The Province of Logic by William Kneale; An Introduction to Deductive Logic by HuguesLeblancReview by: Maurice L'AbbéThe Journal of Symbolic Logic, Vol. 23, No. 2 (Jun., 1958), pp. 210-212Published by: Association for Symbolic LogicStable URL: http://www.jstor.org/stable/2964402 .

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schemata of Kneale's first calculus which differ from Gentzen's schemata are the following:

PvQ *p _p v-elim.: p Q --intr.: p - , -,-elim.: *

* Q {Fx} 3xFx -intr.. P P D) Q. P D) Q.

I V-intr.. vxFx v

-elim.: {Fx}

The rules of proof for using Kneale's calculi are not explicitly stated, but can easily be supplied by following his examples. A more difficult point is the interpretation of expressions like {Fx}. Contextually the author explains {Fx} as a (possibly infinite) set of propositions 'FxL', 'Fx2'% ..., i.e., of "all possible values of the function 'Fx'." He tries to prove the usual schemata for the functional calculus, for instance P Fx P ~ VxFx' x not free in P, but the proof is not clear without a further rule con-

cerning the connection between expressions like 'Fx' and those like '{Fx}'. This does not mean that Kneale's calculus is necessarily incomplete. However, we have to realize that in applying the author's calculi to a special mathematical theory, for

instance to a form of number theory, the use of the schema { } in Kneale's sense yxFx

amounts to the acceptance of a rule of infinite induction, i.e., of an extra-logical principle. In favor of Kneale's attempt it must be remarked that he altogether avoids

axioms (like Gentzen's P v -nP), as well as the constant A, with the rule -, and that

[PI furthermore there occurs no entailment relation like Gentzen's . On the other hand Q

the author consciously forgoes the easy separation between the intuitionistic and the classical part of logic that is present in Gentzen's calculus.

The author's second calculus is similar to Gentzen's treatment of logic by means of Sequenzen. The schemata are rules for getting an involution relation from given

involution relations, for instance r,/ /,/, P, A where the Greek letters stand for

r, E/A&, A whrI h re etr tn o (possibly void) sequences of propositions. Some of the rules of this calculus provide us with the usual liberties concerning order and repetitions of propositions occurring above or below the principal line of a schema, two other rules are similar to Gentzen's Schnittregel, and six special rules (applicable in both directions) have the role of

r/P v Q. A r, VxFxl,& definitions of the logical signs v, & D _ v3, for instance Q

r/P, Q, A' r, {Fx}/A In conclusion the author maintains that the character of logic as a pure involution

calculus is lost the moment we introduce further relations, as for instance the 6-

relation (or also the equality-relation), so that set theory and arithmetic should not be considered as parts of logic. Moreover phrases such as "the logic of colors" or the "logic of obligation" and so on, occurring in some recent philosophical writings, are in the author's opinion simply a sort of abuse of the word "logic."

GERT H. MfTLLER

WILLIAM KNEALE. The province of logic. Mind, n.s. vol. 66 (1957), p. 258. Corrections to the preceding.

HUGUES LEBLANC. An introduction to deductive logic. John Wiley & Sons, New York 1955, and Chapman and Hall, London 1955, xii + 244 pp.

This manual is designed primarily for philosophy students who have no previous

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REVIEWS 211

knowledge in mathematics. For such students, this book should prove very satisfactory. Its content has been carefully chosen so as to give an introduction to the fundamentals of modem elementary logic. A list of exercises and a selected bibliography are included.

In the first chapter, sentential logic is introduced as a system of interpreted signs. The main purpose is the presentation of the concept of sentential deduction. The cou rse followed by the author, similar to the one already adopted by Cooley (VIII 80), is as follows: once the sentential connectives are introduced, a list of the most im- portant valid sentential schemata is given; a certain number of rules of deduction are stated; then, a sentential deduction is defined as a series of statements, each member of which is either a premise or an instance of a recorded schema or follows from pre- vious members of the series through one of the recorded rules of deduction. The truth- table method is presented as a means of identifying valid sentential schemata. The chapter closes with a brief sketch of many-valued sentential logics and modal sentential logics. The second chapter is constructed on a similar pattern. Quantificational logic is introduced from the semantical point of view. Once the notion of quantifier is introduced, a list of the most important valid quantificational schemata is given; then, rules of deduction are stated, and finally the concept of quantificational deduction is defined. The notion of quantificational validity is briefly discussed and a test of validity in a finite domain is given. The last section deals with the intuitionist logic.

