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：认知情态词具有独特的逻辑特征，这在广义经典框架中难以解释。例如，尽管形如$p\wedge\Diamond\neg p$（"$p$，但可能非$p$"）的语句看似矛盾，但$\Diamond\neg p$并不蕴含$\neg p$——后者在经典逻辑中本应成立。同样，分配律和选言三段论等经典逻辑定律对认知情态词失效。现有解释方案往往矫正过度或不足：有些预测$p\wedge\Diamond\neg p$（所谓认知悖论）仅具有弱化矛盾性——在某种不总能将$p\wedge\Diamond\neg p$替换为矛盾式的蕴含概念下成立，这类理论低估了嵌入认知矛盾的语句不当性；另一些方案则过度修正经典逻辑，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直觉上对认知情态词仍有效的规则。本文旨在寻求折中方案，为认知情态词构建语义与逻辑系统：既将认知矛盾视为真正的逻辑矛盾，又使分配律与选言三段论失效，同时保留直觉上仍成立的其他经典定律。我们从基于正交格（而非布尔代数）的代数语义出发，提出更具体的可能性语义——以通过相容性关联的部分可能性为基础。两种语义系统导出相同的后承关系，我们对此进行公理化，并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知情态词语言的可能性模型。