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MANY -VA L U E D LOGICS J. B A R K L E Y R O S S E R Professor of Mathematics, Cornell University Ithaca, N. Y.,U.S.A. ATWELL R . T U R Q U E T T E Assoc. Professor of Philosopliy, University of Ninois Urbana, Ill., U.S,A. 1952 N O R T H - H O L L A N D P U B L I S H I N G COMPANY AMSTERDAM Copyright 1952 by N. V. Noord-Hollandsche Uitgevers Maatschapkv Amsierdam PRINTED IN THE NETHERLANDS DRUKKERIJ HOLLAND N.V.. AMSTERDAM ACKNOWLEDGMENTS The authors of the present work wish to express their appreciation to the North-Holland Publishing Company and to Editors L. E. J. Brouwer, A. Heyting, and E. W. Beth for their kind invitation to contribute this volume to the series Studies in Logic. Acknowledgment is due also to Lucille LeRoy Turquette for her invaluable aid in preparing the manuscript for publication. J. B. R. and A. R. T. I INTRODUCTION It has become a truism that every statement is either true or false. It might be supposed that this principle must be disproved before one can write a serious work on many-valued logic. This is by no means the case. In fact, the present volume will not constitute a disproof of such a principle. However, in the following chapters, systems of many-valued logic through the level of the first order predicate calculus will be constructed in such a manner that they are both consistent and complete. The tools of construction will include the logical procedures of ordinary English and those of some formalized systems of two-valued logic. Thus, in effect, we shall use the truism in constructing many-valued logics. It does not follow from this that ordinary two-valued logic is necessary for the construction of many-valued logic, but it does follow that it is sufficient for such constructions. The ability to establish such a sufficiency is certainly no more mysterious than the fact that a Harvard graduate can learn Sanskrit using his native English. At this point, however, a word of warning is in order, for the treatment of many-valued logic which follows is concerned with the behavior of many-valued statements and not with their meaning. This indifference toward the meaning of many-valued statements indicates that we have no prejudices regarding the possible interpretations of our systems of many-valued logic. As far as our treatment is concerned, the meaning of a many-valued statement could be a linguistic entity such as a many-valued proposition or a physical entity such as one of many positional contacts. 2 For that matter, the meaning of a many-valued statement might be quite different from either a proposition or a positional contact. I n any case, regardless of the possible interpretations of many-valued logical systems, it is our opinion that 1 For example, see Bochvar 1939 and Rpichenbach 1944. For example, see Shannon 1938 and SBstakov 1946. 2 INTRODUCTION most interpretations that have so far been proposed can not be taken too seriously until the precise formal development of such systems has been carried to a level of perfection considerably beyond that which is reached even in the present work. Of course, we do not wish to deny the possibility of finding interpretations for subsystems of many-valued logic such as the statement calculus or predicate calculus of first order, but we do consider various recent proposals for interpretations of many-valued logic definitely premature. 3 Typical of such interpretations are those which concern recent physical theories that involve a theory of measurement requiring the use of numbers. Since a theory of many-valued numbers has not yet been constructed, it is not possible at present to show that the proposed interpretations actually apply for systems of many-valued logic incorporating a theory of numbers. 4 Since the present work does not complete the task of con