The third chapter presents both sentential logic and quantificational logic as systems of uninterpreted signs, that is, as calculi. Two formulations of both calculi are given and developed. The first one is the well-known formulation found in Principia mathematica, the system of axioms for the sentential calculus being simplified as by Bemays (with the second axiom slightly modified). The second formulation is obtained from the first by simply replacing the axioms by corresponding axiom schemata and by dropping the rules of substitution. In the last two sections, a few alternative formulations are stated, among them a formulation of the calculus of natural de- duction of Gentzen.

In chapter four, the calculus of identity is obtained by adding to the quantificational calculus the primitive sign of identity and the usual corresponding axiom schemata. Within this new calculus there are developed an elementary calculus of classes and an elementary calculus of relations, class abstracts and relation abstracts being introduced by non-contextual definitions. No applications to any concrete deductive theory are made.

The metamathematical properties of consistency, completeness, decidability, and in- dependence, as applied to the sentential calculus and the quantificational calculus presented in chapter three, are studied in the last chapter. The method used to prove the completeness of the sentential calculus is adapted from Rosser (XVIII 326). The completeness of quantificational logic is proved by the original argument of G6del; however, the lemma about the possibility of finding a formula in Skolem normal form corresponding to any given formula is assumed without proof. It seems difficult to motivate the choice of G6del's proof rather than that of Henkin, which, by the sim- plicity of its structure and its elegance, should prove more accessible to a philosophy student.

The text is remarkably clear and rigorous. To the reviewer, this seems to be one of the best textbooks of its kind.

The relatively few typographical errors should not present serious difficulties to the careful reader. We mention the following (from a list transmitted by the author): page 62, lines 22, 23, and 24, replace "G" by "F"; page 69, line 23, insert " -" before

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212 REVIEWS

the first "G"; line 31, replace "infallible" by "fallible"; page 101, line 21, insert " " before the last "q"; page 118, line 23, replace the first "q" by "(p v q"; page 207, line 5, replace the first "0" by "3"; line 6, replace "2" by "0".

MAURICE L'ABBE

ALAN H. GARDINER. The theory of proper names. A controversial essay. First edition, Oxford University Press, London 1940, 67 pp. Second edition, Oxford University Press, London, New York, and Toronto 1954, viii + 76 pp.

Except for minor verbal changes, suppression of the earlier Preface, and the addition of a six page "Retrospect 1953," the second edition is identical with the first. The author is severely critical of Russell's view of proper names, and makes lesser criticisms as well as emendations of the views of Mill, Keynes, Bertelsen, Funke, Br6ndal, and others. His own view is expressed in this tightly packed definition (p. 73): "A proper name is a word or group of words which is recognized as having identification as its specific purpose, and which achieves, or tends to achieve, that purpose by means of its distinctive sound alone, without regard to any meaning possessed by that sound from the start, or acquired by it through association with the object or objects thereby identified."

The author explains (p. 41) "that when I dwell upon the 'sound' of proper names, I am referring only to the preponderating attention paid to their distinctive sensible externals as opposed to the associated meanings." He does not deny that proper names sometimes have originally, or may acquire, "meaning." But, he urges, it is in the nature of their function as referring that they are proper names, and this they do in logical independence of their meaning. Names may be of individuals, of groups of individuals, e.g., the Azores, or of "collectives," e.g., the Duma. Psychologically they name by calling to mind an idea, thought, impression, or memory of the thing named. Something of this mental sort exists whenever a word is used as a name, whether the "object" named be real, fictional, imaginary, or mythical.

On most of these points the author's view is common-sensical, and is illustrated by many examples from established language, which he distinguishes sharply from temporary speech improvisations. His account is consistent with the view that general terms are names of universals or other abstract entities, but he does not discuss this possibility. Had he done so his study might have led him to modify some of his other statements. Thus he might have distinguished more sharply between reference by proper names and reference by means of descriptions, for the latter requires, as the former does not, reference to characteristics possessed by the entity referred to. If universals can be referred to by proper names, does this also require the psychological mediation of images of memory or imagination? But presumably the images one happens to have when one thinks, say, of the cardinal number three have little to do with the nature of that number. Similarly, the images we happen to have when we think of the proper name, 'Vercingetorix,' may be quite unnecessary to our using that word as a proper name. Perhaps the naming relation is a direct one between name and object named, not an indirect one mediated by imagery. Were he to admit this to be so he would have to face the question: What is the situation when one uses a word with the intent to name, when what one intended to name does not, or no longer, exists.

CHARLES A. BAYLIS

MARTIN A. GREENMAN. A Whiteheadian analysis of propositions and facts. English with Spanish Extracto. Philosophy and phenomenological research, vol. 13 no. 4

(1953), pp. 477-486. In a strictly historical paper the author summarizes Whitehead's views as to the

relations of propositions and facts as follows: "... a proposition which is neither signified by a sentence nor realized in any actual world has the ontological status of a

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