Der Aufbau Einer Höheren Logik. by Wilhelm AckermannReview by: Kurt SchütteThe Journal of Symbolic Logic, Vol. 40, No. 3 (Sep., 1975), p. 458Published by: Association for Symbolic LogicStable URL: http://www.jstor.org/stable/2272186 .

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458 REVIEWS

The following result for classical logic is proved in Section 2, although it is not stated in this form. LEMMA: For any 31 sentence a (consisting of a first-order formula preceded by second- order existential quantifiers) in a language whose only non-logical symbols are individual con- stants, there exists a quantifier-free sentence that is equivalent to a in all infinite structures. This lemma then forms the crucial step in an elimination of quantifiers result for the extended logic.

The failure of interpolation is then shown by giving two sentences such that although the first implies the second, the two have no non-logical symbols in common. The results mentioned above then preclude having an interpolant in the language of equality. H. B. ENDERTON

A. N. PRIOR. Existence in Lesniewski and in Russell. Formal systems and recursive func- tions, Proceedings of the Eighth Logic Colloquium, Oxford, July 1963, edited by J. N. Crossley and M. A. E. Dummett, Studies in logic and the foundations of mathematics, North-Holland Publishing Company, Amsterdam 1965, pp. 149-155.

The author's aim is to explain certain "peculiarities" of Legniewski's system of logic known as ontology. He suggests that ontology should be viewed as a "broadly Russellian theory of classes deprived of any variables of Russell's lowest logical type." Thus the variables of Legniewski's lowest logical type are to be treated as standing for class-names. An interpretation of this sort seems to commit one to the existence of classes, but this, we are assured, is not the case since by class-names we are to understand common nouns. In Legniewski's logic common nouns "apply to" individuals, which are "named" by Russellian individual names.

As Russell himself has pointed out, his system implies that there exists at least one individual. No such proposition can be proved in Legniewski's ontology, which is a mark of logical purity or ontological neutrality. Prior sees little merit in this, and to him the choice of axioms which carry with them no existential import is "strangely arbitrary." One may concede that there is no harm in having a system of logic with a built-in assumption that there are individuals, but it would be rather embarrassing if instead of an ontologically neutral logic we were offered a logic with a built-in commitment to the existence of entities other than those belonging to the ontological category of individuals. For there is no unanimity among philosophers or logicians as regards the existence of such entities, and a logic which prejudges the issue one way or the other would in fact be useless.

In Legniewski's systems expressions of the form "a Eb" are usually read as "a is b" or " a is a b ". According to Prior they should be read " the a is a b ". In the reviewer's opinion this suggestion is misleading because on putting a common noun which applies to several individuals in the place of "a" in "a E b" we get a false proposition whereas the same sort of substitution may turn "the a is a b" into a proposition that is true. "The" in "the a is a b" goes with "a" to form a compound noun. It does not go with "is a" to form a functor.

The name of Legniewski is spelled correctly throughout the paper except in its title. C. LEJEWSKI

WILHELM ACKERMANN. Der Aufbau einer hoheren Logik. Archiv fur mathematische Logik und Grundlagenforschung, Bd. 7 (1965), S. 5-22.

Der Verfasser geht von einem typenfreien System der Logik aus, das er in einer friuheren Arbeit XVI 72 entwickelt und als widerspruchsfrei nachgewiesen hat. Dieses System El, in dem der Satz vom ausgeschlossenen Dritten nicht allgemein gilt, wird in der vorliegenden Arbeit durch zusatzliche SchluBregeln, die sich ausschlieBlich auf "eigentliche" Pradikate beziehen, zu einem wesentlich starkeren System Z2 erweitert. Dabei ist die "Eigentlichkeit" eines Pradikates A durch die Formel (Ex)A(x) A (x)(y)(A(x) A A(y) F x = y v -x = y) definiert. Die erste zusatzliche SchluBregel des Systems Z2 besagt im wesentlichen: "Ist A ein eigentliches Pradikat und gilt der Satz vom ausgeschlossenen Dritten im Bereich der Dinge, auf die A zutrifft, fur ein Pradikat B, so gilt er entsprechend auch fur das Pradikat Ax1 ... Xn(Ey)(A(y) A B(xl , * * *, xn, y))." Weitere zusatzliche SchluBregeln des Systems Z2 sorgen fur die Extensionalitat und fur ein Auswahlaxiom der eigentlichen Pradikate. Das Hauptergebnis dieser Arbeit besteht in dem Nachweis, daB sich die klassische Analysis im System 2 entwickeln laBt. KURT SCHUTTE

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