Modal Logic

© 2011 By Paul Herrick

A Survey of the Main Branches of Logic

Modal logic is “the study of the modes of truth and their relation to reasoning.” The modes of truth are the different ways that a proposition can be true or false. We have already met some of these notions above. The most important are these:

  • Necessary truth. A proposition is necessarily true if it is true and cannot possibly be false.
  • Necessary falsity. A proposition is necessarily false if it is false and cannot possibly be true.
  • Contingent truth. A proposition is contingently true if it is true and in addition there are possible circumstances in which it would be false.
  • Contingent falsity. A proposition is contingently false if it is false and in addition there are possible circumstances in which it would be true.
  • Possible truth: A proposition is possibly true if it is true in at least one possible circumstance.
  • Possible falsity. A proposition is possibly false if it is false in at least one possible circumstance.

Recall that in logic a circumstance counts as “possible” as long as its description is not self-contradictory. Keep in mind that possible does not mean the same as “probable.” For example, the following circumstance is possible, although quite unlikely: someone wins a million dollars in the New York Lottery six days in a row. Possible but not probable. However, the following is not even possible, since its description is self-contradictory: there is a man who is taller than himself.

With some elementary modal concepts defined, some principles of elementary modal logic may be stated with precision. Where P is any declarative sentence:

  • If P is necessarily true, then P is also possibly true.
  • If P is necessarily true, then P is not contingently true.
  • If P is necessarily true, then P is not possibly false.
  • If P is contingently true, then P is also possibly true.
  • If P is contingently true, then P is also possibly false.
  • If P is necessarily false, then P is not possibly true.
  • If P is possibly true, then P is not necessarily false.
  • If P is possibly false, then P is not necessarily true.

And where P and Q stand for any declarative sentences:

  • If P is necessarily true and Q is necessarily true, then P and Q are equivalent.
  • If P is necessarily true and Q is necessarily true, then P and Q are consistent.
  • If P is necessarily false and Q is necessarily false, then P and Q are inconsistent.
  • If P is necessarily false and Q is necessarily false, then P and Q are equivalent.

Aristotle discovered the following interesting and useful modal principles and stated them in one of his logic texts, the first work of modal logic in history:

  • Saying “It is false that it is necessary that P” is equivalent to saying, “It is possible that it is false that P.”
  • Saying “It is necessary that P is false” is equivalent to saying, “It is false that it is possible that P is true.”
  • Saying “It is possible that P” is equivalent to saying, “It is not necessary that it is false that P.”
  • Saying “It is necessary that P” is equivalent to saying, “It is not possible that it is false that P.”

Letting the symbol ☐ (named “box”) stand for “It is necessarily true that,” and letting the symbol ◊ (named “diamond”) stand for “It is possible that,” and letting the symbol ≡ (called “triple bar”) represent the relation of logical equivalence, these principles go into the standard notation of modal logic as follows:

~☐P ≡ ◊ ~P

☐~P ≡ ~◊ P

◊ P ≡~ ☐~P

☐ P ≡ ~◊ ~P

As noted, Aristotle is the founder of modal logic, but we owe the first modern system of modal logic to the Harvard logician, C. I. Lewis (1883-1964). The work started by Lewis was greatly advanced in the 1960s and ’70s by Saul Kripke, Alvin Plantinga, and David Lewis, using an idea that had first been introduced into logical theory by the great German philosopher, logician, and mathematician, Gottfried Leibniz (1646-1716): the notion of a “possible world.” Using the Leibnizian concept of a possible world, Kripke formulated a brand new semantics for modal logic, “possible worlds semantics.” Today, virtually all advanced work in modal logic and on the frontiers of logic rests on one version or another of possible worlds semantics. It may sound surprising, but the notion of a possible world—of a way things might have been or might be—can be used to illuminate the whole of logical theory, and it can resolve many theoretical problems that might not otherwise be solved. See the Useful Links for more on this fascinating and illuminating logical idea—the idea for which this Web site is named.

While predicate logic is especially interesting to mathematicians, modal logic is especially interesting to philosophers because many of the most interesting arguments in the history of philosophy—arguments about the nature and existence of God, free will, the soul, and much more—are modal in nature and can only be analyzed in a deep way using the techniques of modal logic.

For example, in one of the most important works of modal logic ever published, The Nature of Necessity (Oxford University Press, 1974), after systematically defending the modal logic of de re necessity, Alvin Plantinga presented a new version of Anselm’s classic ontological argument for the existence of God, translated it into the precise terms of quantified S5 modal logic, showed that it is perfectly valid, and defended the argument against objections. Modal logic is a fascinating branch of logical theory. For more, see Useful Links.

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