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MATHEMATICAL LOGIC IN LATIN AMERICA Proceedings of the IV Latin American Symposium on Mathematical Logic held in Santiago, December 1978
Edited by
A. I. ARRUDA Universidade Estadual de Campinas Brazil R.CHUAQUI Universidad Cat6lica.de Chile Santiago, Chile N. C. A. DA COSTA Universidade de Sao Paulo Brazil
1980
NORTHHOLLAND PUBLISHING COMPANY AMSTERDAM. NEW YORK. OXFORD
© NORTHHOLLAND PUBLISHING COMPANY, 1980 All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without the prior permission of the copyright owner.
ISBN: 0444 85402 9
Publishers: NORTHHOLLAND PUBLISHING COMPANY AMSTERDAM. NEW YORK. OXFORD
Sole distributors for the U.S.A. and Canada: ELSEVIER NORTHHOLLAND, INC. 52 VANDERBILT AVENUE NEW YORK, N.Y. 10017
Library of Congress CatalogIng in Publication Data
LatinAmerican Symposium on Mathematical. Logic, 4th, Santiago de Chile, 1978. Mathematical. logic in Latin America. (Studies in logic and the foundations of mathematics v. 99) Bibliography: p. Includes indexes. 1. Logic, Symbolic and mathematical.Congresses. I. Arruda, Ayda I. II. Chuaqui, R. III. Costa, Newton C. A. da. IV. Title. V. Series. QA9.A1L37 1978 511' •3 7920797 . ISBN 0444854029
PRINTED IN THE NETHERLANDS
to
ALFRED TARSKI teacher and friend
PREFACE
This volume constitutes the Proceedings of the Fourth Latin American Symposium on Mathematical Logic held at the Catholic University of Chile, Santiago from December 18 to December 22, 1978. The meeting was sponsored by the Pontifical Catholic University of Chile, the Academy of Sciences of the Institute of Chile, the Association for Symbolic Logic, and the Division of Logic, Methodology and Philosophy of Science of the International Union of History and Philosophy of Sci ence. The Organizing Committee consisted of Ayda I. Arruda, Rolando B. Chuaqui (chairman), Newton C.A. da Costa, Irene Mikenberg, and Angela Bau (Executive Secretary). Most of the sponsors were represented at the opening session. The Catho 1i c University was represented by its Rector Jorge Swett, its ViceRector for Academic Affairs Fernando Martinez, and its Dean of Exact Sciences Rafael Barriga who gave an address. The President of the Chilean Academy of Sciences, Jorge Mardones,a1so said a few words. Representing the Association for Symbolic Logic, Newton da Costa, Chairman of its Latin American Committee, opened the meeting. In preparation for the Symposium there was a logic year at the Catholic University. Advanced courses and seminars were given by Ayda I. Arruda (Universidade Estadua1 de Campinas, Brazil), Jorge E. Bosch (Centro de Altos Estudios en Ciencias Exactas, Buenos Aires, Argentina), Rolando Chuaqui (Universidad Cato1ica de Chile), Newton C.A. da Costa (Universidade de S~o Paulo, Brazil), and Irene Mikenberg (Universidad Catolica de Chile). Preceding the Symposium, there was a twoweek Seminar consisting of short courses. Below are reproduced the Scientific Programs of the Seminar and the Symposium. A look at these programs shows the progress in research in Logic in Latin American in the last few years. (1) The papers which appear in this volume are the texts, at times considerably expanded and revised, of most of the adresses presented by invitees to the meeting. Also included are two papers by Australian logicians (Bunder and Routley) who could not come because of difficulties in booking space in airlines. Expanded versions of a few short communications are also included. This volume is dedicated to Professor Alfred Tarski. In his previous visit to Chile and Brazil, he stimulated the development of Logic and encouraged the Organization of the III and IV Symposia. His influence was decisive in getti ng the sponsorship of the Association for Symbolic Logic. Many of the participants from Latin America and the United States can claim him directly or indirectly as their
(1) For a history of the previous Latin American Logic Symposia see: A short history of the Latin American Logic Symposia in NonClassical Logics, Model Theory and Computability, NorthHolland Pub. Co. 1977, pp, lx xv i l , v
vii
viii
PREFACE
teacher. Although he could not be physically present at the Symposium, he followed the proceedings with great interest. The Organizing Committee would like to acknowledge the financial support giv_ en to the meeting and the publication of these proceedings by the following institutions: the Catholic Unjversity of Chile, the Academy of Sciences of the Institute of Chile, the Fundacion de Estudios Economicos del Banco Hipotecario de Chile the Comision Nacional de Investigaciones Cientificas y Tecnologicas, the Interna: tional Union of History and Philosophy of Science, and the CocaCola Export Co. The editors would like to thank Irene Mikenberg, who was instrumental in the preparing of the cameraready copy. Most of the typing was done by M. Eliana Cabanas assisted by Rosario Henriquez. The editors wish to express their appreciation. The editors would also like to thank NorthHolland Publishing Co. for the inclusion of this volume in the series Studies in Logic and the Foundations of Mathematics.
The Editors
Instituto de Matematica Pontificia Universidad Catolica de Chile June 1979.
PROGRAM OF THE SEMINAR
Ayda I. Arruda and Newton C.A. da Costa, (Brazil), TopiC6 on Modal LoMc. (Si x 1ectures). Jorge Bosch, (Argentina), TopiC6 in the
Phito~ophy
PanaCOn6~tent
06 $eience. (Six lectures).
Luis F. Cabrera, (Chile), Equivalence Retation6 and the Continuum (Three lectures). Ulrich Felgner, (West Germany), The Continuum
Hypoth~~.
Gen~zed
Hypoth~~.
(Two lectures).
Ulrich Felgner, (West Germany), Apptieation6 06 the Axiom 06 Algeb~ and Topology. (Ten lectures). Jerome Malitz, (U.S.A.),
and
Contnuetib~y
to
Quanti6ie46. (Four lectures).
PROGRAM OF THE SYMPOSIUM
DECEMBER 18. 9,30  12,00
Opening Session.
15,15  16,05
N.C.A. da Costa, (Brazil), A Model Theo~etieal Ap~oaeh to Vbto~. 'I R. Chuaqui, (Chile), Foundatiol~ 06 S~tieal Metho~ U~ing a Semantieal VeMnition 06 P~obab~y.
16,30  17,20
J.R. Lucas, (England),
14,00  14,50
T~h, P~obab~y
and Set
Theo~y.
DECEMBER 19. 9,00  9,20 9,20 
9,40
9,40  10,00
~1.G.
ment
Schwarze, (Chile), AuomatizatiOn6 604
a AdcU:Uve MeMMe
Sy~t~.
M.S. de Gallego, (Brazil), The Lattice StAuetMe 06 4Vafued LukMie.wi.cz Afgeb~. A. Figall 0 ,(Argent ina), The Vet~nant 'Sy~tem 604 the MO!Lgan Afgeb~M ov~ a Finite O~d~ed Set.
10,00  10,20
A.M.Sette, (Brazil), A Funetonial
10,30  11,,00
I. Mikenberg, (Chile), A Clo~Me 60~ P~al Algeb~.
ix
App~oach
to
F~ee
Ve
Int~p~etab~y.
x
PROGRAM OF THE SYMPOSIUM
11,15  12,05
U. Felgner, (West Germany), The Model and Undeeidab~lj.
ab~lj
Theo~lj
06
FCG~oup~, Ve6~
DECEMBER 20. 9,00 
9,50
E.G.K. LopezEscobar, (U.S.A.), Tnuthvalue Semantico on.b.,Uc. Logic..
60~
Intuit
10,15  10,45
A.I. Arruda, (Brazil), On
11,00  11,50
W. Reinhardt, (U.S.A.), S~6ac.Uon Ve6~on and Axio~ 06 InMni.tlj in a Theo~lj 06 P~opMUu wUh Nec.u~Ulj OpeMtM.
14,00  14,50
J. Bosch, (Argentina), To~d a Conc.ept 06 Seienti6ic. TMough Spec.ltLe. RelaUvUlj.
15,15  16,05
O. Chateaubriand, (Brazil), An ExamlnaUon 06 06 MathemaUco.
P~c.o~~tent
Set
Theo~lj.
Theo~y
GOdel'~ Phieo~ophlj
DECEMBER 21. 9,00 
9,20
E.H. Alves, (Brazil), Some
9,20 
9,40
M. Corrada, (Chile), A Fo~a1lzaUon 06 the Imp~edic.aUve Theo~lj 06 CW~U U~ing Z~elo'~ AUMond~ng~auom Wlthout PMameteM.
9,45  10,35 11,00  11,50 14,20  14,40
R. Vaught, (U.S.A.), Model M. Benda, (U.S.A.), On
Theo~lj
on the Logic. 06
Vaguenu~.
and A~6ibte Set6.
Pow~6ut Auo~
L. F. Cabrera, (Chile), UnlVeMat Set6 ~e1.
15,00  15,30
Rem~k6
Set6.
06 Induc.Uon. 60~
Sel6duat
CW~U
06 80
H.P. Sankappanavar, (Brazil), A C~ct~zaUon 06 P~nc.lpat 06 Ve MM~an Atgeb~ and lU AppUc.aUo~.
Cong~enc.u
15,45  16,30
C.C. Pinter, (U.S.A.), Topo.togic.a.t Vua.tUlj TheMlj.in Log.ic..
A.e.geb~aic.
DECEMBER 22. 9,00 
9,20
L. Flores, (Chile),
9,20 
9,40
M. Manson, (Chile), Veontic., Manljvalued and No~ve Log.ico.
9,45  10,45 11, 00  11, 50
Hempel'~
Nomo.togical Veduc.Uve Model.
X. Caicedo, (Colombia), Bac.Izand60IL.th Slj6tein6 6M A~b~lj Quan
UMeM.
J. Malitz, (U.S.A.), Compact
F~gment6
06
High~ O~d~
Log.ic..
MATHEMATICAL LOGIC IN LATIN AMERICA A.I. Arruda, R. Chuaqui, N.C.A. da Costa (eds.) © NorthHolland Publ ishing Company, 1980
A SURVEY OF PARACONSISTENT LOGIC
C*)
AlJda I. AMuda
ABSTRACT. This paper constitutes a first attempt to sistematizethe present stateof thedevelopment of paraconsistent logic, as well as the main topics and open questions related to it. As we want this paper to have mainly an expository character, we wil I not in general be rigorous, especially when an intuitive presentation is better for a first understanding of thequestions under consideration, as, for example, in Section 1. Section 6 is perhaps the only one where the reader wi 11 find some original results. The bibl iography, though large, is of course not intended to be complete. A general idea of the content of this paper is given by the Index.
INDEX,
1.
Informal
1
Introduction.
2.
Paradoxes, Antinomies, and Hegel's Thesis.
3
3.
Historical Development of Paraconsistent Logic.
6
4.
Objectives and Methods of Constructijn of Paraconsistent Logics.
11
5.
Da Cos ta 's Pa racons is ten t Log i c.
13
6.
Paraconsistent Set Theory.,
17
7.
Miscellaneous Topics.
22
8.
The Philosophical Significance of Paraconsistent Logic.
24
9.
Open Questions.
26
10.
Bibl iography.
27
L
INFORMAL INTRODUCTION,
Let £ be a language and IF the set of formulas of £; then any non empty set of F i s i sa'id to be a
ptWPO.6~UOl1at
.6lJ.6tem 06 £.
sub
We say that a propositional
(*) This paper was partially written when the author was Visiting Professor at the Catho1 ic University of Chile, with a partial grant of the Fundacao de Amparo a Pesquisa do Estado de Sao Paulo (FAPESP), Brazi 1.
2
AYDA 1. ARRUDA
system S is ruv thUL.6 (or HeJUl.c..UtlL6Hege£'J.> thuLJ.» is the statement that there are true contradictions (cf. Petro v 1974). Sometimes, Hegel's thesis is also formulated as to imply that consistency is a sufficient but not necessary condition for the existence of abstract objects; concerning the existence of concrete objects, consistency is neither necessary nor sufficient. Clearly, Hegel's thesis can only be supported with the help of a paraconsistent logic. We shall show that at the abstract level Hegel's thesis is in fact true: there are paraconsistent theories (like the set theories described in Section 6) in which certain objects have inconsistent properties; for example, they belong and
A SURVEY OF PARACONSISTENT LOGIC
5
simultaneously do not belong to the same class. Therefore, one of the rna i n achievements of paraconsistent logic is to have proved that Hegel's thesis is true at the formal and abstract level. This means that an antinomy from the point of view of classical logic may be surprising enough  a veridical paradox from the standpoint of paraconsistent logic. (Such result seems to signify that in the field of logic there is a certain kind of relativism, an issue which we will not discuss here.) Now, what can we say about the validity of Hegel's thesis in connection with concrete objects? It seems to us that paraconsistent logic is unable to settl e such problem. Only special sciences and epistemology can establish the truth or falsity of Hegel's thesis at the level of real, concrete objects; or, equivalently, if the real world is consistent or not. Anyway, we believe that Petrov is right in his interpretation of existing antinomies: they do not prove that Hegel's thesis is true for concrete objects, but at least they give us some hints on the plausibility of it. According to Petrov 1971, p. 388: "No elimination of fallacy in scientific knowledge has negative consequences for the adequacy or the completeness of knowledge. "Certain ... antinomies (as the classical ones originated in quantum physics by the wave and corpuscular aspects of elementary particles) , however, can in principle be eliminated only with the aid of theories and methods the acceptance of which encroaches too much upon the adequacy or the completeness of knowledge. "The conclusion is therefore plausible that certain ... antinomies are not fallacies as standard logic wants us to believe, but are peculiar obj ective truths." Therefore, we can conclude that the fundamenta 1 si gnifi cance of paraconsi stent logic in connection with the common antinomies is that we are now able to accept Therefore, most of them as veridical paradoxes, at least at the abstract level. from now on we should not try to exclude antinomies a priori, because contradictions are forbidden by logic. Only a posteriori elimination of antinomies is legitimate depending on logical, scientific, and epistemological reasons. Finally, it is worthwhile to remark that, among contemporary phi 1osophers, Wittgenstein maintained original views about contradiction and logic: "Indeed, even at this stage, I predict a time when there will be mathematical investigations of calculi containing contradictions, and people will actually be proud of having emancipated themselves from contradiction." (Wittgenstein 1964, p. 332.) "If a contradiction were now a cctually found in arithmetic, that would only prove that arithmeti~ with such a contradiction in it could render a very good service. "
AYOA I. ARRUDA
6
(Wittgenstein 1956, p. 181e.)
3.
HISTORICAL DEVELOPMENT OF PARACONSISTENT LOGIC.
Several philosophers since Heraclitus, including Hegel, until Marx, Engel s, and the present day dialectical materialists, have proposed the thesis that contradictions are fundamental for the understanding of reality; in other words, they claim that real ity is contradictory, that is to say, that Hegel's thesis is' true in the real world. Clearly, if one accepts Hegel's thesis, one has to employ a new kind of logic (a paraconsistent logic), in order to study inconsistent but nontri vi al theories. Strangely enough, philosophers who accept Hegel's thesis have not estab· lished any formal system of paraconsistent logic. Instead of this, some of them have proposed the socalled dialectleal tagle, whose nature is rather obscure. Therefore, in our account of the historical development of paraconsi stent logic, we shall take into consideration only the work of logicians. According to ~ukasiewicz 1971, Aristotle had already an idea of the possibility of derrogation of the principle of contradiction, and consequently, of the possibility of paraconsistent logic. In effect, tukasiewicz writes: "Now according to Aristotle the principle of contradiction is not the highest law, at least not in the sense that it yields a neaesry pressuposition for all other l.oqi c a.l. axioms. In particular the principle of the syllogism is independent of the principle of contradiction. This is gotten from a Long overlooked and misunderstood passage in the Second Analytics: An. Post. All, 77a 1022 ...  'The impossibility of joint affirmation and denial is pressuposed by no proof (syllogysm) unless the conclusion itself was also to have demonstrated such. Then it is demonstrated insofar as one accepts that it is true to predicate the major term of the middle term and not true to deny it. But as far as concerns the middle term and likewise the minor term, it makes no difference to hold that it is and is not. If, for instance, an object is given (e. g., Callias) of which one can truthfully predicate that it is man and insofar as man just a living creature and not also not a living creature; so will it be true to predicate that Callias is a living creature and not also not a living creature, even if man were not man and Callias not Callias. The reason for this lies in the fact that the major term holds not only of the middle term but also of other objects as well because it has a greater range (than the middle term); so that it makes no difference in the conclusion, if the middle term is the
A SURVEY OF PARACONSISTENT LOGIC
7
same and not the same.' "According to Aristotle this syllogism is valid (A = living creature, B =man, C = Callias): B is A (and not also notA) C, which is notC, is Band notB C is A (and not also notA) "However, if a syllo gism remains val id when the pri ncipl e of contradiction doesn't, then the principle of syllogism (and indeed the dictum de omni et nullo) is independent of the principle of contradiction. " (cf. tukasiewicz 1971, pp. 503504.) Hence, perhaps Aristotle would deserve the title of founder of paraeonsis tent logic. Leaving out Aristotle, the two real forerunners of paraconsistent logic are J. tukas i ewi cz and N. A. Vas il 'ev. Both, i ndependently each other, argued between 1910 and 1911 that similarly to nonEuclidean geometry, a revision of the ba sic laws of Aristotelian logic would yield nonAristotelian logics. And both suggested to eliminate the principle of contradiction (it is interesting to note that Vasil'ev also proposed the el imination of the te.ilium VtOVt da.tuJt, thus being sometimes considered as a precursor of manyvalued logic, a field created by Post and lukasiewicz seve~al years later). tukasiewicz published in 1910, in Polish, a book and a Note (see tukasiewicz 19l0a and 19l0b), in which he tries to show that the principle of contradiction is not so fundamental as one usually thinks. He writes: "(A) The principle of contradiction cannot be proven by proclaiming it directly evident. For: (a') evidence does not appear to be a permissible criterion of truth; it turns out that false propositions as well are held to be evident (cf. the Cartesian proof of God). (b') the principle of contradiction does not appear to be evident to everyone: for the old eristic thinkers of Megara or for Hegel it was in all probability not evident. "(B) The principle of contradiction cannot be proven by setting it up as a natural law determined by the psychical organization of man. For: (a') it is possible td determine false proposltlons by our psychical organization (cf. e. g., many sensory hallucinations); (b') it is questionable whether the principle of contradiction can be validated as a iaw determined by the psychical organization of man. "(C) The principle of contradiction cannot be proven on the basis
8
AYDA 1. ARRUDA
of the definition of statements or negations. .•. For: (a') If one accepts that the negation "A is notE" means the falsity of the affirmation "A is B", then the principle of contradiction is not to be deduced therefrom. The notion of logical multiplication is not contained in the definition of negation, respectively falsity, and it is this notion which directly bestows on the principle of contradiction its characteristic imprint . . . . In terms of the definition of falsity or negation, however, it would still be possible to accept that the assertions "A is B" and "A is not B" hold at the same time in that they are both true and false at the same time. (b ") of course, if one prefers rather to avoid designating one and the same proposition as true and false, another definition of falsity can be set up which is of much greater account than the usual definition in terms of the basic thought in the concept, in that it is much more carefully formulated. The basic notion of falsity is, namely, that false propositions are no representation of the objective, or  in other words  that false propositions
correspond
to
If the principle of contradiction fails to hold now, there will be cases in which A is and is not Bat the same time. The principle of contradiction can in no way be derived from this definition of falsity." (eL tukasiewicz 1971, pp. 505506.) N. A. Vasil 'eV was a medical doctor who becomes Professor of Philosophy atthe University of Kazan, Russia. His ideas abo~t the possibility of derrogation of the 1aws of contradi ction and exc1uded mi ddl e were pub1i shed ina seri es of papers between 1910 and 1913. In these' papers he proposed some views for the construction of logical systems in which these laws are not valid. "Vas il 'ev int ended to cons t r uc t a nonAris tote l i an and universa l. logic, universal in the sense that it might cover an infinite number of logical systems. For him, a logical system is composed of two parts: that which he called Metalogic, i. e., 'an indispensable oore nothing objective.
of laws related to thought whioh are neoessary for any thinking and whioh oannot be eliminated from logio without its losing its logical charaoter' (Comey 1965); and a second part which we call here ontological basis of logie, i. e.,
'a varying range of laws which
are
funotions of the properties of the known objeots.' (Comey 1965.)
"Vasil' ev intended to see which postulates of logic could be changed or eliminated from logic without its losing itslogicalcharacter. Thus he was led to drop the (ontological) law of excluded middle and also the LAW OF CONTRADICTION which he took in the Kantian
A SURVEY OF PARACONSISTENT LOGIC form: ev
9
'no object can have a predicate which contradicts it'. Vasil'
distinguished the law of contradiction from the LAW
CONTRADICTION: 'one and the same judgement cannot ly true and false'. 1965.)
OF NONSELF
be s nul taneous
Vasil' ev took these to be different
The latter helong to me t a l o g i c , and the first,
aws'. (Corney retained,
L
would belong to the ontological basis of logic. "Vasil' ev considered worlds in connection to which t h e re are only three sorts of different basic (predicative) judgements: tive, "S is P"; negative, "S is not P"; and indiferent
affirma
(or contra
dictory ), "S is and is not P", such that only one of these
ments can be true for a given ohject and predicate.
judge
From these hy
potheses he delineated an Imaginary Logic with an ontological law of excluded fourth substituting the ontological law of excluded middle.
