Operators in the mind: Jan Lukasiewicz and Polish notation

TL;DR

This paper explores the historical and conceptual connections between Lukasiewicz's Polish notation for logic and modern neurally inspired matrix operators that process logical functions, highlighting their formal similarities and potential implications.

Contribution

It establishes a formal parallel between Polish notation and neural matrix operators, linking three-valued logic with neural computation models.

Findings

Matrix operators reproduce Polish notation structure

Logical matrices generate three-valued logic similar to Lukasiewicz's

Formal parallels suggest new perspectives in logic and neural computation

Abstract

In 1929 Jan Lukasiewicz used, apparently for the first time, his Polish notation to represent the operations of formal logic. This is a parenthesis-free notation, which also implies that logical functions are operators preceding the variables on which they act. In the 1980s, within the framework of research into mathematical models on the parallel processing of neural systems, a group of operators emerged -- neurally inspired and based on matrix algebra -- which computed logical operations automatically. These matrix operators reproduce the order of operators and variables of Polish notation. These logical matrices can also generate a three-valued logic with broad similarities to Lukasiewicz's three-valued logic. In this paper, a parallel is drawn between relevant formulas represented in Polish notation, and their counterparts in terms of neurally based matrix operators. Lukasiewicz's three-valued logic, shown in Polish notation has several points of contact with what matrices produce when they process uncertain truth vectors. This formal parallelism opens up scientific and philosophical perspectives that deserve to be further explored.