Vector logic is a mathematical model of the propositional calculus in which the logical variables are represented by vectors and the logical operations by matrices. In this framework, many tautologies of classical logic are intrinsic identities between operators and, consequently, they are valid beyond the bivalued domain. The operators can be expressed as Kronecker polynomials. These polynomials allow us to show that many important tautologies of classical logic are generated from basic operators via the operations called Type I and Type (...) II products. Finally, it is described a matrix version of the Fredkin gate that extends its properties to the many-valued domain, and it is proved that the filtered Fredkin operators are second degree Kronecker polynomials that cannot be generated by Type I or Type II products. (shrink)

In this article we describe the logical performances displayed by a context-dependent associative memory model. This model requires the existence of a network able to construct the Kroneker product of two vectors, and then to send the composed vector to a correlation distributed memory. This system of nets is capable to sustain all the operations of the classical propositional calculus. This fact implies the existence of vector logics where the logical functions are displayed by matrix operators constructed using the properties (...) of the Kronecker product. When the basic binary matrix operators act on fuzzy inputs, a probabilistic many-valued logic emerges. The present approach implies the potential existence of alternative vector logics. We describe a vector Shefferian “Iogic”, and we comment the potentialities of multidimensional vector logics. (shrink)

In this work, we investigate the representation of counterfactual conditionals using the vector logic, a matrix-vector formalism for logical functions and truth values. Inside this formalism, the counterfactuals can be transformed in complex matrices preprocessing an implication matrix with one of the square roots of NOT, a complex matrix. This mathematical approach puts in evidence the virtual character of the counterfactuals. This happens because this representation produces a valuation of a counterfactual that is the superposition of the two opposite truth (...) values weighted, respectively, by two complex conjugated coefficients. This result shows that this procedure gives an uncertain evaluation projected on the complex domain. After this basic representation, the judgement of the plausibility of a given counterfactual allows us to shift the decision towards an acceptance or a refusal. This shift is the result of applying for a second time one of the two square roots of NOT. (shrink)

Natural languages can express some logical propositions that humans are able to understand. We illustrate this fact with a famous text that Conan Doyle attributed to Holmes: “It is an old maxim of mine that when you have excluded the impossible, whatever remains, however improbable, must be the truth”. This is a subtle logical statement usually felt as an evident true. The problem we are trying to solve is the cognitive reason for such a feeling. We postulate here that we (...) accept Holmes’ maxim as true because our adult brains are equipped with neural modules that perform naturally modal logical computations. (shrink)