May 1, 2018 15:49 History and Philosophy of Logic MereologyAssertoricSyllogistic HISTORY AND PHILOSOPHY OF LOGIC, 00 (Month 200x), 1–10 Mereology in Aristotle’s Assertoric Syllogistic Justin Vlasits Received 00 Month 200x; final version received 00 Month 200x Abstract How does Aristotle think about sentences like ‘Every x is y’ in the Prior Analytics? A recently popular answer conceives of these sentences as expressing a mereological relationship between x and y: the sentence is true just in case x is, in some sense, a part of y. I argue that the motivations for this interpretation have so far not been compelling. I provide a new justification for the mereological interpretation. First, I prove a very general algebraic soundness and completeness result that unifies the most important soundness and completeness results to date. Then I argue that this result vindicates the mereological interpretation. In contrast to previous interpretations, this argument shows how Aristotle’s conception of predication in mereological terms can do important logical work. 1. Introduction At the heart of Aristotle’s syllogistic are four kinds of predication: • bAa Universal Affirmative: b belongs to (or is predicated of) every a. (Equivalent to saying ‘Every a is b’). • bEa Universal Negative: b belongs to no a. (‘No a is b’). • bIa Particular Affirmative: b belongs to some a. (‘Some a is b’). • bOa Particular Negative: b does not belong to some a. (‘Not every a is b’). In the Prior Analytics, Aristotle’s investigations of logical relations between these kinds of propositions are primarily proof-theoretic. However, we get a hint of the semantics of these propositions in the famous dictum de omni et nullo: We use the expression ‘predicated of every’ when nothing can be taken of which the other term cannot be said, and we use ‘predicated of none’ likewise.1 The dictum gives the meaning of Aristotle’s technical phrases ‘predicated of every’ and ‘pred- icated of none’. The basic idea is that ‘a is predicated of every b’ means that for every z, if b is predicated of z, then a is also predicated of z. Similarly, ‘a is predicated of no b’ means that for every z, if b is predicated of z, then a is not predicated of z.2 Because the dictum is used to justify the syllogisms of the first figure to which all the others are reduced, it carries enormous weight in understanding Aristotle’s semantics. It is, as Morison 2015 says, ‘a governing principle in Aristotle’s logic’ (p. 112). Moreover, while it directly explains the meanings of universal affir- Thanks to Tim Clarke, Klaus Corcilius, Alex Kocurek, John MacFarlane, Marko Malink and two anonymous referees for valuable comments on this paper. 1lègomen δὲ τὸ κατὰ παντὸς κατηγορεῖsvjai ὅtan mηδὲν ® λαβεῖn kaj’ οὗ θάτεροn oÎ λεχθήσvετai· kaÈ τὸ κατὰ μηδενὸς ὡσvαύτwc (APr 24b28-30). With Ross in excising ‘tοῦ Ípokeimènou’. Although it is present in all the manuscripts, it is absent in Alexander’s citation and an otherwise unparalleled use of the term in the Analytics. 2It is disputed whether the phrases ‘a is predicated of every b’ and ‘a is predicated of no b’ are defined in terms of a notion of predication different from universal affirmative. See Corcoran 1972, 1973, Barnes 2007, for the view that they are different, but Malink 2013, Morison 2008, following Michael Frede for the view that they are the same. While I will stay neutral with respect to the dispute between these two readings, the former reading is usually associated with a view of predication akin to Frege’s, which takes predication to be a relation between two different syntactic types. This view has been discredited by Mignucci 2000, Malink 2009, Corkum 2015. See Crager 2015 for a version of the Corcoran/Barnes reading that does not use the Fregean conception of predication. In this paper, I will not take a stand on whether the two kinds of predication are different. Everything in the semantics will be neutral between those two interpretations. History and Philosophy of Logic ISSN: 0144-5340 print/ISSN 1464-5149 online c 200x Taylor & Francis http://www.informaworld.com DOI: 10.1080/0144534YYxxxxxxxx May 1, 2018 15:49 History and Philosophy of Logic MereologyAssertoricSyllogistic 2 mative and universal negative propositions, it indirectly explains the meanings of particular propositions, which are each the contradictory of a universal proposition.3 How are we supposed to understand this notion of predication? My aim is to argue that the relation of predication defined or elucidated by the dictum de omni et nullo is a mereological relation between universals. This position has been gaining in popularity in recent years. In the early days of mathemati- cal reconstructions of Aristotle’s logic, the schematic letters (which are placeholders for terms) were taken to denote non-empty sets of