CS(n): An Extension of CS by Ivo ThomasReview by: Alonzo ChurchThe Journal of Symbolic Logic, Vol. 15, No. 2 (Jun., 1950), pp. 141-142Published by: Association for Symbolic LogicStable URL: http://www.jstor.org/stable/2266990 .

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REVIEWS 141

(here called "rule of derivation"). Twelve tautologies of the propositional calculus are introduced as axioms, and besides these the four axioms: Aaa, Iaa, CKAmaAbmAba (the leading principle of Barbara), CKEnaIbmOba (the leading principle of Ferio). From these are derived as consequences the traditional laws of opposition, and the leading principles of the traditional conversions and syllogisms.

According to the author, the method and the last four axioms of the system are sub- stantially due to Lukasiewicz (1866), with a change in the last axiom-Lukasiewicz uses the leading principle of Datisi in place of that of Ferio. The idea of using the axioms Aaa and Iaa is attributed to Leibniz (1007, p. 9). Many details of the presentation, and the discussion of the system are new in the present paper.

As unsolved problems regarding the system the author lists: possible connections with many-valued or intuitionistic propositional calculi; independence of the axioms; com- pleteness in various senses; proof of consistency.-As to the last, the reviewer would re- mark that the system possesses an interpretation according to which the values of the nominal variables are the non-empty subclasses of a fixed finite class, the other notations having their obvious meanings. In the light of this interpretation, a consistency proof by means of finite truth-tables is readily constructed.

The author discusses the question of allowing the null class (in addition to other classes) as a value of the nominal variables, and he points out that, by omitting the axiom Iaa, just those theorems are canceled which have to be canceled for this purpose. Of the two systems, with and without the axiom laa, it is said that it is a pseudo-problem to ask which is the true one, and that the choice between them is a matter of practical usefulness and convenience, in a particular connection or for a particular development. The reviewer would add that, for representing in formalized fashion the elementary reasoning of every- day discourse, the system without the axiom Iaa is clearly the more convenient-since it is often impracticable, or undesirable to demand, that it be ascertained in advance that a particular class is not null before applying such reasoning to it. (Compare the review X 133(2), as well as, e.g., Lewis and Langford 4561, pp. 62-66.)

Some other partial systems are briefly discussed, and some alternative choices of axioms. The paper contains also comment on the usefulness and admissibility of the formal

method, the method in which abstraction is made from the meaning of the formulas of a system in order to study their syntax; further some discussion of the relationship of the theory of the assertoric categorical syllogism to other parts of logic.

Author's correction: In D7, on page 37, for "two propositions" read "two nominal vari- ables. " ALONZO CHURCH

Ivo THOMAS. CS(n): An extension of CS. Ibid., vol. 2 (1949), pp. 145-160. Bochenski's system (see the foregoing review) is extended by the introduction of a new

notation n, "interpreted as the infinitizing negative of traditional logic." A new rule of inference allows the interchangeability of nna and a, where a is any nominal variable (the author attempts to express this by an equation nna = a and to regard it as an axiom or a definition, but this is open to some objection in the present connection, and it would seem to be preferable to use a rule of inference). The definition of the notation E is changed to Eba = Abna, the primitive notation I is replaced by the definition Iba = NAbna, while Bochenski's definition for 0 remains unchanged. The only other change is to drop the leading principle of Barbara from among the axioms, since it has become non-independent. Besides the traditional laws of opposition, conversions, and syllogisms, this system now yields also the usual obversions and contrapositions (or, more correctly, their leading principles).

The author writes that his system still awaits proof of consistency. However, the sys- tem has an interpretation in which the values of the nominal variables are the non-empty proper subclasses of a fixed finite class, and from this a consistency proof is readily ob- tained. In particular, if we take the fixed finite class to have exactly two members, there results Wedberg's consistency proof (see below).

Neither Bochenski nor Thomas has any notation for the logical product of classes. If this were to be introduced, no doubt some restriction would be required in order to allow

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142 REVIEWS

for pairs of classes having no (i.e., no non-empty) logical product. This remark is in one sense a sufficient reply to the objection of Popper (XIV 143(25)). The reviewer has not seen the treatment of Wiegner (XV 65), which allows both the logical product and the logical sum of classes, as well as the negation n, in connection with the traditional syl- logismns and immediate inferences.

Author's correction: In the table on page 160, after l in the third line of the table, for 1011 read 1110. ALONZO CHURCH

ANDERS WEDBERG. The Aristotelian theory of classes. Ajatus (Helsinki), vol. 15 (for 1948, pub. 1949), pp. 299-314.

The system treated in this paper is very similar to that of Thomas (see the two pre- ceding reviews). The negation or complement of a class is indicated by placing an accent after a class variable. Of the four notations A, E, 1, 0, only the first is taken as primitive, aid the other three are defined by means of it. Propositional variables are not used, but there is a rule according to which every substitution instance of a tautology of the proposi- tional calculus is a theorem. Other rules of inference are a rule of substitution for class variables, and miodus ponens. There are five axioms, which we may rewrite as follows, using Lukasiewicz's notations for the connectives of the propositional calculus in place of those used by Wedberg: Aaa", Aa"a, CAabAba', CKAabAbcAac, CAabNAab'.

(In the intended interpretation of the system, the last axiom has the effect of excluding the null class and the universal class as values of the variables; if the null class and the universal class are to be allowed, the axiom may be replaced by CAaa'Aab, as the author suggests in a footnote.)

The consistency of the system is established by referring to an interpretation in which the range of the variables consists of two elements 1 and 2, where 1 and 2 are taken as complements of each other, and Aab has the meaning of a = b. As the author points out, this amounts to considering the formulas which are valid in a universe of discourse con- taining exactly two individuals. The independence of the five axioms is established by means of simple independence examples.

With the aid of a decision procedure, it is shown that the system is complete, in the sense that a formula is a theorem if (and only if) it is valid in every universe of discourse containing at least two individuals. That the system is not complete in a stronger sense is shown by giving an example of a formula which is valid in a universe of discourse contain- ing exactly two individuals, but is not valid in a larger universe.

The inference from a major premiss CPQ and a minor premiss P to the conclusion Q is the same which is traditionally known as modus ponens, or modus ponendo ponens. Wed- berg retains the traditional name for this inference (as has been urged by Quine), remain- ing in this respect nearer to the traditional terminology than Bochenski. On the other hand Bochenski seems to be more faithful to the traditional doctrine in that he does not assume all the laws of material implication for the connective C, but only such laws as are needed for his purpose and which for the most part may reasonably be considered as implicit (if not explicit) in the traditional doctrine of conditional propositions.

Wedberg's proof of consistency can be applied also to the systems of Thomas and Bochenski. But the same is not true of Wedberg's proof of completeness, because of the difference just explained as to laws of the propositional calculus (Thomas following Bo- chenski in this regard). ALONZO CHURCH

Ivo THOMAS. Logic and theology. Dominican studies (Oxford), vol. 1 no. 4 (1948). Off- print, 22 pp.

The main part of this paper, and the most interesting, is devoted to advocating and defending the use of formal logic in the study of theology. This is supported both by direct argument and by many references to the precept and practice of Thomas Aquinas. The author quotes further a passage from the concluding paragraphs of the Introduction to Quine's XII 56 (or V 163)-in which hope is expressed for the fruitful application of mathe- matical logic to science, especially to non-quantitative aspects-and adds, "If these hopes are justified, then they may surely be justified for theology, where non-quantitative tech-

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