Pigmental Paradox

From Raymond Smullyan’s Logical Labyrinths (2009):

We now visit another knight/knave island on which, like on the first one, all knights tell the truth and all knaves lie. But now there is another complication! For some reason, the natives refuse to speak to strangers, but they are willing to answer yes/no questions using a secret sign language that works like this:

Each native carries two cards on his person; one is red and the other is black. One of them means yes and the other means no, but you are not told which color means what. If you ask a yes/no question, the native will flash one of the two cards, but unfortunately, you will not know whether the card means yes or no!

Problem 3.1. Abercrombie, who knew the rules of this island, decided to pay it a visit. He met a native and asked him: “Does a red card signify yes?” The native then showed him a red card.

From this, is it possible to deduce what a red card signifies? Is it possible to deduce whether the native was a knight or a knave?

Problem 3.2. Suppose one wishes to find out whether it is a red card or a black card that signifies yes. What simple yes/no question should one ask?

Shick Shtick

Slightly adapted from Joseph Degrazia’s Math is Fun (1954):

Six Writers in a Railway Car

On their way to Chicago for a conference of authors and journalists, six writers meet in a railway club car. Three of them sit on one side facing the other three. Each of the six has his specialty. One writes short stories, one is a historian, another one writes humorous books, still another writes novels, the fifth is a playwright and the last a poet. Their names are Abbott, Blake, Clark, Duggan, Eccles and Farmer.* Each of them has brought one of his books and given it to one of his colleagues, so that each of the six is deep in a book which one of the other five has written.

Abbott reads a collection of short stories. Clark reads the book written by the colleague sitting just opposite him. Blake sits between the author of the short stories and the humorist. The short-story writer sits opposite the historian. Duggan reads a play. Blake is the brother-in-law of the novelist. Eccles sits next to the playwright. Abbott sits in a corner and is not interested in history. Duggan sits opposite the novelist. Eccles reads a humorous book. Farmer never reads poems.

These facts are sufficient to find each of the six authors’ specialties.


*In the original, the surnames were Blank, Bird, Grelly, George, Pinder and Winch.

The Brain in Train

I feel odd when I consider this possibility: that all my thoughts are strictly determined, no more under my control than a straw in a gale or a stone in an avalanche. It seems paradoxical to have strictly determined thoughts about strictly determined thoughts. But is it? And is strict determinism fatal for finding the truth? I don’t think so. In fact, I think that strict determinism is essential for truth. But irrelevant associations get in the way of our understanding this. If our thoughts are determined, they seem like automatic trains running on rigid tracks. We might want to go to the station marked “Truth”, but if the switches are set wrong, the train will never get there. Or it will thunder through and never stop.


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The Brain in Pain

You can stop reading now, if you want. Or can you? Are your decisions really your own, or are you and all other human beings merely spectators in the mind-arena, observing but neither influencing nor initiating what goes on there? Are all your apparent choices in your brain, but out of your hands, made by mechanisms beyond, or below, your conscious control?

In short, do you have free will? This is a big topic – one of the biggest. For me, the three most interesting things in the world are the Problem of Consciousness, the Problem of Existence and the Question of Free Will. I call consciousness and existence problems because I think they’re real. They’re actually there to be investigated and explained. I call free will a question because I don’t think it’s real. I don’t believe that human beings can choose freely or that any possible being, natural or supernatural, can do so. And I don’t believe we truly want free will: it’s an excuse for other things and something we gladly reject in certain circumstances.


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Neuclid on the Block

How many blows does it take to demolish a wall with a hammer? It depends on the wall and the hammer, of course. If the wall is reality and the hammer is mathematics, you can do it in three blows, like this:

α’. Σημεῖόν ἐστιν, οὗ μέρος οὐθέν.
β’. Γραμμὴ δὲ μῆκος ἀπλατές.
γ’. Γραμμῆς δὲ πέρατα σημεῖα.

1. A point is that of which there is no part.
2. A line is a length without breadth.
3. The extremities of a line are points.

That is the astonishing, world-shattering opening in one of the strangest – and sanest – books ever written. It’s twenty-three centuries old, was written by an Alexandrian mathematician called Euclid (fl. 300 B.C.), and has been pored over by everyone from Abraham Lincoln to Bertrand Russell by way of Edna St. Vincent Millay. Its title is highly appropriate: Στοιχεῖα, or Elements. Physical reality is composed of chemical elements; mathematical reality is composed of logical elements. The second reality is much bigger – infinitely bigger, in fact. In his Elements, Euclid slipped the bonds of time, space and matter by demolishing the walls of reality with a mathematical hammer and escaping into a world of pure abstraction.

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Ear Will An Thee

(This is a guest-review by Norman Foreman, B.A.)

Yr Wylan Ddu, Simon Whitechapel (Papyrocentric Press, ?)

If, like me, you froth at the mouth and roll on the floor biting the carpet when you hear the phrase “Pre-order now”, then relief is at hand. You might have thought that “pre-ordering now” was as logical as “ordering pre-now”. You were wrong. Here is a book that really can be pre-ordered now, because it doesn’t exist yet. If it ever does exist, it will cease to be pre-orderable now. In the meantime, you’re pre-ordering it whether you know it or not. In fact, the less you know, the more you’re pre-ordering it. All life-forms in the Universe, actual and otherwise, are pre-ordering it at this very moment, from the humblest virus to the mightiest hive-mind.

Front cover of yr wylan ddu by slow exploding gulls

Yr Wylan Ddu (2003) by Slow Exploding Gulls

There’s no escape, in other words. And no more review, you might think, given that the book doesn’t exist yet. True, but I can review the title. It’s Welsh, it means “The Black Gull”, and it’s pronounced something like “Ear Will An Thee”. It was also originally the title of an album in 2003 by the Exeter electronistas Slow Exploding Gulls. Whether S.E.G. will object to the appropriation remains to be seen. If they do, it can be pointed out that Dirgelwch Yr Wylan Ddu, or Secret of the Black Gull, was the title of a children’s book by Idwal Jones (1890-1964) published in 1978.

Front cover of Dirgelwch Yr Wylan Ddu by Idwal Jones

Idwal Jones’ Secret of the Black Gull (1978)

There is nothing corresponding to “of” in the original title of that book, but then Welsh grammar doesn’t work like that. Yr Wylan Ddu contains some good examples of how it does work. It’s an active, almost clockwork or organic, phrase compared to its static English equivalent. In isolation, the Welsh words for “the”, “black” and “gull” would be y, du, and gwylan, pronounced something like “ee”, “dee” and “goo-ill-an” in southern Welsh. But put them together and they mutate in more ways than one: Yr Wylan Ddu (adjectives generally follow the noun in Welsh). The similarity between gwylan and “gull” isn’t a coincidence: the English word is borrowed from Celtic.

However, it is unlikely that Yr Wylan Ddu will actually be written in Welsh or any other Celtic language. First, Whitechapel doubtless feels that this would reduce his already small audience. Second, he doesn’t speak Welsh. Or write it. So the book will probably follow past trends and be written in English. It’s also safe to predict that it will refer to at least one black gull. So: pre-order now. And please carry on doing so until further notice.