Semantic Foundations of Reductive Reasoning

TL;DR

This paper provides a semantic and mathematical analysis of reductive logic, focusing on the representation and validity of reduction operators used in backward reasoning processes.

Contribution

It introduces a formal semantic framework for representing reduction operators and analyzes their validity within reductive reasoning, expanding the theoretical foundations of this paradigm.

Findings

Mathematical foundations for reduction operators

Semantic criteria for validity of reductive inference

Framework applicable to diverse reasoning activities

Abstract

The development of logic has largely been through the 'deductive' paradigm: conclusions are inferred from established premisses. However, the use of logic in the context of both human and machine reasoning is typically through the dual 'reductive' perspective: collections of sufficient premisses are generated from putative conclusions. We call this paradigm, 'reductive logic'. This expression of logic encompass as diverse reasoning activities as proving a formula in a formal system to seeking to meet a friend before noon on Saturday. This paper is a semantical analysis of reductive logic. In particular, we provide mathematical foundations for representing and reasoning about 'reduction operators'. Heuristically, reduction operators may be thought of as `backwards' inference rules. In this paper, we address their mathematical representation, how they are used in the context of reductive reasoning, and, crucially, what makes them 'valid'.