This paper proposes natural deduction systems for the representation of inferences in which several agents participate in deriving conclusions about what they believe or know, where belief and knowledge are understood in an intuitionistic sense. Multi-agent derivations in these systems may involve relatively complex belief (resp. knowledge) constructions which may include forms of nested, reciprocal, shared, distributed or universal belief/knowledge as well as attitudes de dicto/re/se. The systems consist of two main components: multi-agent belief bases which assign to each agent a subatomic system that represents the agent’s beliefs concerning atomic sentences and a set of multi-agent labelled rules for logically compound formulae. Derivations in these systems normalize. Moreover, normal derivations possess the subexpression property (a refinement of the subformula property) which makes them fully analytic. Relying on the normalization result, a proof-theoretic approach to the semantics of the intensional operators for intuitionistic belief/knowledge is presented which explains their meaning entirely by appeal to the structure of derivations. Importantly, this proof-theoretic semantics is autarkic with respect to its foundations as the systems (unlike, e.g. external/labelled proof systems which internalize possible worlds truth conditions) are not defined on the basis of a possible worlds semantics. Detailed applications to a logical puzzle (McCarthy’s three wise men puzzle) and to a semantical difficulty (Geach’s problem of intentional identity), respectively, illustrate the systems. The paper also provides comparisons with other approaches to intuitionistic belief/knowledge and multi-agent natural deduction.

中文翻译：

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信念和知识的直觉多智能体亚原子自然演绎

本文提出了用于表示推论的自然演绎系统，其中几个代理参与得出关于他们相信或知道的结论，其中信念和知识在直觉意义上被理解。这些系统中的多主体推导可能涉及相对复杂的信念（或知识）构造，其中可能包括嵌套、互惠、共享、分布式或普遍信念/知识的形式以及自言自语/re/se 的态度。该系统由两个主要组成部分组成：多智能体信念基础，它为每个智能体分配一个亚原子系统，该系统表示智能体对原子句子的信念，以及一组用于逻辑复合公式的多智能体标记规则。这些系统中的推导归一化。而且，正态推导具有子表达式属性（子公式属性的改进），这使得它们完全解析。依靠归一化结果，提出了一种直觉主义信念/知识的内涵算子语义的证明理论方法，该方法完全通过诉诸推导结构来解释它们的含义。重要的是，这种证明理论语义就其基础而言是自给自足的，因为系统（与例如将可能世界真值条件内在化的外部/标记证明系统不同）不是基于可能世界语义来定义的。逻辑谜题（麦卡锡的三个智者谜题）和语义难题（Geach 的意向身份问题）的详细应用分别说明了这些系统。