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

TL;DR

This paper develops a new modal logic framework for reasoning about hypotheses and conjectures, avoiding modal collapse through non-bivalent semantics and introducing dynamic operators for conjectural transitions.

Contribution

It introduces conjectural modal logics with Axiom C, demonstrates their properties, and proposes a non-bivalent semantic framework to prevent modal collapse.

Findings

Axiom C does not cause modal collapse without Axiom T or bivalent logic.

The systems KC and KDC are sound, complete, and non-trivial.

A dynamic operator models the transition from conjecture to fact.

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.