Book contents
- Frontmatter
- Contents
- Preface
- Prologue: Hilbert's last problem
- 1 Introduction
- PART I PROOF SYSTEMS BASED ON NATURAL DEDUCTION
- 2 Rules of proof: natural deduction
- 3 Axiomatic systems
- 4 Order and lattice theory
- 5 Theories with existence axioms
- PART II PROOF SYSTEMS BASED ON SEQUENT CALCULUS
- PART III PROOF SYSTEMS FOR GEOMETRIC THEORIES
- PART IV PROOF SYSTEMS FOR NON-CLASSICAL LOGICS
- Bibliography
- Index of names
- Index of subjects
2 - Rules of proof: natural deduction
Published online by Cambridge University Press: 07 October 2011
- Frontmatter
- Contents
- Preface
- Prologue: Hilbert's last problem
- 1 Introduction
- PART I PROOF SYSTEMS BASED ON NATURAL DEDUCTION
- 2 Rules of proof: natural deduction
- 3 Axiomatic systems
- 4 Order and lattice theory
- 5 Theories with existence axioms
- PART II PROOF SYSTEMS BASED ON SEQUENT CALCULUS
- PART III PROOF SYSTEMS FOR GEOMETRIC THEORIES
- PART IV PROOF SYSTEMS FOR NON-CLASSICAL LOGICS
- Bibliography
- Index of names
- Index of subjects
Summary
This chapter gives, first, the calculus of natural deduction, together with its basic structural properties such as the normalization of derivations and the subformula property of normal derivations. Next, the calculus is extended by mathematical rules, and it is shown that normalization works also in such extensions. The theory of equality is treated in detail, as a first example. Finally, predicate logic with an equality relation is studied. It is presented as an extension of predicate logic without equality, and therefore normalization of derivations applies. The question of the derivability of an atomic formula from given atomic formulas, i.e., the word problem for predicate logic with equality, is solved by a proof-theoretical algorithm.
Natural deduction with general elimination rules
Gentzen's rules of natural deduction for intuitionistic logic have proved to be remarkably stable. There has been variation in the way the closing of assumptions is handled. In 1984, Peter Schroeder-Heister changed the rule of conjunction elimination so that it admitted an arbitrary consequence similarly to the disjunction elimination rule. We shall do the same for the rest of the elimination rules and prove normalization for natural deduction with general elimination rules.
Natural deduction is based on the idea that proving begins in practice with the making of assumptions from which consequences are then drawn. Thus, the first rule of natural deduction is that any formula A can be assumed. Formally, by writing
A
we construct the simplest possible derivation tree, that of the conclusion of A from the assumption A.
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- Proof AnalysisA Contribution to Hilbert's Last Problem, pp. 17 - 38Publisher: Cambridge University PressPrint publication year: 2011