Necessity and Possibility

Some things could have been different.

examples: your hair color, your place of birth, who is President, whether there are any people, whether there are any unicorns.

Alternative statement: Some propositions that are true could have been false, and some propositions that are false, could have been true.

Some things couldn't have been different. They have to be the way they are.

examples: the number of sides of a triangle, the sum of 7 + 5, that if p is true and q is true then (p&q) [the conjunction of p and q] is true, whether all bachelors are unmarried, whether everything is self-identical, whether there is anything that doesn't exist.

Alternative statement: Some true propositions could not have been false, and some false propositions could not have been true.

Some terms:

A proposition is necessary (or necessarily true) iff it could not be false.

A proposition is possible (or possibly true) iff it could be true.

A proposition is impossible iff it could not be true.

A proposition is contingent iff it is neither necessary nor impossible.

A proposition is contingently true iff it is contingent and it is true.

Some shorthand notation:

Let '□(p)' abbreviate 'p is possible' and let '◊(p)' abbreviate 'p is possible'.

What are some truths that govern necessity and possibility?

Which of the following are true, no matter what propositions p and q are?

If □(p) then ◊(p)
If ◊(p) then □(p)
If □(p) then p
If ◊(p) then p
If p then □(p)
If p then ◊(p)
If □(p) and □(q), then □(p&q)
If ◊(p) and ◊(q), then ◊(p&q)
If □(□(p)) then □(p)
If □(p) then □(□(p))
If ◊(□(p)) then □(p)
If □(if p then q) and p, then □(q)

Possible Worlds: a helpful way to think about possibility and necessity

Discovered by Gottfried Leibniz (1646-1716).

A possible world is a complete way that things could be

One of the ways things could be is the way they really are. Thus, the actual world is that possible world which is the way that things are (in fact). Other possible worlds are just other (complete) ways that things could be instead.

A proposition is necessarily true iff it is true in every possible world.

A proposition is possibly true iff it is true in some possible world.

A proposition is impossible iff there is no possible world in which it is true.