Leçons de Logique Algébrique. by Haskell B. Curry; Robert FeysReview by: Hugo RibeiroThe Journal of Symbolic Logic, Vol. 19, No. 2 (Jun., 1954), pp. 146-147Published by: Association for Symbolic LogicStable URL: http://www.jstor.org/stable/2268922 .

Accessed: 13/06/2014 00:26

Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at .http://www.jstor.org/page/info/about/policies/terms.jsp

.JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range ofcontent in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new formsof scholarship. For more information about JSTOR, please contact support@jstor.org.

.

Association for Symbolic Logic is collaborating with JSTOR to digitize, preserve and extend access to TheJournal of Symbolic Logic.

http://www.jstor.org

This content downloaded from 188.72.126.108 on Fri, 13 Jun 2014 00:26:34 AMAll use subject to JSTOR Terms and Conditions

146 REVIEWS

method is to formalize some of Hull's postulates in symbolic logic and give a formal derivation of a theorem which is interpreted as expressing Tolman's expectancy principle.

The following definition, however, requires revision:

rjpsi =Df (x) : (3z) . xeRj . zeSi . D . Ex D Gz.

The writer interprets the left side of this definition as meaning that the response complex rs "leads to" the stimulus complex si, and he interprets the right side as meaning that "whenever conditioned reactions are evoked, whether reinforced or not, stimulus complexes are generated". But there is the difficulty that the relation rs to

Rj is left obscure, and similarly for si and Si. A more serious difficulty, furthermore, is that the right side of the definition is always true if at least one thing in the universe has the property G, the property of being generated. It is doubtful, indeed, that the relation of "leading to" can be defined in any such simple way as this, since a relation of causation seems to be involved somehow, and causation is a notoriously difficult concept to define. There is also a defect in the formal derivation in connection with subscripts. But Behan is to be commended for making one of the few serious attempts to apply the techniques of symbolic logic to a genuine problem in behavior theory. If he has failed, it is primarily because of the difficulty of what he attempted to do.

FREDERIC B. FITCH

HASKELL B. CURRY. Lepons de logique algdbrique. With a preface by Robert Feys. Collection de logique math6matique, s~rie A. Lithographed. Gauthier- Villars, Paris; E. Nauwelaerts, Louvain; 1952, 163 pp.

The present book contains the lectures given in 1950-51 at the University~of Louvain. For a long time the need, in the French language, of monographs like this one and other of the same collection has been too obvious. The book is readable independently of the well-known author's lectures at the University of Notre Dame (XVI 56), but is partly intended to supplement them, the emphasis being now on the algebraic aspect rather than on the logistic aspect of the most significant systems of propositional calculus. In fact, except for several examples and remarks, it does not presupose any special knowledge in mathematics or in logic, so that it should be understandable by any one who has that "ability of abstract thinking and reasoning with symbols one expects to find in mathematicians and also in logicians of philosophical background." The main body of the book consists of an analysis of the formalization and interrelations of the logical algebras. The algebras dealt with are successively, quasi-ordered algebras (Chap. II), logical groups or semi-lattices, lattices, and distributive lattices (Chap. III), implicative logical groups, implicative and subtractive lattices, classical implicative lattices, and Boolean rings (Chap. IV), minimal, intuitionist, strict, classical (and Bool- lean) algebras (Chap. V). In Chapter VI there is a brief account of the relations between an algebra and a secondary algebra of it (e.g. between the algebra of classes and an algebra of epipropositions of it), of the traditional logic, of modal algebras, and of re- lation algebras. Finally in a short appendix there is a general theory for Lukasiewicz's notation, and an introduction to the author's combinatory logic.