Later he generalized these ideas to a logic with an ontological law of excluded (n+l) t.h , n
> 2.
He also tried to show
that
his
logic
with his law of excluded fourth has a classical interpretation,
as
it is the case with the Imaginary Geometry of Lobachewski." (Arruda 1977, pp. 45.) The first logician to construct a system of paraconsistent propositional calculus was S. Jaskowsk i (see Jaskowsk i 1948 and 1949), following a suggestion of tukasiewicz.
He called his system
diocU6~ive
(or diocounhive) logic.
hensive and developed account of discussive logic may be found in
A compre
da Costa
and
Dubikajtis 1968 and 1977, and in Kotas and da Costa 197+a. Jaskowski motivated his discussive logic referring to several problems which originated the need of paraconsistent logic from Jaskowski 1948): 1)
\
(in the following, quotations are
'
The problem of organizing deductively theories which contain contradic
tions, as it happens with dialectics:
" 'The principle that
dictory statements are not both true and false is the
two
contra
most certain
This is how Aristotle ... formulates his opinion known as
of all'.
the logical principle of contradiction.
Examples of convincing rea
sonings which nevertheless yield contradictory conclusions were the reason why others sometimes disagreed stand.
with the
Stagiri t e ' s
That was why Aristotle's opinion was not in the least
versally shared in antiquity.
His opponents included Heraclitus of
Ephesus, Antistenes the Cynic, and others century Heraclitus ideas were taken up by Hegel, classical
firm uni
In
the early who
logic a new logic termed by him dialectics,
opposed
19 th to
in which co
existence of two contradictory statements is possible." 2)
To study theories where there are contradictions caused
by vagueness:
10
AYDA I. ARRUDA
"The contemporary formal approach to logic increases the precision of research in many fields, but it would not be correct to formulate Aristotle's principle of contradiction as: 'Two contradictory sentences are not both true'. We have namely to add: 'in the same language' or 'if the words occurring in those sentences have the same meaning'. This restriction is not always observed in every day usage, and in science too we often use terms that are more or less vague (in the sense explained by Kotarbinski ... ), as was noticed by Chwistek ... Any vagueness of the term a can result in a contradiction of sentences, because with reference to the same object X we may say that 'X is a' and also 'X is not a', according to the meaning of the term a adopted for the moment." 3) In order to study directly some empirical theories whose postulates or basic assumptions are contradictory. " ... it is known that the evolution of empirical disciplines is marked by periods in which the theorists are unable to explain the results of experiments by a homogeneous and consistent theory, but use different hypotheses, which are not always consistent with one another, to explain the various groups of phenomena. This applies, for instance, to physics in its presentday stage. Some hypotheses are even termed working hypotheses when they result in certain correct predictions, but have no chance of being accepted for good, since they fail in some other cases. A hypothesis which is known to be false is sometimes termed a fiction. In the opinion of Vaihinger ... fictions are characteristic of contemporary science and are indispensable instruments of scientific research. Wheth~r we accept that extremist and doubtful opinion or not, we have to take into account the fact that in some cases we have to do with a system of hypotheses which, if subjected to a too precise analysis, would show a contradiction among them or with a certain accepted law, but which we use in a way that is restricted so as not yield a selfevident falsehood." Jaskowski had already constructed a paraconsistent propositional calculus. but N. C. A. da Costa is actually the founder of paraconsistent logic. Independently of the work of Jaskowski, he started in 1958 (cf. da Costa 1958), to develop some ideas which led him to the construction of several systems (see Section 5) 0 f paraconsistent logic. including not only the propositional level but also the predicate level (with and without equality). the corresponding calculi of descriptions, as well as some applications to set theory. Da Costa systems were extended and studied by several authors as. for example, J. E. de Almeida Moura. E. H. Alves, M. Fidel, M. Guillaume, A. Lopari c , D. Marconi. R. Raggio, etc.
A SURVEY OF PARACONSISTENT LOGIC
11
Da Costa and his collaborators investigated also several other systems of paraconsistent logic, some of them having intimate connections with relevant logic (see Arruda and da Costa 1965). In the last years many logicians contributed to the development of paraconsistent logic (some of them quite independently of the works of Jaskowski and da Costa), for example: F.G. Asenjo, L. Dubikajtis, W. Dziobiak, T. Furmanowski, J. Kotas, L. H. Lopes dos Santos, R. K. Meyer, A. Neubauer, J. Perzanowski, G. Priest, R. Routley, and J. Tamburino. It deserves to be mentioned that D. Nelson in a pioneering paper (Nelson 1959) studied a system of paraconsistent logic and applied it to arithmetic. The term pakaeon6i¢tent log~e was coined by the Peruvian philosopher F. Mira Quesada, and was employed by the first time in a lecture delivered by him at the Third LatinAmerican Symposium on Mathematical Logic, held at the University of Campinas, Brazil, in 1976 (cf. Mira Quesada 1976). Today, paraconsistent logic is a growing field of logic, and is being cultivated especially in Brazil (Campinas and Sao Paulo). Poland (Torun and Katowice), Australia (Camberra), and U. S. A. (Pittsburgh). We shall end this historical sketch by also observing that nowadays paraconsistent loqic is beinq employed in the formalization of dialectical logic (da Costa and Wolf), and to study aspects of the thought of some dialectical materialists (Apostel). Although dialecticians think that dialectics is in principle unformalizable, the fact is that formalization constitutes a powerful tool to make dialectical logic, and dialectics in general, more understandable. Here we have a situation similar to intuitionism: although intuitionists think that intuitionistic logic is in principle not. formalizable, they do not deny that formalization is important for a better understanding of their logic.
4. OBJECTIVES AND METHODS OF CONSTRUCTION OF PARACONSISTENT LOGICS. In order to better motivate the study of paraconsistent logic  its objectives and methods of construction  we refer to other problems besides those mentioned by Jaskowski. 1) To study directly the socalled logical and semantical paradoxes. For example, if we want to study directly the paradoxes of set theory. not trying to avoid them (as usual), we need to construct set theories ifl which such paradoxes are derivable, but are not formal antinomies. In this case we need a paraconsistent logic. 2) To have logical systems in which paraconsistent theories may be based. For example. logical systems to found certain paraconsistent reconstruction of the theory of Meinong, different and possibly stronger versions of the usual set the
12
AYDA I. ARRUDA
ories, naive set theory, dialectics, and certain physical theories which perhaps are inconsistent (certain versions of quantum mechanics). 3) To study certain principles in their full strength as, for example, the principle of comprehension in set theory or in higherorder predicate logic. 4) To understand better the concept of negation. It seems that the notion of negation is precise only when we consider negations of simple propositions, as, for example, in the following sentences: "This sheet of paper is not white", or "These bottles are not cold". These simple and intuitive uses of negation can be extended in several ways, originating different categories of negati on, among which there will be the classical negation, and some paraconsistent negations. As we shall see, all negations that will be treated in this paper have some intuitive nucleus which is extended in several distinct ways. The creation of nonEuclidean geometries was one of the most important steps in the evolution of the human thought. Leaving aside their rel evance a s new mathematical disciplines, they have a philosophical significance. In effect, after the i r discovery, we understand better the meaning of some fundamental notions of science, and even the real meaning of scientific knowledge. We believe that the same happens in connection with paraconsistent logic: the construction of such logics, and of some heterodox logics, constitutes a fundamental experience of thought, whose byproducts are of basic importance for the understanding of the true meaning of logicity. There are two main methods of construction of paraconsistent logics: 1) Firstly, the syntactical method. By an appropriate modification of a given system of logic we can get an intuition of what would be a "good" paraconsistent logic. In this way, the systems C n , 1';;; n .;;; w , and the systems P and P* (see Arruda and da Costa 1965) were obtained. Of course, after a system of paraconsistent logic is constructed, it is reasonable and convenient to look for a semantics for it. This was the case concerning the above mentioned systems, as well as many others, for instance, the intuitionistic system. 2) Secondly, the semantical method. Given a well developed semantics which satisfies the conditions for being a semantics for a paraconsistent logic, we try to obtain the axiomatic system determined by it. This method was used by da Costa and Dubikajtis to construct higherorder discussive logics (cf. da Costa and Dubikajtis 1977). Evidently, the classification of the methods of construction of paraconsistent logics just sketched is neither rigorous nor complete. The fact is that the construction of paraconsistent logics ;s done in the same way as that of mathematical and logical structures. There is no "royal road" for this endeavour, the most important to achieve this construction is to have a certain intuition of the fecundity of the structure to be obtained. The truth of this is showed by the fact
13
A SURVEY OF PARACONSISTENT LOGIC
that paraconsistent logics are intimately connected with other branches of logic and mathematics; for example, with intuitionistic and relevant logic, manyvalued logic, algebra and topology.
We may say that one of the characteristics
0
f
the
importance of paraconsistent logic is its interconnection with various branches, already well established, of logic and mathematics.
In this way,
paraconsistent
logic has roots not only in the cultural tradition but also firm roots
in logic
and mathematics, from which it gets some of its inspiration, and for which it will contibute with new basic and interesting ideas.
5. DA COSTA'S PARACONSISTENT LOGICS. Loosely speaking, a paraconsistent logic is a logic in which a contradiction,
A & lA, is not in general an antinomy.
The propositional calcul i Cn
,
1 (B::> A)
3)
A::> (B::> A
6)
(A::>C) ::> ((B::>C) ::>(AVB::> C))
9)
A,A::>B/B
&B)
12)
B(n) ::> ((A::>B) ::> ((A::> IB) ::> IA))
13)
A(n)
&B(n)
::> (A::> B) (n)
The postulates of Cw DEF I NI TI ON.
THEOREM 1. 
2)
(A ::>B) ::> ((A::>(B::>C))::> (A::>C))
4)
A &B ::> A
5)
A &B ::> B
7)
A::> AVB
8)
B ::> AV B
10)
ilA::> A
11)
& (A &B) (n) & (A V
B) (n)
are 111.
In Cn , 1 .;; n < w, 1*A is an abbrevi ati on for
16
AV iA
I A & A(n) .
1 A in the .tYltuUi.onv..uC' po~ilive pftopol>ilional C'a.f.cu.f.uI>,
then, I A in Cw •
THEOREM 2. 
AU the we and valid l.lC'hemata 06 the uaM.tC'al pOl.lilive pftOp
ol.lilional Calc.utu. one. a.f.!.lo valid in Cn,
1';; n
C I 3x A(x) :J C
IV)
C:J A(xl / C:J 'tJx Af xl ,
V) In A and B Me. C'ongI1.Ue.11.t n0ltmulcu, (cf. Kleene 1952), 011. one. Ls obtaine.d nl1.om the. othe.l1. by the. eLUrvLnation 06 vaC'uou.6 quanti6.{.elrJ.>, then A '" B iJ., an auom.
\Jx(A(x))(n):::> (\JxA(x))(n)
VI)
The postulates of CC,)* are those of Cw plus IV above. The postulates of
c;; , 1';; n ';;·W,
are those of the corresponding
C:
pl us
C:
(C;),
the fo 11owing: I
)
x=
II"')
x
C:
I
\
Theorems 17 are easi ly extended to
x = y:J (A(x) :::> A(yl).
(C~ ).
The semantics for
n < w , is an extension of that of Cn • 1 ~ n < w (see Arruda and da Costa 1977), but a valuation semantics for C~ (C::O) is still an open problem.
1
~
THEOREM 10. 
The. C'atC'u£.{.
THEOREM 11.
In r IA .in Cn*, them aU the k:t!l.a.n6n0ltm.o On A Me deduuble in
C:
(C;;),
Cn , 1 .;; n ~ w, nl1.om the. fztMn.o noJtm.o
THEOREM 12. Cn
the n0ltmulcu, in
r.
In the. .cymbal = doe..o not OC'C'U!l. in the 60ltmula A, them 1 A in
i.n, and only in, 5.3.
on
1';; n .;; w , Me undeudable.
1 A
in C: , 1';; n ~ w .
THE CALCULI OF DESCRIPTIONS D n , 1 .;; n
The calculi of descriptions Dn , 1
~
~
w.
n c w, are obtained from Cn,l';;;
n~
w,
17
A SURVEY OF PARACONSISTENT LOGIC
introducing the description symbol
t,
and the postulates DID5 below. The symbol
ism and conventions are borrowed from Rosser 1953, with clear adaptations. If FIx) is a formula, them "the object x such that FIx)" is denoted by txF(x). If there is one, and only one, object which satisfies F{x), this object; otherwise,
rx
Ff x) will
txF{x) will denote an arbitrary object.
denote
The semantics
of Cn , 1";; n < w , can be extended to D n , 1";; n < w , as mentioned in and da Costa 1977; but D w still lacks a good semantics.
Arruda
The postulates of D n are those of C n ' 1";; n ..;; w, plus the following (where the restrictions are the usual ones):
Fxl
Dl.
IJx
D2.
IJx (PIx)
D3.
t x F[x) = t y F{y)
:J F[t Y Q{y))
== Q{x))
:J tx PIx) = t x Q[x)
D4. P[ty Q{y)):J 3x PIx) 55. 3jx Pix)
[lJx([tx P{x) =x)
:J
THEOREM 13. 
in
r
Dn ,
1";; n
Let A j ' A 2' . . . ,
Then
U {AL
Am be the pJUme componen:t6 06 :the
IA in Do i6, and only i6,
A~n)
,".,
A~n)
r
r
in
IA
w •
THEOREM 14.
Let F be a 60tunuLa 06 Do, and
wbf.>:tduting ,* 601t "l ,
Then,
F* the 60tunuLa obtained
IF in Do i6, and only i6,
6ltom F
1 F* in D n,
1";;
n<w. THEOREM 15. 
6,
Dn
if.>
a COnf.>eJtvative edenf.>ion 06 C n ' 1";; n ..;; w .
PARACONSISTENT SET THEORIES.
One of the aims of paraconsistent set theories is the study of the
conse
quences of the schema of separation when we employ as subj acent log i c a pa raconsistent logic.
In the usual set theories one weakens the schema
and maintains classical logic as subjacent logic.
of separation
In paraconsistent
set theo
ries, we try to weaken the subjacent logic and to strengthen the usual tions of the schema of separation, in order to obtain
"inconsistent
The basic problem of the construction of strong paraconsistent set which the schema of separation can
formul asets"
theories
. in
be formul a ted without restrictions to avoid
antinomies, but not formal paradoxes, has not been solved yet.
Particularly, be
cause there has not been found adequate paraconsistent logics to attack the problem.
Nonetheless, while
not even trying to solve the basic problem of paracon
18
AYDA 1. ARRUDA
sistent set theories, we can investigate the following two problems, whose solutions maygive some hints on the heterodox properties of paraconsistent set theories. PROBLEM 1. 
Admitti ng the exi stence of some sets wh i ch do not
usual set theories, to study their properties. ties of Russell's set. Ro =
x ixif x) .
ex i s tin the
For example, to study the proper
PROBLEM 2. To investigate the conjecture according to which when we weaken the subjacent logic we can obtain set theories ex.Wtenc..i.aU'1 .6bwngeJ1.
th a n the
usual ones. DEFINITION. guage. £ Fix).
Let T and T' be two nontrivial set theories having the same lan
T is said to be ex.WtenuaU'1 .6.tJz.ongeJ1. than T', if, t 3£F(x) in T' implies that
for every abstract
t 3£F(x) in T. and there
exists at
least one abstract £G(x) such that t 3xG(x) in T but not in T'. 3£F(x) means that £F(x)
exists (cf. Rosser 1953, pp. 219220).
Of course, (Informally
speaking, we say that T is existentially stronger than T' if all sets which exist in T' do also exist in T, and there exists at least one set in T that does not exist in T' .) For the study of these two problems, the systems Cn, i. c n c c , are adeOn the other hand, if we want to obtain paraconsistent set theories exis
quate.
tentially stronger than the usual ones, it is easier to start with NF good development of NF, see Rosser 1953). first approach to the study of
Problem~
(for a
Since we are interested only
1 and 2, we consider here only a
version of NF. whose postulates are given in Rosser 1953, pp. 212213. = as a primitive symbol.
ina weak taking
Since we have already mentioned that Do is essentially
equivalent to the calculus of descriptions given in Rosser 1953, then we axiomatize this weak version of NF. here denoted by NF o • in the following way. The postulates of NFo are those of Do plus the following: EXTENSIONALITY: \;/x\;/'1\;/z((XE'1=XEZ) :l'1=zl. SEPARATION: 3'1 \;/x(XE'1 == Fix)), in the case x and fJ are different variables. fJ does not occur free in F(x), and F(x) is stratified.
The conventions and set theoretical notations are like those of Rosser 1953, with clear adaptations. Starting with NF o we construct in this section a hierarchy of set theori es
19
A SURVEY OF PARACONSISTENT LOGIC
w, having the corresponding Dn , 1';;; n';;; w, as subjacent logics. Before sketching these theories it is worthwhile mentioning the problems re
NF
, 1.;;; n';;;
n
lated to the formulation of the schema of separation in NFn , 1';;; n';;; w. Many forms of the schema of separati on for NFn , 1';;; n .;;; w, have been proposed, but most of them are proved to trivialize the corresponding NF n (see, for instance Arruda 1975b and 197+). tion for NF n
An apparently sure formulation of the schema of separa
' 1.;;; n';;; w,
is proposed in Arruda 197+; nonetheless, this fonnula
tion is not adequate for the study of Problem 2.
Taking into account this fact,
we present a weak form of NFn , 1';;; n < w , but a strong version of NFw • It is also convenient to clarify the meaning of the phrase "Russell set" that The Russell set for NF o is Ro =
will be used in this section.
xl x f/
xl .
Th e
Russell set for NF n is Rn = x(x f/ x & (x E xl (n)). 1 .;; n < w. Since Rn trivializes the corresponding NFn • it cannot exist in NFn , but can exist in NFm , m> n.
In order to have a better understanding of the properties of Russell sets, it is convenient to strengthen the weak version of NFn , 1';;; n < w, by the introduction of Quine individuals (x is a QtL 1 < n < w .
von Neumann 
5) To see if the systems P and P* (Arruda and da Costa 1965a), plus the schema of separation without restrictions to avoid formal antinomies, but not formal paradoxes. are trivial or not. 6) To develop paraconsistent higherorder modal and tense logics. 7) To find a simplified formulation of the axiomatics of discussive presented in da Costa and Dubikajtis 1977.
logic
8) To try to adapt world semantics for the calculi Cn • 1< n<w. and to verify if the resulting semantics are more intuitive than those already known (as for example, those of da Costa and Alves 1976, and Loparic 1977).
9) To develop modal and tense paraconsistent logics (some hints are given in Alves 1976). 10) To complete the algebraic study of the systems Cn , 1 < n < co , in the various versions as, for example, those of Fidel 1977, da Costa 1966, and Sette 1971) . ~
11) To continue the study of higherorder paraconsistent logic began in Alves and Moura 1978. 12) To study systematically the relations among relevant and paraconsistent logic.
10.
BIBLIOGRAPHY.
In the following bibliography we list papers on paraconsistent logic and related topic~. Of c~urse, due to the vastness of the subject (especially the related topics), this bibliography is not intended to be complete. We no not mention, for example. abstracts of published papers, neither many papers on dialectical logic, relevant logics, and nonclassical logics in general. The works with
28
AYDA I. ARRUDA
an * are not on paraconsistent logic but are mentioned in the text as references for notations, terminology, and axiomatizations of classical logical systems. G.
Achtelick, L. Dubikajtis, E. Dudek, and J. Konior.
197+.
On .the independence 06 ax,[omb 06 JaifzOWO/U'6 di6cu66ive
p'wp06ilional
ca.lculu!>, to appear. A. R. Anderson and N. D, Belnap. 1976. Entailment, The Logic of Relevance and Necessity, vol. 1,
Princeton University Press (vol. II, to appear). E. H. Alves.
1976.
1978.
Logica e Inconsistencia: Um Estudo dos Calculos Cn,
l';; n .;; w
(Master Thesis), preprint by the Institute of Mathematics, Universidade Estadual de Campinas, Campinas, Brazil. On .the decUdabUillj 06 a blj6.tem 06 d£a.lecUcal ptwpo6ilional £.ogic, Bulletin of the Section of Logic, Polish Acad. of Sciences,7, pp. 179184.
E. H. Alves and J. E. de Almeida Moura. 1978.
On !>Ome h£ghVtOtLdVt pMaCOYl!.lib.te.nt calcuU, in
Mathematical Logic,
Proceedings of the First Brazilian Conference (Eds. A. I. Arru
da, N. C. A. da Costa and R. Chuaqui), Marcel Dekker Inc., New York, 18.
pp.
L. Aposte1. 1967.
Logique. e.t: d£ale.cUque., in Logique et Connaissance Scientifique,
197+.
Log,[ca e d£a.le.t:Uca In. Hege..t, in La Formalizazzione della Dialet
(Ed. J. Piaget), Ga11imard,
P~ris,
tica (Ed. D. Marconi), Rosemberg
pp. 357374.
&Se11ier, Torino, to appear.
