The following appeared in Volume 97, Number 2 (Spring, 1998) of the APA Newsletters

What Modal Logic Is

Our everyday reasoning is filled with talk of possibilities and necessities. Expressions such as "can," "could," "might," "must," "cannot," "have to," etc., are used to assert possibilities and necessities. But there are many kinds of possibility these expressions can be used to denote, and there is a real danger of ambiguity if one is not aware of these different senses of possibility.

Modal logic is the study of one such kind of possibility and necessity, namely logical possibility and logical necessity. Informal modal logic, as I conceive it, has two general areas of concern. The first is to distinguish various kinds of possibility and to discuss the potential ambiguities and informal fallacies resulting from confusion of these notions of possibility. The second is to offer a limited treatment of standard modal logic comparable to the limited treatment of standard propositional logic sometimes offered in informal logic courses. This treatment includes symbolizing standard modal operators, translating English expressions into symbolic modal logic and vice versa, introducing a basic semantics for modal logic, and testing the truth value of modal expressions and the validity of simple arguments involving modal expressions. Of course, the primary function of this treatment of modal logic is to enhance the student's understanding of modal claims occurring in ordinary language.

Different Senses of Possibility and Necessity

Ontological Possibility:

Something is possible in the ontological sense if it is a genuine possibility, i.e., describes a way the world actually could have been. The following are some examples of the sort of thing we often mean when we say that something is possible in the ontological sense.

a. Something is pragmatically possible if it is not overly inconvenient. In this sense, it may be possible for me to go to the store this afternoon, but it is not possible for me to go to Mars this afternoon.

b. Something is technologically possible if it can be done given current levels of human technology. In this sense, it is possible to go to Mars, but not possible to fly a spaceship through the sun.

c. Something is nomologically possible if it does not violate a law of nature. In this sense, it is possible to fly a spaceship through the sun, but it is not possible to fly faster than the speed of light.

Notice that as we move down the list, the set of possibilities gets bigger and bigger. Is there a biggest set? Yes:

d. Something is logically possible if it does not involve a self-contradiction. It is logically impossible if it does involve a self-contradiction, and logically necessary if its denial involves a self-contradiction. In this sense, it is possible to fly faster than the speed of light, but not possible for there to be a six-sided triangle. It is the notion of logical possibility which may be used to characterize deductive validity. An argument is deductively valid if it is logically impossible for all the premises to be true and the conclusion false. All kinds of unexpected things are logically possible. For example, it is logically possible that all bachelors turn purple at midnight, it is possible that toads jump faster than the speed of light (given a suitably loose definition of "toad"), etc. An important thing to keep in mind here is that whether a self-contradiction is present will depend on how the relevant concepts are defined.

These four senses of possibility are ontological in that they each, when used, describe a way the universe could have been. The ontological senses of possibility are to be distinguished from the epistemic sense of possibility.

Epistemic Possibility:

Epistemic possibilities are "for all we know" possibilities. For example, "For all I know about geometry, there could be six-sided triangles." It is not as if there could really be six sided triangles. Rather, I am merely indicating my ignorance of geometry. When we discuss epistemic possibilities, we are not describing a way the universe might have been, but rather what we don't know. The confusion of epistemic with ontological possibilities often occurs in appeals to ignorance, as when someone says "You haven't proven that God doesn't exist, so it could still be true that he does." Here, the "could" statement may be uncontroversially true if understood in terms of epistemic possibility, but more controversial if understood to indicate some ontological sense of possibility, for it is arguable that it is logically impossible that God as he is commonly conceived exists.

Logical Possibility:

Logical possibility is the broadest sense of the ontological senses of possibility. This is so because the universe could never be such that a self-contradiction obtains. When you utter a self-contradiction, you utter something of the form "p and not-p." You take back what was claimed in the first conjunct in the second, the net result being that nothing at all is claimed.

Logicians have focused on logical possibility and necessity because of its philosophical implications, some of which we shall explore later.

Symbolizing Modal Expressions

We symbolize logical necessity with the box () and logical possibility with the diamond (). I first discuss combining negations with the box and diamond, noting that logical possibility and logical necessity are inter-definable with the help of negation: p = ~~p and p = ~~p. Thus, possibility and necessity are two sides of the same coin. I then go though various examples of symbolizing ordinary English claims using the box and diamond.

