Definitions

# Raymond Smullyan

Raymond Merrill Smullyan (born 1919) is an American mathematician, logician, philosopher, and magician.

Born in Far Rockaway, New York, his first career (like Persi Diaconis a generation later) was stage magic. He then obtained a BSc from Chicago in 1955 and a Ph.D. from Princeton in 1959. He is one of many outstanding logicians to have studied under Alonzo Church.

## Life

While a Ph.D. student, Smullyan published a paper in the 1957 Journal of Symbolic Logic showing that Gödelian incompleteness held for formal systems considerably more elementary than that of Gödel's 1931 landmark paper. The contemporary understanding of Gödel's theorem dates from this paper. Smullyan later made a compelling case that much of the fascination with Gödel's theorem should be directed at Tarski's theorem, which is much easier to prove and equally disturbing philosophically. The culmination of Smullyan's lifelong reflection on the classic limitative theorems of mathematical logic is his quite readable:

• Smullyan, R M (2001) "Gödel's Incompleteness Theorems" in Goble, Lou, ed., The Blackwell Guide to Philosophical Logic. Blackwell (ISBN 0-631-20693-0).

Smullyan is the author of many books on recreational mathematics, recreational logic, etc. Most notably, one is titled Booksources/0139550623.

Many of his logic problems are extensions of classic puzzles. Knights and Knaves involves knights (who always tell the truth) and knaves (who always lie). This is based on a story of two doors and two guards, one who lies and one who doesn't. One door leads to heaven and one to hell, and the puzzle is to find out which door leads to heaven by asking one of the guards a question. One way to do this is to ask "Which door would the other guard say leads to hell?". This idea was famously used in the 1986 film Labyrinth.

In more complex puzzles, he introduces characters who may lie or tell the truth (referred to as "normals"), and furthermore instead of answering "yes" or "no", use words which mean "yes" or "no", but the reader does not know which word means which. The puzzle known as "the hardest logic puzzle ever" is based on these characters and themes. In his Transylvania puzzles, half of the inhabitants are insane, and believe only false things, whereas the other half are sane and believe only true things. In addition, humans always tell the truth, and vampires always lie. For example, an insane vampire will believe a false thing (2 + 2 is not 4) but will then lie about it, and say that it is. A sane vampire knows 2 + 2 is 4, but will lie and say it isn't. And mutatis mutandis for humans. Thus everything said by a sane human or an insane vampire is true, while everything said by an insane human or a sane vampire is false.

His book Forever Undecided popularizes Gödel's incompleteness theorems by phrasing them in terms of reasoners and their beliefs, rather than formal systems and what can be proved in them. For example, if a native of a knight/knave island says to a sufficiently self-aware reasoner "You will never believe that I am a knight", the reasoner cannot believe either that the native is a knight or that he is a knave without becoming inconsistent (i.e. holding two contradictory beliefs). The equivalent theorem is that for any formal system S, there exists a mathematical statement which can be interpreted as "This statement is not provable in formal system S". If the system S is consistent, neither the statement nor its opposite will be provable in it.

Inspector Craig is a frequent character in Smullyan's "puzzle-novellas." He is generally called into a scene of a crime that has a solution that is mathematical in nature. Then, through a series of increasingly harder challenges, he (and the reader) begin to understand the principles in question. Finally the novella culminates in Inspector Craig (and the reader) solving the crime, utilizing the mathematical and logical principles learned. Inspector Craig generally does not learn the formal theory in question, and Smullyan usually reserves a few chapters after the Inspector Craig adventure to illuminate the analogy for the reader.

His book To Mock a Mockingbird (1985) is a recreational introduction to the subject of combinatory logic.

Apart from writing about and teaching logic, Smullyan has recently released a recording of his favourite classical piano pieces by composers such as Bach, Scarlatti, and Schubert. Some recordings are available on the Piano Society website, along with the video "Rambles, Reflections, Music and Readings". He has also written an autobiography titled Some Interesting Memories: A Paradoxical Life (ISBN 1-888710-10-1).

In 2001, documentary filmmaker Tao Ruspoli made a film about Raymond Smullyan called This Film Needs No Title.

## Quotations

• I now introduce Professor Smullyan, who will prove to you that either he doesn't exist or you don't exist, but you won't know which. --Melvin Fitting

## External references

• Is God a Taoist? by Raymond Smullyan, 1977.
• Planet Without Laughter by Raymond Smullyan, 1980.
• An Epistemological Nightmare by Raymond Smullyan, 1982.