Aristotle. 1955. The Works of Aristotle, vol. 1 (logical works), Oxford University Press. A. I. Arruda. 1964. Considerac;oes sobre os Sistemas Formais NFn (Thesis), Universidade Federal do Parana, Curitiba, Brazil. 1967. SUIt WlC h£Vuvr.ch£c de cafcu..t6 ptwp06ilionne..t6, C. R. Acad. Sc. Paris 265, pp. 641644. 1968a. SUIt Wle. h£Vr.aJr.ch£e. de cafcu..t6 p~op06£t:,[onne..t6, C. R. Acad. Sc. Paris 266, pp. 3739. 1968b.
SUIt WlC
h£Vuvr.ch£e de ca.tcu..t6 ptwpo6iliorme..t6,
C. R. Acad.
Sc .
Paris
A SURVEY OF PARACONSISTENT LOGIC
29
266, pp. 897900. 1969a. SWt une hi~c.hie de c.alc.tLt6 de p!CecUc.a.t6, C. R. Acad. Sc, Pari s 268, pp. 629632. 1969b. SWt c.eA:tainu algeb!Cu de UiL6.6U vwnuiL6.t>lj.6teme NFw ' C. R. Acad. Sc. Paris. 270 A, pp , 11371139. 1971. La mathemat. on the log.LC 06 vaguenco.6, to appear. 197+b. A.6emanUcal .6tudy 06 .6Qme .6Y.6te~ 06 vaguenco.6 log.Lc, to appear. 197+c. On the log.LC 06 vaguenu.t>, to appear. A. 1. Arruda and N. C. A. da Costa. SWt une hi~hie de .6y.t>temu 60!lJrlm, c. R. Acad. Sc. Paris 259, pp,
1964.
29432945.
o pateadoxo
de CU!C!Cy  Moh ShawKwe..L, Boletim da Sociedade Matematica de Sao Paulo 18, fascs. 19 e 29, pp. 8389. 1968a. On the po.t>tulate 06 .6epMation, Notices AMS 15, pp. 399400. 1968b. FUl!.thete COn.6..Ldeteation.6 on the pO.6tulate 06 .6epMation, Notices AMS 15', p. 555. 1970. SWt le .6chema de la .6epMat.{on, Nagoya Mathematiaal Jurnal 38, pp. 7184. 1974. Le .6chema de la .6epMat.{on dan.6 lu .6Y.6teme.6 ]n' Mathematica Japonicae 19, pp, 183186.
1965.
30 1977.
AYDA I. ARRUDA Une 4emantique po«4
te catQUt C~, C. R. Acad. Sc. Paris 284 A, pp. 279
282. F. G. Asenjo. 1965. Viatect[c !og~c, Logique et Analyse VIII, pp. 321326. 1966. A catcutu4 06 antinom~u, Notre Dame Journal of Formal Logic VII, pp. 103105. 1972. On dialect[c tog~c, Teorema II, pp. 133134. F. G. Asenjo and J. Tamburino. 1975. Log~c 06 ant{Y!om~e6, Notre Dame Journal of Formal Logic XVI, 278.
pp. 272
N. D. Belnap Jr. 1977. A Me6ull 6oU/tvatued !og~, in Modern Uses of MultipleValued Logic (Eds. J. M. Dunn and G. Epstein), Reidel, Dordrecht, pp. 537. J. Bl aszczuk. 1978. weakut Y!OJunat catcu.U wdh fLupea to MY!  counte.JtpCVtt6, Bulletin of the Section of Logic, Polish Acad. of Sciences, 7, pp. 102106. J. Blaszczuk and W. Dziobiak. 1975a. RemafLk4 on Pe.fLzanoW4k~'.6 moM .0y.o tem6 , Bulletin on the Section of Logic, Polish Acad. of Sciences, 4, pp. 5764. 1975b. Moda! .oy.otem.o fLe!ated to S4 n 06 Soboc.~i1.6Iz~, Bulletin of the Section of Logic, Polish Aca~. of Sciences, 4, pp. 103108. 1976a. Moda! .oy.6tem.o ptac.ed ~n the "tfL~ang!e" S5T 1 T, Bulletin of the Section of Logic, Polish Acad. of Sciences, 5, pp. 138142. 1976b. An a)(~omat~zat~on 06 Mn  c.ountefLpafLt.6 60Jt .oome moda! ca.ecu.e~, Reports on Mathematical Logic 6, pp. 36. 1977. Moda! !09~C.o connected w~th S4 n 06 Soboc~i1.6Iz~, Studia logica XXXVI, pp. 151164. B. Bosanquet. 1906. ContfLad~ct~on and JteaLi..ty,
Mind 15, pp. 112.
D. D. Corney. 1965.
Review of V. A. Smirnov 1962, The Journal of Symbolic Logic 30, pp. 368370.
A SURVEY OF PARACONSISTENT LOGIC
31
N. C. A. da Costa. Nata ~ob~e a ~on~eito de contnadi~o, Anuario da Sociedade Paranaense de Matematica 1, nova serie, pp. 68. 1959. Ob~~va~ou ~ob~e a ~on~Uto de ewt~nua em matemiiU.~a, Anuari 0 da Sociedade Paranaense de Matematica 2, pp. 1619. 1963a. Sistemas Formais Inconsistentes (Thesis), Universidade Federal do Parana, Curitiba, Brazil. 1963b. ca£cuto p!!.Opo~ili.onnet6 pou): tu ~y~temu nOJlmet6 in~o~i6tanU, C. R. Acad. Sc. Paris 257, pp. 37903793. 1964a. ca£~uto de pJtedi~aU poun: tu ~yMemu 60~et6 in~o~~tanU, C. R. Acad. Sc. Paris 258, pp. 2729. 1964b. ca£~uto de p~edi~ ave~ egaLUe poun. tu ~y~temu no~et6 in~~~ ~, C. R. Acad. Sc. Paris 258, pp. 11111113. 1964c. ca£c.uto de ducJL.iptio~ poWt tu ~ yMimu no~et6 inc.o~~tanU, C. R. Acad. Sc. Paris 258, pp. 13661368. 1964d. SWt un. ~y~teme in~o~i6tant de theoJL.ie du e~emblu, C. R. Acad. Sc. Paris 258, pp. 31443147. 1965. SWt lu ~y~te.mu no~e~ Cl ' C;, C~, Vi e.t NFi' C. R. Acad. Sc. Paris 260, pp. 54275430. 1966a. Algebras de Curry, Sao Paulo, Brazil. 1966b. Op~o~ nonmonotonU da~ lu ;fAeJ.UM, C. R. Acad. Sc. Paris 263 A, pp. 429432. 1967a .. Une nouvelle hiVLM~hie de theoJL.iu in~o~i6tanU, Publications du oepartement deMathematiques,Universite de Lyon, France, 4, pp. 28. 1967b. F~u e:t ideaux d'une a£gebJte Cn' C. R. Acad. Sc. Paris 264 A, pp. 549552. ~ 1971. Rema~que~ ~Wt t« ~y~teme NFl' C. R. Acad. Sc. Paris 272 A, pp. 11491151. 1974a. RemaJtque~ ~Wt lu ~al~uto Cn' c~, C~ e:t Dn, C. R. Acad. Sc. Paris 278 A, pp. 819821. 1974b. On the theoJty 06 in~o~~tent 6o~a£ ~y~t~, Notre Dame Journal of Formal Logic XI, pp. 497510. 1975a. RemaJtk.~ on Jaik.ow~H di~cuMive togic, Reports on Mathematical Logic 4, pp. 716. 1975b. Review of Asenjo and Tamburino 1975, MatherRatica1 Reviews 50 # 9545. 1979. Ensaio Sobre as Fundamentos da L6gica, HUCITEC, Sao Paulo, Brazil . 1958.
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N. C.
1976.
AYDA I. ARRUDA A. da Costa and E. H. Alves. Une 6eman..t6ive. plt.opo!.>\.tionaR. eateuR.u!.>, Bulletin of the Section of Logic, Polish Acad. of Sciences, 4, pp. 3347. 1975b. Re.malt.11.6 on di!.>eu!.l6ive. plt.opo!.>I.tI.onaR. eateutu6, Studi a Logi ca 34, pp.3943. Y. Gauthier. 1967.
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to appear.
33
34
AYDA I. ARRUDA
R. Gotesky. 1968.
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The U6C¢ 06 ~nco~~tency,
XXVIII,
pp. 471500.
J. Grant. 1974.
Incomplete model.o,
Notre Dame Journal of Formal Logic XV,
pp.
601607. 1975.
Incon!>~!>tent and ~ncomplete logic!>,
Mathematics Magazine 48,
pp. 154159. 1978.
Notre Dame Journal of Formal
CI'.lu.!>i~ca.aon 60Jt incon!>i!>tent theoJtic¢,
Logic XIX, pp. 435444.
G. Gunther. 1958.
Me
V~e a.Jti.otot~che Log~11. dc¢ SUn!> und
dCJt Re6lexion.
~chta.Jti.otot~c.he Logil1.
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pp. 360
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1964.
Va.!> PMb.e.em unCJt FoJtm~~CJtUJ1g dCJt .:tJta.lt6zcYldenta1.Ma1.e/i..t.U,chen Log~l1.,
Hegel Studien I, pp , 65130.
1971.
Vie ~to~che KategoJtie de.o NeueYl,
1972.
NatUnliche Za.h.e und V~el1.til1.,
HegelJahrbuch,
pp. 3261.
HegelJahrbuch, pp. 1532.
B. Hartmann.
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Vie BegJti66I>loMI1. aU ~ek.:t;i!>che,
•
HegelJahrbuch,
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K. G. Havas. 1974.
V~e
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HegelJahrbuch,
362265. S. 1. Hessen. 1910.
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Logos 2,
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S. Jaskowski.
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Rachunell. z:dan d.e.a !>y!>temow dedul1.cyjnyeh !>pueez:nych,
Societatis Scientiarun Torunensis, Sectio A, I, n95, 1949.
Studi a
pp.5577.
0 Konjunkcj~ dy!>ku!>yjnej w Jta.chunl1.u z:dan dla !>y!>temow dedul1.cyjnuch !>pJtz:ecz:cych,
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Studia Societatis Scientiarun Torunensis
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A SURVEY OF PARACONSISTENT LOGIC 1969.
Pl1.opo6iliona.f. ea.f.c.u1.u6 6011. corWi.a.di.etOI1.Ij deductive 61j6tem6, Studia Logica XXIV, pp. 143157. (English translation of Jaskowski 1948.)
H. W. Johnstone. 1960.
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pp. 310.
Van Nostrand.
G. K1 ine. 1965.
N. A. VMU'ev and the de.vetopme.nt 06 manljva.f.ue.d tog.ic, tributions to Logic and Methodology in Honor Bochenski (Ed. A. T. Tymieniecka),
of
in Con
J. M.
NorthHolland, Amsterdam,
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39
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Cambridge Uni vers i ry Press,
A SURVEY OF PARACONSISTENT LOGIC
Universidade Estadual de Campinas Departamento de Matematica Campinas, SP., Brazil a~
Universidad Catolica de Chile Instituto de Matematica Santiago, Chile.
41
MATHEMATICAL LOGIC IN LATIN AMERICA A.I. Arruda, R. Chuaqui. N.C.A. da Costa (eds.) © NorthHolland Publishing Company, 1980
ON STRONG AXIOMS OF INDUCTION IN SET THEORY AND ARITHMETIC MbtO.6.ta.v Benda.
ABSTRACT.
Althought Godel showed arithmetic incomplete, his work did not point out a natural extension of it which would make it more complete. This changed with Paris' discovery and Harrington's simpl ification of Paris' work. We continue in this direction by considering schemas which are more akin to general principles than the ad hoc sentences of Paris and Harrington. They can be shown to be a natural strengthening of the axiom of induction and their interesting feature is that they apply to set theory as well. The lecture shan illuminate the main points in proving their independence, shall outl ine their relative consistency strength and raise the issue of their appl icabil ity in arithmetical situations.
I NTRODUCTI ON. A lot of activity has occurred since J. Paris (see Paris 197+) discovered a mathematical sentence independent of the axioms Peano's arithmetic, PA.Harrington simplified Paris' sentence making it close to Ramsey's Theorem (see Harrington and Paris 1977). In Solovay 197+ , Solovay placed a function connected with the new Ramsey relation within a hierarchy of recursive functions which gave a new proof of independence for HarringtonParis sentence. In Silver 197+, Silver gave an exposition of HarringtonParis to which we owe some debt in that our combinatorial background resembles his, however, what we needed to prove flowed naturally to Silverlike partition result. More recently, alternative sentences appeared (see Ketonen 197+ and Pudlak 197+). The paper of McAloon, McAloon 1978, is a nice exposition. In another direction the combinatorial aspects of the strong Ramsey theorems were treated in Erdos and Mills 197+ and denda 1979a . In our view there are two issues of paramount philosophical importance which come out of Paris' result. One of them is the fact that we have a true sentence on the basis of which we can prove by purely finitistic means the consistency of arithmetic. In view of Godel's theorem, this is probably the ultimate we can achieve in proving the consistency of arithmetic. The other issue con ce rn s axiomatization of arithmetic. Since Pea no nobody came with an axiom which was forgotten by him. Godel did find a sentence without derivation from Peano's axioms but it does not seem to be relevant for purely arithmetical questions. The sentences we have mentioned above are much closer to being applicable in arithmetic hence Paris' discovery may be regarded as a first step in finding a schema describing a universal method of proof much like the schema of induction does. We think that this is the most exciting open problem in this area. We would like to think that the present paper is a step in this direction. We 43
44
MIROSLAV BENDA
define certain axiom schemas AD, AI' ... which are stronger and stronger, the step from An to An+1 being an addition of the combination if 3 to the axioms in An" The schema AO is equivalent to the induction schema. The schemas are formulated in the language of set theory. It is wellknown that PA is (up to relative interpretability) Zermelo's set theory with the axiom that everything is finite replacing the axiom of infinity. It turns out that the finite Ramsey theorem plays, in PA, the same role as the ErdosRado theorem in ZFC. In order to have the full benefit of this theorem in Zerme10's set theory, we take as our basis the following theory: T = Zermelo's set theory without the axiom of infinity and with the axiom "if K is a cardinal so is 2K " • Then by PA (an abuse of notation) we denote the and by ZP (p for power) we denote T + "there is much weaker than ZFC, since VK where K =]Lw which we prove essentially in T demonstrating with the schemas, are: 1.
PA + Al
~
Con (PA)
2.
ZP + Al
1
Con (ZF)
theory T + "everything is finite" an infinite set". Note that ZP is is a model for it. Our results, that infinity has not much to do
Item 1. is not really new, we have shown the connection of Al to HarringtonParis elsewhere (see Harrington and Paris 1977). Item 2 is new, we were surprised that we could get by with ZP only. If we regard Al as true (see the last section for a discussion of this point) and think that the chances of consistency of ZP are better than those of ZF we see from 2, that, as in arithmetic, we have improved the chances of the consistency of ZF. As already mentioned, T denotes the Zermelo's set theory with the axiom "if K is a cardinal so is ZK (= Ip(K)I)" and without the axiom of infinity. In this theory, we have the usual notions as in ZF. We shall often refer to classes of T, these should be though of as their defining formulas. By C we denote the class of cardinals. If we say "for arbitrarily large x" we mean (ifKE C)(3x) [ixl "'K ... J. V is the class of all sets, N is the class (it is not a set in T) of natural numbers. Vex where ex E On (ordinal s) is the set of sets rank < ex. If R is a class then IIRII =
{Ixl:
xER}.
By Pw ( X) we denote the finite subsets of the set X. relations
~Je
use the Erdos partition
K ....
t
which mean that if 6: [K)t .... c (where [XJ is the set of subsets of X of size t) then there is If ~ K, Iff 1'" A which is 6 homogeneous that is 6 is constant on [ff)t. The finite Ramsey theorem is (vt)(lfc.) (ifm)(3n) [n .... (m)~) quantifiers limited to N) and the ErdosRado theorem is
(all
ON STRONG AXIOMS
OF INDUCTION
45
1. COMBINATORIAL PART. DEFINITION 1.1.  let R be a class of sets.
Then by R' we denote the class
{ftER I for arbitrarily large 6 ER,
ft~6}.
We call R' the derivative of R. The derivative may be iterated: RCl+ 1 (R Cl), and RCl = n{R !iJO is limit. iJ The definition of the derivate is related to the definition of a derivative of a modeloid (see Benda 1979a). In one version of exposition a modeloid is a set of partial onetoone functions on a set and its derivative consists of functions which can be extended within the modeloid to functions whose domain or range contain prescribed element, or, speaking loosely, which can be extended to a lot of functions in the modeloid. This analogy is not artificial for, as we shall see, there is a link between derivatives of classes of sets and derivatives ofmodeloids and the way we chose the definition of a derivative of a class gives the right analogue to the modeloid derivative. On the other hand, a slightly more cumbersome but weaker definition of a modeloid would yield most of the results in this paper; we shall state this precisely later. DEFINITION 1.2. let mEN and R a class of sets. Ramsey if for some cardinal n a)
ft~6ER
b)
litl>m and
c)
if 16
I;;;.
We say that R is m
impli'es ftER (\16)[
6
C
It. ....
6ERj
implies ftER
n then there is ftE R such that
lit I;;;. m and it C 6.
Definitions 1.1 and 1.2 are of paramount importance for this paper. definition enables us to state the axioms more neatly.
One more
DEFINITION 1.3. If R isa class of sets then IIRII = {1ft
I: ftER}.
The axioms we are going to investigate have this form
First of all, let us get some examples of Ramsey sets: PROPOSITION 1.4..sezs homogeneoU6 6O!t eVe!ttj
Let
6.£
:to
6.: [Vj..t .... C. for ..t .t
.£.;;;~EN.Then:thec..fM606
Ls Ram;.,etj (6O!t 6ome m).
PROOF: let m=max{:t.I'£';;;Q}+1. We show that R is mRamsey. Condi.c tion (a) is clearly satisfied. For (b) let lit I > m and assume that any set 6~ft
MIROSLAV BENDA
46
t of smaller size is in R. We show that It is nohomogeneous. Let a, bE[t] 0 If au b has size smaller than It we are done, i.e. nOla) = nO(b). If not then lau b 1= It. Since lit I> to + 1 there are x E a  band yE b  a and tQe sets au{y} and bu{x} are in R so they are nO homogeneous. Let cE[It] 0 be a set which does not contain x nor y. Then
because Cu (a  {xl ) U {y} is a proper subset of for the other equality. But for the same reason
n(a) showing n(a)
=
It,
n«a  {x}) u {y}) = n«b 
so is in Rand simil arly {y})U
Ixl ) = neb)
= neb).
For condition (c) the existence of n follows, in PA, from the finite Ramsey theorem, and in ZP, from the ErdosRado theorem. The sketch of proof in PA is, we take n suchtthat n til
nQ _ 1
~ (m)~Q
to o
~ (no)c
where nO is such that nO
Similarly, in ZP.
~ (n
t1 1 )c etc., un1
Alternatively, one can splice together results with
the functions n{ into a single function and use the combinatorial out iteration. • We shall now state precisely the axiom schema AO: (A For: eVeJLy O):
mE
{n
N
R J.i, Rll.»U>ey then IIRII = N.
Clearly, this is a schema; note that even though m does not appear in the scope of the quantifier one should remember that R stands for a formul a which may have m as a variable besides, of course, some other variables which are quantified over universally. PROPOSI TI ON 1.5.  The naUaw.£ng (i) (t t )
The auom M.hema
(Aolt.
The {nduc.ti.an .t>chema {n
PROOF:
Me equi.valent:
PA.
For convenience we write the induction schema a follows: \;/x (\;/y (If
C
x
~ ¢(y))~ ¢(x))~ \;/x
¢(x)
This is easily seen to be equivalent to the usual induction schema axiom of foundation in this setup). Now define ~(lf) by \;/x (Ixl.;;
Iyl ~
(which is the
¢ (x)).
We claim that ~ defines a Ramsey set with m = n = 0 and, indeed, it is easy to check the conditions (a)  (c). We now use (A O) to conclude that
ON STRONG AXIOMS OF INDUCTION
47
IJn IJx([xl < n .... ¢(x))
which is the same as IJ x e . To prove the converse, we use the finite Ramsey Theorem which is a consequence of the axiom of induction. Say ~(x, m ••• ) is a formula and m is given such that R=
Ix
11/1 (
x, m... )}
is Ramsey. From this, we get n satisfying condition (c). let be such that
n
Let q;;>max(m, n}, and
n .... (q)~.
Take a set t
of size
n
and for .6E[t]m
define
Let H~t be homogeneous for 6 of size IHI;;> n ; it follows from (c) that we have .6E[H]m such that 6(.6) = 0, therefore 6 is constantly 0 on [H] m. Now by (b) this implies that [H]m+l CR, which in turn implies that [H jm+2 C R etc , , finally getting HER. Therefore, qEIIRIi. • Having identified' (A O)' let us now discuss (AI)' It turns out that (AI) is independent of both arithmetic and set theory for the reason that it implies their consistency. We shall prove its consistency for arithmetic (in ZFC) in fact, we shall show that (An) is true. If we use the language Lw w in which the schema (A 0: ) may be formal ized for 0: < wI then even (Ao:) is true I for any 0: < wI'
Note that if 0: < "' each of the sentences in (A 0: ) is firstorder. Thei r complexity, as far as alternation of quantifiers is concerned, increases, each step adding a combination IJx 3y. (A) then may be formulated as countable conjunctions of first order sentences etc. In general, (A) may be formulated in L 10: [ + ,w (0: > w) . 0: PROPOSI TI ON 1. 6. (ZFC). Fatz. Ra.m.6ey Theatz.em i..6 eqtu:va1.ent to ,i6atz.