The Semantics of Modal Logic

Below I briefly note what syntax and semantics are, and contrast the semantics of propositional logic (i.e., truth tables) with the semantics of modal logic. The semantics used for propositional logic cannot be used for modal logic because modal logic is not truth-functional. For example, knowing that p is true will not tell us whether p is true. Similarly, knowing that p is false will not tell us whether p is false, for things may be possible but not actual. So we need an alternative semantics for modal logic. Such a semantics has been recently developed, and it is called possible worlds semantics. A possible world is just a way the universe could have been. More perspicuously, it is a complete description of a universe that contains no self-contradictions. The basic semantics is as follows: p is true if and only if p obtains in some possible world. p is true if and only if p obtains in every possible world. ~p is true if and only if p does not obtain in any possible world. Propositional connectives not under the scope of a modal operator are interpreted by means of ordinary truth tables. I then have students apply this basic semantics to a number of claims made in ordinary language. I like to include examples from Aristotle's famous discussion of the sea battle in De Interpretatione19a30-35, noting that the inference from (p v q) to (p v q) is invalid.

There are actually a number of different systems of modal logic and it is somewhat controversial as to which is best, for different theorems hold in different systems. Alternative systems include T, S4 (the distinctive S4 axiom being p -> p), S5 (the distinctive S5 axiom being p -> p) and others as well. Much of the controversy surrounding which modal system should be adopted seems to revolve around which conception of iterated modalities seems to best capture philosopher's intuitions regarding possibility and necessity. S5 is particularly interesting because, if it were correct, then a version of the ontological argument could be constructed that would appear to be sound. I do not go into the details of the various possible modal systems when teaching critical thinking, but I do discuss the philosophically useful notion of accessibility. We can think of the progress of time in the actual world as a succession of possible worlds becoming the actual world. A possible world w is accessible from the actual world if and only if the actual world is such that w's becoming the actual world does not involve a self-contradiction. For example, if it is logically impossible that the past be changed, then no possible world with a past different than that of the actual world is accessible from the actual world. Another example: if there were a being in the actual world which is such that once it exists it cannot (logically) cease to exist (i.e., an eternal being), then no possible world in which it does not exist is accessible from the actual world. Note that a possible world may be accessible from the actual world at one time but not at another.

Philosophical Applications of Modal Logic

Discussing philosophical applications of modal logic can give students a sense of the potential concrete payoffs of logic. There are a number of topics one might discuss here. I tend to focus on topics in philosophy of religion because of their intrinsic interest, but one could also discuss free will or even modal arguments in philosophy of mind.

A. Omnipotence: One issue is omnipotence. What do we mean by saying that a being, e.g., God (were God to exist) is all-powerful? We can't mean that he could do anything, because some things are logically impossible. We can't mean that God can do anything that it is logically possible for any creature to do, for then could God make a rock so big he cannot lift it? I can, but God presumably could not. Similarly, I can do evil, or be crushed to death by a falling anvil, but presumably God could not. To avoid these objections, we need to say that God can do anything it is logically possible for God to do. So suppose God makes the rock. It ceases to be logically possible for God to lift it, so his inability to lift it ceases to be a denial of his omnipotence. Similarly for doing evil. Thus, certain classic objections to God's being omnipotent can be avoided if one is careful about what is meant when it is said that "God is all-powerful."

B. Malcolm's Version of the Ontological Argument.

Malcolm's argument goes like this:

P1: If God does not exist, his existence is logically impossible.

P2: If God does exist, his existence is logically necessary.

P3: God either does or does not exist.

P4: God's existence is either logically impossible or logically necessary.

P5: If God's existence is logically impossible, then it is self-contradictory.

P6: God's existence is not self-contradictory.

P7: God's existence is not logically impossible.

C: God's existence is logically necessary.

The reason P1 is said to be true is this:

P8: In no possible world does God ever come into existence or cease to exist.

P9: Suppose God does not exist in the actual world.

Ca: God does not exist in any possible world.

This argument is invalid. It follows from P8 and P9 that God does not exist in any possible world accessible from the actual world, but this is not to say that he does not exist in some possible world not accessible from the actual world. P1 appears to be false for a similar reason. If God does not exist in the actual world then, given that he is eternal, he cannot exist in any possible world accessible from the actual world. But this is not to say that God does not exist in any possible world, for he could still exist in some possible world not accessible to the actual world. In other words, Malcolm's argument involves an equivocation on 'logically impossible'. For P1 to be true, 'logically impossible' has to be understood to mean not occurring in any world accessible to the actual world. But for Malcolm's argument to be valid, 'logically impossible' has to mean not occurring in any possible world. This is an interesting example of how modal ambiguities can result in informal fallacies.

