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