Logic as a tool a guide to formal logical reasoning ( PDFDrive ) 210

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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 6 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