INCOMPLETABILITY, WITH RESPECT TO VALIDITY IN
EVERY FINITE NONEMPTY DOMAIN, OF FIRST ORDER
FUNCTIONAL CALCULUS
WILLIAM CRAIG
First order functional calculus is complete with respect to validity in every
nonempty domain, i.e., every formula which is valid in every nonempty domain
is a theorem. Moreover, given any positive integer k, first order functional calculus can, by additional axioms or rules of inference, be made complete, in a
similar sense, with respect to validity in every nonempty domain of not more
than k individuals. In contrast to these well-known results, a proof is given of the
following Theorem: There exists no constructive logistic system whose theorems
are exactly those formulas of first order functional calculus which are valid in
every finite nonempty domain. By a constructive logistic system is meant a logistic system such that there is a general recursive process whereby, given any sequence of formulas, it can be determined whether or not the sequence constitutes a proof in the system.
Introducing the notion of a general model, Henkin has proved that, for n = 1,
n = 2, and n = co, the customary systems of functional calculus of order n are
complete in the following sense : Every formula which, roughly speaking, is true
for any assignment with respect to any nonempty general model is a theorem.
[Leon Henkin, Completeness in the theory of types, J. Symbolic Logic vol. 15
(1950) pp. 81-91.] In contrast, from the Theorem the following result is derived:
There exists no constructive logistic system whose theorems are exactly those
formulas of first order functional calculus which, roughly speaking, are true for
any assignment with respect to any finite nonempty general model.
The Theorem is shown to follow readily from the following weaker Lemma:
There is no general recursive process whereby, given any formula of first order
functional calculus, it can be determined whether or not there exists a finite
nonempty domain in which the formula is satisfiable. The Lemma is proved by
methods similar to those which Turing [A. M. Turing, On computable numbers,
with an application to the entscheidungsproblem, Proc. London Math. Soc. (2)
vol. 42 (1936) pp. 230-265, especially pp. 261-262] uses to demonstrate that
there is no general recursive (computable) process for determining provability
in first order functional calculus. (A theorem which is equivalent to the Lemma
has recently been proved by a different method. See B. A. Trahténbrot, NévozmoznosV algorifma dlâ problêmy razrêsimosti na konéënyh klassah (Impossibility
of an algorithm for the decision problem of finite classes), C. R. (Doklady)
Acad. Sci. URSS. vol. 70 (1950) pp. 569-572. The work of Trahténbrot was
called to my attention only after I had completed the proof of the Theorem.)
PRINCETON UNIVERSITY,
PRINCETON, N. J., U. S. A.
721
722
SECTION VI. LOGIC AND PHILOSOPHY
THE INFERENTIAL THEORY OF NEGATION
HASKELL B.
CURRY
In lectures delivered in 1948 (see A theory of formal deducibility, Notre
Dame University, Mathematical Lectures, no. 6, 1950) I proposed what is essentially a semantical definition of the propositions formed from the elementary
propositions of a formal system by propositional connectives and quantifiers.
The theory was based on Gentzen's inferential rules. So far as negation was
concerned, four different systems were considered, viz.: LM, the minimal system;
LJ, the intuitionist system; LD, the minimal system with excluded middle;
and LK, the classical system. Each of these systems had a semantical interpretation: LM, when negation is interpreted as refutability in the sense of Carnap,
i.e., as implying one of a given set of "directly refutable" propositions; LJ for
absurdity, i.e., implying every proposition; LD for refutability with excluded
middle, leading to a system of strict implication; and LK for the interpretation
by truth tables. This paper reports—using the previous notations—some recent
improvements in the theory of negation as follows :
1. If 36, §J, and A are positive (i.e., formed wholly without negation), and if
3£, "1 §) 11- A holds in LD*, then 36 11- A holds in LA*. Hence the positive propositions A such that | \- A holds in LD* are the same as those for LA*; and if
such an A is elementary, it is derivable in the original system ©.
2. If 36 11- A is derivable in LD (LK), then 36, ~1 A \\- A is derivable in
LM (LJ). This is an extension of the Glivenko theorem. It shows that LD is decidable. The proof does not hold if quantifiers are present.
3. If "1 A is defined as A 3 F, where F is a new primitive proposition, and
blank right prosequences are replaced by F, then each system LX is transformed
into a system LXF, and every proposition, prosequence, etc. of LX is changed
into an "^-transform" indicated by a subscript F. A necessary and sufficient
condition that 36 11- g) hold in LX is that 36F | h §)F hold in LXF. Further, the
elimination theorem holds for the system LXF if it holds for LX.
4. The possibility of other formulations of LD, LK has been examined. Reasons r
for the failure of a multiple-consequent formulation of LD are given. A singleconsequent formulation of LK (and of the positive system LG) has been found
and proved equivalent to the original one.
