05 Modal Logic
Modal Logic¶
it extends propositional and predicate logic
World¶
is similar to state
it’s like a reality in Rick and Morty. We can assume every thing that can/cannot happpens as part of infinite realities.
Symbols¶
| Symbol | Meaning | Interpretation | CTL Equivalent |
|---|---|---|---|
| \(\Box\) | Necessarily | All worlds | \(AX\) |
| \(\Diamond\) | Possibly | Some world | \(EX\) |
Scenarios¶
| Type | Representation | Interpretation |
|---|---|---|
| Possibility | \(\Diamond \phi = \lnot \Box (\lnot \phi)\) | possibly true; not necessarily false |
| Necessity | \(\Box \phi = \lnot \Diamond (\lnot \phi)\) | necessarily true; not possibly false |
| Uncertainity | \(\lnot(\Box \phi) = \Diamond (\lnot \phi)\) | not necessarily true; possibly false |
| Impossibility | \(\lnot (\Diamond \phi) = \Box (\lnot \phi)\) | not possibly true; necessarily false |
Notes¶
-
Necessity requires possibility, impossibility requires uncertainity
-
Necessity \(\implies\) possibility, impossibility \(\implies\) uncertainity
-
Necessity and impossibility are not symbolically contradictory (look at the position of the \(\lnot\) symbol)
\[ \begin{aligned} &\Box(\phi) \\ \underbrace{}_\text{not here} &\Box ( \lnot \phi) \end{aligned} \]