Proclus and the Neoplatonic Syllogistic

J. of Philosophical Logic 30 (2001), 187-240

Abstract

This paper investigates the logic of Proclus as set forth in his three main metaphysical treatises Elements of Theology, On the Parmenides, and Platonic Theology. Its specific object is to explain how Proclus could simultaneously subscribe to Platonic diairesis and its Aristotelian expression in the tree of Porphyry and at the same time hold that all reality falls in a linear order emanating from the One. It is argued that Proclus' logical method proceeds by first discovering the tree of diairesis by dividing "concepts" using hyper and privative negation, operations common employed by Neoplatonic writers but first clearly described by Proclus. The negations not only divide a point into its immediate successors, they also define a total order among them. These orderings together with the structure of the tress determine a total order for the entire set of points on the tree.

It is shown that the vocabulary Proclus uses to describe causation is essentially that of scalar adjectives and that his hyper and privative negations are standard operations on scalars. It is explained how Prolcus uses syllogisms interpreted over such total orderings in a way that validates the moods of Aristotle's logic which he regularly invokes as an metatheoretic tool. Syllogistic grammar and natural deduction proof theory (in the manner of Corcoran and Smiley) is extended to include the scalar negations. It is shown to be sound and complete relative to an abstract set of structures that includes Prolcus' total orderings. It is also explained how the logical theory may be expanded further to incorporate a third scalar negation and predicate operators for conjunction and disjunction that are generalizations of Kleene's strong connectives.

Proclus' dictum that affirmation generates negation is shown to be an expression of the isotonic property of scalar hyper and privative negation, and that the via negativa for knowing the One to be a special case. Modus pones and modus tollens are explained to be for Proclus special instances of the syllogisms Barbara and Baroco. Contrariety receives an new analysis in which its terms are simultaneously satisfiable. The canons attributed to Proclus by Ammonius that validate replacing both terms of a proposition by their negatives without altering the proposition's quantity are shown to be valid under Boolean interpretations by non-empty sets if the negation is question is set complementation, but invalid in scalar structures. However, the canons are valid under Proclus' non-Boolean scalar interpretations in which the negation is either hyper or privative. It is pointed out that Proclus' is committed to a kind of linear density in the causal order by his doctrines that any point is susceptible to further dialectical analysis (into triads), and that causation is infinite but partitioned into taxa that are finite, equipollent, and isotonic.

The paper concludes that Proclus preserves much of Aristotle's logic while adapting it to the quite different linear metaphysics of Neoplatonism. In the process he makes use of varieties of negation that though non-standard in both Aristotelian and Russellian logic are nevertheless well defined operations of natural language. Their orientation in a semantic structure with a distinguished "positive" supremum is, moreover, required by purely logical considerations resulting from their combinations with the syllogisitic.