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American Postulate Theorists and Alfred Tarski

2003; Taylor & Francis; Volume: 24; Issue: 4 Linguagem: Inglês

10.1080/01445340310001599588

1464-5149

Michael Scanlan,

History and Theory of Mathematics

Abstract This article outlines the work of a group of US mathematicians called the American Postulate Theorists and their influence on Tarski's work in the 1930s that was to be foundational for model theory. The American Postulate Theorists were influenced by the European foundational work of the period around 1900, such as that of Peano and Hilbert. In the period roughly from 1900 – 1940, they developed an indigenous American approach to foundational investigations. This made use of interpretations of precisely formulated axiomatic theories to prove such metatheoretic properties as independence, consistency, categoricity and, in some cases, completeness of axiom sets. This approach to foundations was in many respects similar to that later taken by Tarski, who frequently cites the work of American Postulate Theorists. Their work served as paradigm examples of the theories and concepts investigated in model theory. The article also examines the possibility of a more specific impetus to Tarski's model theoretic investigation, arising from his having studied in 1927 – 1929 a paper by C. H. Langford proving completeness for various axiom sets for linear orders. This used the method of elimination of quantifiers. The article concludes with an examination of one example of Langford's methods to indicate how their correct formulation seems to call for model-theoretic concepts. Acknowledgements I have received valuable suggestions on the first version of this article from George Weaver and John Corcoran. I want to also acknowledge the invaluable contributions of Ivor Grattan-Guinness to the study of both the history and the philosophy of logic. These contributions are embodied in his own scholarly work, in his role as founding editor of History and Philosophy of Logic and in other leadership roles he has taken on in the field. Notes For additional information on the work done in this seminar see also Vaught Citation 1974 (pp. 159 – 63). More information on American Postulate Theorists can be found in Scanlan Citation 1991 . They also appear in a number of contexts in Ivor Grattan-Guinness' history of logicism (Grattan-Guinness Citation 2000 ). This literature can be taken to extend up to the discussion of "Postulate Theory" in Church Citation 1956 (§55). For instance, (W. V. Quine ( Citation 1985 ), in his memoirs, recalls the influence of Huntington. The Philosophy Department course listings for 1937 included for graduate students the mathematics course entitled 'Postulate Theory' which was offered by Huntington. For a more complete treatment of E. H. Moore, see Parshall Citation 1984 . The later Bourbaki project might be an example of the sort of non-philosophical systematization impulse that I have in mind here. Category theory might be as well. I use the term 'semantic interpretations' here to distinguish the assignment of values in a domain to vocabulary elements of a language from an interpretation of one theory in another, by means of definitions of the terminology of the one theory in that of the other. The latter sort of interpretation is involved in relative consistency proofs, for example of non-euclidean geometries in euclidean geometry. Many aspects of Tarski's textbook Introduction to Logic and to the Methodology of Deductive Sciences read like a course in postulate theory. His suggested readings give prominence to works by E. V. Huntington, C. I. Lewis, C. H. Langford and J. W. Young. Some points that can be cited here are the coincidence of dates of the Langford publication with the seminar and results on dense orderings that are in the appendix to Tarski Citation 1936 , which he says date directly from the work of the seminar. I will say more on this later. There is a letter, in the Sheffer papers at Harvard, of 17 March 1925 in which Langford (writing from Cambridge, England) tells Sheffer that he has succeeded in completeness proofs for serial order, betweenness, and cyclic order. For more about Sheffer see Scanlan Citation 2000 . In Lewis and Langford Citation 1932 ch. XI is titled 'Postulational Technique: Deduction' and ch. XII is titled 'Postulational Technique: Deducibility'. In the opening paragraph of ch. XII 'deduction' is aligned with specific deductions while 'deducibility' is said to concern questions about 'logical consequence'. Hodges attributes the first use of 'first-order' in the modern sense to Langford Citation 1926 , cf. Hodges Citation 1983 (p. 3). 'Series' is a term for linear orderings that Langford adopts from Principia Mathematica. This is sometimes described as the 'one world' framework for logic or, somewhat misleadingly, as the 'one universal language' framework for logic. This is sometimes called the normal form for satisfiability, because the conditions for the satisfiability of one of these formulas in an interpretation is rather clear. An assignment of variables to elements of the interpretation satisfies the unquantified matrix iff it satisfies one or more of the disjuncts. For any of these disjuncts to be satisfied, each formula that is a conjunct must be satisfied.