Abstract
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.
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Notes
Diaconescu (2015) studies several metalogical Aristotelian diagrams for abstract (Tarski-style) consequence relations, which transcend the distinction between syntax and semantics. However, he does not deal with any Aristotelian diagrams for syntactic consequence in particular.
Note that this argumentation for an analogy (between the metalogical notion of semantic consequence and the categorical statements from syllogistics) is itself perfectly in line with the methodological/heuristic perspective on Aristotelian diagrams.
Throughout this paper I will make use of the well-known mnemonic vowels (A, I, E, O) for the categorical statements. These are the first two vowels from each of the Latin verb forms ‘affirmo’ and ‘nego’.
The precise interpretation of the principle of existential import in syllogistics has been a matter of substantial scholarly debate (Parsons 2017), but this need not concern us here.
In particular, these models can be first-order models, relational structures, etc. These details do not matter here.
In particular, the system can be a Gentzen-style or a Fitch-style natural deduction system. Again, these details do not matter here. (Also see Footnote 5.).
The problematic nature of this quantifier mismatch should not be exaggerated. Ultimately, \( \Gamma \,{ \vdash }\,\varphi \) and \( \Gamma \,{ \vdash }\,\neg \varphi \) are effectively contrary to each other, regardless of whether their definitions can be characterized as universal (A- and E-) statements. In the literature there exist other, object-logical examples of squares of opposition that exhibit a similar quantifier mismatch. For example, in the square of opposition for public announcement logic, the formulas <!p> q and <!p> \( \neg \) q are contrary to each other, although the semantics of these formulas involves an existential, rather than a universal quantification (over public announcements of p) (Demey 2012, 2017b).
Thanks to an anonymous reviewer of Demey (2017a) for some useful discussion on this.
Furthermore, Theorem 1 cannot be used to provide a definition of syntactic consequence, since that would clearly result in circularity: the theorem characterizes syntactic consequence in terms of MCS, and thus of consistency, but as we have seen in the beginning of Sect. 3, the latter notion is itself defined in terms of syntactic consequence.
The sets \( \Gamma \) for which it does hold that \( \Gamma \,{ \vdash }\,\varphi \) or \( \Gamma \,{ \vdash }\,\neg \varphi \) for all \( \varphi \), are called deductively complete.
Many logicians have a rather ‘asymmetric’ perspective on soundness and completeness: soundness is usually taken to be a kind of minimal criterion that has to be met by any serious candidate proof system, while failure of completeness is seen as much more tolerable. However, based on certain technical results on so-called Scott consequence relations, one can argue for a more ‘symmetric’ perspective, which treats soundness and completeness more on a par with each other (Scott 1971, 1974; Brown 2015). In particular, a logical system that is complete but not sound is deemed equally worthy of attention as a system that is sound but not complete. Thanks to an anonymous reviewer for some interesting remarks about this issue.
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Acknowledgements
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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Demey, L. Aristotelian diagrams for semantic and syntactic consequence. Synthese 198, 187–207 (2021). https://doi.org/10.1007/s11229-018-01994-w
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DOI: https://doi.org/10.1007/s11229-018-01994-w