Herbrand's Fundamental Theorem - an encyclopedia article

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 Recherches sur la th\'eorie de la d\'emonstration --- submitted by Jacques Herbrand (1908-1931) in 1929 at the University of Paris. Herbrand's Fundamental Theorem is, together with G\"odel's incompleteness theorems and Gentzen's Hauptsatz, one of the most influential theorems of modern logic. Because of its complexity, Herbrand's Fundamental Theorem is typically fouled up in textbooks beyond all recognition. As we are convinced that there is still much more to learn for the future from this theorem than many logicians know, we will focus on the true message and its practical impact. This requires a certain amount of streamlining of Herbrand's work, which will be compensated by some remarks on the actual historical facts.