Around and Beyond the Square of Opposition

Part of the series Studies in Universal Logic pp 53-71

Leibniz, Modal Logic and Possible World Semantics: The Apulean Square as a Procrustean Bed for His Modal Metaphysics

  • Jean-Pascal AlcantaraAffiliated withCentre Georges Chevrier-UMR 5605, Université de Bourgogne Email author 

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Even if Leibniz didn’t have the opportunity to actually conceive an explicit modal logic system, remains the fact that he had worked out a modal metaphysics, of which the inaugural act, in his Elementa juris naturalis (c. 1671) was an obvious reference to the Apulean square of opposition. Later, scholars acknowledged in this passage probably one of the first sketch of deontic logic of norms. His modal metaphysics rather deals with the so-called alethic modalities, sometimes expounded through a language such as R.M. Adams wondered whether Leibniz could be “a sort of grandfather of possible worlds semantics for modal logic”. In the following study, the Apulean square is used as a hermeneutic tool: however, is the square really well-fitted to express some basic implications of the modal metaphysics? The most remarkable point is that Leibniz was well aware of the K-distributive axiom □(pq)⊃(□p⊃□q) common to the main modal systems today. This awareness dissuaded him to trust the too easy solution of the necessitarianism issue grounded on the well-known distinction coming from Boethius between a “necessitas consequentiae” and a “necessitas consequentis”.

Thus Leibniz must consider a proof-theory solution style (for a contingent proposition, the reducibility of the predicate to its subject cannot be achieved). A new square could be making up, however implying a difficulty: if the modal metaphysics cannot admit the logical convertibility of the contingent propositions of which the dictum of one is exactly the negation of the dictum of the other, known since Aristotle. This is the reason for what the Apulean square could represent a Procrustean bed for Leibniz’ modal metaphysics. At the end, a new modality square may be drawn, according to which each modality in the corner is expressed by quantifications on possible worlds. Whether possible worlds semantics could supersedes Leibniz’ own explanation of contingency thanks to this new square of modalities, i.e. a possible worlds square of modalities, will be the topic of a second part of this study.


Deontic logic K-Distributive axiom of modal logic systems Blanché’s hexagon Necessitarianism Proof theory Modal metaphysics Algebra of concepts Convertibility of the contingent propositions Possible worlds semantics Boethius Calvin Leibniz

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