178 Logic as a Tool |= ∀x(x = f (x) → (P (f (f (x))) → P (x))) ∀x∀y (f (x) = y → g (y ) = x) |= ∀z (g (f (z )) = z ) ∀x∀y (f (x) = y → g (y ) = x) |= ∀z (f (g (z )) = z ) ∀x∀y (f (x) = y → g (y ) = x), ∀x∀y (g (x) = g (y ) → x = y ) |= ∀z (f (g (z )) = z ) For more exercises on derivations with equality, on sets, functions, and relations, see Section 5.2.7 (b) (c) (d) (e) Raymond Merrill Smullyan (born 25.5.1919) is an American mathematician, logician, magician, Taoist philosopher, and piano player, best known as an avid popularizer of logic through his series of scientific (mathematical logic) and recreational books with enchanting logical puzzles In his childhood Smullyan showed passion and strong talent for music (as a piano player), chess, and mathematics, but was not quite happy at school so he left to study on his own He was keen on problem solving, both in maths and chess where he composed many problems of his own He also started learning magic tricks and became a very good magician, earning money by performing magic acts in nightclubs in Greenwich Village During the early 1950s he studied at the University of Chicago (where one of his teachers was the prominent philosopher and logician Rudolf Carnap), while himself teaching at Dartmouth College He achieved his PhD at Princeton University under Alonzo Church and published his first paper on logic in 1957, where he showed that Gödel’s incompleteness phenomena based on self-reference also apply to much simpler formal systems than Peano arithmetic and gave new, very intuitive explanations of Gödel’s theorems Smullyan has been for many years a Professor of Mathematics and Philosophy at Yeshiva University, New York, and at Lehman College and the Graduate Center of the City University of New York, where he is still a Professor Emeritus He is also an Oscar Ewing Professor of Philosophy at Indiana University Smullyan has published several well-known books on mathematical logics, including Theory of Formal Systems (1959), First-Order Logic (1968), Gödel’s Incompleteness Theorems (1992), Recursion Theory for Metamathematics (1993), Diagonalization and Self-reference (1994), Set Theory and the Continuum Problem (1996, together with Melvin Fitting), and A Beginner’s Guide to Mathematical Logic (2014) In his book First-Order Logic Smullyan developed the method of analytic tableaux as a variant of the Semantic Tableaux of Beth and Hintikka, and related it to Gentzen’s proofs systems Smullyan also wrote a few books on Taoist philosophy, but he is most popular for his fascinating books with logical, mathematical, and chess puzzles, including What Is the Name of This Book? (1978), The Chess Mysteries of Sherlock Holmes (1979)