Modern Formal Logic Research Articles

La théorie probabiliste de Carnap et son application aux problèmes de l’économétrie

Carnap's theory makes possible a more thorough discussion of the problem of induction and of certain problems in mathematical statistics. Carnap distinguishes inductive, or hypothetical, probabilities and empirical or statistical probabilities. His theory of inductive probability, which is based on modern mathematical logic, opens up considerable possibilities for its application in mathematical statistics and econometrics.

Leon Henkin. Some interconnections between modern algebra and mathematical logic. Transactions of the American Mathematical Society, vol. 74 (1953), pp. 410–427.

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Translations from the Philosophical Writings of Gottlob Frege ed. by Peter Geach, Max Black

104 BOOK REVIEWS cause present for grave punishment they can never directly harm or tamper with the integrity of the body either for the reasons of eugenics or for any other reason." Indeed, many writers who uphold the liceity of punitive sterilization make much capital out of this statement. Yet the Pope seems to have, implicitly at least, described punitive sterilization as a direct interference with bodily integrity. And if punitive sterilization is direct it would fall under the condemnation of the Holy Office which in 1931 declared that direct sterilization of man or woman, whether perpetual or temporary, is contrary to the natural law. Is punitiv~ sterilization direct or indirect? Does it not, to use the words of Pope Pius XII, aim at making procreation impossible? What is the object of this sterilizing act? And even if we must make a gesture in this context to the intention of the State authorities in imposing sterilization as a punishment-must we not say that: they intend to make procreation impossible, at least as a means? We have queried a number of points in Fr. Kenny's work. But we would wish that our doing so should be interpreted as an expression of our belief that the work is a valuable contribution to the literature on medical ethics and of our hope that there will be further editions in the preparation of which the objections we have raised may be kept in mind. And if it is not unpardonable to end on a slightly facetious note we, of this eastern hemisphere , should love to know what the " skedger wegers " (p. 4) really are! St. Patrick's College, Maynooth, Eire JoHN McCARTHY Translations from the Philosophical Writings of Gottlob Frege. Edited by PETER GEACH and MAX BLACK. New York: Philosophical Library, 1952. Pp. ~44 with index. $5.'75. Mr. Geach and Mr. Black have rendered a signal service in translating and editing, and the Philosophical Library in publishing, these scattered fragments from the works of the great German mathematician, Gottlob Frege. For it was Frege who exercised a profound and decisive influence on men like Russell and Wittgenstein, and of whom it may be said that his was perhaps the most profound and seminal influence leading to the development of modern mathematical logic. And yet despite his greatness, essays such as these which are indispensable for an understanding of the semantical and even logical import of Frege's work, "have," in the words of the editors, "long been buried in various German periodicals (mostly now defunct)." Nevertheless, merely bringing these essays to light, and even bringing BOOK REVIEWS 105 them to light in such remarkably clear and smooth translations as these, will doubtless not in itself guarantee that Frege will come to be readily understood and appreciated. For like so many great seminal thinkers, Frege is a man whose thought would seem to be quite as ponderous as it is important, and quite as crabbed and involved as it is suggestive. Accordingly, in attempting to survey and appraise this somewhat difficult material, this reviewer wonders if it might not be illuminating, even if somewhat unorthodox, to try to bring out the contrast between Frege's notion of logic and some of the dominant ideas that might be said to be operative in a more traditional, Aristotelian type of logic. That such an approach is unorthodox may be seen from the fact that it is not Frege's own approach. Instead, his initial concern is with mathematics and with providing an adequate account of basic mathematical concepts; and it would always seem to be from some such point of departure that he moves on to logical and semantical questions. Thus in none of the selections here given does Frege attempt anything like a direct or developed account of how his view of logic differs from the traditional one. Nevertheless, suppose for purposes of argument that we project a possible interpretation of Aristotelian logic and then consider some of the points of contrast between it and Frege's view of the subject. Thus on the Aristotelian view it might be supposed that logic is concerned with the tools or instruments of knowledge. Moreover, as...

