意义的几何化
The Geometrization of Meaning
摘要：
哲学中最大的问题之一是意义问题。意义思考的转折点是塔斯基对真理的定义，逻辑语义学和模型论的快速发展是这一成就的结果。也许在古典逻辑学家和哲学家中不太为人所知的是，正是范畴论提供了足够的数学工具来研究形式化理论的语法和它们的语义之间的关系。本文旨在改变这一状况，并对迄今为止所取得的成果进行初步的哲学分析。它们涉及形式化的代数理论与公理的形式方程的法律，理论的基础上命题逻辑和连贯的布尔逻辑，以及可判定的逻辑，这不一定是布尔。这些理论的句法-语义关系采取各自句法和语义类别之间二元对立的形式。这些对偶由适当的伴随函子给出。在所有考虑的情况下，语法-语义二元论对应的代数几何二元论。我们分析了这些结果的哲学意义。它们使我们能够以新的眼光看待意义问题，并制定一个区分形式理论和经验理论的标准。曾经为塔斯基的真理定义而争论不休的争论被转移到了一个新的背景下。关于塔斯基是否真的成功地把意义问题归结为纯粹的句法术语的争论已经被取代了：在句法范畴和语义范畴之间，不存在一种归约关系，而是一种相互作用关系，而这种关系是由伴随函子给出的。我们还触及了对意义问题的绝对方法的其他哲学方面。
英文摘要：
One of the greatest problems in philosophy is that of meaning. The turning point in thinking on meaning was Tarski's definition of truth, and the rapid development of logical semantics and model theory was a consequence of this achievement. Perhaps less well-known among classical logicians and philosophers is that it is category theory that provides adequate mathematical tools to study the relationship between the syntax of formalized theories and their semantics. The aim of this article is to change this situation and make a preliminary philosophical analysis of the results obtained so far. They concern formalized algebraic theories with axioms in the form of equational laws, theories based on propositional logic and coherent Boolean logic, as well as decidable logic which is not necessarily Boolean. The syntactic-semantics relation for these theories takes the form of dualisms between the respective syntactic and semantic categories. These dualisms are given by the appropriate adjoint functors. In all the considered cases, the syntax--semantics dualism corresponds to the algebra-geometry dualism. We analyze the philosophical significance of these results. They allow us to look at the problem of meaning in a new light and formulate a criterion distinguishing formal theories from empirical theories. The disputes that were once fought over Tarski's definition of truth are transferred to a new context. The polemics as to whether Tarski actually succeeded in reducing the problem of meaning to purely syntactic terms has been superseded: between the syntactic and semantic categories there is not a relation of reducibility but rather that of interaction, and this relation is given by adjoint functors. We also touch upon other philosophical aspects of the categorical approach to the problem of meaning.
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