These results are established in the basic L-systems without recourse to the
il-systems, or to technical restrictions necessary in the lectures.
PENNSYLVANIA STATE COLLEGE,
STATE COLLEGE, PA., U. S. A.
LOGIC AND PHILOSOPHY
723
RELATIVELY RECURSIVE FUNCTIONS AND T H E E X T E N D E D
KLEENE HIERARCHY
MARTIN DAVIS
The properties of Kleene's notion of a function being recursive in other functions is developed in a manner entirely analagous to his Recursive predicates and
quantifiers, Trans. Amer. Math. Soc. vol. 53, pp. 41-73. In particular direct analogues of Kleene's Theorems I, II, IV, VII, and VIII are proved. I t is proved that
the set S is recursive in the set T if and only if both S and B are T-canonical
in the sense of Post, thus correlating our point of view with Post's. (Cf. Bull.
Amer. Math. Soc. Abstract 54-7-269.)
Next, making use of Post's idea of representing an entire family of predicates
by a single "complete" set, a generalized theory of the Kleene hierarchy of
predicate forms is developed. The results of Post's abstract cited above are
then easily obtainable by specialization. The precise degree of unsolvability of
certain actual decision problems is determined. The generalized theory of the
Kleene hierarchy is then used to extend the ordinary Kleene hierarchy into the
constructive transfinite. It is shown that for ordinals <co2 the same predicates
are obtained from different integers representing the same ordinal. Moreover
the predicates corresponding to ordinals <u>2 are all shown to be definable in a
second order functional calculus with Peano's postulates.
UNIVERSITY OF ILLINOIS,
URBANA, I I I . , U. S. A.
ON DISPLACEMENTS OF SYSTEMS OF DATA AND
STRUCTURES
MARIO DOLCHER
A first outline of the foundations which I have posed for a general structure
theory is given in the paper: M. Dolcher, Nozione generale di struttura per un
insieme (Rendiconti del Seminario Matematico della Università di Padova
(1949) pp. 265-291). My program, aiming to reach a well-founded systematics
of mathematical theories by means of an intrinsic (group-theoretic) characterization of these theories, has been exposed at the Innsbruck Mathematical
Congress (resumed in M. Dolcher, Sur Vaxiomatisation de la systématique des
tliéories mathématiques par une théorie générale des structures, Nachrichten der
Oesterr. Mathematischen Gesellschaft, Dez. (1949) p. 33).
Starting from a system of data S (i.e. a set of "sentences" as the definition of
operations, relations, • • • ) on a set / (briefly: a system {I \ S}), we consider the
autogroup T of the system, i.e., the [sub]group [of the symmetric group (I!)] of
the automorphisms of {/ | S}. We assume V to characterize S on I, assuming as
equivalent two systems Sf, S" such that T' = r " ; this equivalence implies the
724
SECTION VI. LOGIC AND PHILOSOPHY
possibility of converting the data of £' into terms of S". Then, we assume the
classes of similar groups in (I !) as the different structures which are possible on
7; we say, the system S is a representation of the corresponding structure on 7.
Besides the autogroup T, it is important to consider the [sub]group A (including T) [of (7!)] of the substitutions which take S into a system equivalent
to S (displacements allowed by S). It is easy to establish that A is the largest
group in which T is invariant.
The extreme cases are A = (7!) (invariant system) and À = T(fixed system).
To be mentioned is the case in which T is a (invariant) subgroup of index 2;
then {7 | S} may be said to be an orientable system; the various cases of "orientability" fall under the above general definition.
Furthermore, we can avail ourselves of the notions given for a system in order
to get at analoguous ones for structures.
On the other hand, we can arrive at "relative" (i.e. immersion-) notions by
considering instead of (71) a subgroup of (7!).
It is easy to give examples of the described notions by assuming for 7 a set
of 4, 5, or 6 elements. More interesting are the ones we can easily find in classical
geometry: the similitudes of a space are "displacements" even if they are not
automorphisms ( = congruences) ; the affine geometry is a fixed structure if
conceived as "relative" to the projective structure. The investigation of classical
geometry seems to be fairly interesting from this point of view; philosophical
considerations are possible.
I expect such considerations would be of interest for topology, in connection
with Wiener's problem and the research of topological invariants.
For the ideal-theoretic investigation of algebraic, geometry, the considerations of the autogroup and of the displacement group of the lattice of ideals
seems to lead to a criterion—perhaps the only one—in order to ascertain the
correspondence of ideal-lattice-theoretic propositions to the geometrical ones.
UNIVERSITY OP TRIESTE,
TRIESTE.
ONTOLOGICAL POSITIVISM
JAMES K.
