Differential and integral calculus for logical operations. A matrix-vector approach

Abstract

A variety of problems emerged investigating electronic circuits, computer devices and cellular automata motivated a number of attempts to create a differential and integral calculus for Boolean functions. In the present article, we extend this kind of calculus in order to include the semantic of classical logical operations. We show that this extension to logics is strongly helped if we submerge the elementary logical calculus in a matrix-vector formalism that naturally includes a kind of fuzzy-logic. In this way, guided by the laws of matrix algebra, we can construct compact representations for the derivatives and the integrals of logical functions. Inside this semantic-algebraic calculus, we obtain expressions for the derivatives of some of the basic logical operations and show the general way to obtain the derivatives of any well-formed formula of propositional calculus. We show that some of the basic tautologies (Excluded middle, Modus ponens, Hypothetical syllogism) are members of a kind of hierarchical system linked by the differentiation algorithm. In addition using the logical derivatives we show that relatively complex formulas can collapse in simple expressions that reveal clearly their hidden logical meaning. The search for the antiderivatives produces naturally an integral calculus. Within this logical formalism an indefinite integral can always be found for any logical expression. Moreover, particular integrals can be constructed based on detachment properties that lead to logical expressions of growing complexity. We show that these particular integrals have some similarities with the 'generalizing deduction' procedures investigated by Lukasiewicz.