Some Interconnections Between Modern Algebra and Mathematical Logic

Some Interconnections Between Modern Algebra and Mathematical Logic

Some interconnections between modern algebra and mathematical logic

Mathematical logic originated when mathematical methods were brought to bear on traditional questions of logic, especially the problem of what constitutes a valid proof. For the most part work in this field has remained close to foundational questions. Recent results, however, have indicated the possibility that the methods of mathematical logic may be of use in tackling specific problems in other branches of'mathematics. This possibility is illustrated in the present paper, which presupposes no knowledge of mathematical logic. 1. Some concepts of logic. We shall first focus our attention on algebraic structures consisting of a set D on which two binary operations are defined. Rings are familiar examples of such structures; but to begin with we do not make any restrictions on the nature of the binary operations. Instead, we proceed to construct a simple, specialized language, L, which can be used to refer to any one of our structures. First we provide two symbols, + and ., to be used as names of the operations. Next we wish to have symbols (called individual constants) which can be used as names of particular elements in D. In the case of rings, for example, it is customary to employ the symbols O and 1 in this way; for our more general purpose, however, we shall reserve the letter v, to be used with various distinguishing subscripts. In addition to referring to particular elements of D we wish to make general statements about all elements of D, and for this purpose we introduce further symbols x, y, z, X1, Yi, (called individual variables), which will be used as variables whose range is D. Additional symbols are to include parentheses, an equality sign, and the following words: not, and, or, if, then, all, exists. The symbols of L are used in constructing formulas and sentences as follows. An individual symbol (constant or variable) is called a term, and the result of putting a + or between two terms and enclosing the result in parentheses is also a term. Basic sentences are formed by placing an equality sign between two terms, and further sentences are built up from these basic ones in the following ways. If A and B are sentences, so are (not A), (A and B),

Aristotle's Syllogistic from the Standpoint of Modern Formal Logic.

Aristotle's Syllogistic from the Standpoint of Modern Formal Logic.

Hans Reichenbach. The verifiability theory of meaning. Proceedings of the American Academy of Arts and Sciences, vol. 80 no. 1 (1951), pp. 46–60. - Carl G. Hempel. The concept of cognitive significance: a reconsideration. Proceedings of the American Academy of Arts and Sciences, vol. 80 no. 1 (1951), pp. 61–77. - Gustav Bergmann. Comments on Professor Hempel's “The concept of

In 1923 I was asked by the editors of the der Physik to write an article about the philosophy of physics. Since one of the points I wanted to discuss in this article was the reality of the physical world, I used the occasion of a trip to the Bavarian mountains to see Moritz Schlick at his summer vacation place and to talk with him about this problem. I told him that I had recently discussed these questions with a young philosopher who was writing a book entitled Der logische Aufbau der Welt, and who advanced the idea that Mach's positivism could be given a strict logical form by regarding any statement about the physical world as logically equivalent to a combination of statements about sense data. I myself objected to this equivalence ; I thought that only a probability relation ship could be maintained. Schlick supported me in this view; he argued that the positivistic equivalence was untenable. That was in 1923. A few years later, after Schlick had invited Rudolf Carnap to become a member of his department and had succeeded in attaching Ludwig Wittgenstein to his circle in Vienna, Schlick changed his views completely and turned posi tivist. In this environment, those views were elaborated which culminate in the thesis that the meaning of a sentence is the method of its verification. Mainly as the achievement of Carnap,1 the verifiability theory of meaning was thus given an elaborate logical form ; for the first time in its long history, which goes back to William of Occam and Leibniz, this theory was set forth with the precision tools of modern symbolic logic. My article written in 1923 had to wait for publication until 1929, when the volume of the Handbuch finally came out.2 My refusal to accept the equivalence interpretation was based on the distinction between a complex (Gebilde) which is artgleich to its elements from a complex which is artverschieden to its elements, a thesis which I took over into a later book of mine as the distinction between a reducible and a protective complex of elements.3 But I knew that it was vain to go into a further elaboration of my conception before I had been able to con