A distinction between philosophical and mathematical logic is explained in the Introduction. Chapter I is devoted to describing the general concept of a formal system, after indicating the evolution of this concept and exhibiting an example of it for the case of the system of natural numbers. There are the language of the formal system, the non-rigid current language one uses in particular to communicate the construction of the formal system, and the (syntactic or semantic) meta-language. Depending on the predicates of the meta-language, one has or has not a completely formal system. Epimethods and epitheorems such as those which state consistency, completeness, and decidability of the formal system are discussed and their importance

This content downloaded from 188.72.126.108 on Fri, 13 Jun 2014 00:26:34 AMAll use subject to JSTOR Terms and Conditions

REVIEWS 147

emphasized. A formal system S can take the form of a "logistic system" T if the "obs" and operations of T are those of S and if, for each predicate / of S. there is an operation F of T such that f(a1, . . ., aJ) is identified with F Fla, . . ., a"), where Frege's F is the only predicate of T; and conversely one can get an S from T, so that logical algebras have both "relational" and "logistic" forms. The quasi-order < is the fundamental relation in logical algebras and D is the corresponding operation in the logistic form, x = y being defined by x ? y & y S x. The method of treating the several algebras is an epitheoretical one - in particular the author uses a general re- placement theorem in connection with the (monotonic relatively to ? ) operations of those algebras. The algebras are introduced in relational or logistic form as appears more natural. As a rule, every concept is preceded by an informal discussion including significant examples. There are at the end of each chapter many valuable remarks or just references to the bibliographic list, and throughout the book complementary and detailed explanations (e.g. the decision method for free semi-lattices and for free distributive lattices, a proof of the representation theorem for Boolean algebras in the finite case, the treatment of equations in Boolean algebras). HUGO RIBEIRO

IRVING M. COPI. Introduction to logic. The Macmillan Company, New York 1953, xvi + 472 pp.

This primer opens with four chapters (Part I) on language and closes with four chapters (Part III) on induction. The middle six chapters (Part II) are devoted to deduction, with equal emphasis on Aristotelian and on modem techniques.

Part I studies briefly, but effectively, the various uses of language (identified as information, expression, and direction), the various types of agreement and disagree- ment, fallacies (grouped under two convenient headings: fallacies of relevance and fallacies of ambiguities), and finally definitions. Part III includes a novel and welcome element, a chapter on probability. It also has chapters on analogy, Mill's celebrated canons of causation, and science and hypothesis. The reviewer regrets that the fifty- four pages devoted to Mill have not been spared for statistics. Statistical inference is induction of a sort, and an outline of its workings might prove more beneficial to the undergraduate than a detailed analysis of Mill's canons.

Part II has three chapters on Aristotelian logic, covering the A, E, I, and 0 propo- sitions, various immediate inferences, and four types of mediate inferences: the syl- logism, the enthymeme, the sorites, and the dilemma. As a test of syllogistic validity Copi offers both the Venn diagrams and the six rules bequeathed by medieval logicians. These three chapters are traditional in content and form, but they are extremely lucid and provide an excellent testing ground for the student.

Of the last three chapters of Part II, two deal with truth-functional inferences, one with monadic quantificational inferences. The former are carried out with the aid of ten rules, among which the reader will recognize the familiar Gentzen rules of con- ditional elimination, alternation introduction, conjunction elimination, conjunction introduction, and alternation introduction. The remaining five rules are modus tolens, hypothetical syllogism, disjunctive syllogism, destructive dilemma, and a weak variant of the familiar rule of interchange of equivalents. Notable is the absence of the rule of conditional introduction (or conditionalization) and, for that matter, of any rule discharging premisses. What sort of completeness the rules may enjoy is not discussed.

Monadic quantificational inferences are carried out with the aid of four extra rules: universal generalization (UG), existential generalization (EG), universal instantiation (UI), and existential instantiation (EI). The author's version of UG: "From 'IY' to infer '(x)gx', where 'y' denotes any arbitrarily selected individual," is not effective. As a working substitute use: "From '9y' to infer '(x)Tx' where 'y' is not free in any line of the proof which has been inferred by EL." The thus amended rules may be

This content downloaded from 188.72.126.108 on Fri, 13 Jun 2014 00:26:34 AMAll use subject to JSTOR Terms and Conditions