Citation: Falessi, D.; Schang, F. A Relational Semantics for Ockham’s Modalities. Axioms 2023, 12, 445. https://doi.org/10.3390/ axioms12050445 Academic Editors: Lorenz Demey and Stef Frijters Received: 25 March 2023 Revised: 22 April 2023 Accepted: 25 April 2023 Published: 30 April 2023 Copyright: © 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/). axioms Article A Relational Semantics for Ockham’s Modalities Davide Falessi 1, * and Fabien Schang 2,3, * 1 École Pratique des Hautes Études-PSL, University of Lucerne, 6002 Lucerne, Switzerland 2 Lycée Fabert, 57000 Metz, France 3 Lycée JeanZay, 54800 Jarny, France * Correspondence: davide.falessi@outlook.it (D.F.); schangfabien@gmail.com (F.S.); Tel.: +33-06-38-33-27-49 (D.F. & F.S.) Abstract: This article aims at providing some extension of the modal square of opposition in the light of Ockham’s account of modal operators. Moreover, we set forth some significant remarks on the de re–de dicto distinction and on the modal operator of contingency by means of a set-theoretic algebra called numbering semantics. This generalization starting from Ockham’s account of modalities will allow us to take into consideration whether Ockham’s account holds water or not, and in which case it should be changed. Keywords: numbering semantics; contingency; de re–de dicto; logical relations; modalities; Ockham 1. Introduction The article is structured as follows. A first, more historical part will be entirely dedicated to the set-up of Ockham’s account of modal propositions and their possible readings. It will be considered as the de re–de dicto distinction, providing some informal rules regarding this distinction that can be deduced from Ockham’s account (Section 2, pp. 1–4, written by D. Falessi). Then, Ockham’s account of contingency and its application to the modal squares provided in Ockham’s commentary on Aristotle’s De Interpretatione is presented. This brings us to two modal hexagons that will be drawn as generalizations of those modal squares by means of the application of contingency, as Ockham defines it (Section 3, pp. 4–9, written by D. Falessi). Finally, a formal section will be devoted to a formal semantics of Ockham’s modal statements. More especially, it will consist of a set of two kinds of logical forms, whether de re or de dicto, and a corresponding second-order logic where modalities are viewed as a dyadic predicate including properties and worlds. After devising a set-theoretical semantics, according to which the meaning of formulas corresponds to their model sets or ordered truth-conditions, Ockham’s statements of (non-)contingency will then be redefined by means of an external use of negation, and our algebraic translation of logical relations will result in a complex structure, i.e., a logical icosagon (Section 4, pp. 9–16, written by F. Schang, including both Appendices A and B dedicated to a logical reformulation of modal statements starting from Ockham’s account). Needless to say, the conclusion and all the sections are the result of a common work of discussion and sharing opinions and ideas. 2. Ockham’s Account: De dicto/De re Distinction In medieval logic, there are two possible readings of a modal proposition. A modal proposition can be taken either in sensu compositionis (compound sense) or in sensu divisionis (divided sense) (see also [1,2] for the medieval theories of modal logic). For a fully-fledged explanation of Ockham’s account of modalities, see [3,4]. We shall consider here just the de re and de dicto readings, the status of contingency as a modal operator, and the modal squares. Ockham defines the compound sense as follows: Axioms 2023, 12, 445. https://doi.org/10.3390/axioms12050445 https://www.mdpi.com/journal/axioms