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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, may, must, ought, and others. Whereas more traditional
forms of first-order logic work only 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 such as "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 set of modal operators are usually given to be possibility,
actuality, and necessity,. 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); and a
necessary statement is one which could not possibly be false. Also important is the term "contingent":
a contingent statement is one which is not necessarily true, i.e., is possibly true, and possibly
false; a contingent truth is one which is actually true, but which could have been otherwise.
Metaphysical and other modalities
Modal logic is most often used for talk of the so-called alethic modalities: "it is necessarily the case that..." or "it is possibly the
case that...." These (also called metaphysical modalities or subjunctive modalities) need to be
distinguished from various similar-sounding claims using 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. Conversely, we might say "Goldbach's Conjecture is possibly true, but possibly it is false." Here we mean that it
is epistemically possible that it is true or false. At the same time if the sentence is true it is logically
necessary that it is true, and if it is false it is logically impossible.
There are several analogous modes of speech, which though less likely to be confused with alethic modalities are still closely
related. One is talk of time. It seems reasonable to say that possibly it will rain tomorrow, and possibly it won't; on the other
hand, if it rained yesterday, if it really already did so, then it cannot be quite correct to say "It may not have rained
yesterday." It seems the past is "fixed," or necessary, in a way the future isn't. Many philosophers and logicians think this
reasoning isn't very good; but the fact remains that we often talk this way and it is good to have a logic to capture its
structure. Likewise talk of morality, or of obligation generally, seems to have a modal structure. The difference between "You
must do this" and "You may do this" looks a lot like the difference between "This is necessary and this is possible." Such logics
are called deontic, from the Greek
for "duty"
Significantly, modal logics can be developed to accommodate most of these idioms; it is the fact of their common logical
structure (the use of "intensional" or non-truth-functional sentential operators) that make them all varieties of the same thing.
Epistemic logic is
(arguably) best captured in the system "S4" ; deontic logic in the system "D", temporal logic in "t" (sic:lowercase) and alethic logic S5.
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 say 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).
Whatever the notation, the two operators are definable in terms of each other:
- []p (necessarily p) has the same meaning as ~<>~p (not possible that not-p)
- <>p (possibly p) has the same meaning as ~[]~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 []p.
- Distribution Axiom: If [](p → q) then ([]p → []q) (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:
- []p → 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 []p implies [][]p, 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.)
Development of the field of modal logic
Although Aristotle's logic is almost entirely concerned with the theory of the
categorical syllogism, his work also contains some
extended arguments on points of modal logic (such as his famous Sea-Battle Argument in De Interpretatione § 9) and
their connection with potentialities and with time. Following on his works, the Scholastics developed the groundwork for a rigorous theory of modal logic, mostly within the context of
commentary on the logic of statements about essence and accident. Among the medieval writers, some of the most
important works on modal logic can be found in the works of William
of Ockham and John Duns Scotus.
The contemporary logical analysis of modality can be traced to C. I.
Lewis's "A Survey of Symbolic Logic" (1918), in which he he developed the logical systems S1-S5. J. C. C. McKinsey used
algebraic methods (Boolean algebras with operators) to prove the decidability of Lewis' S2 and S4 in 1941. Saul Kripke developed the relational semantics for
modal logics (1959, 1963). Vaughan
Pratt introduced dynamic logic in 1976. Amir Pnueli proposed the use of temporal logic to formalise the behaviour of continually operating
concurrent programs in 1977.
Temporal logic is closely related to modal logic, as adding modal
operators [F] and [P], meaning, respectively, henceforth and hitherto leads to a system of temporal logic.
Flavours of modal logics include: propositional dynamic logic (PDL), propositional linear temporal logic (PLTL), linear temporal logic (LTL), computational tree
logic (CTL), Hennessy-Milner logic, S1-S5, and T.
References
- Robert Goldblatt, "Logics of Time and Computation", CSLI Lecture Notes No. 7, Centre for the Study of Language and
Information, Stanford University, Second Edition, 1992, (distributed by University of Chicago Press).
- Robert Goldblatt, "Mathematics of Modality", CSLI Lecture Notes No. 43, Centre for the Study of Language and Information,
Stanford University, 1993, (distributed by University of Chicago Press).
- G.E. Hughes and M.J. Cresswell, "An Introduction to Modal Logic", Methuen, 1968.
- E.J. Lemmon (with Dana Scott), "An Introduction to Modal Logic", American Philosophical Quarterly Monograpph Series, no. 11
(ed. by Krister Segerberg), Basil Blackwell, Oxford, 1977.
See also
External links
This article contains some material originally from the Free On-line Dictionary of Computing which is used with permission under the GFDL.
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