186 Logic as a Tool 4.3.5 Prove Theorem 137 for equivalent replacement in Natural Deduction with equality by structural induction on the formula A 4.3.6 Derive each of the following in Natural Deduction with equality (a) ND x1 = y1 ∧ x2 = y2 → g (f (x1 , f (x2 ))) = g (f (y1 , f (y2 ))) (Universal quantification is assumed but omitted.) (b) ND ∀x(x = f (x) → (P (f (f (x))) → P (x))) (c) ∀x∀y (f (x) = y → g (y ) = x), ∀x∀y (g (x) = g (y ) → x = y ) ND ∀z (f (g (z )) = z ) 4.3.7 Prove again, now using Natural Deduction with equality, that the line in the old jazz song “Everybody loves my baby, but my baby don’t love nobody but me” implies that “I am my baby,” that is, ∀xL(x, MyBaby) ∧ ∀y (¬y = Me → ¬L(MyBaby, y )) ND MyBaby = Me For more exercises on derivations with equality, on sets, functions, and relations, see Section 5.2.7 Dag Prawitz (born 16.05.1936) is a Swedish philosopher and logician who has made seminal contributions to proof theory as well as to the philosophy of logic and mathematics Prawitz was born and brought up in Stockholm He studied theoretical philosophy at Stockholm University as a student of Anders Wedberg and Stig Kanger, and obtained a PhD in philosophy in 1965 After working for a few years as a docent (associate professor) in Stockholm and in Lund, and as a visiting professor in US at UCLA, Michigan and Stanford, in 1971 Prawitz took the chair of professor of philosophy at Oslo University for years Prawitz returned to Stockholm University in 1976 as a professor of theoretical philosophy until retirement in 2001, and is now a Professor Emeritus there While still a graduate student in the late 1950s, Prawitz developed his algorithm for theorem proving in first-order logic, later implemented on one of the first computers in Sweden (probably the first computer implementation of a complete theorem prover for first-order logic) and published in his 1960 paper with H Prawitz and N Voghera A mechanical proof procedure and its realization in an electronic computer In his doctoral dissertation Natural deduction: A proof-theoretical study Prawitz developed the modern treatment of the system of Natural Deduction In particular, he proved the Normalization Theorem, stating that all proofs in Natural deduction can be reduced to a certain normal form, a result that corresponds to Gentzen’s celebrated Hauptsatz for sequent calculus Prawitz’ Normalization Theorem was later extended to first-order arithmetic as well as to second-order and higher-order logics As well as his pioneering technical work in proof theory, Prawitz has conducted important studies on the philosophical aspects of proof theory, on inference and