi are sound, complete, conservative, and enjoy cut elimination and subformula property. Our proposal applies the multi-type methodology in the design of proper display calculi, starting from a semantic analysis which motivates syntactic translations from single-type non-normal modal logics to multi-type normal poly-modal logics.

A Logic for Aristotle's Modal Syllogistic

Abstract: We propose a new system of modal logic to interpret Aristotle's theory of the modal syllogism which while being inspired by standard propositional modal logic is also a logic of terms and which admits a (sound) extensional semantics involving possible worlds. Although this logic does not allow a fully faithful formalisation of the entirety of Aristotle's syllogistic as found in the Prior Analytics it sheds light on various fine-grained distinctions which when made allow us to recover a fair portion of Aristotle's results. This logic allows us also to make a connection with the various axioms of modern propositional logic and to perceive to what extent these are implicit in Aristotle's reasoning. Further work wil involve addressing the question of the completeness of this logic (or variants thereof) together with the extension of the logic to include a calculus of relations, some instances of which are found, as Slomkowsky has shown, in the Topics..