Modality

Possible Worlds

Possibility and Necessity

Triangles have three sides. The Wassermans have two kids.

Possibility and Necessity

It is necessary that: Triangles have three sides.

It is possible that: The Wassermans don’t have two kids.

De Dicto and De re

It is necessary that God is good. God is necessarily good. A sentence expresses a de dicto modal

statement iff the modal operator modifies the sentence.

A sentence expresses a de re modal statement iff the modal operator modifies the predicate.

Essential and Non-Essential Properties

Object o is essentially F iff

i. o is F; and

ii. it is necessary that: If o exists, then o is F.

Object o is contingently F iff

i. o is F; and

ii. it is possible that: o exists and o is not F.

The Linguistic Mistake

We could have used the world ‘triangle’ to mean ‘square’. If we had done that, triangles would have had four sides, rather than three. So it’s not necessary that triangles have three sides.

The Epistemic Mistake

Suppose that someone asked you whether

329*747=254,763. Before checking the calculation, you would say that it’s possible that 329*747=254,763 and it’s possible that it doesn’t. But if it’s possible that it doesn’t equal 254,763 then it’s not necessary that it does.

Possibility and Necessity

It is necessary that: Triangles have three sides.

It is possible that: The Wassermans don’t have two kids.

Possibility and Necessity

□ (Triangles have three sides.) ◊ (The Wassermans don’t have two kids).

Possibility and Necessity

□ (Triangles have three sides.) ~◊~(Triangles have three sides.) ◊ (The Wassermans don’t have two kids). ~□~(The Wassermans don’t have two kids).

Possible Worlds

A statement is something that is true or false. A set of statements is consistent iff all of the

members could be true together. A set of statements is complete iff, for any

statement S, either S or not-S is a member of the set.

A possible world is a consistent and complete set of statements.

The actual world (wa) is the possible world consisting of all true statements.

Possible Worlds

Predicates ‘W’ = ‘is a possible world’ ‘T’ = ‘is true in’

Definitions □ p = (x) (Wx Tpx) ◊ p = (x) (Wx & Tpx)

Colloquially Necessity is truth in all possible worlds Possibility is truth in some possible world

Possibility and Necessity

It is necessary that: Triangles have three sides.

It is possible that: The Wassermans don’t have two kids.

Triangles have three sides.