Mateo Sanabria Ardila
MODAL logic 101
Logic + \{\diamond,\square\}
"modal" refers to there being different modes or ways in which something can be true or false
Modal Logic
A Modal Logic formula is define inductively as follow:
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Atomic propositions are formulas.
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If P and Q are formulas then ¬P,P∧Q,P∨Q,P→Q are formulas too.
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If P is a formula then ☐P and ⃟ P are formulas too.
\tiny \square(\ \neg have\_passaport \ \vee \ \neg have\_ticket \rightarrow \neg board\_flight \ )
Modal Logic Formulas
MODAL LOGIC semantics
There are many ways to give meaning (endow) to modal logic; each one results in a different modal logic. Let's focus on Kripke semantics (relational semantics).
\mathcal{M} = < W,R,\pi >
KRIPKE semantics
Formulas need a context in order to evaluate its truth value.
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There are two black cars outside.
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The variable x is always an integer.
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Always y is greater than x.
Kripke structure requiere:
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W: a set of worlds/states
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R: a relation the set of worlds
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π: a function to evaluate the value of formulas
\tiny \mathcal{M},w \models \neg \phi \ \text{ iff is not the case that} \mathcal{M},w \models \phi
\tiny \mathcal{M},w \models \phi \wedge \tau \ \text{ iff both } \ \mathcal{M},w \models \phi \ \text{ and } \ \mathcal{M},w \models \tau
\tiny \mathcal{M},w \models p \ \text{ iff } p \in \pi(w)(p)
KRIPKE semantics
\mathcal{M} = < W,R,\pi >
\tiny \mathcal{M},w \models \lozenge \phi \ \text{ iff } \ (\exists w' | w R w' : \mathcal{M},w' \models \phi)
\tiny \mathcal{M},w \models \square \phi \ \text{ iff } \ (\forall w'| w R w' : \ \mathcal{M},w' \models \phi)
Modal Logic
By Mateo Sanabria Ardila
