Table Of ContentSTUDIES IN LOGIC
AND
THE FOUNDATIONS OF MATHEMATICS
VOLUME 74
Editors
A. HEYTING, Amsterdam
J. KEISLER, Madison
A. MOSTOWSKI, Warszawa
A. ROBINSON, New Haven
P. SUPPES, Stanford
Advisory Editorial Board
Y. BAR-HILLEL, Jerusalem
K.L. DE BOuvERE, Santa Clara
H. HERMES, Freiburg i, Br.
J. HINTIKKA, Helsinki
J. C. SHEPHERDSON, Bristol
E.P. SPECKER, Ziirich
NORTH-HOLLAND PUBLISHING COMPANY
AMSTERDAM • LONDON
LOGIC, METHODOLOGY AND
PHILOSOPHY OF SCIENCE IV
PROCEEDINGS OF THE FOURTH INTERNATIONAL
CONGRESS FOR LOGIC, METHODOLOGY
AND PHILOSOPHY OF SCIENCE,
BUCHAREST, 1971
Editedby
PATRICK SUPPES
Stanford University,Stanford, US4
LEON HENKIN
University ofCalifornia, Berkeley, USA
ATHANASE JOIA
Academie Roumaine, Bucarest, Roumaine
GR. C. MOISIL
Universite de Bucarest, Bucarest, Roumaine
~c
~
~
1973
NORTH-HOLLAND PUBLISHING COMPANY
AMSTERDAM • LONDON
AMERICAN ELSEVIER PUBLISHING COMPANY, INC.
NEW YORK
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North-Holland Publishing Company, Amsterdam-1973
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North-Holland ISBN for the volume;0720422744
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PREFACE
This volume constitutes the Proceedings of the Fourth International
Congress for Logic, Methodology and Philosophy of Science held in
Bucharest, Rumania, from August 29 to September 4, 1971, under the
auspicesof the Division of Logic, Methodology and Philosophy of Science
of the International Union of History and Philosophy of Science. The
papers published in the present volume are texts of addresses given by
invitation of the Program Committee.
The Program Committee was an international committee appointed by
the Executive Committee of the Division of Logic, Methodology and
Philosophy of Science. Its members, representing a number of different
countries,consistedofthefollowingindividuals: PatrickSuppes(Chairman),
Yehoshua Bar-Hillel, Leon Henkin, Mary Hesse, Athanase Joja, Stephan
Korner, A. A. Markov, Grigore Moisil,Nicholas Rescher, J. F. Staal, and
Wolfgang Stegmilller.The editors of this volume served as the Executive
Committee of the Program Committee.
In accordance with previous congresses sponsored by the Division,
the work of the Congress was divided into a number of sections. The
various section committees, who worked with the Program Committee in
organizing the activities of each section, including both the invited ad-
dressesand invited symposia as wellas the selectionof contributed papers,
weremadeup ofthefollowingindividuals,withthe chairman being the first
named individual: Section I (Mathematical Logic), Andrzej Mostowski,
Y. L. Ershov, Kurt Schutte; Section II (Foundations of Mathematical
Theories), A. A. Markov, Ronald B. Jensen, Laszlo Kalmar, Joseph
Shoenfield;SectionIII (Automata and Programming Languages), Michael
Rabin, V.M. Glushkov, Grigore MoisH, John C. Shepherdson; SectionIV
(Philosophy of Logic and Mathematics), Georg Kreisel, Hans Hermes,
DagPrawitz;SectionV(GeneralProblemsofMethodology and Philosophy
ofScience),WolfgangStegmiiller,PavelV.Kopnin, Imre Lakatos; Section
VI(Foundations ofProbabilityand Induction), Jaakko Hintikka, Bruno de
Finetti, WesleyC. Salmon; Section VII (Methodology and Philosophy of
PhysicalSciences),Adolf Griinbaum, Brian Ellis, Martin Strauss; Section
x
PREFACE
VIII (Methodology and Philosophy ofBiologicalSciences), MortonO. Bec-
kner, Aristid Lindenmayer; Section IX (Methodology and Philosophy of
PsychologicalSciences),AlekseyN. Leontiev, F. Bresson,R. DuncanLuce;
Section X(Methodologyand Philosophy ofHistorical and SocialSciences),
HermanWold, MironConstantinescu, W. H. Dray; Section XI (Methodo-
logyand Philosophy ofLinguistics),Solomon Marcus, John Lyons; Section
XII (History of Logic, Methodology and Philosophy of Science), Mary
Hesse,Athanase Joja, Gunther Patzig.
