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.
