When pure logic is plain absurd

A man sells off his second-hand television set to three impoverished old ladies for pounds 30, towards which each old lady contributes exactly one third - or pounds 10 apiece. After pocketing the money, he belatedly decides that the television's value is in fact no more than pounds 25. About to reimburse the extra five, however, he realises that he himself will need pounds 2 for a parking meter and so returns only three. Thus, each of the three ladies has paid him not pounds 10 but pounds 9 for the set: that is, pounds 27. He himself retains pounds 2 for the parking meter. Which makes pounds 29 in all. What has become of the missing pound?

That, you are most likely thinking, is a logical paradox. But it isn't, since the fallacious conclusion derives not from any inherent illogicality in the story's premise but from its deliberately garbled presentation. When the thickets of distracting marginalia have been cleared to lay bare the essential facts, the apparent paradox instantly vanishes. By the end, after all, the three ladies are left with pounds 1 apiece, there are pounds 2 in the meter and the vendor has pounds 25 in his wallet. Which adds up to the pounds 30 we started with.

Authentic paradoxes are a different matter. Even when they've been exposed for what they are, something still continues to feel not quite right about them. (For a few readers, no doubt, something continues to feel not quite right about the little fallacy recounted above.) We may logically grasp why Achilles eventually must overtake the tortoise, but the way Zeno originally told the tale is not without a peculiar logic of its own, one that is somehow not conclusively invalidated by the abstract mathematics of converging infinite series, even when we're capable of understanding them.

Above all, real paradoxes don't go away. The immemorial Cretan paradox - when a Cretan says: "I am a liar", is he lying or telling the truth? - could be detected in the antinomies which undermined set theory in the 19th century, and its irksome self-reference was at the heart of Kurt Godel's two celebrated theorems of undecidability, dating from the early 1930s, which devastated the mathematical community by proving that maths raised questions that it could never hope to answer.

What this is all leading up to is that I would now like to propose an entirely original paradox. At least, I hope it's a paradox and I hope it's original. A total amateur where mathematical logic is concerned, I know I run the risk of being informed by some disdainfully incredulous reader that it's just a paraphrase of some dog-eared puzzle settled by the Pythagoreans two millennia ago. Yet, as I tell myself, nothing ventured, nothing gained. In any event, it goes, as cocktail pianists say, something like this:

A businessman decides to have his office renovated and invites two interior decorators to present him with comprehensive estimates for the job. He tells them that he will accept the lower of the two estimates, that they be confined to round figures, and that neither is to exceed pounds 10,000.

When one of the designers asks what will happen if both come up with an identical figure, he replies that in that eventuality he'll simply take his custom elsewhere.

The first decorator endeavours to figure out the best options open to him in the light of his rival's correspondingly best options. He immediately rules out submitting an estimate for the full pounds 10,000 on the grounds that the second decorator will propose either less, in which case he will get the job, or the same, in which case neither will get it. So he considers pounds 9,000. But there too, as he soon realises, he'll get the job only if the second decorator proposes pounds 10,000, which he, the second decorator, will rule out for the very same reason as the first.

Eight thousand pounds, then? But wait. If (thinks the first decorator) I submit an estimate for pounds 8,000, I'll get the job only if the second decorator proposes pounds 9,000 (pounds 10,000 remember, has been rejected). But the second decorator won't propose nine because in that case he'll get the job only if I propose pounds 10,000, which I obviously won't do for the reason already stated. pounds 7,000? No again, since I can undercut my rival only if he proposes pounds 8,000 (now that both pounds 9,000 and pounds 10,000 have been rejected). But he won't propose pounds 8,000 because he in his turn knows he'll get the job only if I propose pounds 9,000 and he must equally know that I won't because in that case my getting the job would depend on his proposing pounds 10,000, which patently he won't. So what about pounds 6,000? If I propose pounds 6,000, I undercut him only if he proposes pounds 7,000, but he won't because that would mean he undercuts me only if I propose pounds 8,000, which of course I won't do because winning the contract would then depend on his proposing pounds 9,000, and he naturally won't propose nine because he must realise that there is absolutely no chance of my proposing pounds 10,000.

You get the picture. No matter how low his estimate, the first decorator is forced to the conclusion that no decision procedure exists to allow him to maximise the likelihood of his successfully undercutting his competitor. The more scrupulously rational he is in his attempts to second-guess the second decorator's reasoning, the less chance he has of formulating the most advantageous course of action. Logic has, in short, an ironically paralysing effect on his train of thought.

So is this a genuinely new paradox? If so, it would appear to me to have real applications in such areas of vexed and vigorous human competitiveness as politics, business, diplomacy, etc. If not, I would be interested to know why not.

Three Famous Paradoxes

n ZENO'S SECOND-best-known paradox (after the one about Achilles and the tortoise) is that of the Arrow. Its premise can be stated very simply: consider an arrow shot into the air. At any given instant - which might nowadays be captured by a camera - it occupies one, and only one, position in space. But any object occupying one, and only one, position in space cannot be in motion: at that particular instant, it must be stationary. Since, in the course of its flight, the arrow can be shown to traverse an unbroken sequence of such instants, it is therefore never in motion.

n The philosopher Eublides once "proved" that there could never be such a thing as a heap of sand. For, as he said, a single grain of sand certainly could not be said to constitute a heap. Nor would adding a second grain do the trick, since no one has ever thought of two grains of sand as a heap. And nothing changes no matter how many grains are added to the original one. There appears to be no one specific stage of the process at which, by the addition of a single grain, a non-heap is suddenly transformed into a heap.

n Perhaps the most notorious paradox of self-referentiality is Bertrand Russell's Barber Paradox. In a village there is only one barber, who shaves all the inhabitants who don't shave themselves. So who shaves the barber? If he shaves himself, then, in conformity with the above definition, he doesn't shave himself, since he shaves only those who don't shave themselves. But if he doesn't shave himself, then, by the same definition, he does after all shave himself, since his brief is to shave all those inhabitants who, like him, don't shave themselves. It was by discrediting the fundamental set-theoretical concept of the set of all sets that Russell eventually escaped such vicious circularity.