Proclus and the Neoplatonic syllogistic

AN investigation of Proclus' logic of the syllogistic and of negations in t he Elements of Theology, On the Parmenides, and Platonic Theology. It is sh own that Proclus employs interpretations over a linear semantic structure w ith operators for scalar negations (hypernegation/alpha-intensivum and priv ative negation). A natural deduction system for scalar negations and the cl assical syllogistic (as reconstructed by Corcoran and Smiley) is shown to b e sound and complete for the non-Boolean linear structures. It is explained how Proclus' syllogistic presupposes converting the tree of genera and spe cies from Plato's diairesis into the Neoplatonic linear hierarchy of Being by used of scalar hyper and privative negations.