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## Modal logic
The basic
## Metaphysical and Epistemic Modalities
Within modal logic, claims about
On the other hand, suppose that someone asks you if 54 squared is 2926 and you stammer, "I don't know, I suppose it's possible." Here you are using an
Epistemic possibilities also bear on the actual world in a way that metaphysical possibilities do not. Metaphysical possibilities bear on ways the world
The vast bulk of philosophical literature on modalities concerns ## Possible Worlds and the Interpretation of Modal LogicIn the most common interpretation of modal logic, one considers "all logically possible worlds". If a statement is true in all possible worlds, then it is a necessary truth. If a statement happens to be true in our world, but is not true in all possible worlds, then it is a contingent truth. A statement that is true in some possible world (not necessarily our own) is called a possible truth. Whether this "possible worlds idiom" is the best way to interpret modal logic, and how literally this idiom can be taken, is a live issue for metaphysicians. For example, the possible worlds idiom which would translate the claim about Bigfoot as "There is some possible world in which Bigfoot exists". To maintain that Bigfoot's existence is possible, but not actual, one could say, "There is some possible world in which Bigfoot exists; but in the actual world, Bigfoot does not exist". But it is unclear what it is that making modal claims commits us to. Are we really alleging the existence of possible worlds, every bit as real as our actual world, just not actual? David Lewis infamously bit the bullet and said yes, possible worlds are as real as our own. This position is called "modal realism". Unsurprisingly, most philosophers are unwilling to sign on to this particular doctrine, seeking alternate ways to paraphrase away the apparent ontological commitments implied by our modal claims. ## Formal rulesThe concepts of necessity and possibility enjoy the following de Morganesque relationship:
- "It is
**not necessary that***X*" is equivalent to "It is**possible that not***X*. - "It is
**not possible that***X*" is equivalent to "It is**necessary that not***X*.
well formed formulae of propositional logic operators for necessity and possibility. In some notations "necessarily p" is represented using a "box" ([]p), and "possibly p" is represented using a "diamond" (<>p). The notation we will use here uses the operator "Lp" for "necessarily p" and "Mp" for "possibly p." Whatever the notation, the two operators are definable in terms of each other:
- Lp (necessarily p) has the same meaning as -M-p (not possible that not-p)
- Mp (possibly p) has the same meaning as -L-p (not necessarily not-p)
- Necessitation Rule: If p is a theorem of K, then so is Lp.
- Distribution Axiom: If L(p → q) then (Lp → Lq) (this is also known as axiom K)
- Lp → p (If it's necessary that p, then p is the case)
The system most commonly used today is ## External links- Stanford Encyclopedia of Philosophy entry
- A discussion of modal logic by John McCarthy
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