The Congresswasjointlysponsored byIUHPS and the Academy ofthe
Socialist Republic of Rumania. The arrangements for the holding of the
Congress in Bucharest were carried out by the Organizing Committee,
which consisted of the following individuals: Athanase Joja (President),
Petre Botezatu, Georg Ciucu, Dumitru Ghise, Crizantema Joja, Stefan
Milcu, Grigore Moisil, Mihai Neculce, Radu Negru, Octav Onicescu,
Dionisie Pippidi, Gabriel Sudan, Serban Titeica, Solomon Marcus, and
Constantin Popovici (General Secretary).
The editors have confined themselves to arranging the volume and
handlingvarious technical matters relating to publication, without attempt-
ing detailed editorial treatment. By and large, the choice of notation and
symbolismhas been left to the individual authors. The full program of
invited addresses and symposia is printed at the end of the volume.
On behalf of the Program Committee and the Organizing Committee,
the editors wishto thank the many persons bothin Rumania and in other
parts ofthe world who contributedtheir generousassistancein determining
the programand arrangingfor the holding ofthe Congress.
The Editors
September 1972
ELEMENTARY LOGIC
GR. C. MOISIL
Institute ofMathematics, Rumanian Academy, Bucharest, Rumania
1. We shall consider a typified logic of propositions.
A well-formed formula of Type I (I-wff)is defined recursively:
1. Propositional variables P, q, r,pl' ..., are I-wff's;
2. IfA, B, are I-wff's then (A & B), (A vB), (IA), (A ~ B), (A +-+B),
etc., are I-wff's.
Among the wff's of Type I the valid well-formed formulas (vl-wff)
are defined syntactically and semantically.
The syntactical definition. Suppose given the definition of a demonstra-
tion; i.e., a sequence of I-wff's
El , ..., E; (1)
suchthateach E,isan axiom or results from the precedentonesbya transi-
tion rule.
A vl-wff is a I-wff which can be imbedded in a demonstration.
The semantical definition. With every I-wff E we associate a functionIE:
IE: L~ ~ L2,
where L = {O, I}. A vl-wff is a I-wff whose associated function is the
2
constant 1.
2. Using the metalinguistic variables we can avoid the substitution rule.
Among the transition rules we note the modus ponens
A,A~B
----- (2)
B
and the rule of adjunction, used by A. Heyting
A,B
(3)
A&B
4 GR.C. MOISIL
but some other rules are used in the nonformalized reasoning. Let us
consider the rule
A,B,A ~ (B~ C)
(4)
C
Such a rule seems to be valid; for instance the tree
has the structure
and the three I-wff's on the top of the tree (*)are instances ofthe axioms;
the transition rule used in (*)is the modus ponens. When we are saying
that a rule of transition
(5)
is valid, we think of the following definitions.
Metatheoretical definition. For each substitution if transforms
(1, (1
AI, ... ,An into vI-wff's then transforms B into a vI-wff.
(1
First syntactical definition. One can construct a tree like (**) having
at the top Au ...,An; each I-wffresults from the I-wff's which are above
it by one of the transition rules.
The metatheoretical definition givesthe followingsemantical definitions.
First semantical definition. For each substitution of the variables of the
functions fAt' ...,fA by functions of new variables, for which the func-
n
tions fAt' ... .I»,become the constant 1, fB becomes the constant 1.
Second semantical definition. This definition differsfrom the first by the
fact that the substitution is not arbitrary, the functions defining being
(1 (1
not arbitrary but associated with I-wff's.
Third semantical definition. For each assignment of valuesin L to the
2
variables, for which the values of rA , ...,fA are 1, the value of fB is 1.
J, 1 n
ELEMENTARY LOGIC 5
Fourth semantical definition. The associated inclusion
fAt n ... nfA CfB (6)
n
istrue in L
2•
Following G. Gentzen we shall write (5) in the form
(AI, ...,An)It (B) (7)
and we shall call (7) a II-wff. With each II-wff SIl, for instance (7), we
will associate a function
(8)
or
(9)
,-+' being the implicational connective in L2•
Fifth semantical definition. fSIl is the constant 1.
Given a Boolean algebra B, we can associate with a l-wff a function
IE.B:
fE.B:Bn -+ B.
