Table Of ContentLOGICAL WRITINGS
JACQUES HERBRAND (in the center)
On the expedition during which he fell to his death
(Photo Courtesy of Suzanne Lalltman)
JACQUES HERBRAND
LOGICAL WRITINGS
Edited by
WARREN D.OOLDFARB
Harvard University
A Translation of the'E crits logiques·. edited by
JEAN VAN HEIJENOORT
D. REIDEL PUBLISHING COMPANY/DORDRECHT-HOLLAND
JACQUES HERB RAND: ECRITS LOGIQUES
First published by Presses Universitaires de France, 1968
Library of Congress Catalog Card Number 74-146963
ISBN-13: 978-94-010-3074-8 e-ISBN-13: 978-94-010-3072-4
001: 10.1007/978-94-010-3072-4
All Rights Reserved
Copyright C 1971 by D. Reidel Publishing Company, Dordrecht, Holland
Softcover reprint of the hardcover I st edition 1971
No part of this book may be reproduced in any form, by print, photoprint, microfilm,
or any other means, without written permission from the publisher
PREFACE
In 1968 Jean van Heijenoort published an edition of Herbrand's collected
logic papers (Herbrand 1968). The core of the present volume comprises
translations of these papers and of the biographical notes also appearing
in that edition. With two exceptions, this is their first appearance in
English; the exceptions are Chap. 5 of Herbrand's thesis and Herbrand
1931c, both of which appeared in van Heijenoort 1967, the former trans
lated by Burton Dreben and van Heijenoort, and the latter by van
Heijenoort. These two translations have been reprinted here, thanks to
the permission ofthe Harvard University Press, with only minor changes.
The remainder of the present translations are my own; I am grateful to
van Heijenoort for providing an English draft of 1931, which forms the
basis of the translation appearing here.
In these translations, the bibliographical references have been stan
dardized (see p. 299 below) and the notation has been changed so that
it is fairly uniform throughout (any differences from Herbrand's original
notation are mentioned in footnotes). Herbrand's technical terminology
is not always translated literally; the principal instances of this are
'reduite', translated 'expansion' (except in 1930, Chap. 3, § 3, where it
is translated 'relativization'), 'champ', translated 'domain', and 'symbole
de variable apparente', translated 'quantifier'. In other cases of this sort,
the French terms appear in double brackets immediately following the
English renderings. In addition, Herbrand's texts contain numerous over
sights and typographical errors; these are corrected without note when
Herbrand changed them himself in published Errata, and are noted
otherwise. Herbrand also tended to express himself rather hastily, result
ing in many obscurities; in these translations an attempt has been made to
balance the demands of literalness and clarity. In Chap. 5 of 1930, since
the subject matter is more difficult, the translators have departed some
what more from the original than is the case in the rest of this volume.
VI LOGICAL WRITINGS
Moreover, the present volume contains many footnotes and more
extended notes which clarify or correct the text and compare Herbrand's
investigations with other work in logic. Each Note is connected with a
particular passage of the text, and would, were it not for its length, appear
there as a footnote. These Notes follow the appropriate paper, except
in 1930 where they follow the appropriate chapter; the passage with
which a Note deals is indicated by a footnote and this footnote is men
tioned below the title of the Note. There are also cross-references between
Notes and footnotes. Notes B-1 are Dreben's (they are slightly modified
versions of Notes C-G in van Heijenoort 1967; Note E is a corrected
version of the original Note C, and is to appear in this revised form in
the second printing of van Heijenoort 1967). The Note to 1928, Note J,
and Note N are by Dreben and myself, and Notes A, B, C, D, K, L, M,
and 0 are mine, Note D based in part on Dreben's Note B in van Heijen
oort 1967. In those Notes closely connected with Herbrand's text,
Herbrand's terminology is for the most part employed; in those of a
broader nature, more modern terminology is used.
Immediately preceding 1930, 1931, 1931b, and 1931c there are brief
introductory notes; the last two are van Heijenoort's (the one to 1931b
appeared in French in Herbrand 1968, and the one to 1931c in van
Heijenoort 1967), the second mine, and the first is mine, in large part
based on van Heijenoort's introduction to Chap. 5 in van Heijenoort 1967.
Footnotes not Herbrand's own are enclosed in double brackets (as
are editorial clarifications in the text); in 1930, Chap. 5, they are Dreben's,
except for those numbered 4,5,7,8, 18, 19,21,23,36,46,57,89, and 94,
which are van Heijenoort's, and 1,2, 10,51,55, and 86, which are mine;
in 1931c, 2 is van Heijenoort's and the rest mine (these footnotes of
Dreben and van Heijenoort appeared - except for minor editorial
changes - in van Heijenoort 1967); the remainder are all mine, but for
number 4 of 1930, Chap. 2, which is Dreben's.
I am deeply grateful to Burton Dreben for his inestimable assistance
in every phase of the preparation of this volume. Jean van Heijenoort
was also of great help, as was A. Paul Phillips, who suggested many
improvements and did a large part of the proofreading. I would also
like to thank Daniel Garber, W. V. Quine, Kenneth Ribet, Eira Ruben,
Thomas M. Scanlon, Frank W. Thompson, Maria Tymoczko, and A.