FEIBLEMAN
The aim of the positivists is to systematize scientific knowledge. For this they
need not only logic, mathematics, and the empirical sciences but also metaphysics and particularly ontology. Their anti-metaphysical thesis has been
aimed at the uncontrolled extrapolation of transcendental metaphysics. They
have identified metaphysics with the unbridled invention of metaphysical entities
which it was not considered necessary to submit to the test of sense experience.
The possibility of metaphysics was condemned for the errors of some metaphysicians. But a finite ontology consistent with the aims of positivism is pos-
LOGIC AND PHILOSOPHY
725
sible. Accordingly, the postulate-set of logical positivism is adopted, with two
exceptions: the anti-metaphysical postulate is denied and one more postulate
is added. The result is a new postulate-set from which we can deduce a finite
ontology. Examples from the latter are given. Universals, no longer claimed to
be ubiquitous and eternal, are nevertheless retained as widespread and persistent. They are not created at will but must be discovered by means of the
mathematical and empirical sciences. Ontology is defined by consistency-rules
between divergent sets of empirical data. Thus, as in transcendental metaphysics,
two orders are required; but in place of realms of essence and of existence, we
have persistent and transient orders; and in place of destiny, intent. The remainder of the paper is concerned with a reinterpretation of the old postulates
which are retained in the new postulate-set, and with the establishment of
ontology as a speculative field of operations.
TULANE UNIVERSITY OF LOUISIANA,
N E W ORLEANS, LA., U. S. A.
SUR LES BASES PHILOSOPHIQUES D E LA
FORMALISATION
F é L I X FIALA
Les méthodes formelles des algébristes contemporains constituent un remarquable instrument d'analyse et d'expression des démarches fondamentales du
mathématicien et semblent parfois en épuiser l'essentiel. La technique du calcul
formel est en tout cas suffisamment élaborée pour qu'il puisse être utile d'en
dégager les bases méthodologiques, ou ce que nous appellerons la métaphysique.
Or si les exigences techniques du formalisme mathématique sont clairement
explicitées, ses prétentions métaphysiques, souvent encore implicites, ne sont
pas moins grandes; elles sont de plus souvent exclusives et peuvent devenir un
obstacle au développement des mathématiques.
Dans un récent congrès de philosophie des sciences (Paris, octobre 1949), on
entendit, par la voix de M. Dieudonné, le formalisme bourbakien exclure de son
rayon d'intelligibilité l'intuitionisme brouwerien, représenté par M. Heyting,
et affirmer l'existence d'une cloison étanche entre deux manières de penser
incommunicables. Ceci nous apparut l'expression même d'une métaphysique
exclusive, forte et fermée, plutôt qu'une exigence strictement technique. L'origine
philosophique de l'incompréhension réciproque fut d'ailleurs immédiatement
dénoncée par M. Bernays.
Les mathématiciens ont pourtant déjà eu à eclaircir de semblables situations.
Lors de l'invention des geometries non-euclidiennes, la synthèse de deux théories
apparemment irréductibles exigea un effort sur deux plans: sur le plan technique
(surfaces à courbure négative constante, modèles de Poincaré et de Klein) et sur
le plan philosophique (abandon de la métaphysique de l'évidence et de l'adéqua-
726
SECTION VI. LOGIC AND PHILOSOPHY
tion totale de la géométrie à l'espace réel ou de celle des jugements synthétiques
a priori).
La réconciliation entre les points de vue formaliste et intuitioniste, désirable
pour la cohérence de la science mathématique et de la connaissance en général,
doit être poursuivie sur deux plans: sur le plan technique (correspondances
entre logiques classique et intuitioniste) et sur le plan philosophique.
C'est là que certaines philosophies à base métaphysique ouverte et affaiblie
(philosophie du non, de Bachelard, dialectique idonéiste de Gonseth, etc.) jouent
un rôle efficace. En reconnaissant la légitimité de différents niveaux de formalisation, la relativité d'une démonstration et peut-être même la contingence de la
méthode hypothético-déductive, ces philosophies ne s'opposent nullement à
l'exigence technique de formalisation. Elles cherchent au contraire à en exprimer mieux les principes méthodologiques, tout en garantissant la plus vaste
extension de son champs d'action. L'affaiblissement de postulats métaphysiques
peut être la source de progrès analogues à ceux qui ont parfois accompagné
l'abandon de positions techniques trop fortement déterminées.
(Résumé d'un travail à paraître dans Les études de philosophie des sciences,
Ed. du Griffon, Neuchâtel, 1950.)
UNIVERSITY OF NEUCHâTEL,
NEUCHâTEL, SWITZERLAND.
ARE WE LACKING WORDS?
JACQUES HADAMARD
Humanity does not cease to gain new ideas. How will it find words to express
them?
In many cases, new words are created, which, however, does not take place
without bringing some scandal among our grammatists and other purists.