Sixth-tenth semantical definitions. These definitions are to be obtained
from the first-fifthsemanticaldefinitions usingfAt.B' ...,fAn.B,fA.Binstead
offAt , •••,IAn,fn·
It is possible to give different syntactical definitions to the validity of
II-wff; we will call vll-wff''s the valid II-wff's.
Second syntactical definition. We will call demonstration any sequence
ofII-wff's:
(10)
inwhicheachSpisan axiom or resultsfrom the precedent onesbyatransi-
tion rule. A vII-wffis a II-wffwhichmay be imbedded in a demonstration.
The third,fourth andfifth syntacticaldefinitions for the validityofII-wff's
require for the validity of (7) the validity of the I-wff's:
(AI & ... & An) -+ B (11)
(AI -+ (A2 -+ ..• -+ (A" -+ B) ...) (12)
iA v ... viAnvB. (13)
1
G. Gentzen introduced another kind of II-wff's, which he called 'Se-
quenzen'
(14)
6 GR.C. MOISIL
the inclusion associated with such a form being
J1~',(1 (] ••• (] fAn C fBl U ...U fBm. (15)
The function associated with (14) and the different semantical definition
for the II-wff's having the structure (14) are obtained from those related
to the Form (7) by writing BI v ... vBm and fB:U ...UJ»; instead of B
andfB, respectively.
3. The transition rules of a demonstration of Type (10) were given by
G. Gentzen.
We shall write these rules in the form
(SP, ...,S:I) itx (TIl) (16)
Sp, ...,S:I, T" being II-wff's. Let us give some examples:
«SL ,SDIt (TI»)iiI «S~(l)'...,S;(S» It (TI») (17)
«SL ,SDIt (TI»)iiI«SA, Sf, ,SDIt (TI») (18)
«SA, SA, Sf, ,SDIt (TI»)iiI «SA, SL ,sD It (P») (19)
«Sf, , S~) It (TI (T!, Uf, ... , U~)It (WI»)
),
itx «Sf, ... , S:, U], ..., U~)It (WI»). (20)
These III-wff's will be called Permm , RenfnH Replll and Enchlll·
For a better understanding of the idea of vIII-wff's, one must think of
the Ill-wff's which will be called Dedm :
«AI' ...,An,B) It (C») itx«AI' ...,An)It (B --+ C»).
This is the famous Deduction Theorem. The validity of this III-wff
in classicalsentence calculus must be proved.
4. It is easy to extend these ideas, defining the (N+l)-wff's by
(S~, ...,S~)N+I (T~) (21)
si.,...,S;, TNbeingN-wff's.
With each N-wff may be associated a function and an inclusion; it is
easy to define the valid N-wff's, the ideas of demonstration, demonstra-
tion from hypothesis, etc.
5. The usefulness of the calculus of the second type was clearly shown
by the work of G. Gentzen for the classical and the intuitionistic propo-
sitionallogic.
ELEMENTARY LOGIC 7
The use of propositional logic of higher order is very fruitful for de-
veloping nonclassical propositional calculi too.
There exist logics in which no I-wff is valid. These logics may be de-
veloped as calculus of vll-wff's.
The calculus whose models are all general lattices, which willbe called
the strictly positive logic, has the axioms
(A & B) It (A) (A) It (A vB)
(A & B) It (B) (B)It (AVB) (I)
(A, Bht (A &B) (AvB)It (A, B)
using II-wff's of the Form (14).
The strictly symmetric logic introduces a negation, that is a connective
N with (see Section 6 below):
(A) t{(NNA)
(II)
(A) It (B)) itr(NB) It (NA)).
The logic ofquantum theory having the models introduced by Garrett
Birkhoff and J. von Neumann has the Axioms (I), (II) and one of the
following
It (A, NA)
(III)
(A, NA) It
that express the principle of excluded middle and the principle of contra-
diction.
The pure positive logic which introduces the implication in two forms:
'rr' ...,
asarelation ofconsequenceofanytype 'N" ...and asaconnective
'--+', related by the modus ponens and the deduction theorem:
It is the implicationallogic ofHenkin; it is the Heyting logic, restricted
to the connective '--+'alone and has as models the Hilbert algebras of the
school of Bahia Blanca.
Thepure modal logicis obtained from the pure positivelogicby dualiza-
tion, introducing a new connective '-' which is the exception; A- B
will be read "A excepted B." The axioms of this logic are (IV) and (V):
