Herbrand's Fundamental Theorem - an encyclopedia article
Claus-Peter Wirth

TL;DR
Herbrand's Fundamental Theorem offers a constructive link between predicate and propositional logic, playing a central role in modern logic and impacting proof theory and computational logic.
Contribution
This article clarifies the core message and practical significance of Herbrand's Theorem, simplifying its complex history and highlighting its influence on logic.
Findings
Herbrand's Theorem is central to predicate logic.
It has significant implications for proof theory.
The theorem's practical impact is greater than commonly recognized.
Abstract
Herbrand's Fundamental Theorem provides a constructive characterization of derivability in first-order predicate logic by means of sentential logic. Sometimes it is simply called "Herbrand's Theorem", but the longer name is preferable as there are other important "Herbrand theorems" and Herbrand himself called it "Th\'eor\`eme fondamental". It was ranked by Bernays [1957] as follows: "In its proof-theoretic form, Herbrand's Theorem can be seen as the central theorem of predicate logic. It expresses the relation of predicate logic to propositional logic in a concise and felicitous form." And by Heijenoort [1967]: "Let me say simply, in conclusion, that Begriffsschrift [Frege, 1879], L\"owenheim's paper [1915], and Chapter 5 of Herbrand's thesis [1930] are the three cornerstones of modern logic." Herbrand's Fundamental Theorem occurs in Chapter 5 of his PhD thesis [1930] --- entitled…
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Taxonomy
TopicsLogic, programming, and type systems · Logic, Reasoning, and Knowledge · Computability, Logic, AI Algorithms
