0809.5: Reflexicon

Logic Puzzle #5: The Self-Referential Crossword Puzzle (March 2009)

Self-referential sentences have long puzzled philosophers. Epimenides’ 6th century B.C. poemCretica, contained the line:The Cretans are always liars, evil beasts, slow bellies.

Epimenides himself was from Crete. If we interpret his claim as meaning that all Cretans lie all the time, then his statement, if true, entails that it is false; and if it is false, then it must be true.

The so-called Epimenides paradox is most simply presented in a form called the liar: “This statement is false.” The liar was well-known in ancient and medieval times. The epitaph of the third-century poet Philetus reports that he died worrying about it:

Philetus of Cos am I/ ’Twas The Liar who made me die/ And the bad nights caused thereby.

Bertrand Russell’s paradox for set theory, discovered in 1901, relies on considering the set of all sets that do not contain themselves. Russell’s paradox, which devastated Gottlob’s Frege’s attempt to reduce mathematics to logic, spurred a century’s worth of research on the foundations of mathematics. Today, the liar and other self-referential sentences are hot topics in philosophy and logic journals because of their ramifications for mathematics, language, and metaphysics.

Every entry in the self-referential crossword puzzles below contains a number spelled-out, followed by a blank space, a letter, and an ‘S’ where a plural is appropriate. All entries accurately describe the completed puzzle. Here is a simple puzzle, and its solution.

Note that the solution to Puzzle #1 contains the (lower horizontal) entry ‘Four Os’; there are in fact four ‘O’s in the puzzle. There is also one ‘F’, three ‘E’s and and two ‘H’s, just as the other entries say. Here is another simple puzzle, for you to complete.

Call one of these self-referential puzzles complete if, for every letter token in the puzzle, there is an entry which states how many instances of that letter type appear in the completed puzzle. Thus, if there are any ‘Z’s in the puzzle, there will be one (and only one) entry which states exactly how many ‘Z’s there are. The following puzzle is complete. I believe that it is the only complete one of its type possible. (While there is only one entry for each letter, some numbers may appear more than once.)