Arthur C. Clarke’s tertium non datur

Saturday

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Arthur C. Clarke is best remembered for this science fiction stories, but many of his dicta and aphorisms have become common currency and are quoted and repeated to the point that their connection to their source is sometimes lost. (Clarke’s thought ranged widely and, interestingly, Clarke identified himself as a logical positivist.) Recently I quoted one of Clarke’s well-known sayings in Happy Birthday Nicolaus Copernicus!, as follows:

“Two possibilities exist: either we are alone in the Universe or we are not. Both are equally terrifying.”

quoted in Visions: How Science Will Revolutionize the Twenty-First Century (1999) by Michio Kaku, p. 295

In so saying, Clarke asserted a particular case of what is known as the logical law (or principle) of the excluded middle, which is also known as tertium non datur: the idea that, given a proposition and its negation, either one or the other of them must be true. This is also expressed in propositional logic as “P or not-P” (“P v ~P”). The principle of tertium non datur is not to be confused with the principle of non-contradiction, which can be formulated as “~(P & ~P).”

Even stating tertium non datur is controversial, because there are narrowly logical formulations as well as ontological formulations of potentially much greater breadth. This, of course, is what makes the principle fascinating and gives it its philosophical depth. Moreover, the principle of the excluded middle is subtly distinct from the principle of bivalence, though the two usually work in conjunction. Whereas the law of the excluded middle states that of a proposition and its negation, one of the other must be true, the principle of bivalence states that there are only two propositional truth values: true and false.

To get started, here is the principle of the excluded middle as formulated in The Cambridge Dictionary of Philosophy edited by Robert Audi:

principle of excluded middle, the principle that the disjunction of any (significant) statement with its negation is always true; e.g., ‘Either there is a tree over 500 feet tall or it is not the case that there is such a tree’. The principle is often confused with the principle of bivalence.

THE CAMBRIDGE DICTIONARY OF PHILOSOPHY second edition, General Editor Robert Audi, 1999, p. 738

And to continue the Oxbridge axis, here is the formulation from Simon Blackburn’s The Oxford Dictionary of Philosophy:

excluded middle, principle (or law) of The logical law asserting that either p or not-p. It excludes middle cases such as propositions being half correct or more or less right. The principle directly asserting that each proposition is either true or false is properly called the law of bivalence.

The Oxford Dictionary of Philosophy, Simon Blackburn, Oxford University Press, 1996, p. 129

For more partisan formulations, we turn to other sources. Mario Bunge formulated a narrowly syntactical conception of the law of the excluded middle in his Dictionary of Philosophy, which is intended to embody a scientistic approach to philosophy:

EXCLUDED MIDDLE A logical truth or tautology in ordinary (classical) logic: For every proposition p, p v ~p.

Dictionary of Philosophy, Mario Bunge, Prometheus Books, 1999, p. 89

By way of contrast, in D. Q. McInerny’s Being Logical: A Guide to Good Thinking we find a strikingly ontological formulation of the law of the excluded middle:

“Between being and nonbeing there is no middle state. Something either exists or it does not exist; there is no halfway point between the two.”

D. Q. McInerny, Being Logical: A Guide to Good Thinking, Part Two, The Basic Principles of Logic, 1. First Principles, p. 26

What these diverse formulations bring out for us is the difficulty of separating logical laws of how formal systems are to be constructed from ontological laws about how the world is constructed, and in so bringing out this difficulty, they show us the relation between the law of the excluded middle and the principle of bivalence, since the logical intuition that there are only two possible truth values of any one proposition — true or false — is so closely tied to our logical intuition that, of these two values, one or the other (but not both, which qualification is the principle of non-contradiction) must hold for any given (meaningful) proposition.

The powerful thing about Clarke’s observation is that it appeals to this admixture of logical intuitions and empirical intuitions, and in so doing seems to say something very compelling. Indeed, since I am myself a realist, and I think it can be shown that there is a fact of the matter that makes propositions true or false, I think that Clarke not only said something powerful, he also said something true: either there are extraterrestrial intelligences or there are not. It is humbling to contemplate either possibility: ourselves utterly alone in a vast universe with no other intelligent species or civilizations, or some other alien intelligence out there somewhere, unknown to us at present, but waiting to be discovered — or to discover us.

Although these logical intuitions are powerful, and have shaped human thought from its earliest times to the present day, the law of the excluded middle has not gone unquestioned, and indeed Clarke’s formulation gives us a wonderful opportunity to explore the consequences of the difference between constructive and non-constructive reasoning in terms of a concrete example.

To formulate the existence or non-existence of extraterrestrials in the form of a logical law like the law of the excluded middle makes the implicit realism of Clarke’s formulation obvious as soon as we think of it in these terms. In imagining the possibilities of our cosmic isolation or an unknown alien presence our terror rests on our intuitive, visceral feeling of realism, which is as immediate to us as the intuitions rooted in our own experiences as bodies.

