The Birth of Model Theory: Lowenheim's Theorem in the Frame of the Theory of Relatives - Hardcover

Badesa, Calixto

Hardcover

ISBN 10: 0691058539 ISBN 13: 9780691058535

Verlag: Princeton University Press, 2004

Inhaltsangabe

Löwenheim's theorem reflects a critical point in the history of mathematical logic, for it marks the birth of model theory--that is, the part of logic that concerns the relationship between formal theories and their models. However, while the original proofs of other, comparably significant theorems are well understood, this is not the case with Löwenheim's theorem. For example, the very result that scholars attribute to Löwenheim today is not the one that Skolem--a logician raised in the algebraic tradition, like Löwenheim--appears to have attributed to him. In The Birth of Model Theory, Calixto Badesa provides both the first sustained, book-length analysis of Löwenheim's proof and a detailed description of the theoretical framework--and, in particular, of the algebraic tradition--that made the theorem possible.


Badesa's three main conclusions amount to a completely new interpretation of the proof, one that sharply contradicts the core of modern scholarship on the topic. First, Löwenheim did not use an infinitary language to prove his theorem; second, the functional interpretation of Löwenheim's normal form is anachronistic, and inappropriate for reconstructing the proof; and third, Löwenheim did not aim to prove the theorem's weakest version but the stronger version Skolem attributed to him. This book will be of considerable interest to historians of logic, logicians, philosophers of logic, and philosophers of mathematics.

Über diesen Titel

Über die Autorin bzw. den Autor

Calixto Badesa is Associate Professor of Logic and History of Logic at the University of Barcelona.

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"A first-rate contribution to the history and philosophy of logic, this is scholarship at its best. It is, to my knowledge, the first book in the history of logic that focuses completely on a single result. Very original in approach and conception, it goes against the grain of much recent scholarship. Given the complexity of the subject, Badesa could not have done a better job of being clear and making the presentation accessible."--Paolo Mancosu, University of California, Berkeley

"The Birth of Model Theory represents a long overdue, in-depth analysis and exposition of one of the most important results in mathematical logic. There are hardly any informed, sustained treatments of Löwenheim's work to be found in the literature. This well-written book should fill this gap."--Richard Zach, University of Calgary

"This book will be extremely useful to those seeking to make sense of Löwenheim's work and those seeking to put it into its historical context. Calixto Badesa draws well-supported conclusions that contradict the entire modern body of scholarship on the topic."--Shaughan Lavine, University of Arizona

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The Birth of Model Theory

Lwenheim's Theorem in the Frame of the Theory of RelativesBy Calixto Badesa

Princeton University Press

Copyright © 2004 Princeton University Press
All right reserved.
ISBN: 978-0-691-05853-5

Chapter One

Algebra of Classes and Propositional Calculus

1.1 BOOLE

1.1.1 George Boole (1815-1864) is justly considered the founder of mathematical logic in the sense that he was the first to develop logic using mathematical techniques. Leibniz (1646-1716) had been aware of this possibility, and De Morgan (1806-1878) worked in the same direction, but Boole was the first to present logic as a mathematical theory, which he developed following the algebraic model. His most important contributions are found in The mathematical analysis of logic [1847], his first work on logic, and An investigation of the laws of thought [1854], which contains the fullest presentation of his ideas on the subject. In what follows I will focus solely on the latter work, to which I will refer as Laws.

Boole's aim is to examine the fundamental laws (i.e., the most basic truths from which all the other laws are deduced) of the mental processes that underlie reasoning. Boole does not challenge the validity of the basic laws of traditional logic, but he is convinced that they are reducible to other more basic laws of a mathematical nature; it is these basic laws that he sets out to find.

In Boole's opinion, the mental processes that underlie reasoning are made manifest in the way in which we use signs. Algebra and natural language are systems of signs, and so the study of the laws that the signs of these systems meet should allow us to arrive at the laws of reasoning. The question of whether or not two different systems of signs obey the same laws can only be answered a posteriori. Applied to natural language-the commonest system of signs-Boole's idea implies that the laws by means of which certain terms combine to form statements or other more complex terms are the same as those observed by the mental processes that these combinations reveal. Thus, Boole believes that it is possible to establish a theory of reasoning by examining the laws by means of which the terms and statements of language are combined.

