Conventional Logic and Modern Logic:
A Prelude to Transition

Joseph T. Clark, S.J.


"If one is limited to three books, choose Bochenski's Precis, Quine's Method's, and Tarski's Introduction."

"Bochenski's Precis and Quine's Methods make an excellent tandem. If one had to choose two books and only two books, these are my choice."


A. General Bibliographical Sources

Alonzo Church, "A Bibliography of Symbolic Logic," The Journal of Symbolic Logic 1 (1936) 121-218; 3 (1938) 178-212. And current bibliographies regularly thereafter. The standard source. There is no better. The Journal is published by the Association for Symbolic Logic. Current Secretary-Treasurer: Mr. Robert E. Luce, Rutgers University, New Brunswick, New Jersey. Latest information discloses that back issues are available and new subscriptions welcome. The articles are in the main professional. But the reviews of current literature by experts contribute accurate estimates of published elementary introductions to the subject. An indispensable tool for any serious work.
E. W. Beth, "Symbolische Logik and Grundbelegung der exakten Wissenschaften," Bibliographische Einfuhrungen in das Studium der Philosophie 3 (Bern: A. Francke Verlag, 1948) . An accurate map of the field. Nice combination of topical and historical considerations. Indispensable for correct orientation of new students of the subject.

B. A Working Bibliography

Alice Ambrose and Morris Lazerowitz, Fundamentals of Symbolic Logic. New York: Rinehart and Company, 1948. Some like this work. It is incomplete. But there are efforts at significant correlation between "conventional" logic and modern logic. There is a modest attempt at "axiomatic formulation." It is worth an examination.
P. Banks, "On the Philosophical Interpretation of Logic: An Aristotelian Dialogue," Dominican Studies 3 (1950) 139-153. An "atmosphere" piece. Accurate in the main. But no specific details and no documentation. Will help to orient one who has little time for anything more complete or more profound.
J. Bar-Hillel, "The Revival of 'The Liar,"' Philosophy and Phenomenological Research 8 (1948) 245-253. A critical reply to Koyre, ibidem 6 (1946) 344-362. The two articles offer an interesting case-in-point and example of the need for logical competence in the discussion of logical problems. One need not agree completely with either author to observe (1) how ignorance of modern techniques obfuscates and confuses issues, and (2) how some ability in modern techniques clarifies issues.
Johannes Bendiek, O. F. M., "Scholastische und mathematische Logik," Franziskanische Studien 31 (1949) 31-48. A well informed scholar enters a well balanced plea for the juncture of the two. His case is competently prepared. His arguments deserve consideration.
E. W. Beth, "Hundred Years of Symbolic Logic," Dialectica 1 (1947) 331-345. The century is: 1847-1947. A neatly integrated piece. A map for the period that it covers and excellent for orientation to newcomers.
E. W. Beth, "The Origin and Growth of Symbolic Logic," Synthese 6 (1947-1948) 268-274. Similar to the above and of about equal worth and utility.
E. W. Beth, "Critical Epochs in the Development of the Theory of Science," The British Journal for the Philosophy of Science 1 (1950) 27-42. Notable for a succinct and masterful x-ray analysis of the Aristotelian ideal of science and the logic adapted to its purposes. A systematic gem.
Max Black, "A New Method of Presentation of the Theory of the Syllogism," Journal of Philosophy 42 (1945) 449-455. A man who knows modern logic very well makes a few suggestions from pedagogical experience. Worth investigation and may be helpful to teachers.
Max Black, The Nature of Mathematics. London: Routledge and Kegan Paul, 1950. One of the standard discussions that appeared after Principia Mathematica of Whitehead and Russell. Recently reprinted and available to build up a working library.
Max Black, "Wittgenstein's Tractatus," Language and Philosophy (Ithaca: Cornell University Press, 1949), pp. 139-165. A critical discussion of a classic in the field. Agreement is not necessary. But illumination will surely be given to the tyro in understanding and appreciating Wittgenstein's dense style and perhaps unfamiliar purposes.
Max Black, "The Semantic Definition of Truth," Language and Philosophy (Ithaca: Cornell University Press, 1949), pp. 89-107. A critical discussion of Tarski's famous essay and related ideas. Same comments as above.
Leonard Bloomfield, "Linguistic Aspects of Science," International Encyclopedia of Unified Science, Volume 1, Number 4. Chicago: The University of Chicago Press, 1947. An introduction to the relevance of an analysis of language to the logic of scientific methodology. Distinguish philosophic attitude of the author from what he is trying to say. It may be that the same things can be truly said from a distinct and different philosophic position.
M. Bochenski, O. P., "Logistique et logique classique," Bulletin Thomiste 10 (1934) 240-248. An introduction to a master in the field, trained in the Polish (Warsaw) school, but perfectly conversant with the Anglo-American tradition. Impartial inquirers who desire to know what a man who knows both modern logic and Scholastic tradition thinks of their correlation, should begin here. Note that Bochenski later alters his opinion that the calculus of propositions should follow the theory of the syllogism.
