Core Concepts
Non-classical modal logics can be represented algebraically, with interactions between necessity and possibility explored in various contexts.
Abstract
The content discusses the algebraic representation of non-classical modal logics using lattice structures. It delves into the semantics of fundamental logic, exploring interactions between necessity and possibility. The article presents representation theorems for lattices with weak negations and independent operations. Various axioms and frame conditions are discussed to establish soundness and completeness in modal logic systems.
Structure:
- Introduction to Non-Classical Modal Logics
- General approach to semantics via algebraic representation theorems.
- Modal Logic Fundamentally Explored
- Study of fundamental logic within intuitionistic and orthologic frameworks.
- Adding Modalities to the Picture
- Definitions of unary operations for multiplicative and additive modalities.
- Interactions Between Necessity and Possibility
- Examination of lattice inequalities and frame conditions for soundness in modal logics.
- Unification in Modal Frames
- Simplifying representations by unifying accessibility relations R and Q.
- Proposed System: Fundamental Modal Logic
- Definition of fundamental modal logic system with additional axioms for completeness.
Stats
In classical modal logic, necessity and possibility are duals: ✷a = ¬✸¬a, ✸a = ¬✷¬a.
Proposition 2.7 states that fundamental logic is sound and complete with respect to bounded lattices equipped with weak pseudocomplementation.
Theorem 2.11 shows that any lattice equipped with antitone operation ¬, multiplicative ✷, and additive ✸ embeds into a relational frame (X, ⊳).
Proposition 4.1 demonstrates the independence of different lattice inequalities over finite lattices.
Quotes
"Adding Reductio Ad Absurdum to fundamental logic yields orthologic."
"In intuitionistic modal logic, ¬✷a does not entail ✸¬a."