(a) Long ago, a wicked king was searching for a new wizard with whom to plot some devious schemes. He summoned to him three wizards who seemed especially promising, and let them into a small room, which was barren except for a lighted candle on a table in the middle of the room. "Listen to me well," he said. "In a few minutes all of you will be blindfolded, and I will paste upon each of your foreheads a uniformly colored spot of black or white paper. At least one spot will be white. The first of you who guesses the color of his own spot will become my new wizard, and ride in his own chariot, with all expenses paid. The other two of you will be sent to a terrible fate that I shall not describe. None of you will be allowed to remove any of the spots, and you will each be allowed only one guess." The king then ordered his guards to blindfold the wizards, proceeded to paste white spots on all the wizards' foreheads, and finally had their blindfolds removed. After a few seconds, one of the wizards correctly identified the color of the spot on his forehead. How did he know it?

(b) Present a predicate calculus axiomatization for the wizard's reasoning.

(c) What sort of thoughts might the other two wizards have been thinking?

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"We could play at questions." -Rosencrantz, in Rosencrantz and Guildensiern Are Dead. (Stoppard, 1967) "Augustine describes the learning of human language as if the child came into a strange country and did not understand the language of the country; that is, as if it already had a language, only not this one." - Wittgenstein, Philosophical Investigations. "I find it difficult to believe that whenever I see a tree I am really seeing a string of symbols." -McCarthy, in a discussion on grammatical inference and pattern recognition. "As a concluding remark: could this art be applied (we put the question in strictest confidence) could it, we ask, be applied to the speeches in Parliament?" -Lewis Carroll, Photography Extraordinary. "There is of course no restriction in the memory format against having concepts without English names, and in fact [its] present memories necessarily include such concepts." -Quillian, describing the structure of the TLC computer program. (Quillian, 1969) "Danger of tumbling upwards be in deep-sea." -Protosynthex III, a computer program. (Schwarcz, Burger, and Simmons, 1970) "The challenge of programming a computer to use language is really the challenge of producing intelligence." -Winograd, 1971. "In any case, these are but steps toward more graphical program-description systems, for we will not forever stay confined to mere strings of symbols." - Minsky, 1970. "What does meaning mean?" -Anonymous. "Imagine a people in whose language there is no such form of sentence as 'the book is in the drawer' or 'water is in the glass', but whenever we should use these forms they say, 'The book can be taken out of the drawer', 'The water can be taken out of the glass'. -Wittgenstein, The Brown Book. "I have traveled more than anyone else, and I have noticed that even the angels speak English with an accent." -Mark Twain.