individuals and syllogistic propositions were under- stood to be about the extensional relations that hold between them.4 Despite its initial promise, the inadequacy of this account has been made clear, by Malink and others.5 In its place, the mereological interpretation of predication has been proposed as a viable al- ternative.6 While this conception does not suffer from the same problems as the set-theoretic interpretation, I will argue that the reasons so far given to support it are not compelling. In- stead, I will give a new argument in support of the mereological interpretation. The core of this argument is a very general algebraic soundness and completeness result for Corcoran’s deduc- tive system RD, the standard natural deduction system used to study Aristotle’s logic. In this proof, I show how RD is sound for the class of Preorders P and complete for the class of Fi- nite Boolean Algebras FBA. This result has the corollary that any class of models M such that FBA ⊆ M ⊆ P is also sound and complete for RD. The systems of Corcoran 1972 and Martin 1997 thus emerge as special cases of the much more general soundness and completeness result proven here. In the final section, I argue 1) that we should interpret the predication relation as a preorder, since no further structure is needed to capture Aristotle’s validities and invalidities, 2) that preorders generally capture the formal structure of part-whole relations, and 3) that the mereological interpretation is thereby vindicated. 2. The Mereological Interpretation of Predication In recent literature on Aristotle’s notion of predication, there have been two arguments for a mereological interpretation.7 The first reason to think that Aristotle has a mereological conception of predication is because of the language he uses. Aristotle’s terminology for predication is steeped in mereological lan- guage. He marks his phrase for universal predication (‘belongs to all’) as equivalent to ‘is in as a whole’ and his terms for different kinds of propositions—‘universal’ (katholou) and ‘par- ticular’ (kata meros/en merei)—are themselves derived from the language of whole (holon) and part (meros). If universal predication is a mereological relation, we would also expect it to be defined in terms of notions related to mereology. While the language is highly suggestive, it does not settle the case in favor of the mereological interpretation. In particular, the linguistic information, on its own, does not tell us that Aristotle isn’t being metaphorical. Aristotle could be reaching for mereological language just because he feels it is the best approximation of what he has in mind. Nevertheless, we are not entitled to conclude that the mereology talk should be taken literally, for the same reason that we should 3The exact manner of the explanation is is a matter of dispute. Some take the dictum to be a biconditional. For Barnes 2007, it is an explicit definition along Tarskian lines. For Malink 2013, take it to give an implicit definition of a-predication. Morison 2015 thinks Aristotle is not giving a definition at all but specifying a rule of inference that ‘characterizes’ the meaning of the universal propositions. In what follows, I will follow the Barnes/Malink biconditional reading, but the account I propose could be translated into a rule-based account along the lines Morison endorses without any serious difficulty. Instead of one rule, we would have two corresponding to the biconditional. This account, unlike Morison’s, would provide both an introduction and an elimination rule for the universal propositions. 4See especially the pioneering completeness results of Corcoran 1972, 1973, Smiley 1973, although see Stekeler-Weithofer 1986 for an early completeness result that does not make this assumption. 5Two areas of particular difficulty are some claims about conversion and Aristotle’s modal syllogistic. See Malink 2009, 2013. 6See especially Mignucci 2000, Malink 2009, Corkum 2015 as well as Malink 2013 for an extension of this basic account to Aristotle’s modal syllogistic. 7See Mignucci 2000, Malink 2009, Corkum 2015. This was also the view of the ancient commentators: Alexander in APr 25.2–4 Wallies 1883, Philoponus in APr. 47.23–48.2, 73.22–3, 104.11–16, 164.4–7 Wallies 1905. May 1, 2018 15:49 History and Philosophy of Logic MereologyAssertoricSyllogistic 3 not take Aristotle’s use of the word ‘ὕλη’ to mean that all matter is timber. What we want, and what the linguistic information does not yet give us, is a reason why Aristotle would want to conceive of predication mereologically. What philosophical work does it do? Without a good sense of its purpose, the metaphorical objection seems hard to answer. One suggestion as to why Aristotle uses the language of part and whole comes from thinking about a specific kind of predication.