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...".Nim said: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".
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.Nim said:What exactly does it mean in modal logic?
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 isNim said:Would a statement like 1+1=2 be considered possible in modal logic or necessary or both?
Modal logic is a type of formal logic that deals with modalities, which are expressions that indicate possibility, necessity, or contingency. It is used to reason about statements that involve concepts such as possibility, impossibility, necessity, contingency, and belief.
Possibility in modal logic refers to the idea that something could be true or exist in a given situation. This is often represented by the modal operator "◊" (diamond) and is typically used to express statements such as "It is possible that..." or "There exists a possible world where...".
Possibility and necessity are two related but distinct concepts in modal logic. While possibility refers to something being true or existing in a given situation, necessity refers to something being true or existing in all possible situations. This is often represented by the modal operator "□" (box) and is typically used to express statements such as "It is necessary that..." or "In all possible worlds, ...".
The principle of modal logic is a set of rules and principles that govern the use of modal operators in logical statements. These principles include the law of non-contradiction, the law of excluded middle, and the principle of necessitation. They help to ensure that modal logic is consistent and valid.
Modal logic has various applications in fields such as philosophy, computer science, linguistics, and artificial intelligence. It is used to reason about concepts such as possibility, necessity, and belief, and has been applied to areas such as modal metaphysics, game theory, knowledge representation, and natural language processing.