The Verifiability Theory of Meaning

In 1923 I was asked by the editors of the "Handbuch der Physik" to write an article about the philosophy of physics. Since one of the points I wanted to discuss in this article was the reality of the physical world, I used the occasion of a trip to the Bavarian mountains to see Moritz Schlick at his summer vacation place and to talk with him about this problem. I told him that I had recently discussed these questions with a young philosopher who was writing a book entitled "Der logische Aufbau der Welt", and who advanced the idea that Mach's positivism could be given a strict logical form by regarding any statement about the physical world as logically equivalent to a combination of statements about sense data. I myself objected to this equivalence ; I thought that only a probability relation ship could be maintained. Schlick supported me in this view; he argued that the positivistic equivalence was untenable. That was in 1923. A few years later, after Schlick had invited Rudolf Carnap to become a member of his department and had succeeded in attaching Ludwig Wittgenstein to his circle in Vienna, Schlick changed his views completely and turned posi tivist. In this environment, those views were elaborated which culminate in the thesis that the meaning of a sentence is the method of its verification. Mainly as the achievement of Carnap,1 the verifiability theory of meaning was thus given an elaborate logical form ; for the first time in its long history, which goes back to William of Occam and Leibniz, this theory was set forth with the precision tools of modern symbolic logic. My article written in 1923 had to wait for publication until 1929, when the volume of the Handbuch finally came out.2 My refusal to accept the equivalence interpretation was based on the distinction between a complex (Gebilde) which is "artgleich" to its elements from a complex which is "artverschieden" to its elements, a thesis which I took over into a later book of mine as the distinction between a reducible and a protective complex of elements.3 But I knew that it was vain to go into a further elaboration of my conception before I had been able to con

Basic Confusions in Current Notions of Propositional Calculi

BASIC CONFUSIONS IN CURRENT NOTIONS OF PROPOSITIONAL CALCULI IN a preceding article,1 we took note of three salient features in the so-called propositional calculi of modern mathematical logic: (I) the use of eonstants and variables to represent the possibility of propositional composition; (2) the tautologous character of all theorems in the calculus; (3) the truth-functionalinterpretation of such propositional compounds. Nevertheless, it may be recalled that of these three features only the last one was the subject of arty real discussion. Thus we sought to show how, when propositional compounds are interpl'.eted truth-functionally, it is impossible to give any sort of adequate account of so-called implicative compounds. And yet at the same time, such compounds, so far from being dispensable or capable of being explained away, are actually presupposed and employed in any conceivable development of propositional calculi. Accordingly, interpreting the types of propositional composition that enter into these calculi truthfunctionally , the calculi themselves are thereby rendered both inadequate and inconsistent. Nor did there seem to be any way out of the impasse, save that of frankly recognizing that these calculi are. not descriptive of logic at all, and that their subject matter consists not of objects of second intention, but of first intention. But why might not this third feature of the calculi be simply omitted? After all, it being only an interpretation, why would it not be possible merely to interpret implicative compounds in a way other than the truth-functional one, and thereby make it assured that the calculi were yeritably proposi,tional calculi? Indeed, the only reason we gave for our suggestion that the 1 Veatch, H., "Aristotelian and Mathematical Logic," The Thomiat, XIII (Jan. 1950)' pp. 50-96. 288 BASIC CONFUSIONS OF PROPOSITIONAL CALCULI fl89 calculi might have to be regarded as pertaining to objects of first intention rather than second intention was that, on the truth-functional interpretation of the theorems, these theorems simply ceased to be either adequately or consistently descriptive of propositional relations. Eliminate the truth-functional interpretation, therefore, and the whole difficulty might seem to vanish. Not only that, but a random glance at the theorems in any one of these calculi would certainly seem to indicate that they were descriptive of propositions. Thus it simply is true of propositions that: p ::) q . ::) . ,.._, q ::) ,,.._, p pVq·::)·qVp p · V · q V r:::) :p V q · V · r ,..._, p v ,.._, q . ::) . ,.._, (p . q) ......., (p . ,.._, p) In other words, upon examination, theorems of this sort do turn out to be tautologous in the sense defined,2 i. e. they are necessarily true, no matter what .proposi,tional values be substituted for the variables p, q, r, etc. Moreover, to clinch the point that it might be possible to interpret the implicative compounds in the calculus in a way other than truth-functionally, .we need only remark that this has actually been accomplished. For Prof. C. I. Lewis, in his so-called calculus of strict implication, has in effect done just this. Thus to follow Prof. Lewis' own account, he says, speaking of the truth-functional interpretation of propositional compounds : " In the usual terms, this means that the truth or falsity of p and q being given, the truth or falsity of,.__, p, of p ::) q, of p V ,....., q, of p · ::) · q ::) p, and of every other function of p and q, which can figure in the system is thereby categorically determined. ·Any system having this character-that there is no function or relation of the elements which is in the system except such as are categorically determined to be true 9 Ibid., p. 60. 240 HENRY VEATCH or to be false by the truth or falsity of their elementary terms-may be called a ' truth-value system.' " 3 On this basis, then, continues Prot Lewis," p :) q definitely holds except when p is true and q false, in which case it definitely fails to hold. This is, in fact, exactly what is expressed by its definition. By contrast p >-3 q 4 is definitely false if p is true and q false, but is not determined to be either true...

