
PROJECTIVELY MODAL ONTOLOGY Vyacheslav Moiseev vimo@vmail.ru Content
Abstract The paper asserts that every ontology presupposes a basic structure, Ontological Tetrade, which consists of source of predications (“modus”), different predications of the source (“modas”), restricted conditions, under which the predications are formed (“models”), and operation of forming of the predications (“projector”). V.Soloviov used projective intuition of Ontological Tetrade comparing predications with projections of the body. It seems, St.Lesniewski also used a similar intuition in a non explicit form. A new axiomatic system, Projectively Modal Ontology (PMO), is offered in the paper. I accept here almost all the logical means of the language of St.Lesniewski’s “Ontology”. Namely I accept Prothotetics without any changes, syntax of expressions of different categorial types, rules of inference with the exeption of Rule of Extensionality. Prothotetical definitions will be used without any changes. Forms of Ontological definitions will be discussed below. Instead of Lesniewski’s functor I shall use a 4placed predicate Mod of the categorial type (N,N,N,(N,N)/N)/S. Expression Mod(a,b,c,f) is read as “a is moda of modus b under the model c with projector f”. Some theorems of PMO are presented with proofs. Some extensions of the primary version of PMO are considered, in particular, a Boolean Algebra on moduses, similar to Mereology of Lesniewski, is investigated. Proof of inconsistency of PMO relatively Prothotetics is considered also.
The paper is devoted to the description of one axiomatical system, which can be called as ^ (PMO). This system has two main foundations: 1) one important philosophical concept from the philosophy of Vladimir Soloviov, and 2) logical form similar to logical form of St.Lesniewski’ Ontology. Breafly speaking PMO = Soloviov ‘ Content + Lesniewski’ Form Therefore, I shall say some words about Soloviov approach first of all. Further I shall explain some logical ideas of PMO.
Soloviov philosophy is a sort of Platonism. There exists a Highest Being (“Unity”) and there exist infinite set of principles, which are different aspects of Unity. Together Unity and its aspects form AllUnity (therefore the title of Soloviov philosophy is also “Russian Philosophy of AllUnity”). This is the case of an ierarchial Ontology with maximum and minimum (nonbeing) elements. Let us see a typical part of the ierarchy: one more ontologically strong principle (S) and, for example, two its aspects (A_{1} and A_{2}) – see fig.1. Soloviov used a projective intuition here, he interpreted aspects A_{1} and A_{2} as “projections” of the principal S (see also my book^{1}). To clear this idea let us see an example of geometrical projections. For example, we have a 3dimensional body B and two 2dimensional projections P_{1} and P_{2} (see fig.2). Every projection P_{i} is made in the framework of a plane: P_{1} in plane _{1}, P_{2} in plane _{2}. We can speak that every projection is the body B under the condition of the plane of projectivity, i.e., P_{i} is Bunderthecondition_{i} “Under the condition” is a functor, which can be called as projector. Finally we obtain P_{i} = B_{i} , where is projector This structure can be generalised and we might to write in general case A_{i} = SC_{i} , where S is a synthesis A_{i} is an aspect of S C_{i} is a restricted condition under which A_{i} is formed is projector, operation of forming of aspects from synthesis and restricted conditions I shall call these four principles, syntesis, aspect, condition and projector, as ^ . One of my basic assumptions is as follows: any Ontology presupposes an Ontological Tetrade in a definite form. I shall use special terms for all elements of Ontological Tetrade: “modus” for synthesis, “moda” for aspect, “model” for restricted condition and “projector” for projector (see fig.3). Modus is a principle of variety, space of possibilities Model is a principle of restriction of variety Moda is an element of variety, one of the possibilities Projector is an act of restriction (transformation) of variety to an element One need to notice that the term “model” is used not in a trivial sense here. I wanted to use one Latin root “mod”: modus, model, moda. Therefore I shall use the term “modal” in the ancient sense of this word expressing an idea of any variation, modification. To differ this sense from the contemporary using of the term “modal” in different modal logics I add word “projectively” to the word “modal”. I think Ontological Tetrade is a very old philosophical structure. We can find it in Plato, in East Philosophy, etc. For example the following realisations of Ontological Tetrade in some philosophical systems can be demonstrated here