More frequently, scientists and especially mathematicians prefer using a
known word in a new meaning. Even if exceptionally useful, it takes place too
often and the number of meanings attributed to the same word increases in an
excessive and disquieting way. "Conjugate" has at least three meanings in
geometry, two in algebra. The case is the same for "pole."
*'Class" is most unfortunate from that point of view. It is applied to algebraic
curves; but its use is also fundamental in logistics; and how many kinds of
"classes" are there in the theory of real functions! There are Baire's classes;
but there are also quasi-analytic classes, and indefinitely differentiable functions are also distributed in classes according to the magnitude of their derivatives of an increasing order.
Those who endow a word with such various meanings are confident that they
belong to separate chapters of science and, therefore, are not liable to come in
conflict with each other; but, precisely, the progress of science more and more
LOGIC AND PHILOSOPHY
727
frequently puts separate chapters in mutual relation, and unexpected conflicts become rather frequent. The author has often been impeded by them.
A useful thing, in order to avoid lack of words, would be not to waste them;
I mean, not to use several words where only one is needed. Why speak of the
"roots" of an equation or of a polynomial while an equation has solutions and
the polynomial has zeros. There is also no reason to denote the solutions of a
differential system as "integrals", an expression which, moreover, is needed in a
different meaning.
COLLEGE OP FRANCE AND POLYTECHNIC SCHOOL,
PARIS, FRANCE,
KONSTRUKTIVE BEGRÜNDUNG DER KLASSISCHEN
MATHEMATIK
PAUL LORENZEN
Praktisch brauchbare Konstruktionsregeln, wie
(1)
1
(2)
x -> x + 1
zur Konstruktion der "Zahlen" 1, 1 + 1, 1 + 1 + 1, • • • (Variable x, y, • • • )
oder
(3)
a ; + l ^ l
(4)
1^ x + 1
(5)
x^y-^x
+
l^y+1
zur Konstruktion von "Aussagen" 1 + 1 - ^ 1 , 1 ^ 1 + 1,1 + 1 + 1 ^ 1 + 1,
• • • (Variable A, B, • • • ) bilden den Ausgangspunkt der konstruktiven Mathematik.
Zur Erleichterung des Ableitens, d.h., des Konstruierens von Zeichen nach
einem System S von Regeln, werden "eliminierbare" Regeln konstruiert. Eine
Regel R heißt eliminierbar (bzgl. S), wenn ein Verfahren bekannt ist, jede
Ableitung, die außer 8 noch R benutzt, umzuformen in eine Ableitung, die nur S
benutzt—aber noch dasselbe Zeichen konstruiert. Ist R eliminierbar bzgl. Ri,
• • • , Rn dann wird Ri, • • • , Rn —» R als Metaregel zur Konstruktion von
Regeln gebraucht, z.B., Ai —> A2 ; A% —> A8 -^-» Ai —» _43. So entsteht die sog.
positive Implikationslogik. Wird zur Zusammensetzung von Aussagen die "Disjunktion" eingeführt durch
(6)
A -> A v B
(7)
B-+AVB
(8)
A(x)-+\rxA(x)
so entsteht die sog. positive Logik.
Nach Einführung der Ungleichheit ^ für Aussagen (entsprechend zu (3) —
(5)) wird die Unableitbarkeit von A (bzgl. S) definiert durch A ^ B für alle
bzgl. S ableitbaren Aussagen B. Die Negation ~1 A wird definiert durch die
728
SECTION VI. LOGIC AND PHILOSOPHY
Eliminierbarkeit von A —» B für eine unableitbare Aussage B. So entsteht die
intuïtionistische Logik. Mit dieser ist die zweiwertige Logik leicht als zweckmäßige Fiktion zu begründen (Kolmogoroff).
Um über die Arithmetik hinaus zu einer Analysis zu kommen, muß zunächst
eine Konstruktion für "alle" arithmetischen Aussagen angegeben werden (Weyl).
Hierzu wird außer der Zusammensetzung durch die logischen Operationen.
—»? A, Ax, v, Va, "1 jedes System
Ai - > p(xi)
An —> p(xn)
als Definition einer Aussage p(x)—entsprechend für mehrstellige Relationen.
p—zugelassen. Mit diesen "Aussagen 1. Schicht" werden "Mengen 1. Schicht"
gebildet (die Menge der x mit A(x)) und dann "reelle Zahlen 1. Schicht". Durch.
Iteration der Schichtenbildung bis co erhält man das Vollständigkeitsprinzip der
klassischen Analysis: "Zu jeder Menge von reellen Zahlen gibt es eine reelle
Zahl als untere Grenze." Diese Abgeschlossenheit der klassischen Analysis entsteht
also durch die Beschränkung auf endliche Schichten. Durch geeignete Definition von "reelle Funktion", "Stetigkeit", usw. läßt sich die Gültigkeit alle
Fundamentalsätze der klassischen Analysis erzwingen.
