Undergrad How Does Modal Logic Interpret Possibility and Necessity?

[Nim]

One of the ways we use the term possibility is "We can go to Mars". Another way is "We may learn how to travel to Mars faster than light as our understanding of physics progresses".

What exactly does it mean in modal logic? Would a statement like 1+1=2 be considered possible in modal logic or necessary or both?

 

[sysprog]

Logically possible means not ruled out by something that is true or is logically necessary. Everything that is logically necessary is also logically possible.

 

[sysprog]

I should have said "true in all possible worlds", rather than simply "true" -- a thing could be false in our world and still be logically possible.

 

[pbuk]

One of the ways we use the term possibility is "We can go to Mars". Another way is "We may learn how to travel to Mars faster than light as our understanding of physics progresses".

Neither of those statements contains the term (or operator of) possibility. The first one is a simple statement of ability "We are able to go to Mars". The second does contain the notion of possibility and can be restated "It is possible for us to learn how to travel to Mars faster than light...".

What exactly does it mean in modal logic?

Possibility has no meaning on its own, it acquires meaning with reference to a frame of worlds. If a proposition is true in any frame that is accessible then it is possible.

Would a statement like 1+1=2 be considered possible in modal logic or necessary or both?

That depends on how you define the accessible frame. If the frame includes normal decimal arithmetic then the statement is clearly possible so the modal proposition "it is possible that 1 + 1 = 2" is true in that model. If the frame includes arithmetic modulo 2 then the modal proposition "it is possible that not (1 + 1 = 2)" is true, which in most formulations of modal logic implies that the proposition "it is necessary that 1 + 1 = 2" is false. Of course one could argue that 2 is not a value of arithmetic modulo 2 and so the proposition has no meaning in that world, but then you are in danger of straying into philosophy which is

• where, in my (limited) experience, modal logic usually ends up
• not allowed on Physics Forums.