Epistemic modals have peculiar logical features that are challenging to account for in a broadly classical framework. For instance, while a sentence of the form $p\wedge\Diamond\neg p$ ('$p$, but it might be that not $p$') appears to be a contradiction, $\Diamond\neg p$ does not entail $\neg p$, which would follow in classical logic. Likewise, the classical laws of distributivity and disjunctive syllogism fail for epistemic modals. Existing attempts to account for these facts generally either under- or over-correct. Some predict that $p\wedge\Diamond\neg p$, a so-called epistemic contradiction, is a contradiction only in an etiolated sense, under a notion of entailment that does not always allow us to replace $p\wedge\Diamond\neg p$ with a contradiction; these theories underpredict the infelicity of embedded epistemic contradictions. Other theories savage classical logic, eliminating not just rules that intuitively fail but also rules like non-contradiction, excluded middle, De Morgan's laws, and disjunction introduction, which intuitively remain valid for epistemic modals. In this paper, we aim for a middle ground, developing a semantics and logic for epistemic modals that makes epistemic contradictions genuine contradictions and that invalidates distributivity and disjunctive syllogism but that otherwise preserves classical laws that intuitively remain valid. We start with an algebraic semantics, based on ortholattices instead of Boolean algebras, and then propose a more concrete possibility semantics, based on partial possibilities related by compatibility. Both semantics yield the same consequence relation, which we axiomatize. We then show how to lift an arbitrary possible worlds model for a non-modal language to a possibility model for a language with epistemic modals.

翻译：认识情态词具有特殊的逻辑特征，这使得在广义经典框架下对其进行解释颇具挑战性。例如，形如 $p\wedge\Diamond\neg p$（'$p$，但可能并非$p$'）的句子看似矛盾，但 $\Diamond\neg p$ 并不蕴含 $\neg p$（后者在经典逻辑中本应成立）。同样，经典逻辑中的分配律和选言三段论对认识情态词失效。现有解释这些事实的尝试往往要么修正不足，要么过度修正。某些理论预测 $p\wedge\Diamond\neg p$（所谓认识矛盾）仅在弱化意义上构成矛盾——其蕴含关系不允许将 $p\wedge\Diamond\neg p$ 直接替换为矛盾式，导致对嵌套认识矛盾的不自然性预测不足。另一些理论则彻底抛弃经典逻辑，不仅删除了直觉上失效的规则，还取消了非矛盾律、排中律、德摩根律及析取引入等对认识情态词仍有效的规则。本文旨在寻求中间道路：为认识情态词构建语义和逻辑系统，使其将认识矛盾视为真正的矛盾，同时使分配律和选言三段论失效，但保留直觉上仍成立的经典规则。我们首先基于正交格（而非布尔代数）提出代数语义，进而构建基于相容偏序可能性的具体可能性语义。两种语义体系得到相同的后承关系，我们对其进行了公理化。最后，我们展示如何将非模态语言的任意可能世界模型提升为包含认识情态词的可能性模型。