Modal logic

Modal logic is a form of logic which deals with sentences that are qualified by modalities such as possibly, necessarily, contingently, actually, can, could, might, etc. Unlike more traditional forms of first-order logic, which can only work with assertoric sentences (such as "Socrates is mortal," "This dog is a terrier," "All cats are reptiles," etc.), modal logic also deals with the logical relationships between problematic statements, such as "It's possible that it will rain on Thursday" or "I can choose to go to the movies tomorrow," and apodictic statements like "Every planet must have an orbit in the form of a conic section" or "if you add 2 and 2, the answer is necessarily 4."

The basic modal operators are usually given to be possibility, actuality, necessity, and contingency. A sentence is said to be actual if it is true; it is said to be possible if it might be true (whether it is actually true or actually false). A necessary statement is one which could not possibly be false; by contrast, a contingent statement is one that might be true and also might be false. (This is not the same, of course, as saying that it is a statement which might be both true and false; there are no statements of that sort.)

Table of contents
1 Metaphysical and Epistemic Modalities
2 Possible Worlds and the Interpretation of Modal Logic
3 Formal rules
4 External links

Metaphysical and Epistemic Modalities

Within modal logic, claims about metaphysical modalities (also known as subjunctive modalities) need to be distinguished from similar-sounding claims about epistemic modalities. For example, when a philosopher claims that Bigfoot possibly exists, he probably does not mean that "it's possible that Bigfoot exists--for all I know." Rather, he is making the metaphysical claim that "it's possible for Bigfoot to exist"--which is a substantive claim concerning ways the world could have been, with apparent ontological commitments.

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 possibility--you are saying that "For all I know, it's possible that 54 squared is 2926." But you are almost surely not making the very hasty claim that it's metaphysically possible for 54 squared to be 2926--which is fortunate, since it turns out that 54 squared is 2916, and it's metaphysically impossible for it to have been otherwise.

Epistemic possibilities also bear on the actual world in a way that metaphysical possibilities do not. Metaphysical possibilities bear on ways the world might have been, but epistemic possibilities bear on the way the world may be (for all we know). Suppose, for example, that I want to know whether or not to take an umbrella before I leave. If you tell me "It's possible that it is raining outside"--in the sense of epistemic possibility--then that would weigh on whether or not I take the umbrella. But if you just tell me that "It's possible for it to rain outside"--in the sense of metaphysical possibility--then I am no better off for this bit of modal enlightenment.

The vast bulk of philosophical literature on modalities concerns metaphysical rather than epistemic modalities. (Indeed, most of it concerns the broadest sort of metaphysical modality--that is, bare logical possibility). This is not to say that metaphysical possibilities are more important to our everyday life than epistemic possibilities (consider the example of deciding whether or not to take an umbrella). It's just to that the priorities in philosophical investigations are rarely set by importance to everyday life--and that should be surprising to no-one.

Possible Worlds and the Interpretation of Modal Logic

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.

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 rules

The 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.

Modal logic adds to the 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)

Precisely what axioms must be added to propositional logic to create a usable system of modal logic has been the subject of much debate. One weak system, named K after Saul Kripke, adds only the following:

  • 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)

These rules lack an axiom to go from the necessity of p to p actually being the case, and therefore are usually supplimented with:

  • Lp → p (If it's necessary that p, then p is the case)

Even with the addition of this axiom, however, K still does not have the rules needed to determine cases where one modal operator ranges over another. For example, K does not determine whether Lp implies LLp, i.e., it does not say whether necessary truths are necessarily necessary, or whether it is possible for them not to be necessary. This may not be a great defect for K, since these seem like awfully strange questions and any attempt to answer them involves us in confusing issues. In any case, different solutions to questions such as these produce different systems of modal logic.

The system most commonly used today is modal logic S5, which robustly answers the questions by adding axioms which make all modal truths necessary: for example, if it's possible that p, then it's necessarily possible that p, and if it's necessary that p it's also necessary that it's necessary. This has the benefit that it fits well with our intuitions about the idiom of possible worlds: if P is true at all possible worlds, then it seems that there can be no possible world at which it is true that there is some possible world where P is false (for if there were such a world, then it would just be the case that P is not true at all possible worlds). Nevertheless, other systems of modal logic have been formulated, in part, because S5 may not be a good fit for every kind of metaphysical modality of interest to us. (And if so, that may mean that possible worlds talk isn't a good fit for these kinds of modality either.)

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