0: 0:
< wI' < wI'
V w 1= (A 0:)'
In 6a.c...t, the in6btUe
VW 1= (Ao:)".
PROOF: Let R be Ramsey for some mEN; it need not be definable withinV w' Define 6: [V w]m .... 2 by if a.E R if a.'1c R
48
MIRaSLAV BENDA
By the infinite Ramsey Theorem, we find H £ Vw infinite homogeneous for 6. It we had 6 equal 1 on H this would contradict condition (c). Thus every set .6EVw' .6E [H jm i s in R. This implies that every .6 EVw' .6~ H, is in R. But then a IIR ll = N for any a, since any subset of H can be extended to arbitrarily large I(. ~ H, I(. E R. We have proved that every (A a) is true in Vw' Conversely, i f 6: [V,l m ~ C (E V ) then the class R of 6  homogeneous sets is Ramsey by 1.4 (,~ W' +1 a so II R II = N for each a < wI' Since R is countable for some a <wI' Tf' =Tf'. Given I(.E Ra = Ra + 1 it can be extended to 1(.1 ERa = Ra + 1, I(. C ILl and so on; we find an infinite H such that every subset of H (from VI) is in is homogeneous for 6. This is the content of the equivalence. •
REMARK 1.7.
R,i .e. H
In the first draft of this paper we posed the following ques
tion: Is it true that for any a < wI there are c, mEN and 6: [Njm ~ c such that if R = {.6 E P W (N) 1.6 6homogeneous} and Ril + 1 = Ril then il;;;. a. R.Laver has answered this question affirmatively. This shows that Proposition 1.6 cannot be improved. that is there is no il < co, such that the infinite Ramsey theorem would be equivalent to "60/l. ,,< il Vw t= (A,,)". is essentially the axiom of induction. Then (AI) In 1.5, we showed that (A a) and its successors may, in this light, be called strong axioms of induction, another (a better) reason 'for this being the fact that their form does resemble the induction schema. However they are called. the important thing is to find a use for them. Below we prove a combinatorial result which will be heavily used in the next section. THEOREM 1.8. t = (a , ... , al ••• l . Let fL be an ordered sequence of 1ength q from H such that for some 13 1,13 2 , [33 E H we have "4 < 13 1 < [32 < 13 3 < fL. Assuming (IJ yEa) (3 zEV )~.6 and realizing that all parameters in this formula are in V an ap"4 "1 plication of 2.2 (i) yields ( II yEa) (3 z The comprehension axiom then gives Formally, we get
E
V
"4
(II yEa)
) q, fL
•
bE V such that 13 1
(II yEa)(3 z
Using 2.2 (i) and the fact that the parameters are limited to
V
"1
E
b) q,fL •
we obtain
MIROSLAV BENDA
54
The comprehension axiom has a simple proof. Let ~(z, v) be a formula of rank from H (6 of length q) and let .tE Vaoand aE Val be given. We should show that
q,
"o < £\'1 < "z < "s < 6
but this is true since any subset of a is in Va
2
The infinity axiom: If we work in ZFC then because H can be extended to arbitrarily large H such that (a) 'V ({3) where a E H, {3 E Fr, we see that all ~E H must be > w (if k ~ 3). The axiom then will be in Til.' On the other hand, if everything is finite, it will be so in all
V~ 6.
~Ie are now ready to prove tha t PA + (AI) 1 Con (PA) and ZP + (AI) 1 Con (ZFC) We take the latter case the former being entirely similar. Let P =(~ 0' ~1"" ~n) be a proof of ZFC. Let kE N be such that the universal closure of any ~,[ in P has quantifier rank';; k. Using (AI), we get H satisfying 2.2 (i), IHI ~ 11.+ 3. By Claims 1, 2 and 3 and induction, we see that every ~,[ET(k., H) and by 2.4 we find that ~n cannot be a contradiction. •
DI SCU SSI ON.
Let us summarize what we have accomplished so far in order to put the results into a broader perspective. We have defined the axiom schemas AO' AI' ... for the theory T, which is essentially Peano's arithmetic when the axiom of finiteness is added to it while it becomes the extended Zermelo's set theory, ZP as we call it, when the axiom of infinity is added to it. The axiom schema AO is provable in T because in PA, it is the finite Ramsey theorem while in ZP it is the ErdosRado theorem. The axiom schema Al is provable in neither of these theories. For ZP, this can be easily seen if we work in ZFC (and assume that it is consistent). A reason for this is that VK where K =.Jw is a model for ZP so if Al were provable in ZP we would obtain VK F Con (ZFC) which is impossible. Now, as far as PA is concerned, the axiom schema Al is true. We can say this because we have a standard of reference for PA, namely the natural numbers. Concerning Al an ZP we see that Al is false in a natural model of ZP and so the question arises whether ZP + Al is even consistent. One way out of thi s difficulty is to point out that Al is a much weaker axiom than, say, the existence of a Ramsey cardinals. In fact, as for arithmetic we just need a cardinal K> w for which the Ramsey theorem is true (i .e. homogeneous sets of power K exist for any 6: [K] n > c}, Although this gives some credence to the consistency of ZP + Al it is not completely satisfying. Let us present a justification of the augmentation of ZP with Al in a way which is based on the uniformity of the proofs of independence we have given. This approach lacks, so far, preciseness however this is, hopefully, compensated by its possible applicability to other question of this sort. The axiom of infinity
ON STRONG AXIOMS OF INDUCTION
55
played such a minor role in the above proof because the sentences Al are formulated using mainly the notion "for arbitrarily large" and because the properties of this quantifier are very similar in PA and in ZP (or ZFC). We wish to make a sweeping generalization of this which consist in: (1) Isolating that which is common to PA and ZP other arguments.
in the above proof and
(2) Defining a class C of sentences which do satisfy the requirements of (1), that is sentences which speak of properties for which the infinity axiom irrelevant. The sentences of AI' A2 ••• should fall in~o this class. (3) Adopting the following General Principle: the hereditary finite sets is true of all sets.
Any sentence from C true of
On the basis of the principle we answer the question whether ZP + Ai is consistent by saying, of course, Al is in fact true of all sets. Some may say that this is the same escape as the one taken when justifying large cardinals (since there is one measurable cardinal (w), why not have two). The principle does differ from this line in that it refers to certain similarities of the respective universes in their totality rather than in what they contain. The goal, and the principle is nothing but a goal, is to bring these similarities out of the shadows.
REFERENCES. M. Benda 1979
a
On H~ngton'~ Pantit£on Relat£on,
1979 b : Modetof.d6 1,
to appear in Combinatorial Theory.
to appear in Trans. of the A.M.S.
P. ErdBs and G. Mills 197+
Some Bound6 nO~ the R~ey  p~  H~ngton Numb~,
to appear.
L. Harrington and J. Paris A Ma.:thcma:ticcU. Incompletene6~ f.n Peano A~hme.:Uc, Handbook of Mathematical Logic, J. Barwise (ed.), NorthHolland Pub. Co. Amsterdam,
1977
pp. 1133  1142.
J. Ketonen 197+
,
Set Theo~y nO~ a SmaLl Unf.v~e,
to appear.
K. McAloon 1978
FO~e6
combf.na.:toULe6
du theMcme d'f.ncomple.tude. Seminaire Bourbaki N° 521.
J. Paris 197+
Independence Re6uL'U
nM
Peano A~hme.tf.c U~f.ng Inn~ Modw,
to appear.
MIROSLAV BENDA
56
P. Pudlak 197+
AnotheJt CombinatoJUal
pear.
Sen;tence I ndependen;t
06 Peano AJU:thme:Ue,
J. Silver 197+
HaNling:ton',5 veMion 06
.the
PaJr.{A lLUuU:.
R. Solovay 197+
Rapidtlj GlLowing
Ram6elj
Function6,
to appear.
Department of Mathematics University of Washington Seattle, Washington 98117
to
a p
MATHEMATICAL LOGIC IN LATIN AMERICA A.I. Arruda, R. Chuaqui, N.C.A. da Costa (eds.) © NorthHolland Publishing Company, 1980
TOWARD A CONCEPT OF SCIENTIFIC THEORY THROUGH SPECIAL RELATIVITY Jo!tge E.
Bo
Otherwise the relativistic LA K W is called
0 ... v(lt, K}
>
O.
11011E{.n6;(:UnLaI1.
It can be shown that axiom (vi) is equivalent to the existence of an invariant velocity, in the following sense: a number c is called an il1vaJr.ia11:t vetoei:tyfor a LA K W = (E, K) iff there exists a schema of uniform motion It in E such that V (It, K) = c nOll evetty K E K. Axiom (vi) is equivalent to the following: (vi')
W hal>
al1
il1vaJr.ia11:t vetoei:ty.
And each of these conditions (vi) (vi")
The eha.ttac:tetti.6tic
k
On
and (vi') is in turn equivalent to W ih "Willy pMUive
(I 0).
It can be shown that in an Einsteinian LA K of (positive) characteristic k, there are exactly two invariant velocities, given by the formula
±. c = 1 Iv'k . In a nonEinsteinian relativistic LA K there is no invariant velocity, as in are the classical case. Naturally, equivalences between (v.i.), (v.i.') and (v.i.") referred to the framework of definition 3, i.e., to a ltetaUvihtic LA K W. We pass now to the problem of units. It turns out that it is possible to establish an adequate concept of equivalence for units without introducing any metrical concept; this will be done, then, in the framework of pure affine geometry. DEFINITION 4. In any LAK we say that the couple (K, K') of allowable frames has eqaivalel1:t time uni.t.6 (respectively "pace uni.t.6), if the time coordinate (respectively space coordinate) in K of the time unit (respectively space unit) of K', equals the time coordinate (respectively space coordinate) in K' of the time unit (respectively space unit) of K. It can be shown that the couple (K, K') has equivalent time units if and only if it has equivalent space units; and each of this conditions is equivalent to the following: the determinant of the matrix K K' is equal to 1. DEFINITION 5. A LAK is 110JrmaUzed iff every couple (K, K') of allowable frames has equivalent time units. For every normal i zed LA K, transforma ti on rna tri ces K K' are of the form 1
Ipeuctl E.£n6:te.£n1ctn I1..£nema.tic..6 is a couple (2, W), where 2 is a PF and W is an EM. The schema (2, W) is val1d if and only if there exists an embedding n: 2 + W. Thus, according to this view, a physical schema may be valid and may be non valid. Observe that if the schema (2, W) is valid, all reallike theoretical physical frames F (belonging to F) are such that the time axis of their corresponding frames in Ware well defined, and consequently we may speak of the time axis (and the space axis) of a reallike theoretical physical frame F. Finally, observe that the notion of schema of uniform motion can be introduced in 2 via the embedding 6, and it results in an invariant concept with respect to the class F in 2.
§3.
STEP III:
EMPIRICAL INTERPRETATION.
From an empirical point of view, a physical schema (2, W) as stated in § 2 is still a rather theoretical entity. What the experimental physicist does is not to perform a global test concerning the whole universe, but to establish the actual empirical conditions in which a test should be relevant. The specification of such conditions constitutes what I call an empirical interpretation. This is formally given in definitions 12, 13 and 14 below. DEFINITION 12.
An ex.peJL£men:t.a..e 6Jtamwonk for kinematics is an
ordered
JORGE E. BOSCH
64
(RE, 1, M), where RE is a well defined class of ~etevant eteme~y eT is a class of measuring instruments, and M is an empirical method (ordered set of rules) such that the following conditions hold: triple
ve~,
(i)
e wUh
06 I and :to any membM e 06 numbeM [:t, x l, c.aU.ed :the c.oMdinateJ.> 06
16 muhod M JA appUed t» any membM i
RE, it pIlOduc.eJ.> an
~eJ.>pec.:t:to
o~dMed p~
L
06
~ea.l
(U) 1 has at .teM:t two di66MeJU: membeM, and 6M eac.h i in 1 :thMe intu.Ui.ve notioVL6 06 .6imu1.:tanUty and c.o.6paUa.U:ty wah lleJ.>pec.:t :to L The.6e UOVL6 genMate ~etmoVL6 SF and CF M in Ve6.[nition 8.
Me no
The relevant elementary events are those which are actually interesting for the experimental physicist: then, the class RE may be finite. All the problem of "equivalent time units" lies in the requirement that the method M must be :the ¢ame for every member of T (system of measuring instruments). In fact, if method ~r involves an explicit use of units, the requirement is such that method M itself must establish the correct way for producing such units. For example, if the unit "meter" is used, method M may contain a rule 1ike thi s: "Take a ugid nod whic.h, at ~e.6:t wah ~e.6pec.:t :to :the 1>:tandMd me:tM 06 PMJA, coinc.ide.6 experY1>:tem 06 meMu.4ng iVL6Uume~ unde!l c.oVL6ide!lation". For time the situation
is similar: if comparison with a standard clock is not judged a satisfactory operation, we can choose a rule like that proposed by Bondi 1965: "A and B can u.6e
M :thm una 06 time :the ha.t6 U6e 06 a nuc1.eu.1>
made up 06 a 1>pec.i6.[c.
numbM 06
It is tacitly intended that this phenomenon of decay must be produced in a box at rest with respect to the corresponding observer. If method M does not involve an explicit use of units, then the units implicitly used are equivalent by definition, due to the uniqueness of method M itself. The idea is that method M should be c.anonic.a! with respect to the elements of 1. and ne.utnoYL.6".
P~OtoVL6
DEFINITION 13.
An experimental framework (RE, T, M) will be called ade
quate if it satisfies the following conditions:
... , in' with n;;' 2, be eteme~ 06 I, and.tu m ... , m bema1, n, 1, 1>uch:that m E i 601L k = 1, •.. , n, Then 601L each k it ill ~e k k q~ed ihas: at .teM:t :tMee due!lminmoVL6 06 m p~oduc.ed by :the 6.[xed iVL6~ument k i mu.1>:t be eve~ betonging:to RE; 1 (U) FOIL each k = 1, ... , n, :the meMMeme~ Ilegill:te!led by i co nce~ning :the de:te!lminmoVL6 06 m Ilevea! :that :the vetoc.i:ty (in :the OILdinM~ .6eVL6 e) 06 m k k JA C.OJt6:tant 601l eVMy meMMed intMVa!; (i)
:te!lia.t
Lu i
poi~
Two remarks are in order: (1) the point mk is supposed to be a representative point of the 1>:ta:te.6 of i k; it may also be said that mk is ~gid.ty belonging to i k ; (2) A due!lminmon of a material point is a primitive notion whose intuitive meaning is the instantaneous state (or apparition) of thi s rna teri a1 point; in Reichenbach's terminology, the determinations of a material point are the genidentical events which constitute this material point. In this sense, the determinations do not depend on any measuring instrument; but, for the sake of briefness, I speak of :the de:te!lminmoVL6 pMduc.ed by an in.6Uument i 1 as signifying :the cOOlLdinateJ.> 06 a due!lminmon e.6:tabffihed by i 1.
TOWARD A CONCEPT OF SCIENTIFIC THEORY DEFINITION 14.
65
The physical framework (in the sense of Definition 9) framework (RE, 1, M) is the quadrupl e of Definition 13) by:
genvwA:ed by the adequate experimental (U, F, R, A) defined (with notations
U = RE
(.i)
at:
U
V, wheJte
V JA the 6e;t 06 aU :the even:t6
pok, p/, p/,
6ueh
Po k
if.. :the event eOl16if..:Ung .in :the (poM.ibly .ideal) de;teJln1.ina.:Uon 06 m k .il16mnt 0 w.i:th ll.e6pee:t:to .[k; PI k JA de6.[ned .in :the Mme way 60Jr. .il16mnt 1
:that :
06 .i p 2k k; mneouf., wUh 6JtOm
Po k, (ti)
JA :the event cOl16JA:Ung .in :the de;teJtm.ina.:Uon (poM.ibly .ideal) , 6.imul
Pok w.i:th ll.e6pee:t:to .i 06 a mctteJUa1 po.int 6Uuctted ctt dJAmnce k, .in :the J.,el16e 06 p06.[:Uve meMWtement 06 .il16:tJtument .i k; k, F JA :the clM6 06 ll.ealUke phy6.ical 6Jtctme6 (P p/' p/);
1
o
(.u.i) FOJr. each F .in F, R(F) JA :the MdeJted pct.iJt (SF' CF), wheJte SFand CF Me th« ll.uct:UOl16 geneJtctted (in the sense of Definition 12 (ti)) by :the meMwUng
.in6:tJtument M60c.ictted:to
F;
(.iv) Fon: each F .in F, A(F) JA :the .6y.6:tem 06 cOOJr.d.inctte6 uemen:t6 06 U by :the .in6:tJtument M60c.ictted:to F.
M.6.igned:to
:the
It is evident that the physical framework generated by an adequate experimental framework is a quite ideal (or abstract) entity. It depends, of course, on some broad spectrum terms and on some primitive ideas, such as "simultaneity with respect to an instrument (or to an object)", etc. DEFINITION 15. Let Y = (RE, 1, M) be an adequate experimental framework, 2 the physical framework generated by it, and W an Einsteinian model. Then we say that the experimental framework Y con6.iJtm.6 the physical schema (2, W), if this schema is valid in the sense of Definition 11. Otherwise we say that Y ll.e6u:te6 the schema (2, W). Until now, no concept of :theOJr.fj has been introduced. In fact the precedi ng machinery suffices to give a formal account of scientific practice (at least concerning relativistic kinematics). But from an epistemological standpoint the following questions are relevant: what kind of object is (if any) the Special Theoll.y of Einsteinian Kinematics? And if such a :theoll.fj exists, does it give some information about reality? Which is its cognitive status? A possible answer is: there is no thing such as the Theory of Special Einsteinian Kinematics; all there is in this connection is a mathematical tool, the Einsteinian model of Definition 6, and the possibility of making up some experimental devices in order to obtain confirmations or refutations according to Definition 15. In a case of refutation, physicists would look for a convenient change in the mathematical tool or in the concepts of physical and experimental frameworks. This is a pragmatic point of view. But there is an alternative, which consists in defining a theory: several ways are possible; it seems to me that one of the most convincing is the following:
tem
DEFINITION 16. The TheOJr.y 06 Spec.ictl E.in6:teinian K.[nemct:UCf.. is the sysin the following entities:
consist~ng
(.i)
The clM.6 '06 aU E.il16:teinian modei..6 .in :the 6en6e 06 Ve6.[n.i:Uon 6.
(ti) The .6:tJtctt.i6.[ed clM.6 a6 aU uemen:tMy evenU: .6:tJtctt.i6.[ect:Uon JA g.iven by :the d.i66eJtent levei..6 06 mean.[ng 06 :the bll.oad .6pec:tJtum :teJtm "uemen:tMy event" . (.u.i)
The clM6 06 aU (a.e:tual 0Jr. .ideal) adequctte expeJt.imen:ta.e. 6JtctmewOJtk.6
.in
66
JDRGE E. BOSCH
;the .6l!JUe 06 VeMnil.iol'l.6 12 and 13: ;thM.u, ah..o a .6.tJLa.:t.i.Med c1.a.6.6, ac.c.olld.ing w.£:th ;the .6.tJLa.:t.i.Mc.at:ion 06 elemen;taJr.1j evenU; (.iv) A .6.tJLa.:t.i.6.ied c1.a..6.6 06 c.on6~a.tion 6unc.;tJ.oYl.6: 601l eac.h un.iVeJr..6e U 06 elemen;taJr.y evenU 06 a g,(.ven level, deno.te by Y ;the c.oMe.6pOncUng c.lau 06 adequa..te expeJUmen.ta.e. 6!tC1.YY1e.w0llk.6; then the c.oMe.6pOncUng c.on6~a.tion 6unc.;tJ.on Ls the 6unc.;tJ.on 6u :Y ..,. {D, l l , wh.ic.h to the adequate expeJUmen.ta.e. 6!tC1.YY1e.wOllk Y E Y M.6.igYl.6 the value 1 (c.on6~a.tion) .i6 thelle ewu an Ee.ln eVeJr.q Jr.ea!.>onable pJr.e!.>ci'l..:ta.tion 06 Cla!.>!.>.lcal K£nematic!'>. More precisely, there is a definition of the TheMy 06 CwMcal K£nema;t[c!.> which is obtained from Definition 16 by changing only part (.l) of this definition: this is accomplished by taking the class of all c£aJ.>!.>.lcal model!.> (easily defined) instead of the class of all Ein!.>.:te.lnlan model!.>. As the experimental framework is the same for both theories, a campa ri son between them is straightforward. Commensurability is thus obtained as a result of formalization, and this may be considered as an advantage (last but not least) of the axiomatic method. §6.
TOWARD A CONCEPT OF SCIENTIFIC THEORY.