I. M. Bochenski, O. P., "Duae consequentiae Stephani de Monte," Angelicum 12 (1935) 397-399. Here the author contributes two more instances of later medieval consequentiae to the growing collection. Examination of his historical method will instruct others how to proceed in this field of research.
I. M. Bochenski, O. P., "Notiones historiae logicae formalis," Angelicum 13 (1936) 109-123. No teacher of logic should fail to consult this excellent piece. The information is accurate, the interpretation and evaluation is correct. Indispensable for orientation.
I. M. Bochenski, O. P., "Notes historiques sur les propositions modales," Revue des sciences philosophiques et theologiques 26 (1937) 673-692. Accurate information and evaluation. Includes Albertus Magnus and Thomas Aquinas. Contains the high water marks of a full historical conspectus.
I. M. Bochenski O. P., Elementa Logicae Graecae. Roma: Anonima Libraria Cattolica Italiana, 1937. An invaluable brochure. Contains systematically arranged and critically edited Greek texts of Aristotle, Theophrastus, and the Stoics, accompanied by a Latin translation after the medieval mode of speaking. The index of terms contains within itself a glossary of Stoic logic terms, available nowhere else, not even in Liddell and Scott's monumental lexicon.
I. M. Bochenski, O. P., Nove Lezioni di Logica Simbolica. Roma: Angelicum, 1938. An excellent introductory text, designed for use at the Angelicum House of Studies in Rome. Particularly sensitive to routine antipathies of conventional realist logicians and attempts to dispel such anticipated fears. Employs the symbolism of Lukasiewicz and the Polish School, not customary in America. But translation into Russellian symbolism is easy. Appendix II contains: "Analisi logica di un testo di S. Tommaso d' Aquino [Summa Theologica, I. 75. 6]," on the immortality of the soul. A most interesting bit of work. See also Bochenski's review in the Bulletin Thomiste 12 (1935) 601-603, of a similar piece of logical analysis, executed by Salamucha in Polish on Thomas' proof for the existence of God "ex motu." Compare too Bochenski, "On Analogy," The Thomist 11 (1948) 424-447. These are indeed samples of how logical techniques can be applied to serious topics of tremendous philosophical and theological import. The surface has as yet scarcely been touched.
I.M. Bochenski, O. P., "De consequentiis Scholasticorum earumque origine," Angelicum 15 (1938) 92-109. A masterful exposition to which this paper is heavily indebted. It is Bochenski's provisional judgment that the technique of the consequentiae was inspired by reflection on the recovered works of Aristotle, independently of historical connection with the previous work of the Stoics, etc.
I. M. Bochenski, O. P., "Sancti Thomae Aquinatis de modalibus opusculum et doctrina," Angelicum 17 (1940) 180-218. A critical text and an illuminating exposition which leave nothing to be desired.
I. M. Bochenski, O. P., Petri Hispani Summulae Logicales. Roma: Domus Editorialis Marietti, 1947. The best text to date of one of the best logical treatises of the Middle Ages. See review of same by Philotheus Bohner, O. F. M. in Franciscan Studies 10 (1950), pp. 74-77, for helpful comments, criticisms and some corrections.
I. M. Bochenski, O. P., La Logique de Theophraste. Fribourg: Librarie de 1' Universite, 1947. A model of historical reconstruction, analysis, and evaluation to which this study is heavily in debt. This monograph is proof enough that only one versed in modern logic can really appreciate the ancient contributions and aspirations.
I. M. Bochenski, O.P., "Non-Analytical Laws and Rules in Aristotle," Methodos 3 (1951) 70-80. An excellent survey that appeared while this essay was in press. Complements where it does not extend the contents of section on Aristotelian logic.
I. M. Bochenski, O. P., Precis de logique mathematique. Bussum, Pays-Bas: F. G. Kroonder, 1948. A first class summary and introduction. Replete with examples and illustrations that really illustrate. Rigorous enough to impart the idea of a logical "system" without overwhelming the neophyte. There are no problems to be tested or solved for practice exercise in acquiring skills. For general accuracy and complete coverage, including relations in outline, the best brief text that has come to my attention.
I. M. Bochenski, O. P., "On the Categorical Syllogism," Dominican Studies 1 (1948) 1-23. It will pay to observe what Bochenski does to and for the syllogism.
I. M. Bochenski, O. P., "L'etat et les besoins de l'histoire de la logique formelle," Proceedings of the Tenth International Congress of Philosophy (Amsterdam: North-Holland Publishing Company, 1949), Volume 1, Part 2, pp. 1062-64. Chairmen of Philosophy Departments who are able and anxious to encourage sound and basic research in their fields will find herein a chart for needed monographic work.
I. M. Bochenski, Ancient Formal Logic. Amsterdam: North Holland Publishing Company, 1951. The book is meant to supply mathematical logicians with a synthetic outline of the main aspects of ancient formal logic which are known in the present state of research. It therefore requires for intelligent comprehension some acquaintance with the symbolism and the state of mind of modern logicians. It does for the initiated what the present essay has attempted to do as a prelude to transition. If this study inspires and prepares a reader to proceed to Bochenski's present brochure, it will have more than fulfilled its function.

