
Projective Modality in the History of Logic Vyacheslav Moiseev, Russia Ancient and Medieval Philosophy had a basic intuition almost missed in the contemporary philosophical tradition. Classical metaphysics disposed all principles (things, ideas, feelings, etc.) within one framework ordering them by degree of unity. Idea of a “Metaphysical Order” can be displayed in the relation of Idea and Matter (Plato), Form and Matter (Aristotle), Essentia and Existentia (Scholastics), Substantia and Modus (Decartes, Spinoza) etc. One basic structure was here. It concluded idea of an Order, from the first side. However, this is not a simple order as reflective, antisymmetrical and transitive relation. It is a “Metaphysical Order” when a relation “less or equal” connects with the idea of a restriction of more united principle to its aspectsprojections like 3dimensional body is restricted to own 2dimensional projections on planes. Therefore, Idea of Order is not independent. It presupposes a more rich structure, in the framework of which Order is a more partial principle. Whenever a non strong order a b is presented, a functor (“projector”) is defined, where (b,c) = a, and c is a principle of restriction of b to a. Therefore, we have: (1) (a b) c((b,c) = a)  being of any order implies existence of a functor and a principle of restriction c, where (b,c) = a. And vice versa, existence of a functor and a principle of restriction c, where (b,c) = a, implies a nonstrong order a b: (2) c((b,c) = a) (a b) (1) and (2) together give (3): (3) (a b) c((b,c) = a) So the non strong order turns out is plunged in a more rich structure with a functor , a principle of restriction c, an equality =, etc. What are these elements? What is a logic of them? It seems classical metaphysical systems knew the answer. I try to reconstruct these ancient philosophical intuitions in a logical system, so called Projectively Modal Ontology (PMO)^{1}, where relation of a non strong order with additional constructions like in (1)(3) are interpreted as a kind of modality, “projective modality”, such that formula “a b” means “a is a mode (“projection”) of b”. Now one can said that classical metaphysics actively used the idea of “projective modality” ordering principles in the Univers. After primary description of PMO, I presuppose to show projectively modal structure of Plato “Parmenides” as one of the most bright examples of “Metaphysical Ordering” in the history of Philosophy. Every part of the dialog is presented as a mode of One receiving by a special projectively modal restriction of One. 1 see Wiaczeslaw I. Moisiejew. Ontologia Stanisława Leśniewskiego i Logika Wszechejedności // Kwartalnik Filozoficzny. Tom XXXII. Zeszyt 1. Przeł. Paweł Rojek. Kraków. Polska Akademia Umiejętności, Uniwersytet Jagielloński. 2004. – pp.101126.; V.Moiseev. Projectively Modal Ontology // Logical Studies, № 9, 2002. – (http://www.logic.ru/LogStud/09/LS9.html). 
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