Modal Logic Flashcards
What are two modalities?
Possibility and necessity
What does the square symbol mean?
It is necessary that…
What does ♦ mean?
It is possible that…
What are the two modal operators?
The square and diamond symbols (necessary and possible)
What are two logical laws for the two modal operators?
If A is necessary, then it is impossible that A is false. If A is necessary, then it is not possible that ¬A:
|= [] A → ¬♦¬A
If A is possible, then A is not necessarily false. If A is possible, then it is not necessary that ¬A
:|= ♦A → ¬[]¬A.
What is Axiom D?
What is necessary is possible.
If []A then ♦A:
|= []A → ♦A
Why is Axiom D labelled D?
It is a characteristic of deontic logic.
What is Axiom T?
Axiom T. What is necessarily true is true: [] A → A
Why is Axiom T labelled T?
Because it connects necessity and truth.
What is the Transmission of necessity?
If a conclusion follows from necessary premises, it is also necessary. Necessity is transmitted from the premises of a valid argument to its conclusion.
If [] B1 … [] Bn |= A, then [] B1 … [] Bn |= [] A
What is Axiom S4?
What is necessary is necessarily necessary: |= []A → [] [] A
What is Axiom K?
|= [] (A→B)→( [] A→ [] B)
Who was Axiom K named after?
Kripke
What does ∆A mean?
A is contingent
When is a proposition contingent?
When it is neither necessary or possible.
What is the logic for A is contingent
C. ∆A↔(¬[] A∨¬♦A)
Use De Morgans
What is the S5 Axiom?
What is possible is necessarily possible: |= ♦A → []♦A
What is Leibniz’ idea of possible worlds?
[] A is true if and only if A is true in all possible worlds.
A proposition is possibly true just in case its negation is not necessary, so a proposition is possibly true just in case its negation is not true in all possible worlds, so just in case its negation is false in some possible worlds, i.e. just in case the proposition is true in some possible worlds. So we have the following truth-conditions for possibility:
♦p is true if and only if p is true in some possible world.
