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.

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 column vectors composed by real numbers s and n, hence:

(where ${\displaystyle q\geq 2}$ 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: ${\displaystyle u^{T}v=\langle u,v\rangle }$: the orthonormality between vectors s and n implies that ${\displaystyle \langle u,v\rangle =1}$ if ${\displaystyle u=v}$, and ${\displaystyle \langle u,v\rangle =0}$ if ${\displaystyle u\neq v}$.

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