Vector logic

Vector logic is an algebraic model of elementary logic based on matrix algebra. Vector logic assumes that the truth values map on vectors, and that the monadic and dyadic operations are executed by matrix operators. "Vector logic" has also been used to refer to the representation of classical propositional logic as a vector space, in which the unit vectors are propositional variables. Predicate logic can be represented as a vector space of the same type in which the axes represent the predicate letters and . In the vector space for propositional logic the origin represents the false, F, and the infinite periphery represents the true, T, whereas in the space for predicate logic the origin represents "nothing" and the periphery represents the flight from nothing, or "something". Classic binary logic is represented by a small set of mathematical functions depending on one (monadic) or two (dyadic) variables. In the binary set, the value 1 corresponds to true and the value 0 to false. A two-valued vector logic requires a correspondence between the truth-values true (t) and false (f), and two q-dimensional normalized real-valued column vectors s and n, hence: and (where is an arbitrary natural number, and "normalized" means that the length of the vector is 1; usually s and n are orthogonal vectors). This correspondence generates a space of vector truth-values: V2 = {s,n}. The basic logical operations defined using this set of vectors lead to matrix operators. The operations of vector logic are based on the scalar product between q-dimensional column vectors: : the orthonormality between vectors s and n implies that if , and if , where . The monadic operators result from the application , and the associated matrices have q rows and q columns. The two basic monadic operators for this two-valued vector logic are the identity and the negation: Identity: A logical identity ID(p) is represented by matrix . This matrix operates as follows: Ip = p, p ∈ V2; due to the orthogonality of s with respect to n, we have , and similarly .