$\rho$-assoc and $\rho$-dist of wfs and f in $\Sigma$ and $\mathscr{L_{HA}}$-theory on 0-OL

TL;DR

This paper introduces pseudo associativity and distributivity properties for certain logical operators within 0-order logic, providing proofs in the Hilbert-Ackermann system.

Contribution

It develops pseudo properties for NFO operators and establishes their proofs within the Hilbert-Ackermann axiomatic framework in 0-order logic.

Findings

Established pseudo associativity and distributivity for NFO operators

Provided formal proofs within Hilbert-Ackermann system

Extended logical properties to 0-order logic context

Abstract

In this paper we create pseudo associativity ($\rho$-assoc) and pseudo distributivity ($\rho$-dist) properties for not fundamental operators NFO $\downarrow$, $\uparrow$, using two semantic rules, also we build proofs for this result in Hilbert-Ackermann ($\mathscr{HA}$) axiomatic system, all in the 0-order logic (0-OL) context.