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Aristotelian diagrams for semantic and syntactic consequence

  • Lorenz DemeyEmail author


Several authors have recently studied Aristotelian diagrams for various metatheoretical notions from logic, such as tautology, satisfiability, and the Aristotelian relations themselves. However, all these metalogical Aristotelian diagrams focus on the semantic (model-theoretical) perspective on logical consequence, thus ignoring the complementary, and equally important, syntactic (proof-theoretical) perspective. In this paper, I propose an explanation for this discrepancy, by arguing that the metalogical square of opposition for semantic consequence exhibits a natural analogy to the well-known square of opposition for the categorical statements from syllogistics, but that this analogy breaks down once we move from semantic to syntactic consequence. I then show that despite this difficulty, one can indeed construct metalogical Aristotelian diagrams from a syntactic perspective, which have their own, equally elegant characterization in terms of the categorical statements. Finally, I construct several metalogical Aristotelian diagrams that incorporate both semantic and syntactic consequence (and their interaction), and study how they are influenced by the underlying logical system’s soundness and/or completeness. All of this provides further support for the methodological/heuristic perspective on Aristotelian diagrams, which holds that the main use of these diagrams lies in facilitating analogies and comparisons between prima facie unrelated domains of investigation.


Metalogic Semantic consequence Syntactic consequence Square of opposition Aristotelian diagram Logical geometry 



Thanks to Hans Smessaert, Margaux Smets and three anonymous reviewers for their valuable feedback on an earlier version of this paper. The author holds a Postdoctoral Fellowship of the Research Foundation–Flanders (FWO). The research reported in this paper was partially carried out during a research stay at the Institut für Philosophie II of the Ruhr-Universität Bochum, which was financially supported by an FWO travel grant.


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© Springer Nature B.V. 2018

Authors and Affiliations

  1. 1.Institute of PhilosophyKU LeuvenLeuvenBelgium

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