PBNF-transform as a formulation of Propositional Calculus, I
Pelle Brooke Borgeke
TL;DR
The paper addresses reformulating propositional calculus as algebraic computations using Boolean polynomials modulo $2$, achieved via the PBNF-transform that maps DBNF into polynomial families within the House of PBNF. By treating logical connectives as polynomial operators and employing a dual operator-space framework visualized on a unit square, the approach eliminates axioms and truth tables in favor of substitution-based algebraic computation. It establishes a bijective correspondence between statement forms and polynomial representations, defines four natural linear/quadratic polynomial families, and demonstrates that the resulting framework yields canonical, testable forms for logical statements. The work has implications for automated theorem proving and offers a compact, algebraic view of propositional logic that can be extended with additional polynomial families and primitive operators.
Abstract
Here, in a series of articles, we show methods for calculating propositional statements using algebraic polynomials as symbols for the connectives, which are named operators. These polynomials originate from the transformation between the principles of duality and the Disjunctive Boolean Normal Form, DBNF, and they appear if we use a geometrization in the unit square and simple algebraic methods, modulo 2. This we call the PBNF-transform. PBNF stands for Polynomial Boolean Normal Form as these families are based on DBNF involved here. In the first paper in this series, we show that statements can be mapped bijectively into different polynomial families g(p,q) belonging to H(g)$, which we call the The House of PBNF. We can also replace the connectives of logic with PBNF, as the polynomials are, in fact, a geometrization of these connectives; the systems are isomorphic. The benefit of this formulation of the Propositional Calculus(PC) is a near trivialization of the methods. No axioms are needed, no truth tables, just a list of polynomials (which in themselves are self-explanatory), the only law of inference is the rule of Substitution.