Bochenski's Ancient Formal Logic is part of a series of Studies in Logic and the Foundations of Mathematics, edited jointly by L. E. J. Brouwer, E. W. Beth, and A. Heyting. Other titles in the series, either in print, in press, or in preparation, are: P. Bernays, A. A. Fraenkel, and A. Borgers, Aziomatization of Set Theory (1952); P. Boehner, O.F.M., Ein Beitrag zur mittelalterlichen Suppositionstheorie in English translation; Haskell B. Curry, Outlines of a Formalist Philosophy of Mathematics, advertised for 1951: K. Durr, The Propositional Logic of Boethius, likewise for 1951; R. Feys and J. C. C. McKinsey, Modal Logics for 1952; E. A. Moody, Truth and Conseguence in Medieval Logic for 1952; A. Mostowski, Sentences Undecidable in Formalized Arithmetic: An Exposition of the Theory of Kurt Godel, ready in 1951; A. Robinson, On the Metamathematics of Algebra, likewise for 1951; J. B. Rosser and A. R. Turquette, Many-valued Logics; B. Sobocinsky, Mereology; G. H. von Wright, An Essay in Modal Logic, announced for 1951. The entire series is well planned, the authors are expert in their fields, and the set Promises to be most valuable.
Philotheus Bohner, O. F. M., "The Centiloquium attributed to Ockham," Franciscan Studies 1 (1941) 58-72; and in the same volume but with discontinuous fascicule pagination: 35-54; 62-70; 2 (1942) 49-60; 146-157; 251-301.  
Philotheus Bohner, O. F. M., "Ockham's Tractatus de praedestinatione et de praescientia Dei et de futuris contingentibus and its Main Problems," Proceedings of the American Catholic Philosophical Association 16 (1941) 171-192.  
Philotheus Bohner, O. F. M., "El sistema de Logica Escolastica: estudio historico y critico," Revista de la Universidad Nacional de Cordoba 31 (1944) 1-24.  
Philotheus Bohner, O. F. M., "The Medieval Crisis of Logic and the Author of the Centiloquium attributed to Ockham," Franciscan Studies 4 (1944) 151-170.  
Philotheus Bohner, O. F. M., "Ockham's Theory of Truth," Franciscan Studies 5 (1945) 138-161.  
Philotheus Bohner, O. F. M., The Tractatus de praedestinatione et de praeseientia Dei et de futuris contingentibus of William of Ockham. Saint Bonaventure, New York: The Franciscan Institute, 1945. Contains an interesting and informative study, constructed along polemical lines (against Michalski), on the medieval problem of a three-valued logic as a tentative solution to theological problems. In this connection see review of the book by Julius Weinberg in The Philosophical Review 56 (1947) 446-449.
Philotheus Bohner, O. F. M., "Ockham's Theory of Signification," Franciscan Studies 6 (1946) 143-170.  
Philotheus Bohner, O. F. M., "Ockham's Theory of Supposition and the Notion of Truth," Franciscan Studies 6 (1946) 261-292  
Philotheus Bohner, O. F. M., "Ockham's Philosophy in the Light of Recent Research," Proceedings of the Tenth International Congress of Philosophy (Amsterdam: the North-Holland Publishing Company, 1949), Volume 1, Part 2, pp. 1113 Father Bohner's work speaks for itself. While not as broad in scope as Bochenski's, it is equally as solid. Almost single-handed in America, he has rescued Ockham from a "black legend" and rehabilitated his reputation more and more toward its just dimensions. Announcement has been made that the Chicago University Press is prepared to issue his Elements and Systems of Scholastic Logic. A work to be anticipated with some excitement in the field.
Philotheus Boehner, O.F.M., Walter Burleigh's De Puritate Artis Logicae. St. Bonaventure, New York: The Franciscan Institute, 1951. Despite numerous misprints this is a valuable contribution. Nothing is more necessary for the history of logic than a competent and careful editing of the relevant texts.
Philotheus Boehner, O.F.M., "Bemerkungen zur Geschichte der De Morganschen Gesetze in der Scholastik," Archiv fur Philosophie (1951) 113-146. Patient research and precise interpretation here reveal beyond all doubt that "De Morgan's Laws" were known in Scholastic tradition from the time of Ockham to the end of the nineteenth century, and have not altogether disappeared from several elementary treatises of the twentieth century. The informed author's final remarks deserve quotation and thoughtful consideration:

