Devlin's Angle

November 2003

Contexts of Paradox

In one popular version, Russell's Paradox asks us to imagine a village where the barber shaves all men who do not shave themselves. The question then is, who shaves the barber? A straightforward attempt to answer this question leads you in a circle from which there seems to be no escape. If the barber shaves himself, then he does not shave himself. But then he does shave himself. But then he does not. And so on ad infinitum. (The possibility of a female barber did not arise at the start of the 20th Century, when Bertrand Russell formulated this puzzle.)

Of considerably greater significance for the development of mathematics was the original set-theoretic version of the paradox: Let R be the set of all sets that are not members of themselves. Is R a member of itself or not? If it is, then it isn't, and if it isn't then it is.

Russell's discovery of this paradox about set formation effectively destroyed the lifetime's work of Gottlob Frege in trying to establish a foundational framework for mathematics based on formal logic. The paradox was resolved -- or some would say sidestepped -- only by the formulation of axioms for set theory a few years later. Russell's Paradox signaled the end of the period of youthful innocence in Cantor's Set Theory, during which it was believed that to every property could be associated a well-defined set, namely the set of all objects having that property.

Many people mistakenly believe that paradoxes such as Russell's are a consequence of self-reference, and that their message is that we should banish self-reference as inescapably paradoxical. True, such paradoxes involve self-reference, but that is merely the weapon used to exploit a loophole that lies elsewhere. Also true is that the standard ways of avoiding such paradoxes do prevent self-reference from occurring, but that is merely a consequence of the real heart of the avoidance strategy.

What the classic paradoxes all exploit is a neglect of context. Their message is that if we ignore context, we can be in trouble -- a moral as significant in mathematics, then, as it is in everyday social life.

The standard axiomatization of set theory replaces the nave (and paradox generating) Axiom of Comprehension, which says that given any property P, there is a corresponding set

{x | P(x)}
by the Axiom of Restricted Comprehension, which says that for each property P and any given set A, there is a corresponding set
{x in A | P(x) }
The set A may be regarded as providing a context for the formation of the set associated with the property P.

Berry's Paradox is another puzzle that seems to hinge on self-reference. It was first published by Russell, who attributed it to the Oxford University librarian Mr. G. Berry. It asks us to consider the following definition of a natural number: "The least natural number not definable in fewer than thirteen English words." Since the sentence itself has exactly twelve English words, this definition seems to be paradoxical. The heart of this seeming paradox is that the concept "definable in English" is not mathematically precise. Making it sufficiently precise may be regarded as imposing a suitable context.

Finally, let's look at the Liar Paradox. This arises when a person utters the sentence "I am lying." (If you want to spell it out more fully, the person can say "The statement I am now making is false." If the person is uttering a truthhood, then the utterance must be false; and if the person is uttering a falsehood, then the utterance must be true.

People sometimes try to wriggle out of this trap by claiming that it simply implies that not all utterances are true or false -- some may be undetermined. But this doesn't make the paradox go away. It returns with a vengeance the moment I replace reference to lying by the lawyer's more cautious version "The statement I am now making is not true." If my utterance is true, then it isn't true. If it is false, then it is true. And if it is undetermined, then it certainly isn't true, which means it is true. Gotcha!

The resolution of the Liar Paradox is to recognize that, just as guns don't kill people, people do (sometimes using guns), so too declarative sentences (of which the various versions of the Liar sentence are examples) do not make claims about the world, people do (sometimes by uttering a declarative sentence).

Explicitly: (i) declarative sentences, being pieces of language, are in themselves neither true nor false; (ii) an utterance of a declarative sentence makes a claim that will be either true or not true; and (iii) uttering a sentence is an action, and performing that action creates a context in which the truth or otherwise of the utterance is determined.

Suppose person a stands up and says "This assertion is false." Do we have a paradox? Well, let's see, paying particular attention to the contexts in which various claims are being made.

Let L denote the sentence uttered.

What does a refer to by the phrase "This assertion"? Not L, that's for sure. Sentences are merely strings of letters; in of themselves they can't refer to anything. Rather, when a utters the phrase "This assertion", he is referring to the assertion (or claim) being made. Call that assertion p (for "proposition").