Boole classifies the propositions of interest to logic into primary and secondary (Laws, pp. 53 and 160). Primary propositions are the ones that express a relation between things. Secondary propositions express relations between propositions, or judgments on the truth or falsity of a proposition. For example, "men are mortal" is a primary proposition (because it expresses a relation between men and mortal beings), but "it is true that men are mortal" is secondary. Propositions that result from combining propositions with the aid of connectives are also secondary. Boole begins his study of the laws of reasoning with the analysis of primary propositions and of the reasonings in which they alone intervene.

1.1.2 According to Boole (Laws, p. 27), in order to formulate the laws of reasoning, the following signs or symbols are sufficient:

(a) literal signs: x, y, z, ...;

(b) signs of operations of the mind: x, +; and -;

(c) the sign of identity: =.

This claim, however, does not have the meaning it would have today. As we will see, Boole uses other signs and operations as well to present and develop his theory.

A literal symbol represents "the class of individuals to which a particular name or description is applicable." Strictly speaking, literal signs stand for classes, but Boole frequently speaks (the definition of product that I will quote later on is an example of this) as if they denoted expressions of the natural language that determine classes (nouns, adjectives, descriptions or even proper names). The reason for this ambiguity is that both literal signs and expressions determining classes are signs of the same conceptions of the mind. For example, the use of the word "tree" indicates that we have performed a mental operation that consists of selecting a class (the class of all trees) that we represent by that word. Now, since the same class can also be represented by a literal sign, Boole sees no substantial difference between saying that x stands for the class of trees and saying that it stands for the word "tree."

Boole defines the product in the following way: "by the combination xy shall be represented that class of things to which the names or descriptions represented by x and y are simultaneously applicable." For example, if x stands for "white" and y for "horse," xy stands for "white horse" or for the class of white horses.

If x and y represent classes that do not have elements in common, x + y represents the class resulting from adding the elements of x to those of y (Laws, pp. 32-33). The sum corresponds to the mental operation of aggregating two disjoint classes into a whole. This operation is performed when we combine two terms by means of "and" as in "men and women," or by "or" as in "rational or irrational." Boole argues for the restriction of the sum to disjoint classes by stating that the rigorous use of these particles presupposes that the terms are mutually exclusive, but, as Jevons observed, Boole himself on occasion analyzes examples with disjunctions whose terms do not exclude each other.

It has been said on occasion that Boole interprets the sum x + y as an excluding disjunction, but, as Corcoran notes, this assertion is incorrect. It is important to distinguish between the definition of sum that Boole adopts and the following one: x + y is the class of objects that belong either to x or to y (but not to x and to y). If Boole had adopted this definition (i.e., if he really had defined the sum as an excluding disjunction), then the sum x + y would be meaningful both if x and y have elements in common and if they do not. However, with Boole's definition, x + y lacks logical significance when x and y have elements in common. In short, Boole's sum is the usual union, but defined only for disjoint classes.

The difference is the inverse operation of the sum, and it consists of separating a part from a totality. Thus, Boole says, if class y is a part of class x, x - y is the class of things that are elements of x and not of y. This mental operation is the one that is expressed by the word "except" when it occurs in expressions such as, for example, "politicians except for conservatives."

The only sign that allows us to form statements is the sign of identity. The equality x = y means that the classes x and y have the same elements; this identification is expressed in language using the verb "to be."

Boole also introduces the symbols 0 and 1, which represent, respectively, the empty class and the class of all the things to which the discourse is limited. As is well known, the idea of limiting the universe to things that are talked about was introduced by De Morgan in [1846]. Boole adopted this idea in Laws, but did not mention its origin.

To be able to refer to a nondetermined part of a class, Boole introduces the symbol v which, he says, represents an indefinite class (Laws, p. 61). The linguistic term that corresponds to this symbol is "some." Now, the expression "some men" is...

Details zur Ausgabe

• Verlag: Princeton University Press
• Erscheinungsdatum: 2004
• Sprache: Englisch
• ISBN 10: 0691058539
• ISBN 13: 9780691058535
• Einband: Gebundene Ausgabe
• Anzahl der Seiten: 256
• Kontakt zum Hersteller: Nicht verfügbar
• Verantwortliche Person: Nicht verfügbar