Die Frage, ob die benutzte Konstruktion "alle" reellen Zahlen der klassischen
Analysis liefert, ist sinnlos, da die klassische Analysis "reelle Zahl" ja nur mit
Undefinierten Begriffen wie "unendliche Folge", "Einteilung der rationalen
Zahlen", usw. definiert.
Die Widerspruchsfreiheit der konstruktiven Mathematik ergibt sich aus der
Widerspruchsfreiheit der einzelnen benutzten Konstruktionsschritte, sie erfordert keine eigene sog. Metamathematik. Ebenso entfällt das Scheinproblem
der Gegenüberstellung von "inhaltlichen" und "formalen" Schließen, da der
Aufbau der konstruktiven Mathematik sprachunabhängig ist, d.h. an keiner
Stelle die Kenntnis einer der traditionellen Umgangssprachen voraussetzt.
UNIVERSITY OE BONN,
BONN, GERMANY. N
EXISTENTIAL DEFINABILITY IN ARITHMETIC
JULIA ROBINSON
A relation p(x\, • • • , xn) among integers is said to be existentially definable
in terms of certain given relations and operations if there is a formula containing
the free variables Xi, • • • , xn , any number of bound variables, and symbols for
particular integers, involving only existential quantifiers, conjunction, disjunction, equality, and the given relations and operations, which holds if and only
LOGIC AND PHILOSOPHY
729
if the relation p(xi, • • • , xn) is satisfied. The range of the quantifiers is to be
the set of all integers. The relation p(x±, • • • , xn) will be called existentially
definable if it is existentially definable in terms of addition and multiplication.
In this case, there is a polynomial R(x\, • - • , xn ; yi, • • - , yk) with integer
coefficients which vanishes for some yi, • • • ,ykii and only if rei, • • • , xn satisfy
the relation p(xi, • • • , xn). Alfred Tarski has raised the question whether every
general recursive relation is existentially definable, but this problem remains
unsolved. It may be remarked that no existential definition of the set of powers
of 2 or the set of primes has yet been found, though the complementary sets
are readily seen to be existentially definable.
Let <j>(u, v) be any relation with the following properties i If <j>(u, v), then u > 0
and 0 < v < uu, but there is no integer n such that v < un whenever cj>(u, v).
The principal result of this paper is that the relation among x, y, z, defined by
y ^ 0 and z — xv, is existentially definable in terms of <j>(u, v). Application is
made of the fact that the quadratic unit ß = b + (b2 — 1) is a power of a =
a + (a — 1)1/2 if and only if b2 — 1 = (a2 — l)c 2 for some integer c. The chief
difficulty is in finding what power ß is of a, and it is in identifying this power
that use is made of the relation <j>(u, v). It is not known whether there is a relation
<t>(u, v) with the desired properties which is itself existentially definable, although
this appears probable. At present, too little is known about the solutions of
Diophantine equations, so that it is not even possible to decide whether the
condition that u and v are positive integers for which u + v is a perfect square
does or does not define a suitable relation. If there is a suitable existentially
definable relation <j>(u, v), Fermat's equation xn + yn = z11 can be replaced by a
Diophantine equation of fixed degree in x, y, z, n, and some additional variables.
Also, it can be shown that the set of primes is existentially definable in terms of
exponentiation, so that under the same hypothesis, the set of primes as well as
the set of powers of 2 would be existentially definable.
BERKELEY, CALIF., U. S. A.
AN ESSENTIALLY UNDECIDABLE AXIOM SYSTEM
RAPHAEL M.
ROBINSON
Mostowski and Tarski have found a finite axiom system, consisting of true
formulas of the arithmetic of natural numbers, which is essentially undecidable
(that is, no consistent extension is decidable). [See J. Symbolic Logic vol. 14
(1949) p. 76.] In this paper, a simpler axiom system is given with the same
properties, and a simpler method of showing the essential undecidability is found.
[Concerning the relation between the two axiom systems, see the following
abstract of Szmielew and Tarski, Mutual interpretability of some essentially
undecidable theories.]
The new axiom system has the primitive concepts 0, S, + , -, and consists of
730
SECTION VI. LOGIC AND PHILOSOPHY
the following seven axioms: If Sa = Sb, then a = b; 0 5^ Sb; if a 7^ 0, then
a = Sb for some b; a + 0 = a; a + Sb = £(a + &); a-0 = 0; a-Sb = a-& + a.
(If any one of these axioms is omitted, the resulting system is no longer essentially undecidable.) Putting 1 = SO, 2 = SI, 3 = S2, •• - , it is readily seen
that all true formulas of the arithmetic of natural numbers not involving any
variables can be proved from the given axioms. Furthermore, if a ^ 6 is defined
to mean that there is an x such that x + a = b, then the following statements
can be proved for a = 0, 1, 2, 3, • • • : For every x, x ^ a or a ^ x; x ^ a if
and only if x = 0 or x = 1 or • • • or x = a. From these results, it is possible
to prove that general recursive functions and sets are formally definable, and
this is all that is needed in the following argument. (On the other hand, many
simple formulas, such as 0 + a = a and a g. a, are not provable from the given
axipms.)