It follows that, in uttering the sentence L, a is making the claim "p is false".

But p itself denotes the claim made by a. Hence:

(1) p = [p is false].

Let c denote the context in which a makes the utterance.

Thus, a's utterance of the phrase "This assertion" refers to the claim that, in context c, proposition p is true. Hence:

(2) p = [in context c, p is true].

Suppose a's assertion is true. In other words, p is true. Using formula (2), this means

(3) in context c, p is true.

By formula (1), we can replace p in formula (3) by [p is false] to obtain:

(4) in context c, [p is false] is true.

Conclusions (3) and (4) are in contradiction. Hence a's claim must be false. In other words, p is false.

But who is making this last statement and in what context?

Is it possible that a himself could make this claim, in context c? (Or, if I am a, when I made that claim just now, could my context have been the same one in which I made my original Liar utterance?) Can we play fast and loose in this way with what we regard as a context?

Suppose a could make that last claim in context c. Then

(5) in context c, [p is false].

By formula (1), this can be rewritten as

(6) in context c, p.

Now we have a contradiction as before. But we don't have a paradox, we have a conclusion, namely, that the statement "p is false" cannot be made in the context c.

In other words, the Liar assertion is false in any context c in which it is uttered, but c cannot itself be the context for making the statement that the Liar assertion is false (in context c).

What seemed at first to be a paradox is in fact a logically sound proof about contexts.

Issues of context are rarely significant in present day mathematics -- indeed, a crucial aspect of the methodology of mathematics is to study various phenomena in a de-contextualized fashion, removed from the contextual complexities that surround them in the real world. But context is fundamental in many areas in which mathematicians find themselves getting involved these days, such as linguistics, communications, HCI, artificial intelligence, information systems design, and business process modeling. There are a number of annual conferences that address issues of context from different perspectives. AI pioneer John McCarthy is just one of several researchers who are trying to develop a logic for contextual inference. And in my 1999 book InfoSense, I discussed the crucial role played by context in studies of information flow and decision making.

It may indeed be the case that, as they say in the X-files, the truth is out there, but you have to remember that "there" is a context, and context matters.


Finally, in response to my column last month about negatively phrased questions, reader Keith Prosser writes to tell me that in Swahili (and related Bantu languages) a negatively phrased question is answered logically. For example, if asked "Don't you like it?" a Bantu will reply "Yes" if he or she does not like it. Prosser goes on to observe that for many East Africans this habit is carried over when speaking in English, sometimes resulting in serious confusion. For example if you were to ask an East African colleague (in English) "Haven't you finished yet?" an answer of "No" probably means s/he has finished. The only solution is to avoid negatively phrased questions in English to native Bantu/Swahili speakers, as one can never be sure if the answer needs to be interpreted Swahili or English style.

Another reader, John Calligy, emailed me the following: "You dealt only with English, but other languages have logical idiosyncrasies. For example, the Korean words "ye" and "anyo" translate into English as "yes" and "no" respectively - usually. But in certain situations, the meanings are reversed. For example, in answer to the question: "You're not a doctor, are you?" a reply of "ye" means "That's right - I'm not a doctor". Koreans are quite consistent about this, which doesn't make it any less confusing.

Oddly enough, I received Calligy's email just after I arrived in Seoul on a business trip to Korea. That evening, over dinner, my local host asked me if I liked jazz, which apparently is very popular in Seoul. I said no, but in the subsequent discussion I had to elaborate that I did not dislike jazz, rather I merely did not actively like it. You understand the distinction, don't you? Or should I have said "do you?"?


Devlin's Angle is updated at the beginning of each month.
Mathematician Keith Devlin ( devlin@csli.stanford.edu) is the Executive Director of the Center for the Study of Language and Information at Stanford University and "The Math Guy" on NPR's Weekend Edition. His most recent book is The Millennium Problems: The Seven Greatest Unsolved Mathematical Puzzles of Our Time, just published in paperback by Basic Books, first published in hardback last fall.