TOWARD A CONCEPT OF SCIENTIFIC THEORY
71
The above treatment of Special Einsteinian Kinematics suggests a generalization to other theories. A more complete account of this program will be developed in another paper: here only a brief sketch of this idea will be proposed. I believe that much of the advanced scientific theories may be presented according to a schema 1 ike tha t of Definition 16 (§ 3). This schema woul d comprise four parts: (~)
({)
A class of mathematical models, defined in a set theoretical framework. A (perhaps stratified) class of entities designated by a broad spectrum
(ill)
A class of adequate experimental frameworks, closely related to enti
term.
ties in ory.
(~).
({v)
A confirmation function depending on the internal structure of the the
It is highly probable that a mathematical model of class ({) will contain more elements than those designated by the broad spectrum term of part (~) : then the definition of a sort of embedding will be in order, and this embedding will supply an interpretation of the theoretical terms of part ({). An experimental framework will be a structure concerning one precise meaning of the broad spectrum terms appearing in (~); and its definition performed in a sufficiently canonical and universal way so as to exclude sibility of adhoe devices. Moreover it would be of a nature appropriate account of aecepted 6a~.
level of would be the posto give
In this manner it would be possible to test theories and even to establ ish comparisons between them. But the experimental meaning will always depend on the choice of one level of'meaning of a broad spectrum term, plus a number of auxiliary hypotheses and empirical statements concerning accepted facts. This is just a program, which has been illustrated by Einsteinian and Classical Kinematics. Its success in a larger domain depends on further research.
REFERENCES. H. Bondi 1965 Some 6peuo.£.
06 the UlUtebuan eqUll'UoM. In Lectures on GenBrandeis Summer Ins titute in Theoreti ca 1 Phys i cs , PrenticeHall, New Jersey. 60,[tI.UOIU
eral Rela tivi ty .
M. Born 1962
Einstein's
theory of Relativity.
Dover Publications, New York.
J. Bosch 1971 On the ax{omat{c 60undat{01U 06 6pe~ ~e.r.at{v{ty. cal Physics, Vol. 45, N° 5, p.p. 1673 1688.
Progress
of Theoreti
A. GrUnbaum 1961 Law and convention {n phy6{co.£. theo~y. In Current issues in the philosophy of sc I ence , H. Feigl and G. Maxwell (edS.). Holt, Rinehart and Winston, New York.
72 1963
JORGE E. BOSCH Philosophical
problems of space and time.
Alfred Knopf, New Yorl :;:HI this lemma means that no term and its negation are provable in the system. SET THEORY. The set theory of Bunder 197+ c based on the higher order predicate calculus outlined earlier, the equality axioms (E1), (E2), (E3), (E4)., (E8) and (E9) and the rule W QX, W QY I H (QXY)
(replacing (E5» was shown to be relatively consistent with the higher order predicate calculus alone. From this we could derive comprehension, pairing, replacement and extensionality. To have the sum set property as well we need: (E6)
I F AH x:>
x
Ax.
We can extend Lemma 6 as follows: LEMMA 7. The .6y.6tem 06 BundeJr. 197+ c. f.nc.tucUng (E1), (E2), (E3), (E4), (E6), (E8), (E9) and WQX, WQ Y I H( QXY)
6.
PROOF.
We add (E6) to the system(s) we considered in Lemmas 2, 3, 4, 5 and
Lemma 2 goes through as before. In the proof of Lemma 3 we need to cons i der the extra case where Z = :;:(FAH)A. T is then Z or AW for some W where F AHW is in a previous step. As any previous step must have a normal form so must T. Thus Z mus t be a hypothesis and the res t of the proof goes through as before. Lemmas 4 and 5 hold as before and in Lemma 6 the extra case again is Z but as :;:HI* :;:(FAH)A or AY for any Y, this case does notarise and so Lemma 6 holds.
:;:(FAH)A,
(E1), (E2), (E3), (E4), (E6), (E8), (E9) and the rule replacing (E5) are relatively consistent with this as in Bunder 197+ b so Lemma 7 holds •• For the set theory of Bunder 197+ b which includes the comprehension, pairing, replacement, sum set and power set properties, we need to add to the higher order predicate calculus: ( A)
I Ax:>
x
FAHx.
If we also want the axiom of infinity to hold in this system we also require: (0)
IAO
A HIGHER ORDER PREDICATE CALCULUS
where O=oBf(WQ1)
and Q
1
=0
81
AXAy(FAHxt\FAHyt\ ::A.AU(XYVlyU)).
We can again extend Lemma 6. LEMMA 8. J.>.{.J.>:te.n:t.
PROOF.
2 to 6.
The. J.>yJ.>:tem
on
BundeA 797+b .{.nc.£u.di.ng (AJ, (0) and (Q1 J
.{.J.>
c.on
We add (A), (0) and (Q1) to the system(s) we considered in Lemmas
Lemma 2 goes through as before. I n Lemma 3 we need to cons i der the extra cases where Z = ::A(FAH) or ::A( BAQ1)' T is then FAHW H(W ( ) or A(QIWI) I 2 I, where WI' (and in the second case W2)' has a normal form as it appears in a previous step AWl (and AW 2)' so only in the second case may T be without norma 1 form. We now consider the proof of AWl (AW The last step in this cannot be by 2). DT:: or DTP'. If t t comes by (0) WI W = I' (Q, W ( which has a normal form, z 2 2) so AWl must come from ZI by a sequence of P,:: and Eq steps, where Z' does not r
come by DTP, DT::, or clearly, by (A) or (0).
If
Z' comes by (QI) i.e., is
FAAQ then WI is Q W for some W in normal form, so WI W2 has normal form. I 3 3 I Thus Z' is a hypothesis. Now Z' can be cancelled only by a DT:: step, so Z' = UV for U E U and V indeterminate which can only lead to a WI = VW 3 W4 ••• Wn for some W3 W4'···'W n in normal form. Then WI W2 is also in normal form, which is impossible. Thus the hypothesis Z' cannot be cancelled and Lemma 3 holds. Lemmas 4 and 5 holds as before and in Lemma 6 we need the extra cases Z ::A(FAH) and Z = FAAQ1' but as ::HI *FAHY, H(YW) or A(Q1Y) for any Y or W, thi s case does not ari se and Lemma 6 holds. Therefore, Lemma 8 holds .• Note that, as is to be expected from the inconsistency proof given in Bunder 197+b, the above proof fails when we have both (A) and (E6) in the system. In the section dealing with Lemma 3, AWl and AW could ~ave been derived by (E6). 2
REFERENCES. M. W. Bunder
on
197+a
Plte.di.c.a:te. c.a1.c.u1.u.J.>
197+ b
A one.auom J.>e.:t :the.OILy bMe.d on Mghe.1t OILdeA plte.di.c.a:te. c.a1.c.u1.u.J.>. published.
aJtb~y
Mgh OILdeA.
Not yet publ i shed. Not yet
82 197+ c
M. W. BUN DER Set theOlLtj in plLecUc.a;te ealeu£.u.6 w.Uh eqWLUty.
Not yet publ ished.
Department of Mathematics The University of Wollongong P.O. Box 1144 Wollongong, N.S.W. 2500 Australia.
MATHEMATICAL LOGIC IN LATIN AMERICA A.I. Arruda, R. Chuaqui, N.C.A. da Costa (eds.) © NorthHolland Publishing Company, 1980
BACKANDFORTH SYSTEMS FOR ARBITRARY QUANTIFIERS Xav.leJL Ca..leedo
ABSTRACT.
Loow(K) is the logic obtained by adding a
KX1 ••• Xk (¢I(X 1)'" ¢k (X k ) ) to the logical operations of L oow' The corresponding
lindstrom's quantifier
finitary logic is L w w (K), and L oo w (Ki.) i.E I is obtained by adjoining a family of quantifiers. In this paper, we give backandforth systems characterizing elementary equivalence in those logics and their fragments of bounded quantifier rank. This general izes work of Fra'l s se and Ehrenfeucht for L w w ' Karp for L oow ' Brown, Lipner, and Vinner for cardinal quantifiers, Badger for MagidorMal itz quantifiers, and others. Our systems apply to higher order quantifiers also.
INTRODUCTION. Ehrenfeucht 1961 and Fraisse 1955 gave backandforth or game theoretical characterizations of elementary equivalence in first order logic, L ww ' later generalized by Karp 1965 to infinitary logic,
L=w' These characterizations were used
to obtain results about definability of ordinals and preservatio~ of elementary equivalence by operations on structures. Lindstrlim 1969 used FraisseEhrenfeucht games to characterize L ww ' Backandforth systems for logics with cardinal quan
tifiers are due to Vinner 1972 and others. Badger 1977 gives systems for 1ogi cs with MagidorMalitz quantifiers (Magidor and Malitz 1977), and shows the fail ure of interpolation and preservation of elementary equivalence by products in these logics. Krawczyk and Krynicki 1976 give systems for certain monotonic quantifiers, without any appl ication. Makowsky 1977 a has similar systems and he studies monotonic quantifiers in detail. Backandforth systems for Stationary Logic, L{aa) (Barwise, Kaufmann and Makkai 1977), were given independently by Kaufmann 1978, Makowsky 1977 b and the author (Caicedo 1977 b). In our doctoral dissertation, we presented backandforth systems characterizing elementary equivalence in logics obtained by adding to first order logic
quantifiers of the form Q; ¢ (x), this means binding one or several variables in and a single formula, and gave various applications, particularly to L w w (Q1 ) L(aa). In this paper, we introduce backandforth systems appropriate for quantifiers binding several formulas: Qx·1· .. ~n(¢1(x·1)' ... , ¢n(Xn)) 83
XAVIER CAICEDO
84
Although the methods may be applied successfully to second and higher order quantifiers, as they were applied to L(aa) in Caicedo 1978, the corresponding results will be published elsewhere. We assume as known the notions of an abstract logic, as well as the extension relation between logics, and the notion of a generalized quantifier (lindstrom 1966, Barwise 1974, Makowski, Shelah, and Stavi 1976). Ol,;;& , ... denote classical structures, and A, B, .•. denote their universes. In Section 1, we introduce quantifier symbols and their interpretations. Instead of considering quantifiers in the sense of Lindstrom 1966 and Mostowski 1957 only, we deal with the more general case of so called "weak models" where the quantifier interpretation forms part of the structure. LindstromMostowski quantifiers are recovered as families of weak models where the interpretations of the quantifiers are determined, up to isomorphism, by the domain of the structure. In Sections 2 and 3, we define the backandforth systems and prove the characterization of elementary equivalence. In Section 4, we consider monadic quantifierS, those where the quantifier binds a single variable in each formula, and extend a result of Friedman 1973 about the failure of Beth's definability theorem in cardinality logics to logics with these quantifiers. Also we show that any extension of Lww(QO) by monadic quantifiers satisfying interpolation must satisfy the downward LowenheimSkolem theorem. In Sections 5 and 6, we give a simpler version of backandforth for cOn~~ quantifiers, which becomes PC definable. The main applications deal with (infinitary) extensions by monadic quantifiers of L,,)w(Ql)' logic with the quantifie r "there are uncountably many". These include an analogue of Lindstrom's theorem for L ww(Q1)' a relative interpolation theorem in L ww(Q1) with respect to such extensions, and the existence of models satisfying few types in those extensions. Makowsky and Stavi discovered independently the relative interpolation theorem in L ww(Ql) and L(aa), with respect to their infinitary extensions. Finally, in Section 7 we show that elementary equivalence is preserved by cartesian products in a natural extension of logic with MagidorMalitz quantifiers.
§1 GENERALIZED QUANTIFIERS. A qua~n~~ ~ymbot is a symbol Q together with a sequence of positive integers (111"'" l1 il > caned the type of the quantifier symbol. Given asetof relation, function and constant symbols, the language Lww(Qj) jEJ is obtained by addi ng to the usua1 forma ti on rul es of Lcos» for atomi c formul as, "l, A, and 3, the new rule: If Q. is a quantifier symbol of type (111' . . . , ttk), ¢1""'¢il are j ~ ~ formulas, and Xl' ••• , x k are lists of ttl' ..• , 11 k variables, respectively, then Q. j xl"'" X. k( ¢ 1"'" ¢ k) is a formula. It is understood that only those free variables of ¢ ~ which appear in the 1ist are bound by the quantifi er.
x~
type
If Ol
is a structure in the ordinary sense and Q. is a quantifier symbol of 11 > , thenan~n;(;eltplte;ta..t[ol1nOJt Q ~11 a is a family k
(111"'"
BACK
Q~
9
"i nR (A ) x .•• x 9(A ).
Qj
where
ANDFORTH SYSTEMS
85
An L ww(Q.j)jE].6t!u.Lc.tuJLe has the form (Ol;Qj)jE]
is an interpretation of
a.
Q. j in
The semantics of Lww(Q.j)jE] is
defined in the usual way, except for the additional clause: (Ol; Qj)jE] 1=
Q. j
x1,···, xn (.
iff
interpreting the symbol
Qj
the language writing
1>
for
Loow(lJ../)jEJ
L+.
Qj is understood, we will abuse
The existential quantifier is the LindstromMostowski quantifier
3(A)
defined by the function well know.
=
{S
C

A
Is*
0}.
of type (1 )
The following quantifiers are
CMd£na.t qua.ntiMe.M, type (1), for each ordinal a: Q.a(A) = {S:=.Allsl ;;;. wa}. for each ordinal a and finite 11., the quantifier of Q.~ (A) = {S :=.An 13 I ~A such that In ~S and III;;;. w } . a
Mag~do4Maiitz qua.nti6~e.M,
type
(11.):
Chang qua.ntiMeJt, type (1) : Q.(A) = {S ~ A I HaJz.t.i.g qua.ntiMeJt, type (1,1): H(A) HenlUn qua.ntiMeJt, type (4) : Hen(A)
lsi = IA\}. {(S, T) I S ~A, T ~A, lsi = ITI}. 4 {S ~ A I 3 6, 9 : A + A such that 6Xg ~S}
Note that the logics obtained from these quantifiers do not incl ude those where the meaning of quantifier is not determined by the domain of the structure, like Sgro's topological logic (Sgro 1977). However, if we consider only logics for classical structures with a finite number of relations, functions, and constants, then any logic is a sublogic of some Lww(Q.j)jEJ' Even more, if the logic L is closed under substitution of relation symbols for formulas then L is equivalent to some Lww (Q.j ) j E ] '
§
2 BACKANDFORTH SYSTEMS,
Through this section (a, q) and (~, 4) will be quantifier structure where q 4 are interpretations for a quantifier symbol of type (11. " " , n ). A and k 1 B are assumed to be disjoint. Sequences in An u s'' (11. E w) wi 11 be denoted
and
a,
a, a', a', T, T ' , h, h'; the value of catenation is denoted by juxtaposition.
11.
will be clear from the context.
Con
DEFINITION 2.1. A baekand6olLth be;tween (a; q) and (~; 4) consists of a linearly ordered set p = (P, b,i isapartial isomorphism from 01 to&. As (U:), interchanging the roles of
(.i..u:)
DEFINITION 2.2.(01; Cij)jEJ
each
jEJ.
(p, {E~I pEP, nEW})
isa
:to ($;Jr.j}jEJ ifitisonefrom (OI;Cij) The existence of such a relation is denoted by
baekand6oJvth
to
(:6';"j)
(a;Cij)jEJ~
61tOm
for
()&..;\)jEJ
REMAR K. We assume that any backandforth sa ti sfi es the extensi on property for the existential quantifier. It is enough to postulate in (,i,i) , for each p' p a 'V T and p < p' the existence of a function 6: A+B such that o a. 'V T 6 (a). Property (B) holds automatically. DEFINITION 2.3. {E I nE w} is a bael 61tOm n ioi, Cij)jEJ to ($; "j )jE] if En is an equivalence relation in An U Bn for each n , and properties (,i) to (,iv) of Def. 2.1 hold, dropping the parameter conditions.
§
Such relation is denoted by (or; Cij) jE]
'V
(~; "j) jE]
.
3 CHARACTERIZATION OF ELEMENTARY EQUIVALENCE.
Let or and!tJ., be classical structures. C = {Cij I jE J} and D = {\ I jE J} are interpretations in 01 and~, respectively, of the quantifier symbols
XAVIER CAICEDO
88
{Q.[ jE]}. j
THEOREM 3.1. 
(ot; C) ~ (~; D) then (ot; C)
I6
Suppose (ot;C) ~ (~; D). Bn define:
E
a
{J
(ot, a; C)
iff
"val
iff
(:e,1';D)
(ot, a; C)
iff
and a, a'EA Yl
For every {J
( ;g,; D) (Xr; D) (Lil).
wheJte P 1.1> nonwell. ondened, (bac.kand6oJtth wUhout Pa.JLamet:eM).
The same as the proof of Theorem 3.1, but
simplifying
the definition of backandforth to: a 'V 0' iff (Ol, a; C) ~ (Ol, 0'; D), etc. Instead of t~ one defines t .. = {4> (y) I (Ol. 0; C ) F 4> (a)}, By Lemma 1. 2, a. a. fI t .. is a sentence of L (Qj') j' E ] ' The rest of the. proof is simpler because a. oow we do not need parameter conditions. Choose any nonwell ordered set P and define 0 ~ t iff a » t , Let PI> P2 > ,., be an infinite descending sequence of P , One shows, as in the proof of Theorem 3.3, by induction on the complexity of 4> (y) , that for all n: => (U).
(Lil)
(U)
=>
(i),
o
Pn 'V
implies:
r
(Ol; C ) F 4>(a)
iff
(~; D ) F 4>(r),
To use the extension property in the inductive step for one chooses
§
4
P n + .:\
< P n +.:\ _ I < .. , < P n ' •
Qj
xl.· ... xm(4) l' ,, . , 4>m )
INTERPOLATION AND MONADIC QUANTIFIERS,
A quantifier symbol of type n(x ,This includes the cardinal quantifiers Qa: "there are at least wa eln ments", as well as Chang's and Hartig's quantifier, but notthe Magidor  Malitz quantifiers. Throughout this section L M will denote the logic L oow (Q.). E ] , j j where Qj runs through all the possible LindstromMostowski monadic quantifiers.
»·
LEMMA 4. I . 
a =L
oow
16 Ol and :t; Me .:\:tJtuc..tu!l.u 06 poweJt at
(Q):tr impUu
I
a
=L;,t
M
mO.6:t
wI' then
•
PROOF. Let 2n = {o I 0 : n ... 2 }, for seA 1et sa = Sand sl = A  S • For each nEw and F: 2n ... {K 'K cardinal, K" wI} define the quantifier:
~(A)={(SI,.. "S)',;/OE2nl n
l~~)I=F(O)}, Since a monadic structure n L« n ,(. (A, SI".'. Sn) is completely determined by the above set of cardinals, there corresponds to it a unique F such that (Tl'".,T n) E Q F (8) i.ff (8, T1,.", Tn) "" (A, SI" ." S). Let 3!K x 4>(x) mean that the truth set of, .4>(x) has exaclty K n elements (K';; wI)' Clearly, for structures of power at most wI' Ol=L (Q);6implies
a
=L
(Q
3,K)
Ir.
and so
oow l' , K definable from the former quantifiers.
oow 1 (Q),(",' because the QF' s are oow F F If ~O is 4> and 4>1 is I 4> :
a
=L
92
XAVIER CAICEDO
() () ) QF"l"'" ( '1>1"1"'1>" rt
rt
rt
=>
f\ I>E 2 rt
1 (,,)1> (i I 1 . [3.' F(I> ) x i Sup {] 1= q,} = x , 16 not have Lowenheim numbeJt such. !>entenc.e haJ> modetJ> 06 MbWuvt.Uy iMge i..ty.
PROOF.
L+ doe!> c..etJLd,[nat
q, with infinite models of size X, X < K
Suppose that there is
,
but not of any size between x+ and K (included). By definition of U5wenheim number, there is a sentence l/J with all its models of power greater than X. Let e be a sentence whose models are the equivalence relations. The classes K 1 (respectively K ) of models of e having as many equivalence classes as a model of 2 q, (respectively, a model of l/J) are PC classes of L+, defined by the projection of the sentences: 8 /\
"6
is a function onto V" /\ 'ifx 'r1 Y (x E Y ++ 6(x) = 6(Y)) 1\
q, V (resp.
1/1 V).
Since these sentences do not have common model s of power 1ess or equal than", K 1 are disjoint PC classes of L+. However, they are inseparable in L+.If 2 has X has equivalence classes of power X', and (A', E ') E K has (A,E) E K 2 1 X' equivalence classes of power X', an easy backandforth argument shows tha t and K
(A, E) =L (A', E'), and so (A, E) =L (A', E'). C M terpolation fails in L+. • COROLLARY 4.5. 
Le;t L+ be a togle between
tivlzaUon and baU66yi.ng i.nteJtpotaUan, then Sk.atem theOllem.