Thomas Tymoczko for their many helpful suggestions with regard to
PREFACE VII
various parts of this volume. Moreover, in van Heijenoort 1967 Dreben
and van Heijenoort thanked Wilbur Hart III, Marc Venne, Daniel
Isaacson, and Dirk van Dalen for their help with the translation of
Chap. 5 of 1930.
Work on this volume was partially supported by N.S.F. grants
# GS-2615 and #GP-4032.
W.D.G.
TABLE OF CONTENTS
More detailed Tables of Contents for 1930 and 1931 can be found on pp. 46 and 216
below.
Preface v
Introduction I
Biographical Note on Jacques Herbrand, by Claude Chevalley
and Albert Lautman 21
The Accident 24
On Herbrand's Thought, by Claude ChevalIey 25
I. On Proof Theory (1928) 29
II. The Consistency of the Axioms of Arithmetic (1929) 35
III. On Several Properties of True Propositions and their
Applications (1929a) 38
IV. On the Fundamental Problem of Mathematics (1929b) 41
V. Investigations in Proof Theory (1930) 44
VI. The Principles of Hilbert's Logic (1930a) 203
VII. On the Fundamental Problem of Mathematical Logic (1931) 215
VIII. Unsigned Note on Herbrand's Thesis, written by Herbrand
himself (1931a) 272
IX. Note for Jacques Hadamard (1931b) 277
X. On the Consistency of Arithmetic (1931c) 282
References 299
Index 310
INTRODUCTION
Herbrand's principal concern in his logical writings was to develop a
unified approach to all questions of proof theory. He based this approach
on his Fundamental Theorem, which is, as Bernays has remarked (1954),
"the central theorem of predicate logic". In addition to providing both
information about the structure of logical proofs and a method for
obtaining consistency and decidability results, this theorem also furnishes
constructive insight into nonconstructive notions such as satisfiability.
However, due in part to the difficulty of Herbrand's texts and in part to
surface similarities with work of Skolem and Gentzen, Herbrand's mode of
investigation is not widely understood today. In order to appreciate fully
the importance of his results and their place in the history of logic, it
is helpful to trace the major influences on Herbrand's work: Whitehead
and Russell's Principia Mathematica, Lowenheim 1915 and Skolem 1920,
and the articles of the Hilbert school which appeared during the 1920's.
Herbrand took the Principia Mathematica to furnish experimental evi
dence that classical mathematics can in fact be codified in a formal
system using a small number of primitive signs and rules of inference
(1930, p. 48 below). The possibility of such a formalization shows the
importance of metamathematical investigations. Herbrand says, "It
becomes natural to study this system of signs in itself, and to pose prob
lems regarding it. The solution of these will have an immediate echo
in our general knowledge of mathematics" (p. 49 below). Herbrand
begins in his thesis (1930) by examining the propositional calculus of the
Principia, and then takes great pains to show that his formulation of
quantification theory is equivalent to that used by Whitehead and Russell
(p. 89 below). Moreover, one of the methods Whitehead and Russell
use to construct quantificational logic seems to be the source of two key
concepts in Herbrand, namely, those of normal identity and property A
(see p. 4 below).
2 LOGICAL WRITINGS
Lowenheim's paper of 1915 contains a number of ideas central to
much subsequent work. He uses the notion of a universe of discourse
susceptible to change, a notion coming from the Boole-Peirce-Schroder
tradition. Lowenheim completely ignores syntactic considerations such
as provability; rather, he assumes a naive notion of a formula's being
true in a universe under a given interpretation of the predicate letters.
Thus, the (set-theoretic) notions of validity and satisfiability come into
play. Using them, Lowenheim proves his well-known theorem: If a
formula of quantification theory with identity is satisfiable, then it is
satisfiable in a denumerable universe. As we shall see, Herbrand rejects
on philosophical grounds the concepts employed here; but much of his
work is concerned with finding unobjectionable surrogates for them. A
key notion in Lowenheim's paper is that of a quantifier-free expansion
of a quantified formula; this notion was used, in various forms, in many
later studies of quantification theory, including Herbrand's. However,
Lowenheim's proof is very informal and difficult to follow. Lowenheim
employs, without justification, several arguments (in particular with
regard to infinitely long formulas) that are extensions to infinite cases
of inferences applicable to finite cases. This sort of extension was common
in the Schroder school, but it leaves the acceptability of Lowenheim's
proof open to doubt (see, for example, van Heijenoort 1967, pp. 228-232).
Skolem later gave several rigorous proofs; both Lowenheim's work and
these improvements are discussed in greater detail below (p. 7).
The method of investigation that Herbrand adopts was essentially due
to Hilbert. Partially in response to Brouwer's critiques of classical
mathematics, Hilbert developed his notion of proof theory. The proof
theorist's task is to examine formal systems as systems of signs, divorced
from any intuitive content. The methods used must be purely finite,
although this restriction is left vague. (In practice, the intuitions under
lying a formal system may play an important role in obtaining results;
but care must then be taken to insure that the use made of them does not
involve any nonfinitistic assumptions.) The most important questions in
Hilbert's metamathematics concern characterizations of provability in
given formal systems; central among these are problems of consistency,
completeness, and decidability. Herbrand's logical work is entirely within
this framework; he agrees that arguments about formal systems must
be finitistic, and is concerned with the problems Hilbert raises. In the
