How Does Modal Logic Interpret Possibility and Necessity?

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In summary, possibility in modal logic refers to the ability for a proposition to be true in any given frame of worlds. Whether a statement like 1+1=2 is considered possible or necessary depends on the frame being used. In most cases, modal logic does not allow for straying into philosophical discussions.
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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?
 
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Logically possible means not ruled out by something that is true or is logically necessary. Everything that is logically necessary is also logically possible.
 
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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.
 
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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".
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:
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.

Nim said:
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
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FAQ: How Does Modal Logic Interpret Possibility and Necessity?

What is modal logic?

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.

What is possibility in modal logic?

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...".

What is the difference between possibility and necessity in modal logic?

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, ...".

What is the principle of modal logic?

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.

What are some applications of modal logic?

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.

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