From this we conclude that in
L ww (12. ) and
L havlng IletaM L+ 6aU6Mu the dOwnwalld Lowenheim
0
PROOF. There is a sentence in Lww(Q.O) ' and therefore in L+, which has models of power w but not larger. By Theorem 4.4 it must have Lowenheim number w .•
§
5
COFILTER QUANTIFIERS, A SIMPLER BACKANDFORTH.
n Let 12. be a quantifier symbol of type (n), q ~ 9(A ) is a eoMUeJr. lnteJr.pllUaUon 06 Q. if it satisfies Monotoni.U.ty: SEq and S ~ s' imply S' E q, and Vf.,f,Wbu.tivi..ty: SuS' E q implies SEq or S' E q. Obviously, q is a
q
cofilter interpretation iff the "dual" interpretation = {S I A  S €I q } is a filter over A. In terms of the language, (m; q) must satisfy the schemata:
\!X(¢4l/J) 4(Q.xq,4Q.xl/J),and
q
tifier is weompte;te if u {Sn I nE w} E q implies
QX(¢V1/I)4(Qx¢VQxl/J).
is an
w complete filter. for some n. •
Sn E q
Acofilter
quan
Equivalently, if
Qa is wco'mplete if its cofinality Ql' MagidorMal itz quantifiers are not
The cardinal quantifiers are cofilter, is greater than
w,
as is the case of
cofilter, however the quantifier
n H
o'
where
H~ xl'"
xn ¢(x 1"'"
eto x
is equivalent to the cofilter
quantifier
means "there is an infinite set n)
1
such
XAVIER CAICEDO
94
tha~
a 1, ..•• an
for all distinct
06 c.G6ille.!L quan..ti..Me.!L btte.!Lp/te:tatiOft6 and a.Mume the numbe.!L 06 /te!atiaft6 in Mc.h 6tJr.uc.tu!Le. is MMte, then: (a)
1'1
(aL; C) == (i:; D)
r 6 in
ad~a1'1
the quan.ti Me.M
1'1
(b)
(al; C ) ==
(e.)
t
6
(z, ;
(oz; C)
(oZ; c)
in6
~
(
.t
D)
S(w)
i66
Me
(1'1, and !J;; if they do not have an interpolant in L w w ( Q1 ) ' then the PC classes K = {OZ r T I or 1= 1>} and 1 K = {tr t T \;e, F 1/1} are inseparable. By Lemma 5.3, there are structures OZ I=(jl 2 and Z, F 1/1 such that (J1.. t r ~ L :t, ~ T. Therefore, 1> and !J; do not have interpolants in
L W
•
M
L ww (12 1 ) , In T ha.6 a (uncountab.tel model., it ha..6 a (u,ncouYl.ta.b.£.e) model. .6aU~nying at mo.6.t coun.tab.£.y n0lt each nEw. many n.typv.. in L W THEOREM 6.5. 
PROOF.
Take
Le..t
K
1
T be. a couYl.ta.b.te. the.oJty in
= K 2 = Mod(T)
in Lemma 5.3.
Theyare
obviously insepa
97
BACKANDFORTH SYSTEMS
rable; then we have of power at most
a6;'.
(a, Q1) If
wI'
t
i~wJ,
(JtJ; Q1) with P nonwell ordered, in
AIt , find
bEBIt such that
ia , a)=\
a
and f(;..
;6;
then
(Q )(i6, b)=L (Q)(Ol, a'); see Theorem oow 1 oow 1 3.6. By Lemma 4.1, ta , =L (m, Therefore, it and satisfy the same M type in L Since the number of equivalence classes of  is countable, the same
Hence,
a)
a').
M.
is true of the number of types.
a'
•
In L w w we have that a theory with infinite models has an uncountable model satisfying at mostcountably many types OVe!L ea.eh eoun;ta.b£.e ~u.b~e.t. Thi sis no't true here as shown by the counterexample:
Q x P(x), , Q x R(x), 1 1 If x If y(P(x) /\
p(y) /\
"
I 1/!)=P (1l)=jl' (Mod ModK(1/!,(1l), ModK(1/!) where the measure u ' defined on subsets of Mod K (1/.) is the restriction of u to Mod K (~ ). A few remarks about this concept are in order. 1) We could be tempted to write P(p IK) instead of PK( (T), then 1);
"'F F
6(0)) ;
otl
x 1/.{ 6] iff X: (T)E R¢; 6 (iii) for negation, disjunction, and conjunction, same clauses as usual; (i v) if l/J = 3 v e, then
"n.
~ F x l/J[ 6J if there is an I'l E W such that Ot. r:F x l/J! n( t1 (~) is the assignment which coincides with 6 in every variable except rly) v where it assigns n .
1\
For each oneone Mod
x E wT , and sentence 1); of ot F F x ~} •
(1j;) = { F :
where (possi
s'
L
r K,x: As for simple probability structures, we define
T
(p ). Mo~ K K, x The afield 11) of subsets of TK where the measure should generated by these sets ModT (1/) • K. x 1j.
r'V ¢ if K, x
A measure on this afield
Mod
r K, x
11
(1jJ)
invariant under
T
K
is
be defined is that
the product measure ~T
of
the measure u defined on the a  field M generated by the sets ModK (¢ ) for 4> a sentence of L,and invariant under "'K' We then defi ne PK, x ll/JJ=
108
ROLANDO CHUAQUI
llT (ModK, x (~.))
for every sentence
\j.
of Ls
.
Bernoulli's law of large numbers can be expressed and proved in this representation. We have, for each real number E > 0, a sentence of LS that expresses that the re 1ati ve frequency of the truth of the sentence jl of Lin the n fi rst terms of the sequence x differs from PK (~) by more than E. This sentence can be written by (*) V (3:mV(v1) " O. K, x These laws can be proved in the usual way, since the measure . ~is the product measure wjl. All other laws of large numbers and central limit theorems also can be obtained. In this type of models, the classical methods of estimationand hypothesis testing can be justified. I shall illustrate this with an example. My discussion follows closely Lucas (1970), Ch. V. Let us suppose we have a coin that looks unbiased and that we have no other relevant information about it. The natural simple probability structure which describes the possible outcomes of tossing the coin is that which assumes the coin's being symmetrical with weight distributed evenly between the faces. Thus, we assume a simple probability structure K defined as KoofExample 5 in Chuaqui 1977, Section 6. The reasons for assuming this model may be varied, just as the reasons for accepting any scientific hypothesis are varied. Among these reasons we may count, for instance, past experiences with similar coins. We now proceed to test this hypothesis. In order to do so, we toss the coin The probability that the frequency of, say,heads in these tossings is computed by building the compound probability structure wK (it is really enough with nK) and proceeding as in the Frequency Model constructed above. If this probability of the actual outcome is very low we reject the orign times for some large number n,
FOUNDATIONS OF STATISTICAL METHODS ina1 hypothesis that K is the adequate simple probability structure, The ability level at which we reject the hypothesis is arbitrary. However, the is justified (cf. Lucas, 1970, Ch. V) because given any significance level obtain an n large enough where this level can be attained. Thus, there is quence of events with probability approaching zero,
109
probmethod we can a se
If we reject the original hypothesis, using methods of estimation, we may assume a new simple probability structure K' as a model for the coin tossing, This K' has to assign a reasonable probabil ity (using again WJ iff T
~S.
R ~ p. and
we define:
113
FOUNDATIONS OF STATISTICAL METHODS (v)
Let t E T, then Tt={~: s ETand~ '" ( T, RI'S > ,then S E F.
T
E
It follows immediately that if T E F and x E T, then T and
x
Ix E F.
DEFINITION 4.6. K =( I, H) is a compound pltOba.b.u..u:y hbtu.cxWte (i)
I =(
(i i )
H is a set of functions such that if 6 E H, then
if:
R> is a causal structure.
F,~
Va
6E
F.
(iii) For each 6 E Hand;t EVa 6=T the set H(6,x) = {g(x) : 9 E Hand 9 I'T " 6 I'T } is a simple probability structure of a fixed simix x larity type v. (f v)
For each T
E
F , there is an 6
E
H, such that Va 6 " T,
is called the causal structure for K and ,I the set
K.
of compound outcomes of
DEFINITION 4.7.
Let K= (I, H> be a compound probability structure, Va 6. Then, 6 : 6 E A and VA 6 z: T J,
A CH, T E F, 6 E if and x E
(i)
AT = {
(i i)
A(T) "
{6 I'T : 6 E A }
I
(iii) let A ::. H T ' then, A(6,;t) " {g(x) : 9 E A and 9 I' T;t " 6 I' T } • x We now assume that for each simple probability structure H (6,x)there is given a symmetry relation between its subsets ~ 6,t. We extend this definition to an equivalence relation v between subsets of H, A, B,such that A ::. ~ and B £ H T" for
T , T'
E
F.
First we introduce a definition of isomorphic simple probability structures. DEFINITION 4.8. Let K and K' be simpl e probabi 1 i ty structures with universes A and A'. Then, (i)
K~
9
K'
iff K'
9
E
AA',
={ g* q{,
:
91
at.
E
A' A and
E K}.
Let ~ K ' ~K' be equi va1ence re 1at ions between subsets of K and K', B £ K and C £ K'. Then B '" C iff K ~ K' and g**B ~ K' C. 9 9 If B"'g C, then Band C are symmetric with respect to '1 O. Let S be a component of T and
e.
E Sn T
with
t~
E '!' with'!' C 1> and y'1 fv 4> : z " {x :
4>} .1.6, and only .1.6,
z = Uy I 'v' X (xE Y
InS we can introduce the fo 11 owi ng objects of (i) (f t )
+>
3 u(x E u ] A 4» ]
S:
a .opec. a, if a" T wi th fv T a coUection 06 cfu6!.>u B, defined by
"
0
B{z) if, and only if, 4>(z) with 4> E '!' and fv 4> (iii) a unaJt.y op~on F defined by x = F(z) if, and only if, x = T with fv T ~ {z} •
~ {z} •
Now, we restrict ourattention to systems S including the impredicative schema of class specification and some assumptions such as extensional ity and others that will be clear from the context. We define the ordered pair of two classes a and b, I a,b] , by z
"[a,b]
if,
and only if, z = a
x
{O}
U
b x {l}
We have the following definition schemata, for every TEA and 4> E 1> : z =
{T :
xl'" .xn
z=u
x , •• • ,x
4>} if, and only if,
z
>
{y :
3 xl'" 3 x~ ..
{T:4>}if,andonlyif, z"{Y:3x •••
n
3x~
,c.
(y " TIl, <j>)}
(yETA4>)}
Collection of classes F defined by F(z)
if, and only if, h(x E f A z = F(x) I,
where F is a unary operation, are called .6upeJt.cfu6!.>u. Superclasses can be represented using the relation It = UX{TX{X} : q,} which represents an operation defi ned by the term T, in the sense that for every 4> E 1> , I 'v'x (q,'" T " It*{x}). S Hence if we define z = x In q,] if, and only if, z =
U
{T
x {x}: q,} with TEA and q, E 1> ,
we can represent the superclasses F by the relation F =i F(x): x E f] in the sense that any class z of this collection is the image under F of {x} for some x E f. F is called a !.>upeJt.c1M.6 ltelat.i.on. F is called the de6.1.ni.ng opeJt.a..Uon of the superclass relation F and is denoted by Oe6 F j f is the set of
124
MANUEL CORRADA
c.odu of F , in symbols
Cd F. Bold face capital letters F, G, H , ... will be used to denote arbitrary superclass relations and the corresponding bold face italic letters the respective superclass. The corresponding light face capital letter will be used to represent its defining operation. If a superclass relation is represented by an arbitrary bold face capital letter, say F, it will be assumed that the correspondi ng sma 11 bo 1d face letter, f, denotes the set of codes of F.
In order to give full proofs of various result stated in the subsequent discussion we would have to use some properties of superclasses. As we have pointed out, superclasses can be represented as a special class of relations, that we have called superclass relations. As a consecuence of this,various results in the theory of superclasses can be obtained from analogous results concerning superclass relations. First at all we introduce the b~nany pte~eate eta between classes and superclass relations, as an abreviation: y 1/ F is an abreviation of 3 z(zEf 1\ F(z)=y). 3y 1/ F will be an abreviation of 3y(y 1/ F). By means of the eta predicate the fact that superclasses can be represented by their corresponding superclass relations can be stated precisely as follows in the equivalence of the following two schematic propositions: (a) F is the superclass relation that corresponds to the superclass F (b)
F(z)
~6,
and only
.(6,
z 1/ F
Due to this fact we will use interchangely the terms superclass and superclass relation. In most of the cases we will say superclass instead of superclass relation. Let ~ be a binary relation. class f~( F) as follows:
For an arbitrary superclass F we define
f~(F) = {<x,y): xEf II yEf 1I  z ={{x}, {x,y}} ,
from (vi). from (vi), (vii) and P.
z e u x v  zEV !\3x3y(X Eu!\y EV!\z = } n A = {x : xEA A <j>} • • An alternative, weaker, formulation of Metatheorem 2 would be {TI,T2}=P+ECEO' Let xE = CEO U {T 3, T4, T5, T6}. By the second part of Metatheorem 2 and Lemma 2
PARAMETERS IN THEORIES OF CLASSES 1, (xii) we have XE2 I P+E 1.
129
Using Metatheorem 1 and 2 we get:
We believe that the above metatheorems, in particular Metatheorem 3, are specially convenient, in the axiomatic foundations of theories of classes that include the impredicati ve comprehensi on axi om schema, such as the Genera 1 Class Theory or oneof its extensions such as the Impredicative Theory of C1asses. This leads to considerable simplifications in the metamathematics of class theories with the schema CEo As examples of this kind, 1et us quote the construction of £ , the constructib1e universe, [cf. Chuaqui 1980 l , and the techniques of replacing classes by numbers in Corrada & Chuaqui 1978. It is known by resu1ts in Gi1more 1974 that the extensionality axiom, that we have called E, is inconsistent with some set theories. Thus, for foundational purposes it will be convenient to eliminate Axiom E. In all the metological relations that implicitely involve the derivability relation I , that appear in the above metatheorems, this re1ations are relativized to sentence E. In order to avoid this relativization there is a device that we sketch here and wi11 prove to be particulary useful in the next section. In xEl +P, the sentences that refer to the existence of some particular" classes and operations between them insure the existence of this classes and operations but not it uniqueness. Consequently locutions such as "y is one of the classes of elements in a such that ~", "y is one of the unions of", "y is one of the universal classes" applies properly in this case. We shall show how to avoid E by an example of the stepbystep procedure involved. By a usual theorem of first order logic [cf.Monk 1976, Theorem 11.32 pp. 2101 even in the absence of extensionality we can introduce denotation symbol s for classes and operations between them. In order to illustrate this step  by  step procedure we take the last part of Metatheorem 2. Here it is used the fact: CEO + I ~ A () V = A. We have: (i) (i i) (i i i) (iv) (v) (vi)
{CEO' {CEO' {CEO' {CEO' {CEO' {CEO'
I}
I
I}
I
I}
I
I}
I
I}
I
3yVx(xEy 3yVx(x Ey \fx(xEA+ Vx(xEAf\ Vx(xEAf\ \fx(xEAnV
...... 3 u Ix e ul f\ x = x) , by CEO' ...... 3u!x EuJ), by (i). 3u(xEu)), logical inference. 3u(x Eu)+x EA), logical inference. 3u(xEu) ...... x EA), by (iii), (iv).
by I, (i i ), (v) and the corresponding denotation of the symb01s. Arguing in this way we obtain the ana1ogous of the previous metatheorems, not invo1vi ng the sentence E: I}
I
METATHEOREM
1'.
METATHEOREM
2'.
METATHEOREM
3'.
CE=pXE2.
A set of sentences
W is ca11ed
...... xEA),
CE=pxEl' HI' T2}1 CEO' CEO I T2 and, {CEO ,I} I T1.
n~~e a~omat£zab!e
OVe4 a
~et
On
~entenc~
MANUEL CORRADA
130
r
if there exists a sentence
such that
~
{r,~}
=W.
It is easily shown that for every n;' 1 in the metatheory, CE +1 is a finite n extension of CEn' This can be proved adding to CE n all the possible classes and operation between them definable in CE n +1 but not in CEn' This large number clearly is finite. The restriction n;' 1 is not essencial; in fact as a consequence of Metatheorem 3 or Metatheorem 3' we obtain:
CE and because CE
,fA
MrU:te1y ax..ioma.t.i.zab!e oveJt CEO
is not finitely axiomatizable we have in consequence: CEO u not 6{.rU:te1y
aJUoma..tizab!e.
To conclude we may mention that in spite of the simplicity of the schema CEO' a very weak theory of classes based upon this schema can be proved to be undecidable. §
2.
THE SETS OF SENTENCES
pr ed
XE1
AND
XE 2pr ed
We denote by Tired Zermelo's predicative Aussonderungsaxiom without rameters, t , e. : Tpred 1
Let
Ifx3yltz[ZEY+>ZEX
XEpred 1
=
{Tpred T l'
2'
A~) where ~ES, fv ~ ~
{x}
pa
and x « y.
T} . 3
It is easily seen that in XEired + p + E one can prove the existence of singletons' the empty class, pairs, ordered pairs of sets and the union of two classes. With this, using Theorem 1 in Levy 1974 with minor modifications we obtain the analogous of Metatheorem 1 for the predicative comprehension axiom: METATHEOREM 4. The analogue of Metatheorem 2 is as follows: METATHEOREM
5.
{Tired, T2} ~ CE6. CEo ~ T2 and {CEO'
I}
~ Tired.
If we set XE~red = {CE6' T3 , T6 } the analogous of Metatheorem 3 can be lished using Metatheorems 4 and 5:
e s ta b
Due to the remarks at the end of the above section it turns out that the sentence E can be eliminated from these metatheorems, and we have: METATHEOREM 4'.
CE S=pX E1pred
PARAMETERS IN THEORIES OF CLASSES
METATHEOREM 6'.
131
CE S;: PX E~red
It is easily verified that for every '1re F),
characteristic of e. We designate by t: the smooth operator that associates to any nonempty the set of all functions 6 of PIA) into A, fulfilling the following
set A
condition:
6(B) f/ B if B * A, and 6(A) is any element of A. t xF [x]
is defined as
'an x such that iF [x] "
and t and r are associated;
in several cases, d(r) may be restricted to functions which satisfy the require
¢ V if V is a definable set different from the universe, d[r) (V) is a fixed element of the universe otherwise. The axioms of rare:
ment that: d(r)[V)
and
rxFz[xl = rYFz[Y]' \lx(P
+7
Q)
> r
xP = r xQ
and the scheme 3x(x = r x F A F) .... \lxF.
THEOREM 6.
vbto.6 b., e 06
Let L e [LI.' L r) be the tanguage L .ill wlUeh one 06 ,U.6 [I., r l , with the eotl.Jl.e.6ponding axiom.6, and T u {F} be a set.
t.,
eto.6ed 60tLmuta.6 06 1)
e (I., r ] b., notLmat;
3)
16
(LL,L
4)
r)
(LI. ,L r
l.
We have: 2)
inn
In L
rlF
1= F
i66
I' 1 F;
PROOF. Obviously,
6O!Lmuta 06 the 60tLm i3xF .... exF=ex(x*x)
L,
e and r are normal.
[rxF *
2 and 3 are consequences of the corWith respect to 4, it suffices to
that if i3xF, then we have \Ix "l F and \lx(LF
THEOREM 7. 
e
b., a theOlLem.
responding results for vbtos in general. Therefore,
I' 1 F in L
in L e (LL,Lr ) without e (L,r);
e [L r ) any
r x (x= x) .... i\lxF 1
axiom.
r
e [I., r ] doe.6 not oeeM in the 60tLmutM r U{F}, then:
\lx(F
Let LL e
+7
x*x), and
x = x), since
u
16
x = x
note
is an
exF= ex(x*x) . •
be the tanguage L in wlUeh two
and e, wah the tLe.6pective axlom.6. theotLy Twho.6e wnguage
+7
amo ng i l l vbto.6 Me I.
I. X (x * x) = e x (x * x) b.,
vaHd in
then LxF=ex«3: xFI\F) V (i3:xFl\x=ex(x* x))) Ll.e,
cs a theOlLem 06 T 601L any 60tunu.ta F. PROOF. Suppose that 3:xF. = eX(3:xF/\F)=exF.
Now. suppose that
Then, ex«(]:xFI\F) V (i 3:xF 1\ x= ex(x *x»)
But exF
and I.xF satisfyF; hence, I.xF = exF.
i 3: xF. Therefore.
ex ( ( 3:xF 1\ F) V ( i 3: xF /\ x =
ex(x*x»)=ex("l3:xF/\ x =ex(x*x)) =ex(x=ex(x*x»=ex(x*x);
a
NEWTON C. A. DA COSTA
140
since LX(X*X)
= ex(x*x),
it follows that
t
xF
=
€xF . •
From now on, when we say that a is a structure for L we suppose characteristic postulates of the vbtos of L are valid in Ol.
tha t
th e
MODEL THEORY AND NORMAL YBTOS.
Now we shall indicate how most results of elementary model theory can be extended to firstorder languages with normal vbtos. First of all, it is immediate that we have the following propositions: THEOREM 8 (Compactness, first form). A 6oltmu£.a A M, a .:theoltem 06 a .:theolty T 06 L i66 A M, a .:theO!Lem 06 bOme 6bu.:te.£.y axi.omatized pM.:t 06 T . THEOREM 9 (Compacteness, second form) • 
Let T be a .:theMy in L;
T hall
a mode.£. i66 evelty MvU.:te.£.y axiomatized pM.:t anT hall a mode.£..