To show that the above axiom system is essentially undecidable, it must be
shown that any decidable extension is inconsistent, and it is sufficient to consider extensions which are logically closed. Suppose that (P is such an extension.
The Godei numbers of the sentences of (P form a general recursive set. It follows
that there is a formula $(x), such that for a = 0, 1, 2, 3, • • • , $(a) or its negation belongs to (P, according as the sentence whose number is a is in CP or not.
Now let Ta be the number of the formula obtained from formula number a
by replacing the variable x (wherever it is free) by a. Then there is a formula
Q(x, y), such that for a = 0, 1, 2, 3, • • • , the sentence expressing the equivalence
of 9 (a, y) and y — Ta belongs to (P. If v denotes the number of the sentence,
"For every y, if Q(x, y), then not $(2/)", then both <b(Tv) and its negation must be
in (P, hence (P is inconsistent.
UNIVERSITY OF CALIFORNIA,
BERKELEY, CALIF., U. S. A.
A LOGISTIC PROOF OF A THEOREM RELATED TO
LANDAU'S THEOREM 4
IRA
ROSENBAUM
Peano's approach to the arithmetic of the positive integers is familiar. Starting
with the undefined concepts, 1, successor of, and natural number, the notions
of addition, multiplication, etc. are introduced recursively and their properties
established. Landau, remarks, "On the basis of his five axioms, Peano defined
x + y for fixed x and all y as follows: x + 1 = x!, x + y' = (x + y)', and he
and his successors then think: x + y is defined generally • • • ." But, as the objections of Landau's colleague, Grandjot, indicated, "x + y has not been defined."
A resolution of the difficulty to which attention had been called by Grandjot
was sought by Landau, in collaboration with von Neumann, and although obtained, was put aside in favor of a simpler solution suggested by Kalmar, which
LOGIC AND PHILOSOPHY
731
is embodied in Landau's proof, in the Grundlagen der analysis, of his Theorem 4.
Because Landau's proof introduces, without comment, entities other than the
undefined concepts, specifically the functions Oy and by, his argument acquires
a certain informality, which it is the purpose of the theorem of the present paper
to correct.
The Peano postulates are used in the forms, 1 e N7 (y)E\ S'y,
~(Ex)(18x),
S e Cls -> 1, and 1 eM.SnM CI M. -> M = N. Here S'x replaces Landau's x'.
Next in place of the usual recursive relations, use is made of a relation 2 pairing
positive integers with ordered couples of positive integers in accordance with
the two conditions: 1) z2xl.<± .z = 8% and 2) zLxS'y. <=± .z e S"2'~*(xy). The
second condition avoids the assumption, which vitiates the usual discussion, that
the couple x, y has a unique sum, so that the successor of this sum is well-defined. A logistic proof is then given,—using quantification theory, the theory of
the definite descriptive operator, and portions of the general theory of classes
and relations,—that (x)(y)El2f(xy). The proof requires the two lemmas, 1)
2 e 1 -> Cls, (i.e., (x)(y)(z2(xy).w2(xy).
-> .z = w)), and 2) (x)(y)(Ez)(z2(xy)).
These two lemmas are a sufficient basis, in accordance with Hilbert and Bernays'
theory of the descriptive operator and Russell and Whitehead's theory of descriptions, to validate the theorem of the present paper, namely, (x)(y)E\ 2(xy).
UNIVERSITY OF MIAMI,
CORAL GABLES, FLA., U. S. A.
TRANSFINITE CARDINAL ARITHMETIC IN QUINE'S
NEW FOUNDATIONS
J.
BARKLEY
ROSSER
In order to deal adequately with the arithmetic of transfinite cardinals in
Quine's New Foundations, it appears to be necessary to have an ordered pair
which is of the same type (in the sense of type theory) as its constituents. Quine
has shown that if the axiom of infinity be assumed, then such an ordered pair
can be defined. We show that if such an ordered pair is available (either by
definition, or assumed as an undefined term), then the arithmetic of transfinite
cardinals is forthcoming in practically its classical form.
This result has two interesting consequences. The first is that the axiom of
infinity suffices for the development of a full fledged theory of transfinite cardinals. The second is that, since the axiom of infinity is an easy consequence of the
theory of transfinite cardinals, it follows 'that one can infer the axiom of infinity
from the assumption that there is an ordered pair which is of the same type as
its constituents. Since this assumption about ordered pairs seems to be a purely
logical assumption, it would appear that in Quine's New Foundations the axiom
of infinity can be derived from purely logical considerations. Quine had originally thought otherwise.
