 # Natural Deduction: The Logical Basis of Axiom Systems

## Excerpt

This book is based upon Gentzen's techniques of natural deduction. The propositional and quantificational rules on which the exposition pivots are stated essentially as Gentzen stated them, and even the names used for them are free translations of his names. Gentzen's techniques constitute a very natural approach to the study of the proofs occurring in axiom systems as well as a sound basis for the analysis of the properties of formal systems as such.

The book is divided into two main parts. In Part One various rules and sets of rules for deduction are presented. Part Two is largely concerned with the metatheoretical analysis of deductions based upon these rules.

Part One begins with a chapter in which a simple set of axioms (those for simple order) is presented and attention is called to the use of logical rules in the deduction of theorems from these axioms. In Chapter Two all the standard propositional rules are given. Here one of the great merits of Gentzen's formulation is obvious: it permits the analysis of proof in an axiom system without presupposing another axiom system--namely, that of the logic itself. The power of this formulation is revealed in the early use of conditional proof, here introduced, of course, as a primitive rule. The analysis of negation into four rules reveals something of its nature and facilitates transition in a later chapter to the intuitionistic logic.

Chapter Three temporarily sets aside the basic exposition of the nature of deduction in an axiom system in order to introduce truth tables and normal forms. This material not only serves the science student interested in computers, but also lays the ground for later metatheoretical discussions. Chapter Four is intended to suggest that the problem of what constitutes a deduction is by no means closed, for alternative logical systems such as those of three-valued logic, intuitionistic logic, and modal logics are available and may be used.

Chapter Five is occupied with the explicit development of monadic predicate logic. The rigorous statement of the rules of predicate logic in a form in which they lead, in Chapter Six, to a final justification for the steps in deductions within first order axiom systems returns the reader to the fundamental problem of axiomatic proof. Chapter Seven introduces the concepts of identity and description, thus completing a standard first course in modern logic.

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