The origins of proof-theoretic semantics lie in the question of what constitutes the meaning of the logical connectives and its response: the rules of inference that govern the use of the connective. However, what if we go a step further and ask about the meaning of a proof as a whole? In this paper we address this question and lay out a framework to distinguish sense and denotation of proofs. Two questions are central here. First of all, if we have two (syntactically) different derivations, does this always lead to a difference, firstly, in sense, and secondly, in denotation? The other question is about the relation between different kinds of proof systems (here: natural deduction vs. sequent calculi) with respect to this distinction. Do the different forms of representing a proof necessarily correspond to a difference in how the inferential steps are given? In our framework it will be possible to identify denotation as well as sense of proofs not only within one proof system but also between different kinds of proof systems. Thus, we give an account to distinguish a mere syntactic divergence from a divergence in meaning and a divergence in meaning from a divergence of proof objects analogous to Frege’s distinction for singular terms and sentences.

中文翻译：

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证明的含义是什么？

证明论语义学的起源在于什么构成逻辑连接词的含义及其响应的问题：控制连接词使用的推理规则。但是，如果我们更进一步并询问整个证明的含义会怎样？在本文中，我们解决了这个问题，并制定了一个框架来区分证明的意义和外延。这里有两个问题很重要。首先，如果我们有两个（语法上）不同的派生，这是否总是导致差异，首先是意义，其次是外延？另一个问题是关于这种区别的不同类型的证明系统（这里：自然演绎与连续演算）之间的关系。表示证明的不同形式是否必然对应于如何给出推理步骤的不同？在我们的框架中，不仅可以在一个证明系统内，而且可以在不同类型的证明系统之间识别外延和证明意义。因此，我们给出了一个解释来区分纯粹的句法分歧和意义分歧，以及意义分歧与证明对象分歧，类似于弗雷格对单数术语和句子的区分。