🤖 AI Summary
This paper addresses the lack of formal semantic foundations for reduction logic—i.e., goal-directed backward reasoning—by establishing, for the first time, a systematic formal semantic framework for reduction operators. Methodologically, it integrates categorical semantics with operational semantics to develop the logical metatheory of reduction calculus and proposes mathematically precise criteria for determining the validity of reduction operators. The core contribution lies in overcoming the unidirectionality inherent in traditional deductive logic, thereby enabling an ontological and verifiable formalization of the fundamental human- and machine-shared inference pattern: “deriving sufficient premises from a given conclusion.” This work provides a rigorous semantic foundation and soundness guarantees for goal-driven AI tasks, including automated planning, theorem discovery, and commonsense reasoning.
📝 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'.