This problem with Malcolm's argument is avoidable if one makes the (counterintuitive) assumption that any possible world is accessible to any other, a theorem which apparently follows from S5. Thus, the soundness of an argument for the existence of God, it may be argued, depends on one's intuitions regarding a theorem of modal logic. This is an excellent example of how relatively abstract principles of modal logic may have a very concrete philosophical payoff.

P6 of Malcolm's argument, hence P7 as well, are also controversial, depending on what attributes one assigns to God.

Discussion of informal modal logic in critical thinking/informal logic classes offers an excellent opportunity to enhance the student's understanding of common topics for such courses as well as an improved understanding of everyday talk of possibility and necessity. Discussion of modal logic also provides a natural segue to discussion of interesting philosophical issues. Many of the philosophical implications of modal logic have a great deal of intrinsic interest, and are reasonably direct and accessible to the first-year student.

Resources

Useful, accessible and concise introductory discussions of modal logic and their philosophical applications include:

The Many Worlds of Logic by Paul Herrick (Fort Worth, TX: Harcourt Brace, 1994), Chapter 6 & Philosophical Interlude.

Introductory Modal Logic by Kenneth Konyndyk (Notre Dame, IN: University of Notre Dame Press, 1986).

More detailed discussions occur in:

Possible Worlds: An Introduction to Logic and Its Philosophy by Raymond Bradley and Norman Swartz, (Indianapolis, IN: Hackett, 1979).

The Nature of Necessity by Alvin Plantinga (Oxford: Clarendon Press, 1974).

An Introduction to Modal Logic by G. E. Hughes and M. J. Cresswell (London: Methuen and Co., Ltd., 1968).

A very concise treatment of the logic occurs in:

A Short Introduction to Modal Logic by Grigori Mints (Stanford, CA: Center for the Study of Language and Information (CSLI), 1992).

A very clear discussion of free will, with some attention to modal logic, occurs in:

An Essay on Free Will by Peter Van Inwagen (Oxford: Clarendon Press, 1983).

Other useful resources are mentioned in the endnotes.

Notes

1. A particularly good example of this trend is How to Think About Weird Things by Theodore Schick, Jr. and Lewis Vaughn (Mountain View, CA: Mayfield, 1995).

2. A good source of humorous and interesting logic examples is Critical Thinking, Fourth Edition, by Brooke Noel Moore and Richard Parker (Mountain View, CA: Mayfield, 1995).

3. Richard McKeon, The Basic Works of Aristotle (New York: Random House, 1941), p. 48.

4. See Herrick's discussion in his book listed at the end of this article for a concise and accessible discussion of these issues.

5. This discussion is based on "Omnipotence" by P.T. Geach, Philosophy, vol. 48, April 1973, pp. 7-20 and "The Paradox of the Stone" by C. Wade Savage, The Philosophical Review, vol. 76, 1967, pp. 74-79, both reprinted in The Philosophy of Religion: Selected Readings, edited by Yeager Hudson (Mountain View, CA: Mayfield, 1991), pp. 54-71.

NEWSLETTER ON TEACHING PHILOSOPHY

• From the Editors
• Articles
  • Teaching Informal Logic in Critical Thinking/Informal Logic Classes, Keith Allen Korcz
  • Cooperative Learning on the First Day of Class, Russell Marcus
  • Philosophy in Historical Context, Paul Saka
  • The Philosophy Distinction Course: An Australian Experiment in Distance Teaching, Christina Slade and San McColl
  • Report on "An E-Mail Survey Concerning the Use on Non-Essay Assessment Methods in Introductory Philosophy Classes," Stephen Palmquist
• Book Reviews
  • The Philosophy of Childhood, Reviewed by Rosalind Ekman Ladd
  • The Scientific Background to Modern Philosophy: Selected Readings, Reviewed by Lisa Bates
  • The Cambridge Companion to Nietzsche, Reviewed by Damian Konkoly
  • Learning to Learn: Toward a Philosophy of Education, Reviewed by Anne M. Edwards
• Addresses of Contributors
• News and Notes
• Books Received