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:

```
Atomic propositions are formulas.
```

If P and Q are formulas then ¬P,P∧Q,P∨Q,P→Q are formulas too.

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.

```
There are two black cars outside.
```
```
The variable x is always an integer.
```
```
Always y is greater than x. 
```

Kripke structure requiere:

```
W: a set of worlds/states
```
```
R: a relation the set of worlds
```

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)