From Knowledge to Conjectures: A Modal Framework for Reasoning about Hypotheses

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 $\mathbf{\text{C}}$ ($\phi \to \mathrm{\square }\phi$), through which established facts are preserved across conjectural layers. While Axiom $\mathbf{\text{C}}$ has often been treated with suspicion because of its association with modal collapse, we show that collapse does not arise from $\mathbf{\text{C}}$ alone, but requires either the presence of Axiom $\mathbf{\text{T}}$ or a concretely bivalent base logic. Accordingly, we avoid $\mathbf{\text{T}}$ 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 $\mathbf{K}\mathbf{C}$ and $\mathbf{K}\mathbf{D}\mathbf{C}$, show that Axiom $\mathbf{\text{C}}$ directly implies $\mathbf{\text{4}}$ and $\mathbf{\text{5}}$, 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, $\mathsf{s}\mathsf{e}\mathsf{t}\mathsf{t}\mathsf{l}\mathsf{e}(p)$, which formalizes the transition by which a conjectural extension becomes designated reality, thereby motivating a corresponding Conjectural Dynamic Logic.