Raymond Smullyan's closest counterpart in English letters is probably Lewis Carroll -- not only in the "Alice" books, which are literarily in a different class from Smullyan's work, but also in the odd little logical and mathematical problems, puzzles and paradoxes that he used to jot down. Carroll was a mathematician living on the edge (the upper edge, we should specify) of what most ordinary people would call "sanity." Smullyan is a logician living on the same edge, and there is a hint of "Alice in Wonderland" flavor about the two collections of very strange chess problems he has published, wrapped in harmless, inconsequential little narratives that help to spin the material out to a decent length for a book.

Smullyan's literary-imaginative world (like Carroll's, but even more so) is one in which everyone behaves logically but not necessarily rationally. That is to say, all of his characters obey established laws to the letter, but only fitfully do they seek reasonably well-defined and worthwhile goals by means that seem fairly appropriate. In a sense, this world reflects the world in which we live; physically, whether we like it or not, we are bound by a whole complex of laws (gravity, thermodynamics, etc.) that we have no choice about obeying, though as our understanding grows we can use them more and more for our own purposes, which seem to contradict their basic premises (i.e., in spite of the law of gravity, we have learned to fly). Psychologically and emotionally, the situation is much closer to pure anarchy -- as in the current case of the air traffic controllers. And so, although the world Smullyan presents in his two collections of chess problems is self-contained, in a sense, it does reflect the real world -- as contemplated by a very lucid, methodical and ruthlessly logical intelligence.

Smullyan has wrapped his second collection in little stories about Haroun Al Rashid, his associates and foes, as he wrapped the first in stories about Sherlock Holmes, but this is a simple sugar-coating on material that might otherwise seem too dryly logical for a general readership. This tactic does allow him to give his work an exotic flavor, with rooks that really are castles and people who have the power to make themselves invisible, others who travel in disguise, military exploits, murders, abductions and the sending of secret messages. All of these narrative trappings disguise a set of chess problems that are not chess problems in the traditional style but rather essays in symbolic logic using the pieces and rules of chess as their language.

As a sort of bonus, the book offers 12 traditional chess problems, in which white must achieve a checkmate (or sometimes a self-mate) in a prescribed number of moves. But its real subject is the much rarer kind of problem called retrograde analysis, in which the reader must examine a diagrammed position not to find a way to victory but to answer a question about what has happened in the game. Most of Smullyan's problems are very complex, involving several closely interlocked steps of deductive logic, but one of the simpler ones will suffice to show the kind of thing he does.

Stripped of its narrative ornaments (it is presented as the key to a code message sent across the lines in a war), the problem is simply to determine which side moved last, and the answer is in two parts. First, the reader must deduce that the board is turned around and the white pawns are moving downward; this is clear because, if the black pawns were in their original position the white king could not have reached the top rank without moving into check. Once it is established that the black pawns are moving up the board, it is clear that white is in check -- and therefore black must have moved last. Since it is white's move, white has a forced mate in two: first the bishop takes the pawn with a check, and then white queens a pawn on square g1, giving mate -- but this is irrelevant to Smullyan's interest and he doesn't even bother to mention it.

These problems are also irrelevant to the game of chess as it is normally played. A position like the one in the diagram would never occur in a serious game between rational players -- but Smullyan is not concerned about that, either; he is simply using chess as a convenient symbolic notation for his little problems in logic. This may be disconcerting to hard-core players who tend to look at a diagram and want to get on with the game, but Smullyan does in passing show a love for the game, too -- particularly its odd rules and practices such as castling, the en passant capture and underpromotion (the promotion of a pawn to knight, bishop or rook rather than queen). If "The Chess Mysteries of the Arabian Knights" will not, ultimately, tell the reader much about chess, it will put him in touch with one of the more interesting minds of our time.