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 . It is important to note that this vector logic identity matrix is not generally an in the sense of matrix algebra.
identity matrix
Negation: A logical negation ¬p is represented by matrix Consequently, Ns = n and Nn = s. The behavior of the logical negation, namely that ¬(¬p) equals p, corresponds with the fact that N2 = I.
involutory
History[edit]
Early attempts to use linear algebra to represent logic operations can be referred to Peirce and Copilowish,[15] particularly in the use of logical matrices to interpret the calculus of relations.
The approach has been inspired in neural network models based on the use of high-dimensional matrices and vectors.[16][17] Vector logic is a direct translation into a matrix–vector formalism of the classical Boolean polynomials.[18] This kind of formalism has been applied to develop a fuzzy logic in terms of complex numbers.[19] Other matrix and vector approaches to logical calculus have been developed in the framework of quantum physics, computer science and optics.[20][21]
The Indian biophysicist G.N. Ramachandran developed a formalism using algebraic matrices and vectors to represent many operations of classical Jain logic known as Syad and Saptbhangi; see Indian logic.[22] It requires independent affirmative evidence for each assertion in a proposition, and does not make the assumption for binary complementation.