10.3 Proof systems for modal logic

Modal logic proof systems build on propositional logic, adding axioms and rules for necessity and possibility operators. These systems, like K, T, S4, and S5, capture different notions of necessity and possibility in formal reasoning.

Proof systems for modal logic are essential for establishing the validity of modal arguments. They provide a framework for deriving theorems and exploring the relationships between different modal concepts, forming the backbone of modal reasoning.

Axiomatization of Modal Logics

Axiom K and Necessitation Rule

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• Axiom K states if a proposition $\phi$ is true, then it is necessarily possible, denoted as $\square\Diamond\phi$
• Axiom K is a fundamental axiom in modal logic that relates necessity and possibility
• Necessitation rule infers the necessity of a proposition from its truth, formally if $\vdash\phi$ then $\vdash\square\phi$
• Necessitation rule allows deriving necessary truths from theorems of propositional logic

Distribution Axiom and Normal Modal Logics

• Distribution axiom, also known as Axiom K, states $\square(\phi\to\psi)\to(\square\phi\to\square\psi)$
• Distribution axiom relates the modal operator $\square$ with implication $\to$
• Distribution axiom ensures the modal operator $\square$ distributes over implication
• Normal modal logics include Axiom K and the Necessitation rule as core axioms
• Normal modal logics form a class of modal logics with desirable properties (soundness, completeness)

Common Modal Systems

System K and System T

• System K is the minimal normal modal logic containing only Axiom K and the Necessitation rule
• System K serves as the foundation for other stronger modal logics
• System T extends System K by adding the axiom $\square\phi\to\phi$, known as the reflexivity axiom
• Reflexivity axiom states if a proposition is necessary, then it is true

System S4 and System S5

• System S4 extends System T by adding the axiom $\square\phi\to\square\square\phi$, known as the transitivity axiom
• Transitivity axiom states if a proposition is necessary, then its necessity is also necessary
• System S5 extends System S4 by adding the axiom $\Diamond\phi\to\square\Diamond\phi$, known as the symmetry axiom
• Symmetry axiom states if a proposition is possible, then it is necessarily possible
• System S5 is one of the strongest and most studied modal logics

Metatheoretic Properties

Soundness

• Soundness is a metatheoretic property that ensures the consistency of a proof system
• A modal logic is sound if every provable formula is valid in the corresponding semantics (Kripke frames)
• Soundness guarantees that the axioms and inference rules of a modal logic do not lead to contradictions
• Proving soundness involves showing that each axiom is valid and inference rules preserve validity

Completeness

• Completeness is a metatheoretic property that ensures the adequacy of a proof system
• A modal logic is complete if every valid formula in the corresponding semantics (Kripke frames) is provable
• Completeness guarantees that all valid formulas can be derived using the axioms and inference rules
• Proving completeness often involves constructing a canonical model that satisfies all consistent formulas
• Completeness is a stronger property than soundness and is crucial for automated theorem proving