The Logical Analysis of Kinship

The present attempt to indicate the general manner in which the kinship system of a people can be stated as an interpreted axiomatic system, with utilization of the symbolism of modern mathematical logic, is intended as a demonstration of the applicability of contemporary logical methods to problems in the social sciences. Aside from the obvious general advantages in the way of clarity and logical rigour to be gained by an application of axiomatic method, there emerges as by-products a convenient system of kinship symbolization and what is believed to be a solution to the long-standing controversy concerning the use of the terms ‘descriptive’ and ‘classificatory’ in reference to kinship systems. Since kinship terms are words in languages, the present exposition is perhaps of interest in that it shows how meaning in a semantic field of related terms can be defined by logical methods. So all the directional or temporal terms in a specific language might be treated as kinship terms are here. The possibility of a systematic treatment of the semantic aspects of linguistics which so far have resisted this type of treatment should be of interest to linguists.

Chapters from Modern Formal Logic.

Chapters from Modern Formal Logic.

The Permanent Significance of Hume's Philosophy

The subject of my lecture is an appropriate one for several reasons. The first is purely chronological. Hume's first and greatest work, the Treatise of Human Nature, was published in 1739, two hundred years ago. Its illustrious author was then quite unknown in the world, and as he tells us himself the book “fell dead-born from the press.” But by the end of the eighteenth century its reputation was securely established, and it has long been regarded as one of the masterpieces of European thought, and as the classical statement of the Empiricist Philosophy. It is true that, like other classics, it has had its ups and downs. During the Absolute Idealist period, which ended early in the present century, Hume was the great bogey-man, and the duty of all self-respecting philosophers was to refute him. In our own day things are different. Empiricism, despite many obituary notices, is very much alive again. And this time it is in close alliance with Natural Science, and has equipped itself with all the technique of modern Symbolic Logic; it is more vigorous in construction and more formidable in criticism than it has ever been before. Consequently Hume is no longer the bogey-man. People now read the Treatise not as an awful warning, but as a source of stimulus and illumination. Incidentally, we can now enjoy his admirable style without any qualms. It is no longer thought that if a philosopher writes in clear and entertaining English, what he writes must therefore be either superficial or false. We regard obscurity and turgidity as demerits, not as signs of profound thinking. Moreover, we have learned to appreciate the eighteenth century, of which Hume was one of the most characteristic products.