The Logical Legacy of Nikolai Vasiliev and Modern Logic by Vladimir Markin & Dmitry Zaitsev

Author:Vladimir Markin & Dmitry Zaitsev
Language: eng
Format: epub
Publisher: Springer International Publishing, Cham

7.4 Logic of n Dimensions

In the paper “Imaginary (non-Aristotelian) logic” Vasiliev advanced an idea of possible development of the logic of n dimensions (Vasiliev 1989, pp. 76–77). For him, such systems differ in a number of types of propositions varying in quality. Aristotelian syllogistic is bidimensional, imaginary logic has three dimensions. In general, a logic of n dimensions must contain n types of propositions with different qualities. Vasiliev himself did not develop these idea into a logical theory.

The reconstruction of the logic on n dimensions was realized by T.P. Kostiouk (2000). She formulated an exact and intuitively transparent semantics for syllogistic language with n types of propositions varying in quality along with the adequate axiomatization.

The system IL can be extended in a natural way to syllogistics I L n with arbitrary number of propositions with different qualities.

There are n syllogistic constants for singular (J 1 , J 2 , …, J n ), universal (A 1 , A 2 , …, A n ) and indefinite particular (I 1 , I 2 , …, I n ) propositions of different quality. Let J i vP means that an individual v stands in i-th qualitative relation to P, A 1 SP – every object from S stands in i-th qualitative relation to P, I 1 SP – some object from S stands in i-th qualitative relation to P. When i = 1 we have a form of affirmative proposition with corresponding quantity. It is convenient to suppose the formulas with i = n to be the forms of negative propositions.

I L n -model is a structure < D, φ, ψ 1, ψ 2, …, ψ n  > , where D ≠ ∅, φ(v) ∈D, ψ i (P) ⊆D, ψ 1(P) ≠ ∅, ψ i (P) ∩ ψ j (P) = ∅, where 1 ≤ i, j ≤ n and i ≠ j; ψ 1(P) ∪ ψ 2(P) ∪… ∪ ψ n (P) = D. In this semantical framework, each universal term is connected with n extensional characteristics.

The truth definitions for atomic formulas are the following:

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