MODAL LOGIC: RULES

 

Modal operators

□p = it is necessary that p

◊p = it is possible that p

 

Truth assignment of □p and ◊p in possible worlds

Necessity: □p is true in world w1 if and only if p is true in every world accessible to w1

Possibility: ◊p is true in world w1 if and only if p is true in some world accessible to w1

 

Accessibility relations between possible worlds

Serial relation: every world has access to at least one world

{w1}→ {w2}

Reflexive relation: every world can access itself

{w1}

Symmetric relation: for all worlds, w1, w2, if w1 has access to w2, then w2 has access to w1

{w1} ←→ {w2}

Transitive relation: For all worlds, w1, w2, w3, if w1 has access to w2, and w2 has access to w3, then w1 has access to w3

{w1} → {w2} → {w3}

 

Rules and Axioms

Necessitation Rule (NEC)

If wff A is a proved theorem (e.g., truth table tautology such as p v ~p), then we may infer □A

Change Modal Operator Replacement (CMO)

◊p :: ~□~p

□p :: ~◊~p

~□p :: ◊~p

□~p :: ~◊p

Major Axioms

AS1: ◊P ↔ ~□~P

AS2: □(P→Q) → (□P → □Q)

AS3: □P→ P

AS4: ◊P → □◊P