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}

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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