Let a = is a weak isomorphism of a and J} iff the following conditions hold: 1) rf> is an isomorphism in the usual sense of A and B; 2) rf>(rJl(vxF» = J}(vxF¢) for any formula F of L{Ol) having no free variables other than x, and any v E V. The bijective mapping ¢ from the universe of Ol to the universe of J} is called a strong isomorphism of ol and J}, which are said to be strongly isomorphic, iff we have: 1) ¢ is an isomorphism of A and B; 2) dry) (K) = k if and only if g[v)(i(K)) = ¢[k), for any v E V, K c loll and kE 1001,where if, is the canoni ca1 extension of ¢ to the power sets of lOll and I;;' I . An isomorphism is a weak or strong isomorphism. Ol and J} are isomorphic if they are weakly or strongly isomorphic. Let rf> be an i6omO!Lp~m 06 Ol and;;'. c1.o6ed .:te.Jtm a 06 L (Ol), and OleA) = J}(A rJ»
THEOREM 10. 
nOlL
eVVLIj
Then
nOlL
¢ (Ol
evVLy
(a»
= J} (a rJ»
c1.O6ed noJtmufu A
06 L (Ol).
PROOF. By simultaneous induction on the lengths of a and A. If ¢ is a weak isomorphism, the proof is as that of Lemma 1 of Shoenfield 1967, p. 172, taking into account that ¢ (Ol (v xF) = .G(vxF¢) by definition. If ¢ is a strong isomorphism, the proof is Similar, but we have to show that ¢(Ol(vxFl) = J}(vxF rJ». By induction hypothesis, we may assume that a (F[a]) = T iff J} (FrJ>[arJ>])= T. Therefore, ¢(d(v)([al rJl(F[a] )=T and a is the name of a l ) = g(v) i( [al
141
VARIABLE BINDING TERM OPERATORS
OZ(F[a]) =T and a is the name of a]»
= g(v)([¢(a)l.fr (F¢[a¢]) =T and a'"
is
the mane of ¢(a)]) = .G (vxF"'), by the definition of strong isomorphism, hence ¢ (07, (vxF)) = fr (vxF'" ) •• COROLLARY. 
I.>bwng LoomOltphfAm Lo a weak LoomofLphfAm
EVVtIj
and
I but not c.on
vVt.Qetlj) .
r is a set of formulas of L, Ol and .fr are structures for L, and lOll subset of I £.1.
is
a
In this case, if a is an element of 107,1, we employ the same name
for a in L(07,) and in L(.fr), as in Shoenfie1d 1967.
We say that Ol is a r sub
structure of.G (and that.f,. is a rextension of Ol) if, for every closed formula A
a. (A) = T
in r (Ol),
imp1 ies that .fr (A)
= T.
It is clear that if r is
of all formulas of L which do not contain quantifiers and vbtos, and and £. = < B, 9
>,
the set
07,= in
If I' is the set of all formulas of L without vbtos, we say
elementary substructure for rsubstructure, and if I' is the set of all formulas of L, then the rsubstructures are called complete substructures. THEOREM 11.
16
e Lo a c.on.atant 06 L and x Lo a valUable, x =e betongl.> to I",
and Ol Lo a p; l.>ubl.>tJtuc.tuJte 06 .fr, thcn 07, [e ) =
PROOF.If
We have:
j} (e J •
See Shoenfie1d 1967, p. 74 . •
a
and .fr are structures for L, and lOll C l.fr I, then the structure
is defined as in Shoenfield 1967, p. 74.
The notion of rdiagram of
.f.r 0l OZ is also
the same of that book, and denoted by Dr ( Ol). THEOREM 12 Me I.>bllic.tWLe.6
.(.oo
(Diagram Lemma) • 
oOlt L
I.>uc.h that
16 r Lo
a
se:
06 60fLmula.a .{.n Land
107, I C liT I, then 07,
iTOlLoamodel06 0r(Ol).
PROOF. 
If
a.
07, and .fr
Lo a r  l.>ubl.>tJtuc.tWLe 06 .fr
is a p; substructure of .fr and A is an axiom of
°r (OZ),
then A is,
by definition, valid in iTOl ' and froz. is a model of 0r(Ol). Conversely, if.f.rOZ is a model of Dr(Ol), then every formula of rrOZ) which is valid in a , is also val id in .fr, and
o:
is a I'  substructure of iT . •
A set of formulas r is said to be regular if x = y and x"* y belong to and for eveiy formula A in r, the formulas of the form A[ xl' X2 ' ••• , xu] belong to I", THEOREM 13
(Keisler I s model extension theorem) • 
r, also
Suppol.>e that 07, .u, a
142
NEWTON C. A. DA COSTA
6vwc.twr.e 601t L, T Ls a .theO!llj wah laYlguage L, aYld I' .u, a lLegu£M 6et 06 601tmulM 06 L. TheYl, a has a rQxteYl6ioYl wlUch.u, a model 06 T i66 eveJtY theOltem 06 T wlUch if.> a dWjUYlCUOYl 06 negatioYl6 06 60ltmu£M in I' .u, va1.id iYl (Jl • PROOF.
The proof of Shoenfield 1967, p. 75, of Keisler'S theorem for standard
model theory remains valid, since all results on which it is based (the reduction theorem for consistency, the completeness theorem, the diagram lemma, etc. ) are true for model theory with vbtos.
•
Let rand t. be sets of formulas. t. is said to be associated to I' i f formula of the form
I;fx 1 I;fx 2
•••
I;f x n A
every
is in t., where A is a dinjunction of
negations of members of r. Bya reasoning similar to that of Shoenfield 1967, pp. 7576,
we establ ish
the following proposition: Let I' and t. be 6W 06 60ltmu£M,
THEOREM 14. 
c.iated to
r',
wdh I' lLegu£M and t. aMO
OZ a 6tltuc.twr.e 60lL L aYld JJ a t. exten6ioYl 06 OZ; theYl, theJte exJA:t6
a I' exteYl6ion t 06 JJ which if.> a complete ef.>teYl6ioYl 06 OZ (and a 60!ttioM an elementMlj exteYl6ioYl 06 OZ). Suppose that T and T'
al6o
are theories of L; it is clear that T' is an exten
sion of T iff every model of T' is also a model of T. If I' is a set of formulas in L, r' is a subset of I' composed of all formulas of I' which are theorems of T, and every structure in which all formulas of
r'
are val id is a model of T, then T is equivalent to a theory having all nonlogica 1 ax i oms in r. THEOREM 15 (LosTarski). Let ~ denote the. se: 06 aU 60ltmu£M 06 L which do Ylot contMn bound vM. A theOltlj T .u, eQu do not COntMYl bOUYld vAAiablef.> i66 eveJtlj ~  6ubJ.>tlLuctlllLe 06 a model 06 T if.> al60 a model 06 T.
PROOF.
If T is equivalent to a theory T' satisfying the conditions of the the
orem, then every definition of
~
~
 substructure OZ of a model .r;. of T' is a model of T', by the
 substructure; thus, every
~
substructure of a model of T is
model of T.
Conversely, suppose that every
mode1 of T;
we have to show that if every theorem of T whi ch does
~
bound variables is val id in the structure C. , then ( Keisler's theorem, also a model of T.
C has a •
Analogously, we have:
~
a
 substructure of a model of T is a is a model
 extension which is a model of T,
no t con t a i n of
T.
and so
r:
By is
VARIABLE BINDING TERM OPERATORS
143
LiU be the. ~iU 06 aU. 6011mulal> in L in wlUch :thel1.e Me no oc06 qua.n:t.LMeM, and T a :theMy wUh £.anguage L. T ~ e.qiUvatelVt t» a :the.oJr.y who~e non1.ogicat axiom~ Me in i66 eve.Jr.y Mb~bwc:tUJte 06 a model 06 T ~ a model 06 T. THEOREM 16. 
CU!rJLenc~
Le:t e be. the: set: 06 aU. 6011mulal> in L wlUch do no:t con;ta.{.n vb:tM.
THEOREM 17. 
FM anlj :theMIj T with .tanguage L, T ~ eqiUvate.1Vt to a :theMIj who~e nonC.ogicat ax.iom~
e i66
belong to
eve.11.1j elemelVtMlj ~ub~:tI1.uc:tUl1.e 06 a mode£. 06 T~ amode.t 06 T.
The sequence of structures O1 j = , 01 2 = < A 2,d 2>, ... is a chain if for every n , An+ 1 is an extension of An in the usual sense of model theory. A union of a chain O1 j.01 2 •... is any structure 01= such that A is the usual union of Aj• A2.... Consequently. a given chain Olj. 012 •... has in general several unions. An elementary (complete) chain is a chain O1j. (Jl2 . . . . in which. for any n , a ; is an elementary (complete) substructure of (Jln+l Clearly. if , ,... is an elementary (complete) chain, then Aj• A2 constitutes also an elementary chain in the usual model  theoretic sense. A strict union of a chain of structures (Jlj, 01 2 " " is any union ot of (Jll' Ol2"" satisfying the extra requirement that if A is a formula of L ((Jln) without free variables, then 01 (A) =Oln(A). for every n; in other words, a strict unionof a chain is any union which is a complete extension of every structure of the chain. Obviously, the next two propositions hold: LiU r devw:te :the ~iU 06 aU. 6011mulal> 06 L without bound van.i.a.nlj union 06 a chain 06 ~:tI1.uc:tUl1.~ 6011. L ~ a r  ex.;teMion 06 eVel1.lj elemen:t a 6 :the chain. THEOREM 18.
ab.t~;
THEOREM 19
(Tarski). 
Any union
06 an
elemen;ta.Jr.y chain Ls an e..teme.1VtM1j ex
.te.n6ion 06 e.Vel1.lj eleme.1Vt 06 :the. chMn. THEOREM 20. 
Anlj comp.tiUe chMn hal> a
~:tJr.,{c:t
union.
PROOF. Let01 j= , Ol2= , ... be a complete chain of structures for L ; L' will denote the language obtained from L by the adjunction of the names of all elements of U= IAjl u IA 2I u ... (the names are supposed to satisfy the conventions of Shoenf te Id 1967). We define the denotation .vu(t) of a closed term t of L' and the value V"U (A) of a closed formula A of L'. relatively to U, as follows: If t (A) is a closed term (formula) of L'. there exists n such that t(A) is a term (formula) of L(Oln); then we pose .vu(t) =
otn ( t)
(V"u (A)
= OZ n ( A)
).
144
NEWTON C. A. DA COSTA
Since Al ,
A2
, •••
A,
constitutes an elementary chain, its usual union
having U as universe, is a structure for L without vbtos and also an el ementary extension of any structure of AI' A2
, ••.
Let d' be a function defined on V and
whose values are functions from the power set of U into U, such that: and there exists a formula A of L', having no free variables having also the property that name of a I, then d'(v)(VI
d' Iv)
V = lal a
=JJu
E
U and
(vxA); if V CU
IV) is an arbitrary member of U.
VulAxla 1)
if
VC U
other than x, and
= T,
where a is the
but there is no formula as A,
Clearly, d' is well defined because
OLI,
ct 2 , ... is a complete chain.
Reasoning by induction on the length of terms and formulas of L',we show that
a' =
( A, d') constitutes a pseudostructure for L.
Fi rstly, one has to prove
that d'(v), for vEV, is such that it makes Iv and II v valid. offers no difficulty.
In connection with I I v
""u ( A x Ia I) = T and a is the name of a
I
'
The proof of Iv
we have to show that i f
I b I ""u( B x Ib I ) = T and b
la I
the OLI , ct2 , ... is a complete chain. Secondly, since for any formula A of L(OL n ) we have ct(A)=T iff OZn(A)=T, the characteristic logical axioms of of v a 1so ho1din OZ' . name of b],
then
vxA = vxB,
=
but this clearly holds by the fact that
Therefore, since the vbtos of L are normal (and axiomatizable), that there exists a structure
the chain Oll
'I
OL z , ...
is
it follows
OZ = (A, d ) for L which is a strict union of
•
An existential formula is a prenex formula which does not contain vbtos,
and
such that all quantifiers in its prefix are existential. As in Shoenfield 1967, p. 77, we can prove the following proposition: THEOREM 21
(Chanq  Los  Suszko) • 
A theMy T .u., equivalent to a theMy
havIng oVJ1.y noVJ1.ogIcal auom/.) whIch Me. e.wte.~ I6, and only ,[6, 06 any c.hMn 06 mode.lJ., 06 T .u., a.t!.lo a mode..[ 06 T . PROOF.tions.
any un,[on
The same as in the case of usual model theory, with obvious modi fica•
The cardinal of a structure OZ is the cardinal of the universe of OZ; the structure
ct is ca 11ed fi nite or i nfi nite accord i ng to whether its car din ali s
finite or infinite; similarly, we define a countable or uncountable structure. Suppose that m is an infinite cardinal.
A firstorder language L with vbtos
is said to be an mlanguage if the set of its symbols has cardinal
.;;;; m.
ory T is an mtheory if L (T), the language of T, is an mlanguage. language (l{o  theory) is called a countable language (theory). ory
without vbtos, we have:
A theAn l{o
As in model the
VARIABLE BINDING TERM OPERATORS
THEOREM 22
16 m v., an e
that T and T' Me srandarui theoJUe!.>. T U T' v., v.,teYlt  a UOI.> ed 60Junuia A, wdhout vbtO!.>, !.>llcch that A v., a theoftem 06 T and "l A ~ a theoftem 06 T'. PROOF. Evidently, if there is such formula as A, then
T U T' is inconsistent.
Now, let us admit that there exists no formula in the conditions of A; under this hypothesis we have to show that Shoenfield 1967, p. 80.
T U T' is consistent.
vJe proceed an in
One has to construct an elementary chain AI, Az, ...
models of T without vbtos and another elementary chain
A;, A; , '"
T' without vbtos , satisfying the condition that Al
A~!L'
/
u
of
of models of
Az!L'
A~!L
is an elementary chain, where L is the language whose nonlogical symbols are those common to the 1anguages of T and of T', both wi thout vbtos.
The c ha ins
AI, Az, ... and A~, A~, ... are easily constructed. If A and A' are the usual unions of AI' Az, ... and of A'l, A~, ... , then A is a model of T without vbtos, and A' is a model of T' without vbtos. In consequence, we can construct a model Shoenfield 1967, p. 80). have to expand
B to a model .G
culty at all, since the vbtos of model and is consistent.
B of
T U T' without vbtos
In order to obtain a model for
•
= (
B, d ) for T
U
T'.
T U T' are all normal.
(see
T U T' (with vbtos), we But th i s offers no diffiTherefore, T U T'
has a
146
NEWTON C. A. DA COSTA
The preceding theorem could also be proved as a consequence of Theorem 4. Precisely as in usual model theory, we deduce from the preceding proposition Craig's interpolation lemma: THEORE~1
25
(Craig interpolation lemma) . 
Let T and T' be two MandaJu:i
theouV>, A a 601Unu1a 06 L(T) wUhouX vbt06 and B a 601Unu1a 06 LIT') without vbt06. 16 A>B J.A a theOltem 06 T U T', then then« J.A a 601Unu1a C, a.L60 withOuX vbtol.> , Mch that 'i A > C and 'if C > B •
PROOF. Shoenfield's proof remains valid, with minor modifications. • Q will denote a set of nonlogical symbols of the language L of a theory T. We say that an nary predicate symbol p not belonging to Q is defi nab1 e in terms of Q in T if there exists a formula A,without vbtos , whose sole free variables are n distinct variables xl' X 2, ••• , x n ' and whose nonlogical symbols belong to Q, such that IT px lx 2" .. x n ++ A. An nary function symbol f wh i c h does not belong to Q is definable in terms of Q in T if there exists a formula A, free from vbtos and satisfying other obvious conditions, such that 'T Y= fXIX2,,,xn ++ A (y ,x] ,x2'''' ,x n are distinct variables). The notion of uweak isomorphism, where u is a predicate symbol or a function symbol, is defi ned as in Shoenfie1d 1967, p, 81. Then, it is immediate that we have: THEOREM 26 (Definability theorem of PadoaBeth) • Let Q be a I.>et 06 noVltogica.t I.>ymboto 06 a I.>tandaJtd theolLY T, and let u be a pftedicate OIL 6unction I.>ymbo£. not in Q. Then u J.A de6inab£.e in teJtml.> 06 Q in T i6 and oVlty i6 60ft eVeJlY two model!.> o; and ;;. 06 T and eVeJlY bijection 1> 6ftom I all to I;;' I which J.A a kweak if.>omoftphiJ.>m 60ft eveJty k in Q, if.> a.L60 a u weak J.AomolLphiJ.>m.
Though we shall not go into detail s , we note that the Theorem of Padoa  Beth has a more interesting formulation (whose proof is based on the main resu1 ts of Corcoran, Hatcher and Herring 1972), as follows: THEOREM 26'. AI.>I.>u.me that Q J.A a se: 06 noVltogica.t I.>ymboll.> 06 the language 06 the theOfty T and that u J.A a rtonlogica.t I.>ymbol 06 T whic.h does not be£.ong to Q. u Is de.6inab£.e. in teJtml.> 06 Q in T i66 60ft any two model!.> al and ;;. 06 T and any bijemve. 6unmon 6ft om Iall to 1;;'1 whic.h J.A a kweak if.>omOftphiJ.>m 60l( eveJly k in Q, 1> J.A alI.lo a uwe.ak J.AomOftpftJ.Am.
Another result which constitutes a complement to Theorem 26', and can be established by the methods of Corcoran, Hatcher and Herring 1972, is the definabi1ity theorem for vbtos below.
VARIABLE BINDING TERM OPERATORS
147
Let at and JJ. be structures for L, v a vbto of L, and is any model of T whose cardinal is m, we can obtain another model if = (B r h > of T of cardinal m, which is not strongly isomorphic to a, by changing the function g(E); this can be done as follows: let V be a subset of the universe of OZ such tha t it is not defi nab1e in the 1aguage L ( a), i , e., there is no formula F of L(a), having at most one free variable, satis
VARIABLE BINDING TERM OPERATORS
fying the condition:
V = [a I Ol(Fx[a] ) =T l .
=
the sole difference that if g(e)(V)
149
Then.& is exactly 1 ike
k, we make
= l,
h(e)(V)
with
l
a,
with
=1= 11. • •
It is a little tedious to check, but Morley's theorem remains true when normal vbtos are added to firstorder languages.
That is to say, if a countable theory
T (with vbtos) is mcategorical for a given uncountable cardinal m,
then it is
also m  categorical for any uncountable cardinal m. The symbol Sn (T) and the notions of type of an ntuple of individuals of the structure Ol for L, n  type of a theory T, principal subset of Sn (T), etc., are defined as in Shoenfield 1967, pp. 9192.
By the same proof as in that book,
we
establish the following proposition: THEOREM 31. T Ls a theoJty and r a non void !>ub!>et 06 5 n (T), !>a;(:i/.,6ying the condiJ:ion!>: 1) T i/., countable; 2) 16 A i/., a cU6junction 06 negation!> 06 60Jtmula!> 06 r, then A it> not a theOltem 06 T. UndeJt thue condiJ:ion!>, theJle ewtt> an n  type K in a countable model 06 T !>uch that r C K.
Any ntype in a countable theOlty
COROLLARY. 
model
i/.,
an n type in a co untable
06 thi!> theMy.
An nul tra filter in T, JF, is a subset of Sn (T) but JF =1= Sn (T);
A, A > B E JF, then
If
THEOREr~
in
T
IT
2) If
A, then A E JF;
B E JF;
PROOF. Let T' be the theory , ••• ,
4)
T
it> an n  type in
T
(and any n type
T to which we have added as new axioms all the for
fnl, where A belongs to JF and fl, ... ,f n are n new constants.
T' is cons i stent, because otherwi se "1 Al V
to IF, would be a theorem of T;
Sn (T), which is absurd. an=
JF C Sn (T),
it> an n  uUJta 6iUeJt in T).
mulas Al r j long
1)
A 1\ B E JF;
5) For any formul a A in Sn (T), A E JF or "1AE JF .
An n  uUJta 6iUeJt JF in
32. 
such that:
3) If A, B E JF, then
Ol(t n),
A I ' ... , A k
be
but in this case IF would be identical
to
...
V lA k '
Therefore, T' has a model
then IF is the ntype of
THEOREM 33 (Ehrenfeucht). 
Let
a ,
where
and if al = Ol(rl)"'"
(al.···' an)' •
T and
r be Jtupec:Uvely a countable con
t>i/.,tent theoJty and a !>Ub!>et 06 Sn (T) which i/., not punupal. UndeJt thue condiJ:io Yl!>, theJte it> a co untable mo del Ol 06 T !>Uch that r i/., not included ina n y n  type
06
oi .
PROOF. The proof of Ehrenfeucht's theorem of Shoenfie1d's book, pp. 9091, mains valid when vbtos are added to firstorder languages.
•
re
150
NEWTON C. A. DA COSTA With exactly the same proof as in Shoenfie1d 1967. pp. 9192. we have:
THEOREM 34 (RyllNardzewski). FoJt a eomple..te and eountable .theoILy T havtng only tnMnLte model6 , :the f,oliowinq pMpe!LUe6 aILe equivatent: I} T.iA K 0  ea:tegofL,{,catj 2) Fon. any na:twe.a1 nwnbe!L n, :the 6et: 06 n  :type6 in T .iA Mnliej 3) FOIL any natWtat nwnbe!L n, ali n  :type6 tn T aILe plLinupat.
Among the
immed i a te consequences of the precedi ng results. we menti on th e
following: THEOREM 35. 
