Modal logic
Modal logic is a form of logic which distinguishes between (logically) "necessary truths" and "contingent truths". Related topics are possibility, impossibility, actuality, and related predicates.
A truth is necessary if it cannot be avoided without logical contradiction, such as "2 + 2 = 4." Necessary propositions either couldn't have been true or couldn't have been otherwise -- perhaps logical and mathematical propositions qualify. By contrast, a contingent truth just happens to be the case, for instance "more than half of the earth is covered by water." Contingent propositions could have been true, but also could have been false -- perhaps "the planet Jupiter exists" qualifies. Possible propositions could have been true -- they include necessary and contingent propositions. Impossible propositions couldn't have been true -- perhaps self-contradictory claims qualify.
In 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.
Modal claims are to be distinguished from similar-sounding epistemic claims. When a philosopher claims that Bigfoot possibly exists, he probably does not mean "Bigfoot might actually exist, for all I know". Rather, he is making a metaphysical claim concerning ways the world could have been, a substantive claim with apparent ontological commitments. 'Epistemic possibility', on the other hand, just traces the confines of our knowledge. "It is possible that p" may be glossed as "I (or we humans) don't know that p is false". It is a claim about those matters about which we have no knowledge one way or the other. When philosophers say "possible", they usually mean the former. An illustration: Someone asks you if 54 squared is 2926 and you stammer, "I don't know, I suppose it's possible". This is 'for all we know' possibility. For, as it turns out 54 squared is 2916 -- and it is metaphysically impossible for it to have been otherwise (say, 2926).
How to best interpret modal claims is a live issue for metaphysicians. Sometimes modal concepts are cashed out in terms of a "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".
This idiom still leaves unclear what we are committing ourselves to when we make modal claims. Are we really alleging the existence of possible worlds, every bit as real as our actual world, just not actual? Renowned philosopher David K. 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.
The concepts of necessity and possibility enjoy the following de Morganesque relationship:
Formal rules
Modal logic adds to the well formed formulae of propositional logic operators for necessity (L) and possibility (M). The two 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)
- Lp → p (If it's necessary that p, then p is the case)
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