Freedom and Rationality Essays in Honor of John Watkins From his Col
x philosophy when he inaugurated a debate about the principle of methodologi cal individualism, a debate which continues to this day, and which has inspired a literature as great as any in contemporary philosophy. Few collections of material in the gener
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BOSTON STUDIES IN THE PHILOSOPHY OF SCIENCE
Editor ROBERT S. COHEN, Boston University
Editorial Advisory Board ADOLF GRONBAUM, University of Pittsburgh SYLVAN S. SCHWEBER, Brandeis University JOHN J. STACHEL,Boston University MARX W. W ARTOFSKY, Baruch College of the City University of New York
VOLUME 117
John Watkins
FREEDOM AND RATIONALITY Essays in Honor of John Watkins From his Colleagues and Friends
Edited by
FRED D' AGOSTINO University ofNew England, Australia and
I.e. JARVIE York University, Toronto, Canada
KLUWER ACADEMIC PUBLISHERS DORDRECHT / BOSTON / LONDON
Library of Congress Cataloging In PubHcatlon Data Freecol'l and rntol'1al1ty : essails 11'1 hQiI:'" of JOhrl Hatk1ns I adHed by Fred O'A;105tlnO and I.e. Jar-vU. p. CII. -- (Boston studtes 1" the ph11osotlhy cf sctenee : v.
,,7)
Bfbl1ogr'lphy: p.
Includes index.
1. Ph1losophy. 2. Sc lence--Phf loscphy. 3. Po t tt lea 1 se tence. T, T::::> Q}. Neither P ::::> T nor T::::> Q has any testable consequences in isolation: {P::::> T}'est = {T::::> Q}'est = 0. A itself entails P::::> Q, which has the form of a 'singular predictive implication'. Thus A test is not the union of {P::::> Tpest and {T::::> Q}test.
A seems to a natural enough axiomatisation of its total content, and it also seems that if any theory in this framework is unified A is. Now consider a reaxiomatisation of A. A *: {((P
1\
-Q)::::> 1), «P 1\ -Q)::::> -1), «-P
1\
Q)::::> 1), «P 1\ Q)::::> n).
As is easily checked, A* satisfies requirements 1-4. But it also satisfies the decomposition requirement. Using only the resources P, Q, and T the weakest statements (those that cannot be further weakened without becoming tautologous) are minimal disjunctions. That is, statements of the form: (±)P v (±)Q v (±)T
where in the place of (±) there is either a negation sign or else nothing at all. A * is tantamount to the following set A ** of minimal disjunctions: {(-P v 1), (-P v Q v
-n, (P v -Q v n, (-P v Q v n).
A** does not satisfy Wajsberg's requirement, because (-P v Q) is both a theorem of A** and a proper part of one of its axioms. Hence A** is not natural. s And because it is not natural its existence is irrelevant to the unity of A. Since every statement is uniquely decomposable into a conjunction of minimal disjunctions, and A* is a four-membered natural class of minimal disjunctions, by the decomposition requirement, any other natural
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GRAHAM ODDlE
reaxiomatisation of A will contain at least four axioms, say Cl' ... C4 • It follows from this (and the independence requirement) that Cl' ... C4 must be logically equivalent to the members of A** and A*.6 Do all the reaxiomatisations of A which satisfy conditions 1-5 also satisfy the organic fertility requirement? Since the organic fertily requirement does not depend on the syntactic form of the axioms, but only on their logical strength, either all these natural axiomatisations satisfy the organic fertility requirement or none do. There is only one singular predictive implication of A and that is (P::::> Q).7 Consi
