Natural Deduction: A Proof-Theoretical Study (Dover Books on Mathematics) by Dag Prawitz (PDF)

Description

An innovative approach to the semantics of logic, proof-theoretic semantics seeks the meaning of propositions and logical connectives within a system of inference. Gerhard Gentzen invented proof-theoretic semantics in the early 1930s, and Dag Prawitz, the author of this study, extended its analytic proofs to systems of natural deduction. Prawitz’s theories form the basis of intuitionistic type theory, and his inversion principle constitutes the foundation of most modern accounts of proof-theoretic semantics.The concept of natural deduction follows a truly natural progression, establishing the relationship between a noteworthy systematization and the interpretation of logical signs. As this survey explains, the deduction’s principles allow it to proceed in a direct fashion — a manner that permits every natural deduction’s transformation into the equivalent of normal form theorem. A basic result in proof theory, the normal form theorem was established by Gentzen for the calculi of sequents. The proof of this result for systems of natural deduction is in many ways simpler and more illuminating than alternative methods. This study offers clear illustrations of the proof and numerous examples of its advantages.

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⭐This is not an introduction to logic. It is a presentation of some analysis of Gentzen’s natural deduction system, based on the author’s PhD thesis about 1965, but it would be useful as a first introduction to the theory of sequent calculus.The majority of logic systems in the last 128 years (since the publication of Giuseppe Peano’s axiomatic logic system in 1889) have used the Hilbert style of logical deduction where every line of every proof must be an unconditional assertion of a true proposition. That sounds like a rational way to proceed until you try to do real mathematics this way. You quickly find that real mathematics works by introducing and discharging assumptions. (For example: “Let epsilon be positive. Let delta satisfy the assumptions of the definition.”) The concept of “natural logic” attempts to represent this kind of real-world mathematical logic in a rigorous and verifiable way by attaching an assumption-list to every assertion.The subject of this book is the first kind of logical deduction system published by Gerhard Gentzen in 1934. In other words, this book is about sequents which have a single succedent formula, whereas Gentzen’s second system used general disjunctions of succedent formulas. It is the first kind of system which this book is about. Prawitz’s purpose is to present various properties of this kind of system, not to teach how to do real-world mathematical logic using such a system. (If you want a practical implementation of a natural deduction system, there are some basic introductions, which do not follow Gentzen’s original rules, in Suppes “Introduction to logic”, Dover, 1957, 1999, Stoll “Set theory and logic”, Dover 1963, 1979, Lemmon “Beginning logic”, Nelson 1965. The only serious practical introduction I know of is in my own book.)One of the reasons why Prawitz’s natural deduction book is not directly applicable as a practical introduction to real-world mathematical logic is the use of tableaux. These are a quaint old method of showing deduction trees graphically. The more practical approach, which you see in the computerised logic systems like metamath.org, Isabelle, Coq, HOL, Mizar and others, uses numbered or tagged lines, not tree-diagrams. But the purpose of the Prawitz book is to present various properties of natural deduction systems, such as completeness and consistency, not to teach practical deduction.

⭐È stato per me un libricino chiarificante su diversi aspetti della deduzione naturale e della teoria del senso.BomÉ um clássico!

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