AMwne :that e .iA :the onty vb:to 06 :the funguage 06 a :theMy T.
16 ali aUOm6 06 T aILe logieat, .then e
e
06
w
no.t deMnable in T.
~:t  OILdeIL COMe6pOnmng aUom6). A .the My T .iA equivatent :to a .theMY WhOM auom6 do not eontain e t66 eVe!Ly e  6ub6UudWte 06 a mode£. 06 T w a mode£. 06 T .
THEOREM 36. 
Let:
be .the MU
ali 60lLmulM w.Uhou.t e in .the
jJfLemea:te ea£.euJ:U6 with equatUy and Hilblli'6 .oymbot e (w.Uh .the
There exist infinitely many vbtos. most of them having interesting properties. For instance, the following ones. mediately
whose characteristic postulates are listed im
after their introductions (they are also governed by the basic
princi
ples Iv and II v; and the smooth operators associated to them will not be exp1icit): en:
If F [xl does not define the universe. en xF [ x I
made
is an individual x
such that x satisfies F iff there are at least n individuals satisfying F. Characteristic postulate of
en:
*"
3xiF>((3x (x = en xF II F) /\ xnl
*" Xu /\
where the variables ew :
+> 3xI 3x2'" 3x n( xl x2 1\ xl F [xI! 1\ F(x2l /\ .,. 1\ F (xu]»,
Xl. x2 ..... xn
and
*"
x3
.,.
x are distinct.
If F[ x ] does not define the universe. then e(,)xF [xl denotes an i n
dividual x such that x satisfies F iff there are infinitely
many individuals
sat; sfy i ng F. Characteristic postulates of e w : (3xiF II 3x(x =€wX F II F)) > 3xI'"
for n = 1. 2 •...
3x n(xI
*" x2
II Xl
*" X3 II ... 1\
151
VARIABLE BINDING TERM OPERATORS
(3 x , FII 3Xl ... 3xn\ly(F[Y)~(Y=Xl VY=X2 V", V y=Xn»))~ I 3x (x = e'J x F /\ F)
for
,
2, '"
n = I,
n Let us admit that F[x] does not define the universe; then, L n F[x) denotes an individual x such that x satisfies F iff there are exactly n i ndinxF ual s satisfying F. If 3n xF and 3 abbreviate respectively the formula stating that there exist at least n individuals satisfying F and the formula stating that there are at most n such individuals, then the characteristic postu1a te of
L
n
is the fo 11owing: 3xIF ~(3x(x= LnxF/\ F) (3nxF /\ 3
Similarly,
enx F [ x ] could be introduced:
nxF».
for F [x J not defining the uni
verse, F[enxF) asserts that at most n individuals satisfy F. We call Frege's symbol the vbto f syntactically defined by Iv' II
v
and
f xF = fxG ~ iix (F  G) .
Consequently, we have: iix (F ++ G)  f xF = f xG THEOREM 37. 
WhateVeJL smooth. openaron.
6 we a.!l.6ouate to
f, f, associated to
6, .u., not nOILmM.. PROOF.
Let us suppose that L is a usual firstorder language having at least n
constants cI , c z , ... , c n , and that
a
is a structure whose universe has n el e
ments.
We shall denote the elements of lOll by al,aZ' ... , an' and admit that
Ol(ci)
=
ai' i
= 1,
2, ... ,n.
able by a fonnula of L.
It is clear that every subset of lOll is defin
Therefore,
a
cannot be expanded to a structure for
L
with f, since the cardinal of loZI is less than the cardinal of the power set of
lOll.
•
When axiomatizable but non nonnal vbtos are included in the set of
vbtos of
L, the 'corresponding' predicate calculus with vbtos is not in general a conserv· ative extension of the predicate calculus without vbtos. The proof of the next theorem, which is a little long,' will be omitted here. We only remark that it is based on an idea of Corcoran, Hatcher and Herring Corcoran, Hatcher and Herring 1972):
(see
in order to prove the completeness theorem
for a particular language with vbtos, they replace it by a convenient usual first
152
NEWTON C. A. DA COSTA
order language, with new furct i on symbols, subjected to appropriate axioms. THEOREM 38. 
r 6 any 1L6u.a.l
6Q!1. a 6illtQ!1.dVL !a.vrguage. £ (waho ut = (A, d ) 60!1. I: ptlL6 the. cou.oma.Uzabte.
f..Vtue.tWte. A Ol:
vbtof..) cart be. e.xpa.vrde.d to a f..Vtue.tWte.
vbto v, in which d ( v) if.> any e.!ement 06 v (IAI) a.vrd the auom6 0 6 v Me v a!i d, the.n v if.> noftma! IwnVLe. v if.> the !.lmooth opeJLatOl1. aMouate.d to v l .
The notion of ultraproduct (Shoenfield 1967, p. 104) can be adapted to structures for L, and its main properties remain true. Of course, generalizations of several other results of standard model theory offer no difficulties; this is what happens, for example, with Lyndon's theorem (cf. Shoenfield 1967, p. 94, exercise 6) .
In Corcoran, Hatcher and Herring 1972 there is a method of converting structures for L in structures for L without vbtos but with extra function symbol s . Witr. the aid of such nethod we can systematically reduce most problems of model theory with vbtos to standard model theory; for instance, Morley's theorem already mentioned, for firstorder languages with vbtos , can be proved by thi s method. It may also be employed to establ ish the soundness and completeness theorems for vbtos in an illuminating way (see Corcoran, Hatcher and Herring 1972). Some of the above results are adaptable to the case of firstorder predicate calculus with vbtos but without equality. Postulates Iv and II v have to be changed as follows: I~.
G[vxP z [xl]
F).
Originally, Hilbert envisaged the symbol e only as a formal device to facilitate metamathematical research; so, the sole scheme which e had to conform
with
was the following: or, better, 3xF > lix(x = e x F > F). Clearly, an interpretation of the e symbol so axiomatized has to be nonstandard. A second nonstandard axiomatization of the e symbo l is obtained if we impose that it has to be governed only by two specific postulates, to wit: exFz[x1= eyFz[yl, and 3xF > lix(x = exF > F) . The third treatment of e so far considered is the standard one , studied above. A variant of it results by requiring that the function d(e) in the structure Ol = (A,d)
be such that d(e)(IOlI)
=d(e){~);
we get a complete system of axioms
for the e symbol so interpreted by the procedure of addi ng to the standard ax ioms for e the fo 11owing new one: ex{x=
x)
= ex{x ,p
xl ,
Nonstandard presentations have also been proposed for the description symbol L •
One of the best known nonstandard handlings of t is that of Hilbert and
Ber
nays (see Hil bert and Bernays 1934, vol. 1, p. 384, and Lei seri ng 1969, p. 101 ) . It is introduced through the
rule:
t
If 3:xF is a theorem, then txFis a term
and one may infer that Ii x (x =
t
defined and the semantics for
so treated cannot be standard.
t
xF > F) .
In consequence,
t
xF is not always
Other trea tments of the descri pt i on symbol are the fo·ll owing, a11 of them in
155
VARIABLE BINDING TERM OPERATORS
vestigated by Hailperin (cf. Hailperin 1954), and some of them by other authors (Liu 1974,
Montague and Kalish 1957,
and Scott 1967):
according to the first
treatment (Russell's), any atomic statement containing an improper l
xF, i. e., such that
description
3:xF is false, is also false; according to the
second
(a modification of Russell's), any atomic statement containing an improper description is true;
finally, according to the third, which is Frege's,
proper descriptions denote a given (fixed) object. second treatments are not standard.
all
im
Evidently, the first and the
In Rosser 1953 there is a calculus of
scriptions to which only a nonstandard semantics can be supplied,
de
i f soundness
and completeness are required. Notwithstanding, for most nonstandard presentations of vbtos nonfunctional semantics can be provided.
In order to give an idea of the method one has to em
ploy to construct such semantics, let us sketch the semantical analysis of a particular version of the e symbol.
We suppose that e has to satisfy only one spe
cific postulate: e)
3xF~\fx(X=EXF~F).
The language to which E is adjoined is a firstorder language with equality, with the notions of term and of formula conveniently modified.
The axioms
and
primitive rules of inference are the usual ones, plus the scheme e. Let us denote the language so obtained by A structure
LE
•
for L E is an ordered pair
waht6uYlkA:ioYleYl,
Grund. d. Math. 3,
Zeitschr. f. math. Logik u.
pp. 3068.
N. Bourbaki.
1966. Theorie des Ensembles, Chaps. 1 and 2,
Hermann, Paris.
J. Corcoran and J. Herring. 1971. Notv., OYl ct !.>emanUcat anaty!.>.i.6 06 vcttUabfe biYlcUYlg tV1m opeJtcttO!L6, Logique et Analyse 55, pp. 644657.
J. Corcoran, W. S. Hatcher and J. Herring. 1972.
VatUctbfe bbrcUYlg tV1m opeJtcttO!L6,
Math. 18,
Zeitschr. f. math. Logik u. Grund. d.
pp. 177182.
N. C. A. da Costa.
1973. Review of Corcoran, Hatcher and Herring 1972, Zentralblatt f. Math. 257, pp. 89. 1975. Review of Liu 1974, Math. Rev. 50, pp. 78. I. F. Druck and N. C. A. da Costa. SuJt fv., " vbt o!.> " !.>efoi'! M. HCltchVL, C. R. Acad. Sc. Paris 281 A, pp. 741743.
1975.
VARIABLE BINDING TERM OPERATORS
161
M. Guillaume. 1964. 
Recherches sur Le symbole de Hilbert,
ClermontFerrand, France.
T. Hailperin. 1954. 
RemaJLquu on ,
Harvard Univ. Press, Cambridge.
Fund. Math. 43,
J. B. Rosser. 1953.  Logic for Mathematicians,
pp.156165.
McGrawHi 11, New York.
D. Scott. 1967. EUf.>tence and duCJl..i.ption in 60nmaf logic, in pher of the Century, Little, Brown & Co., J. Shoenfield. 1967. Mathematical Logic,
AddisonWesley,
B. Russel;L,
Philoso
Boston.
Reading.
A. N. Whitehead and B. Russell. 1925. Principia Mathematica, Cambridge Univ. Press, I (1925), II (1927), and III (1927).
Cambridge, 2nd; vols.
162
NEWTON C. A. DA COSTA
Universidade de Sao Paulo Instituto de Matematica Sao Paulo, SP., Brazil and Universidad Catolica de Chile Instituto de Matematica Santiago, Chile.
MATHEMATICAL LOGIC IN LATIN AMERICA A.I. Arruda, R. Chuaqui, N.C.A. da Co~ta (eds.) © NorthHolland Publishing Company, 1980
THE MODEL THEORY OF FCGROUPS UWc.h FelgneJt
Dem 1.FCKOln gewidmet ABSTRACT. Following R. Baer and B. H, Neumann a group G is called and FC Group if for each 9 E G the conjugacy class {x 1gx;xEG}of 9 is finite. In her celebrated paper W. Szmielew solved the problems of decidability and internal characterization of elementary equivalence for the class of abelian groups. The model theory of abelian groups is by now well established. For the next slightly larger class. i.e. the class of nilpotent groups (even nil2) only a few sporadic results are known. We found it more promising to look at another generalization of commutativity. namely the class of FCgroups. We have the following results: (1)
(2) (3) (4) (5) (6)
For all nEw: the theory of {G; [G:Z(GJ] :::. 11} is· decidable. The theory of {Gj [G: Z(G)] < ~oHs undecidable. The theory of all FCgroups and the theory of all BFCgroups are both undecidable. We classified all stable, wstable and ~o  cat~ gorical groups G for which [G:Z(G)] is finite. We determined the ~2  theory of all periodic FCgroups. We have the surprising result. that the Hypercenter of every FCgroups of finite exponent is firstorder definable without parameters,
It seems to us that it is possible in the near future to get elementary invariants for periodic FCgroups.
§
1.
INTRODUCTION.
The model theory of abelian groups is more or less thoroughly accomplished. Starting from W. Szmielew's fundamental classification of all abelian groups up to elementary. equivalence (1955) a large number of the most important problems in this area has been solved in the last two decades. We may mention here the numerous investigations' concerning the elementary theories of sundry classes of abelian groups, their decision problems, the work on categorical abelian groups, Horn theories of abelian groups and finally results concerning infinitary logic and 1storder logics with generalized quantifiers. In the case of nonabelian groups there are only sporadic model theoretic re163
164
ULRICH FELGNER
sults known. Most of these results, however, are very impressive and substantial. They usually combine deep results and difficult methods from group theory, recursion theory and model theory. We may mention here the work on (1) algebraically closed groups, (2) X categorical stable groups and groups of small Morley rank, (3) the elementary t~eory of free groups, (4) the elementary theory of symmetric groups, (5) the undecidability of the theories of various classes of groups, and finally (6) the work on varieties of groups and verbal subgroups. One may hope for further modeltheoretic results if the groups investigated are not too far removed from abelian groups. Besides the class of nil2 groups another prominent candidate is the class of FCgroups. Following R. Baer (1948) a group G is called an FCgroup if each element has only finitely many conjugates in G. In this paper we start to investigate FCgroups from a modeltheoretic point of view. We also investigate the related class of BFCgroups and the class of groups with finite central factor group. In § 2 we collect some algebraic results on FC9roups from the literature and add a few results which have hitherto been unnoticed. In § 3 we generalize R. Baer's characterization of the hypercentre of a finite group to the case of FCgroups. This has the rather surprising modeltheoretic consequence that the hypercentre of an FCgroup of finite exponent is firstorder definable: In § 5 we show how the algebraic machinery of factor sets (§ 4) can be used to deal with modeltheoretic questions about centrebyfi nite groups (e.g. saturatedness and stability). We also solve the problem which centre byfinite groups are models of the firstorder theory of fjnite groups. The notion of 'FCgroup' intends to generalize the notions of 'abelian group and 'finite group'. In § 6 we therefore compare the 1storder theories of finite groups Th(Fin), of BFCgroups Th(BFC) and of FCgroups Th(FC). These theories are all different. We are mainly interested in questions of the following sort: up to which quantifier complexity are these theories identical. In several cases we shall obtain the exact answer. These are not purely technical questions because the answers tell us how 'close' two theories are. In § 7 we calculate the FCcentre of a restricted wreath product and generalize some earlier results of G. Baumslag. We use the results of § 7 in § 8 where we solve Baldwin's problem whether locallynilpotent stable groups are nilpotent. The answer is negative. Among other things we classify the stable FCgroups and the Xlcategorical FCgroups. We use the following notation. N = {O,l, ... } is the set of all nonnegative integers, Z is the infinite cyclic group of all integers. For n EN, Is. n, 2(n) denotes the cyclic group of order n. (X) denotes the subgroup generated by X wh{le {a,b, •• } only denotes the set of the elements a,b •• ; xY = y 7xy and [x,y] = x 1yl xy. Further [x,y,z] = [(x,y],z] and = {x9 ; 9 E G} is the conjugacy class of x in G. If A and B are subgroups then [A,m denotes the subgroup gener and ated by all commutators [a,b) for a E A and bE B. Thus G' = [G,G) = G(I) G(n+l) = [G(n) G(n)j Put G(W) = n G(n) Let Z (G) denote the terms of the
i
"
,1
EN
ex
transfinitely extended upper central series of G. Thus Zo(G) = 1, ZI(G) = ZIG) is the centre of G and Za+ IIG) = {x E G; 'tJ'J E G: [x,y] E Zex(GJ} and for limit ordinals '" Z"rG) is the union of the previous terms. :IC(G) = if ZorG) is the hypercentre of G (ex runs through all ordinals). The centralizer X in G is denoted by CG(X).
§
w,u., the 6iMt .(.n6.(.nUe otr.d.(.nal and IXI
2. ALGEBRAIC PROPERTIES OF FCGROUPS
.(.0
the cMd.(.naUty 06 X.
165
THE MODEL THEORY OF FCGROUPS In this section we recall some known facts on Fegroups for future We also add a few results which have hitherto been unnoticed.
reference.
Recall that a group, each element of which has only finitely many conjugates, is called an FCgnoup. This terminology is due to R. Baer (1948). Following B.H. Neumann a group G for which there is a positive integer n such that each element of G has at most n conjugates, is called a BFCg~oup (i .e. boundedly FC). A qroup is called toc.illy nMmat if each finite subset is contained in a finite normal subgroup. THEOREM 2. 1.
(J.)Sllbg~Ollp,;
and homomMpYUc ,[mage,; On
FCg~Ollp,;
Me FC
g~oup,;.
(UI J.,;
The weak d~ec.t p~odllet On any numb~ On FCg~oup,; J.,; an FCgnollp.
(ill) 1 n G J.,; a nVii.tety MnUety 9 en~a..ted.
gen~a..ted FCg~oup,
(J.v]
(R. Baer) In G J.,; a.n
(v)
(B.H. Neumann) The
(vJ.)
Pwod,[c FCgnoup,; Me toCilly
toCilly nJ.nUe) •
FCg~oup,
then G/Z (G)
(J.')
Mnile and Z(G)
then G/Z(GI J.,; toCilly no~at (and henc.e
commu.ta..to~ ';llbg~Ollp
G' On an
FCg~oup
G J.,;
pe~odJ.c..
no~at.
(vU] (S.H. Neumann) In G J.,; an FCg~oup, then the On Mnile Md~ no~ a ehMa.d~tic ,;ubgftoup. THEOREM 2.2.
J.,;
(1. SchurR. Baer)
~e.t
t(G) on i l l eteme~
In nM Mme pMJ.tive
J.n.teg~
1>     i »
1
G   ... >:> H
1
,»
1
....;> M ~> G* ~» H       i > 1 PROOF A = {( oJ;(a), aI); aEN} is a normal sub\Jroup of M x G. Put G* " MxG/A. Then m 1+ [m, l)A is an embedding of Minto Z(G*J. If B = {(oJ;{a)b,a I); a EN AbE M} and if A : G* + IMxGJ/B is the canonical projection, then rMxG}/B~ H and the kernel of A is {1m, l)A; mE M} ~ M. Thus G* is a central extension of M by H. The mapping 0: 9 ~ (1, g) A is a homomorphism from G into G* such that the above diagram commutes. It remains to prove that G* can be described as EIM,H,6*J, where ~*" v c «, By definition, G = E(N,H,6J = {(u,al; uEHA aE N} with the multiplication given by 1~,a)o{v,bJ = Iuv, 6(u,v}abJ. Put N = {(I, aJ ; a EN} so that H ~ GIN = {{u,a)N; uEHA aEN}, The natural isomorphism(u,a)H....u from GIN onto H will be called a. Consider the following transversal for the cosets of N in G, Tru,a)NJ = (u,lJ E G. We finally define a factor set F:G/NxG/N+N in the usual way:
F(lu,ct)N, (v,bJN) ",cru,ct)o(v,bINI1o,((u.,a)NiO,((v,bJNJ (uv, 1) 1 0 (u,1) 0 (v,l) (vIu. I, 6(v Iu 1,uv)I) 0 (u, 1) 0 (v,l) II I II II I (v u. uv, 6rv. ,vJ6rv u ,u)6rv u ,uvl )
II, 6(u,vl} EN (the last equation follows from (0) wi t.. a v· 1u l , 0 = u, tv = v) .Thus F(a 1 (u ) , aI (v)} " II, 6Iu,v)). Thus, if Nand N are identified and Hand G/N are identified (via aI), then F and 6 are identical. Using this identification we may, hence, say that 6(u,v) = T(uv)lT{u)T{v) V u,v E H, where T is a transversal for the cosets of N in G. But then oJ; 0 T : H > G* is a trans versa 1 for the cosets of M in G*. Hence, by Schreier's extension theory, G* and E(M,H,1/:o 6J are isomorphic in such a way that the above diagram (with exact rows) commutes, Q.E.D.Lemma 4.1 has a number of consequences.
LEMMA 4.2. Mn..i.te.
16
Let N be a ,c.ent!tal 6ubg!toup 06
oJ; denotu the endomo!tphMm
momo!tphMm o:G
+
a
1+
G 6uc.h
N ffi(G/NJ 6u.c.h that the 6ollowing
d..i.ag!tam commutu:
N >   > G l»G/N
oJ;
J
1
that n" [G:~
an (60!t a E NJ, then thelLe
II
0 N   Nffi(GIN);,>G/N
16 in add..i.tion N..i.6 tOMion6!tee, then 0 ..i.6 an embedd..i.ng.
..i.6
..i.6
a ho
172
ULRICH FELGNER
PROOF. Put H=GIN and choose a factor set tI: HxH"'N such that G= We have \&(tI(u,v))=(tI(u,v) )n=tln(u,v) by the definition of multiplication in the group Cext. Since G/N=H has cardinality n, the exponent of Cext divides n (cf'. M.Hall, "The theory of groups" (1964)p.223, Theorem 15.2.1). The extension G*=E(N,H, W 0 til is hence equivalent to the direct sum N Ell H and the claim follows from Lemma 4.1, Q.E.D. • E(N,H,tI).
LEMMA
4.3.
Let N be a Llbg!l.oup otl G I.>uuA be a factor set. Let e1 be the neutral element of A and e2 the neutral element of H. Intro duce the following ternary relations: ])1 (a,b,c.) x· Y=z holds in H, Ro(x,y,al => o(x,y)=a.