732
SECTION VI. LOGIC AND PHILOSOPHY
An interesting feature of the cardinal arithmetic is that the universe has a
cardinal number, which is necessarily the greatest cardinal. The Cantor paradox
is avoided because Cantor's theorem that 2n is a greater cardinal than n is apparently available only in case n has at least one member whose members are
all unit classes. This additional hypothesis is proved for all the cardinals in
common use in mathematics, so that the classical arithmetic of cardinals breaks
down only for extremely large cardinals of a sort which have no practical use in
mathematics.
NATIONAL BUREAU OF STANDARDS,
Los ANGELES, CALIF., U. S. A.
APPLIED LOGIC AND MODERN PROBLEMS
E. R. STABLER
The main object of this paper is to point out the need of applied logic as an
aid to the solution of current national and international problems.
To emphasize and illustrate the need, a brief postulational study is presented
of a formal statement (S) issued by a group of physical scientists concerning
the problem of control of a new type of weapon. The verdict is reached that the
logical structure of S is seriously deficient, and bordering on inconsistency. It
is noted that the authors of S are presumably in the habit of obtaining assistance
in their own research from applied mathematicians; and that similarly, in
formulating S, they might well have profited by advice from logicians acting in
an applied capacity.
It may be that responsibility lies partially with the logicians themselves,
whose best efforts seem to remain centered on problems of an ultra-pure nature.
It is suggested that some logicians might devote more attention to applications,
and attempt to create a demand for their services as consultants on problems
of the type under consideration. A few directions in which their help might be
especially useful are the following: investigation of postulational questions and
deduction of theorems in connection with formation of new policies* or evaluation of policies already in operation; introduction of abstraction and symbolism
as a means of neutralizing preconceived judgments; detecting pitfalls due to
elementary, but common, logical fallacies; and, in general, creating more precision and consistency in the use of language.
In conclusion, the following hypotheses are presented for consideration: 1.
That the often mentioned lag in the social-political-economic development of
civilization behind its scientific-technological-military development iö explainable in part by the real discrepancy between the intellectual resources which
have been evoked or mobilized for use in the respective areas. 2. That the lag
can be considerably narrowed by cutting down this discrepancy, especially
through the systematic application of logic to crucial problems in the former
LOGIC AND PHILOSOPHY
733
area. 3. That a new and remarkable logical paradox will occur if the time ever
arrives when methods of destruction have been so perfected by aid of logic and
mathematics, and have been confirmed empirically on such a large-scale basis,
that further intellectual activity in the realm of logic and mathematics becomes
paralyzed or impossible.
HOFSTRA COLLEGE,
HEMPSTEAD, N. Y., U. S. A.
ON THE MATHEMATICAL EXISTENCE
ZYOITI
SUETUNA
According to the formalistic point of view, the mathematical existence is what
can be apprehended without contradiction. The existence of something does not
however follow from the mere logical consistency without contradiction. From
the intuitionistic point of view man asserts that the mathematical existence is
what we can construct by our finite acts. But the infinite can never be grasped
adequately by mere finite acts. Indeed to all human knowledge there underlies
our act as foundation. But the real ground of our knowledge is not our act itself;
it is rather our intuition based on acts. The object of our knowledge is formed
and grasped by such a positive, not fictitious, intuition.
The notion of natural numbers is formed by repeatedly adding 1. Every new
addition gives rise to a new number, a process which repeats itself without end ;
and since we have the insight into the whole of such processes, the notion "totality of natural numbers" is formed as mathematical object. Herein lies the
real ground why the notion "arbitrary natural number", which is not a definite
particular number and represents as concrete-universal element all natural
numbers, has a precise mathematical meaning. The mathematical existence par
excellence, I think, is what can be grasped by our positive intuition based on
acts by means of the totality of natural numbers and linear continuum. With
respect to the continuum I lay stress on the fact that the linear continuum as
notion can never be cut off from the continuum of real numbers. I don't think
that the so-called freie Wahlfolge could be a mathematical object. In order to
reconstruct the ordinary analysis as mathematics which is of intuitive significance, we must have the insight into the "contradictory self-identity" of both
continua.
UNIVERSITY OF TOKYO,
TOKYO, JAPAN.
