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. Then we show how to extend our semantics to explain parallel phenomena involving probabilities and conditionals. The goal throughout is to retain what is desirable about classical logic while accounting for the non-classicality of epistemic vocabulary.

翻译：认识模态词具有独特的逻辑特征，在广义经典框架下难以解释。例如，形式为$p\wedge\Diamond\neg p$（“$p$，但可能非$p$”）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，分配律和选言三段论等经典定律对认识模态词也失效。现有解释这些现象的理论往往要么矫正不足，要么矫正过度。有些理论预测$p\wedge\Diamond\neg p$（所谓认知矛盾）仅在弱化意义下构成矛盾——其所依据的蕴含概念不允许将$p\wedge\Diamond\neg p$直接替换为矛盾式；这类理论低估了嵌入认知矛盾的不恰当性。另一些理论则过度修正经典逻辑，不仅删除了直觉上失效的规则，还排除了非矛盾律、排中律、德摩根律和析取引入律等对认识模态词仍直观有效的规则。本文旨在寻求中间立场，为认识模态词建立一种语义与逻辑体系：将认知矛盾视为真正的矛盾，否定分配律和选言三段论，同时保留直觉上仍有效的经典定律。我们首先提出基于正交格（而非布尔代数）的代数语义，进而建立基于相容部分可能性关系的内涵语义。两种语义产生相同的推论关系，我们对此进行公理化。最后展示如何将该语义扩展至概率与条件句中的平行现象。本文的核心目标是：在解释认知词汇非经典性的同时，保留经典逻辑的可取特征。