A Most Ingenious Paradox

by Phil Johnson



ears ago I discovered (the hard way) that the Internet is not the friendliest place for anyone who wants to stand up for logic or defend the coherence of truth.

I've long been concerned about the erosion of rationality in postmodern culture. I've always known there are naive Christians who mindlessly parrot worldly values, and I've been concerned for years about the potential for mischief when spiritual-sounding Christian terminology gets blended with worldly irrationalism. But I had no clue how much and how far irrationalism had already infected the visible church when I posted this article on my website back in 1995.

The article is a defense of the principium contradictionis, or "the law of contradiction"—which says truth is by definition non-contradictory: "A is not non-A." (Some people prefer to speak of it as "the law of non-contradiction." Whatever.)

The law of contradiction is one of three principles classical logicians universally regard as foundational to all human thought. The other two are the law of identity, which states that an object is the same as itself: "A is A." Then there's the law of the excluded middle, meaning that when two propositions directly negate one another, one must be true and the other false; there is no third alternative: "Either A or non-A, but not both A and non-A."

The codification of those three principles is usually attributed to Aristotle. Most philosophers have regarded them as self-evident. (There have been exceptions. Hegel hated classical logic.) But those rather simple principles are the basis of formal logic and rational thought. Without them, rational discourse is simply not possible. In fact, to deny any of those principles is (by definition) irrational.

Anyway, almost 15 years ago I wrote this little essay on the law of contradiction, and it unleashed the fury of several hordes of budding "postmodern Christians." They filled my in-box with protests, solemnly assuring me that human logic is just that: "human," and therefore ungodly. To acknowledge the incomprehensibility of God is to embrace the incoherence of truth, they (illogically) insisted. Illogic? Who cares? They proudly and steadfastly embraced several contradictions in their own worldview. Their theology (early Emergent nonsense) seemed deliberately muddled. Contradictions in one's doctrine are more to be desired than gold, they seemed to be saying. Their whole idea of "faith" was a Kierkegaardian leap into dark nothingness, where (apparently) any and all biblical propositions are fair game for quacks and amateurs to question or contradict, depending on their personal whims.

But the principium contradictionis they could not tolerate, because it contradicted ideas that were frankly more basic to their worldview than the plain statements of Scripture.

So one of these guys—let's call him Daryl—was especially persistent. He e-mailed me again and again, and promised to supply me with "incontrovertible proof that the law of contradiction is false." The irony of his own boast escaped him, and it was clear from the start that his mouth was writing a check his mind could never cash. But he "stayed up half the night" noodling on the problem, then wrote me to say he had two propositions that debunked the principium contradictionis. Here's the salient part of his e-mail:

1. "This sentence is false." 2. And here's one I got from Bertrand Russell: Suppose X is the set of all sets that are not members of themselves. Is the following proposition, "X is a member of itself." true, or false? If it's true, then it's also false. (I hope I'm saying that correctly. I got it from a book about logic.) Anyway, there are two propositions that are both true and false.

My reply:

Daryl, I'm disappointed. This is old stuff. Let's start with Bertrand Russell's "Set of all sets" paradox. You did state it correctly. It is a classic mathematical paradox (one of several.) Those not acquainted with it may have to think a few minutes to see the subtlety of this paradox: Some sets are members of themselves (e.g., the set of all abstract concepts is itself an abstract concept. Therefore it is a member of the set of all abstract concepts.) Some sets are not members of themselves. (e.g., the set of all colors is not itself a color.) Think about the set of all sets that are not members of themselves. Is it a member of itself, or not? If yes, the answer is no, and vice versa. But it reflects a very naive view of logic and math to assert that such paradoxes debunk the law of contradiction. Most of them actually involve the fallacy of equivocation and can be resolved by careful definition. Search and see. The Russell paradox is based on incomplete and outdated set theory. Zermelo-Fraenkel (ZFC) set theory is a system of axioms that avoids the Russell paradox. Jensen, Quine, Hilbert, Frege, and even Russell himself all proposed other more or less successful theories to resolve the "set of all sets" paradox. I'm not much of a mathematician, so I won't attempt to explain all those theories. (But you can Google them.) They all attack the problem by restricting or stratifying the way sets are constructed. They suggest there is a logical gap that must be bridged between the definition of a set and the set's actual construction. In layman's terms: "The set of all sets that are not members of themselves" does not exist. In the real world it reduces to nonsense. The fact that we can define a set does not mean it actually exists. So this involves no actual contradiction and therefore can't be used to disprove the law of contradiction. Here: I'll give you a tougher puzzle. Think about the set of all things that have never been contemplated as a set. The moment you attempt to think of it, it disappears. So technically, you can't even think about such a set, and it can't possibly exist. (But, hey! weren't you in fact thinking about it when it went 'poof'?) Like Russell's "set of all sets" it is an absurdity that can be defined but cannot exist. Such absurdities pose no threat whatsoever to the law of contradiction. Your other paradox, "This statement is false," is a classic conundrum on the order of God and the rock that's too big for him to lift. It doesn't disprove the law of contradiction for three reasons: Statements with self-referent truth assertions are the only kind of statements that lead to such contradictions, because they involve a kind of recursive "logic" that is inherently absurd. Again, the statement commits the fallacy of equivocation. The paradox in "This statement is false" hinges on an ambiguity that can be eliminated with a precise definition of false. The law of the excluded middle means every proposition must be either true or false. That disqualifies "This statement is false" as a valid proposition, because it can't be either true or false. It is an absurd statement, not a real proposition. So the bottom line is that both of your challenges to the law of contradiction boil down to sheer nonsense. And if I may make a friendly observation, Daryl, you are all too prone to assert that nonsense is truth and vice versa. This is a mystery to me, by the way. In [another online forum] you doggedly defended Søren Kierkegaard against the charge that he is irrational. And yet you attack rationality. If you think logic is invalid anyway, why do you object to the assertion that Kierkegaard (or you) are irrational? Far from debunking the law of contradiction, paradoxes such as "This statement is false" have actually elevated the status of the law of contradiction in the study of formal logic and math. Computer scientists, for example, have been forced to grapple with and resolve the absurdity of self-referent truth assertions. Metalanguage is employed precisely to avoid that problem in logic, linguistics and computer programming. Here are a couple of Web sites that have interesting discussions of mathematical paradoxes and computer programming: http://logic.stanford.edu/kif/metaknowledge.html http://logic.stanford.edu/kif/Hypertext/node25.html (The latter has to do specifically with set theory and its relationship to Russell's paradox.) It is perhaps necessary to point out that all true paradoxes can be resolved. If they couldn't be, they would stand as contradictions, and truth itself would be an absurd concept. (That, sadly, is precisely the conclusion some have drawn, both in the secular world and in the visible church.) But while paradoxes are interesting logic puzzles, they do not negate the law of contradiction. Christians would do well to remember this. Too many Christians have fallen into the habit of using the word "paradox" in the neo-orthodox and Kierkegaardian sense of "a flat-out contradiction that we're going to affirm anyway, thus embracing nonsense as truth." But Scripture teaches that God is truth, and He cannot deny himself. So the law of contradiction is established on biblical authority. Yet mathematicians and computer scientists put most theologians to shame when it comes to resolving paradoxes like these. Russell first proposed his set theory paradox in 1901. Mathematicians worldwide—including Russell himself—immediately scrambled to find a way to resolve it. Why? Because they knew if it were irresolvable—if it really debunked the law of contradiction—then it would not only nullify all mathematical knowledge; it would also render truth itself moot. Indeed, overthrow the law of contradiction and everything can be both true and false, thus demolishing the whole concept of truth. In effect, this is precisely what the blending of postmodernism and theology is all about. But it's foolish in the extreme to use the language of "paradox" to make truth itself seem absurd. I'm concerned about Christians who blithely say they are happy to "live with paradox," when what they really mean is that it's OK to have an irrational worldview. Gordon Clark used to define paradox as "a charlie horse between the ears." I don't agree with Clark about everything, but he was right to defend the coherence of truth. When faced with two difficult truths we find hard to reconcile, we ought to view it as an opportunity to work out the kinks in our thinking and try to gain a better understanding of truth. If two biblical truths seem to contradict, we need to take another look and be humble enough to acknowledge that perhaps we have misunderstood one or the other (or both) of the seemingly contradictory ideas. We need to understand—like those mathematicians did at the start of the 20th century—that if "truth" really contradicted itself, the very concept of truth would be moot. Christians of all people ought to stand firmly against every charge that truth is inherently self-contradictory.

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