734
SECTION VI. LOGIC AND PHILOSOPHY
MUTUAL INTERPRETABILITY OF SOME ESSENTIALLY
UNDECIDABLE THEORIES
WANDA SZMIELEW AND ALFRED TARSKI
The theories discussed are formalized within elementary logic. (For notation
see Tarski, J. Symbolic Logic vol. 14, p. 75.) Each theory X has its own (nonlogical) constants and axioms—^-constants and ï-axioms. The ST-variables
range over elements of a fixed set, which is represented by a special ^-constant,
say U (the universe predicate). Consider a constant C not occurring in X, e.g.,
the operation symbol + . A possible definition of C in X is any expression of the
form
/kx, y, z[U(x) A U(y) A U(Z) —> (x + y = z <-> $)]
where <3> represents a formula in X with the free variables x, y, z; the sentence
stating that for any x, y there is just one z satisfying $ is assumed to be provable
in X. A theory SE' is (strictly) interprétable in X if a theory X" can be constructed
for which (i) the set of 5E"-constants consists of all ^-constants and ^'-constants;
(ii) the set of SE"-axioms consists of all SE-axioms and of possible definitions
of ^'-constants in X (one definition for each ^'-constant) ; (iii) all SE'-axioms
are provable in X". (If any ^'-constants are also ^-constants, we first replace
them in X' by symbols not occurring in X.) If X is interprétable in X' and conversely, the theories are called mutually interprétable.
The theories described below are finitely axiomatizable, 5 is a theory of finite
sets. The ^-constants are the universe predicate U and the membership symbol £ . Roughly speaking, the ^-axioms are obtained from Bernays' axiom
system (J. Symbolic Logic vol. 2, pp. 65 ff; ibid. vol. 6, pp. Iff.) by identifying
classes with sets and by eliminating or restricting all axioms which imply the
existence of infinite classes. $' is a small fragment of 3f with three axioms only
—Bernays' axioms 1(1), 11(1), 11(2). 3> is a fragmentary theory of integers.
The ^-constants are U, 0, 1, < , + , •. The 3-axioms characterize the set of
integers as an ordered ring in which 1 immediately follows 0. Sfi is a small fragment of the theory of non-negative integers* It is essentially the theory discussed in a preceding abstract (Robinson, An essentially undecidable axiom
system), but with a universe predicate explicitly introduced.
Theories 3 and 5ft are mutually interprétable; they are both interprétable in
g ' and hence also in g. 3 is known to be essentially undecidable (Mostowski and
Tarski, J. Symbolic Logic vol. 14, p. 76). A theory X being essentially undecidable, the same applies to every theory in which X is interprétable. Hence all
the four theories are essentially undecidable. (For 5ft this has been proved directly
in the preceding abstract of Robinson.) Many other theories are known whose
essential undecidability can be analogously derived from that of 3>; e.g., some
fragmentary theories of concatentation.
UNIVERSITY OF CALIFORNIA,
BERKELEY, CALIF., U. S. A.
LOGIC AND PHILOSOPHY
735
AN INQUIRY INTO THE NATURE OF KNOWLEDGE
G E O R G E C. VEDOVA
In these days of startling and profuse scientific discoveries, partial and conflicting theories, and obliterations of old distinctions, the seeker of knowledge
is often confused and discouraged. This inquiry is intended to provide some
help to such a seeker.
An inquiry into the nature of knowledge must first take into account the
view of "reality" taken by the inquirer. The view of "scientific realism" is
adopted here. This step taken, the next step must be the adoption of a "suitable" logic. The traditional logic, refined by modern standards, and retaining
the "law of the excluded middle"—for Brouwer's views are not generally accepted—is the logic used in science. Adopting this logic we lay down the definition:
Definition. A thing, x, may be said to be known if the truth-value of every
proposition Pi(x) can be determined in this logic,
Gödel's theorem on completability assures us beforehand that no x can be
fully known under this definition. We resign ourselves to this fact since there
are, besides, additional reasons why no x will ever be fully known.
But in science, knowledge is sought not of an individual thing but rather
of a set, d, of physical entities ca, c l2 , • • • connected by a set, Ä., of physical relations rn , r# , • • • apparently obeying a set, L t -, of physical laws la,
l%2 ) ' * ' -
Let this system be denoted by Pì(Cì , Ri, Li), and let M(C, R, A) denote
the mathematical system consisting of a class, C, of undefined terms Ci, <h,
• • • , a class, R, of undefined relations n , r%, • • • , and a class, A, of consistent
assumptions a\, a® , • • • .
Definition. Let Pì(Cì , Ri, Lì) be an interpretation (in the usual sense) of
M(C, R, A) and let Pj(c3-, rj) be a true (or false) proposition of M(C, R, A);
then pj(cij, rij) is a true (or false) proposition of Pi(d , Ri, Lì).
Definition. The system Pi(C,-, Rx, Lì) may be said to be known if the truthvalue of all its propositions PJ(CìJ , r^) can be determined in this way.
THEOREM. If Pi and P2 are interpretations of M, and if the class of assumptions A is complete, then the systems Pi and P 2 are isomorphic (in the usual
sense).
THEOREM. If all interpretations of M are isomorphic, then the class A is
complete.
NEWARK COLLEGE OF ENGINEERING,
NEWARK, N. J., U. S. A.
SECTION
VII
HISTORY AND EDUCATION