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