The constructivist (at least, most species of constructivist, but not necessarily all) must deny the validity of the teritum non datur formulation of the presence of extraterrestrials, and in so doing the constructivist must pretend that our visceral feelings of realism are misleading or false, or must simply deny that these feelings exist. None of these are encouraging strategies, especially if one is committed to understanding the world by getting to the bottom of things rather than denying that they exist. Not only I am a realist, but I also believe strongly in the attempt to do justice to our intuitions, something that I have described in two related posts, Doing Justice to Our Intuitions and How to Formulate a Philosophical Argument on Gut Instinct.

In P or not-P (as well as in subsequent posts concerned with constructivism, What is the relationship between constructive and non-constructive mathematics? Intuitively Clear Slippery Concepts, and Kantian Non-constructivism) I surveyed constructivist and non-constructivist views of tertium non datur and suggested that constructivists and non-constructivists need each other, as each represents a distinct point of view on formal thought. Formal thought is enriched by these diverse perspectives.

But whereas non-constructive thought, which is largely comprised of classical realism, can accept both the constructivist and non-constructivist point of view, the many varieties of constructivism usually explicitly deny the validity of non-constructive methods and seek to systematically limit themselves to constructive methods and constructive principles. Most famously, L. E. J. Brouwer (whom I have previously discussed in Saying, Showing, Constructing and One Hundred Years of Intuitionism and Formalism) formulated the philosophy of mathematics we now know as intuitionism, which is predicated upon the explicit denial of the law of the excluded middle. Brouwer, and those following him such as Heyting, sought to formulate mathematical and logic reasoning without the use of tertium non datur.

Brouwer and the intuitionists (at least as I interpret them) were primarily concerned to combat the growing influence of Cantor and his set theory in mathematics, which seemed to them to license forms of mathematical reasoning that had gone off the rails. Cantor had gone too far, and the intuitionists wanted to reign him in. They were concerned about making judgments about infinite totalities (in this case, sets with an infinite number of members), which the law of the excluded middle, when applied to the infinite, allows one to do. This seems to give us the power to make deductions about matters we cannot either conceive or even (as it is sometimes said) survey. “Surveyability” became a buzz word in the philosophy of mathematics after Wittgenstein began using it in his posthumously published Remarks on the Foundations of Mathematics. Although Wittgenstein was not himself an intuitionist sensu stricto, his work set the tone for constructivist philosophy of mathematics.

Given the intuitionist rejection of the law of the excluded middle, it is not correct to say that there either is intelligent alien life in the universe or there is not intelligent alien life in the universe; to meaningfully make this statement, one would need to actually observe (inspect, survey) all possible locations where such alien intelligence might reside, and only after seeing it for oneself can one legitimately claim that there is or is not alien intelligence in the universe. For am example closer to home, it has been said that an intuitionist will deny the truth of the statement “either it is raining or it is not raining” without looking out the window to check and see. This can strike one as merely perverse, but we must take the position seriously, as I will try to show with the next example.

The day before the Battle of Salamis, Aristotle might have said that there would be a sea battle tomorrow or there would not be a sea battle tomorrow, and in this case the first would have been true; on other days, the second would have been true.

Already in classical antiquity, Aristotle brought out a striking feature of the law of the excluded middle, in a puzzle sometimes known as the “sea battle tomorrow.” The idea is simple: either there will be a sea battle tomorrow, or there will not be a sea battle tomorrow. We may not know anything about this battle, and as of today we do not even know if it will take place, but we can nevertheless confidently assert that either it will take place or it will not take place. This is the law of the excluded middle as applied to future contingents.

One way to think of this odd consequence of the law of the excluded middle is that when it is projected beyond the immediate circumstances of our ability to ascertain its truth by observation it becomes problematic. This is why the intuitionists reject it. Aristotle extrapolated the law of the excluded middle to the future, but we could just as well extrapolate it into the past. Historians do this all the time (either Alexander cut the Gordian Knot or Alexander did not cut the Gordian Knot), but because of our strong intuitive sense of historical realism this does not feel as odd as asserting that something that hasn’t happened yet either will happen or will not happen.

In terms of Clarke’s dichotomy, we could reformulate Aristotle’s puzzle about the sea battle tomorrow in terms of the discovery of alien intelligence tomorrow: either we will receive an alien radio broadcast tomorrow, or we will not receive an alien broadcast tomorrow. There is no third possibility. One way or another, the realist says, one of these propositions is true, and one of them is false. We do not know, today, which one of them is true and which one of them is false, but that does not mean that they do no possess definite truth values. The intuitionist says that the assertion today that we will or will not receive an alien radio broadcast is meaningless until tomorrow comes and we turn on our radio receivers to listen.

The intuitionists thus have an answer to this puzzling paradox that remains a problem for the realist. This is definitely a philosophical virtue for intuitionism, but, like all virtues, it comes at a price. It is not a price I am willing to pay. This path can also lead us to determinism — assuming that all future contingents have a definite truth value implies that they are set in stone — but I am also not a determinist (as I discussed in The Denial of Freedom as a Philosophical Problem), and so this intersection of my realism with my libertarian free willist orientation leaves me with a problem that I am not yet prepared to resolve. But that’s what makes life interesting.

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