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：认知情态词具有特殊的逻辑特征，这在广义经典框架中难以解释。例如：形如 $p\wedge\Diamond\neg p$（"p成立，但可能并非p"）的命题看似矛盾，但 $\Diamond\neg p$ 并不蕴含 $\neg p$（若按经典逻辑则将成立）。同样，经典逻辑中的分配律和选言三段论对认知情态词失效。现有解释这些现象的方法往往矫正过枉或不足：有些理论将 $p\wedge\Diamond\neg p$（所谓"认知矛盾"）仅在弱化意义上视作矛盾——即在一种不允许将 $p\wedge\Diamond\neg p$ 替换为矛盾的衍推关系下成立——导致对嵌入式认知矛盾的不适格性预测不足；另一些理论则完全抛弃经典逻辑，不仅删去直观上不成立的规则，还删除了非矛盾律、排中律、德摩根定律及析取引入律等对认知情态词仍有效的规则。本文旨在寻求折中方案：为认知情态词构建一种语义与逻辑体系，使认知矛盾成为真正的矛盾，同时否定分配律与选言三段论，但保留其他直观成立的经典规则。我们首先提出基于正交格（而非布尔代数）的代数语义，进而构建基于相容性关联的部分可能性模型作为更具体的可能世界语义。两种语义学产生相同的推演关系，我们对此给出公理化刻画，最后展示如何将非模态语言的任意可能世界模型提升为含认知情态词语言的模态模型。