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| Format: | Preprint |
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2025
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| Online Access: | https://arxiv.org/abs/2508.07304 |
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| _version_ | 1866911533168066560 |
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| author | Vitali, Fabio |
| author_facet | Vitali, Fabio |
| contents | 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} ($φ\rightarrow \Boxφ$), 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_07304 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | From Knowledge to Conjectures: A Modal Framework for Reasoning about Hypotheses Vitali, Fabio Logic in Computer Science Artificial Intelligence 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} ($φ\rightarrow \Boxφ$), 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. |
| title | From Knowledge to Conjectures: A Modal Framework for Reasoning about Hypotheses |
| topic | Logic in Computer Science Artificial Intelligence |
| url | https://arxiv.org/abs/2508.07304 |