Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

PBNF-transform as a formulation of Propositional Calculus, II

Pelle Brooke Borgeke

TL;DR

The paper develops Polynomial Boolean Normal Forms (PBNF) as a Boolean polynomial algebra over $\mathbb{Z}_2$ to reformulate the Propositional Calculus, introducing a dual operator space $\mathcal{OP}$ and a polynomial space $\mathcal{H}$ linked by a bijection $g$. It shows that multiple binary PBNF families can represent Boolean statements, derives complete generator sets for all binary connectives from a small primitive basis, and provides extensive tables translating between operator forms and polynomial expressions. The work also analyzes singular Boolean linear polynomials, defines complementary dual spaces, and reveals canonical decompositions of negation within this algebraic framework, setting the stage for future extensions and duality generalizations (Church–Rosser). Practical impact includes a modular, algebraic approach to propositional logic with potential applications in logic theory, automated reasoning, and duality analysis. $\mathcal{P}(p;q)$ spaces, $g(p,q,1)=a_1pq+a_2p+a_3q+a_4$, and the complement/pullback constructions are central to enabling these transformations and extensions.$

Abstract

Here we show, in the second paper in a series of articles, methods to calculate propositional statements with algebraic polyno mials as symbols for the connectives, which here are named operators. In the first article, we explained this formulation of the Propositional Calculus. In short, we transform to a dual space, which we here refer to as a polynomial family, which is another shape of DBNF. We name the polynomial families as PBNF, which stands for Polynomial Boolean Normal Form. We just use the one law of inference, the rule of Substi tution. We can use different polynomial families in the House of PBNF, depending on the statement form, making it even simpler. It is also pos sible to find new theorems and generalize older ones, for example, those given by Church and Barkley Rosser (see follow-up article) concerning duality.

PBNF-transform as a formulation of Propositional Calculus, II

TL;DR

The paper develops Polynomial Boolean Normal Forms (PBNF) as a Boolean polynomial algebra over to reformulate the Propositional Calculus, introducing a dual operator space and a polynomial space linked by a bijection . It shows that multiple binary PBNF families can represent Boolean statements, derives complete generator sets for all binary connectives from a small primitive basis, and provides extensive tables translating between operator forms and polynomial expressions. The work also analyzes singular Boolean linear polynomials, defines complementary dual spaces, and reveals canonical decompositions of negation within this algebraic framework, setting the stage for future extensions and duality generalizations (Church–Rosser). Practical impact includes a modular, algebraic approach to propositional logic with potential applications in logic theory, automated reasoning, and duality analysis. spaces, , and the complement/pullback constructions are central to enabling these transformations and extensions.$

Abstract

Here we show, in the second paper in a series of articles, methods to calculate propositional statements with algebraic polyno mials as symbols for the connectives, which here are named operators. In the first article, we explained this formulation of the Propositional Calculus. In short, we transform to a dual space, which we here refer to as a polynomial family, which is another shape of DBNF. We name the polynomial families as PBNF, which stands for Polynomial Boolean Normal Form. We just use the one law of inference, the rule of Substi tution. We can use different polynomial families in the House of PBNF, depending on the statement form, making it even simpler. It is also pos sible to find new theorems and generalize older ones, for example, those given by Church and Barkley Rosser (see follow-up article) concerning duality.
Paper Structure (4 sections, 16 theorems, 90 equations)

This paper contains 4 sections, 16 theorems, 90 equations.

Table of Contents

  1. Introduction
  2. Short Resume, and Proof of three more binary PBNF-families
  3. The Generators for the Polynomial operators
  4. Boolean Linear Polynomials, Singular Operators and Extensions

Key Result

Lemma 2.1

1. $\emph{Identity}$ for $\lor:$ for $\land$: 2. $\emph{Annihilator}$ for $\lor$: for $\land$: 3. $\emph{Idempotence}$  for $\lor$: for $\land$: 4. $\emph{Absorption}$  for $\lor$: for $\land$: 5. For 1 and 0, we find We also look at the conditional here 6. $(1 \Rightarrow 0) \mapsto p(q+1)+1 = 0$, but $(0 \Rightarrow 1) \mapsto p(q+1)+1 = 1$ and $(0 \Rightarrow 0) \mapsto p(q+1)+1 = 1$ so

Theorems & Definitions (39)

  • Lemma 2.1
  • proof
  • Example 2.2
  • Theorem 2.3
  • Example 2.4
  • Theorem 2.5
  • proof
  • Theorem 3.1
  • proof
  • Theorem 3.2
  • ...and 29 more