arXivDaily arXiv每日学术速递 周一至周五更新
arXiv 2203.02872cs.LO
收藏

认知模态词的正交逻辑

The Orthologic of Epistemic Modals

Wesley H. Holliday, Matthew Mandelkern

备注：
Minor edits for final version forthcoming in Journal of Philosophical Logic
阅读 PDF
arXiv 页面 ↗HTML 原文 ↗

AI总结：

本文研究认知模态词的逻辑特征，提出基于正交格代数语义和部分可能性语义的中间路线逻辑，使认知矛盾成为真正矛盾，同时保留直觉有效的经典定律。

摘要：

认知模态词具有特殊的逻辑特征，在广义的经典框架中很难加以解释。例如，虽然形如 $p\wedge\Diamond\neg p$（“$p$，但可能非 $p$”）的句子看起来是一个矛盾，但 $\Diamond\neg p$ 并不蕴含 $\neg p$，而这在经典逻辑中是成立的。同样，分配律和析取三段论的经典定律对认知模态词失效。现有的解释这些事实的尝试通常要么纠正不足，要么纠正过度。一些理论预测 $p\wedge\Diamond\neg p$（即所谓的认知矛盾）仅仅是在一种弱化的意义上成为矛盾，其蕴含概念并不总是允许我们将 $p\wedge\Diamond\neg p$ 替换为一个矛盾；这些理论低估了嵌入认知矛盾的不恰当性。另一些理论则大肆破坏经典逻辑，不仅消除了直觉上失效的规则，还消除了非矛盾律、排中律、德摩根定律和析取引入等直觉上对认知模态词仍然有效的规则。在本文中，我们旨在寻找一条中间路线，为认知模态词发展一种语义学和逻辑，使得认知矛盾成为真正的矛盾，并使分配律和析取三段论失效，但在其他方面保留直觉上仍然有效的经典定律。我们从一个基于正交格而非布尔代数的代数语义学开始，然后提出一种更具体的可能性语义学，它基于通过兼容性关系关联的部分可能性。两种语义学产生相同的后承关系，我们对其进行了公理化。然后，我们展示了如何将非模态语言的任意可能世界模型提升为带有认知模态词的语言的可能性模型。

英文摘要：

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.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

↑