{"ID":2884129,"CreatedAt":"2026-06-01T04:54:23.091178241Z","UpdatedAt":"2026-06-01T04:54:23.091178241Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2508.07304","arxiv_id":"2508.07304","title":"From Knowledge to Conjectures: A Modal Framework for Reasoning about Hypotheses","abstract":"This paper introduces a new family of cognitive modal logics designed to formalize conjectural reasoning: modal systems in which cognitive contexts extend known facts with hypothetical assumptions in order to explore their consequences. Unlike traditional doxastic and epistemic systems, conjectural logics rely on a principle, called Axiom \\textbf{C} ($\\varphi \\rightarrow \\Box\\varphi$), through which established facts are preserved across conjectural layers. While Axiom \\textbf{C} has often been treated with suspicion because of its association with modal collapse, we show that collapse does not arise from \\textbf{C} alone, but requires either the presence of Axiom \\textbf{T} or a concretely bivalent base logic. Accordingly, we avoid \\textbf{T} and adopt a non-bivalent semantic framework, such as supervaluation-style semantics, Weak Kleene logic, or Description Logic, in which undefined propositions may coexist with modal assertions. This prevents modal collapse and preserves a distinction between factual and conjectural statements. Within this framework we define the modal systems $\\mathbf{KC}$ and $\\mathbf{KDC}$, show that Axiom \\textbf{C} directly implies \\textbf{4} and \\textbf{5}, and prove that these systems are non-trivial, sound, and complete. An inclusion theorem links reality, doxastic states, epistemic states, and conjectural states via set-theoretic inclusion among valuations, providing a unified account of how these layers relate. Finally, we introduce a dynamic operator, $\\mathsf{settle}(p)$, which formalizes the transition by which a conjectural extension becomes designated reality, thereby motivating a corresponding Conjectural Dynamic Logic.","short_abstract":"This paper introduces a new family of cognitive modal logics designed to formalize conjectural reasoning: modal systems in which cognitive contexts extend known facts with hypothetical assumptions in order to explore their consequences. Unlike traditional doxastic and epistemic systems, conjectural logics rely on a pri...","url_abs":"https://arxiv.org/abs/2508.07304","url_pdf":"https://arxiv.org/pdf/2508.07304v2","authors":"[\"Fabio Vitali\"]","published":"2025-08-10T11:37:49Z","proceeding":"cs.LO","tasks":"[\"cs.LO\",\"cs.AI\"]","methods":"[]","has_code":false}