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'.

翻译：逻辑学的发展主要遵循“演绎”范式：从已确立的前提中推导出结论。然而，在人类与机器推理的实践中，逻辑的运用通常采取对偶的“归约”视角：从待证结论出发生成充分的前提集合。我们将这一范式称为“归约逻辑”。这种逻辑表达涵盖了从形式系统中证明公式到试图在周六中午前与朋友会面等多样化的推理活动。本文是对归约逻辑的语义学分析。具体而言，我们为“归约算子”的表征与推理建立了数学基础。启发式地看，归约算子可被视为“逆向”推理规则。本文探讨了其数学表征方法、在归约推理中的运用方式，以及至关重要的——判定其“有效性”的标准。