If we now conclude with a final retrospective survey of our historical comments on the 'laws of De Morgan,' the compelling fact is that 'De Morgan's Laws' were actually known in Scholastic tradition from the start of the fourteenth century up to and including contemporary times. And it must be acknowledged as a considerable achievement of Scholastic logic that it not only discovered these laws but also preserved them continuously intact, even if for the period after 1800 the credit, so it would seem, belongs exclusively to the school of Jesuit logicians.

On the other hand however admission must be made that Scholastic logic managed to maintain its remarkable ascendancy only to the end of the fifteenth century. Soon thereafter ensued a steady decline. Logic became more and more a mere introductory discipline and was on that account adulterated with many extralogical considerations. Most of all, there disappeared the insight, already widely endorsed in the fourteenth and fifteenth centuries, that the doctrine of the consequentiae (which for the most part modern logicians term 'the logic of statements') is the foundation and underpinning of the entire edifice of logic. Scholasticism abandoned this intuition and left it to be rediscovered by modern logicians.

Still another consideration forces itself upon our attention. We prefer to phrase it in the form of a question: How could it happen that the so-called 'Laws of De Morgan' had to be rediscovered? We are inclined to accept easily and as sufficient the explanation that De Morgan just did not know Scholastic tradition at all, or at least not enough. But the real reason rather seems to be that since the seventeenth century and particularly since the eighteenth century modern philosophy and Scholastic philosophy have gone on completely separated ways. There has supervened upon the intellectual world of the West a policy of isolation [Isolationspolitik] with unequivocally grotesque consequences. This separatist program is more accentuated on the side of modern philosophy. For the Scholastics have always and constantly been disposed to come to terms in some way with modern philosophers. Unfortunately they have thereby sacrificed much, perhaps even the best of their tradition to modern philosophy, and certainly not only in the field of logic. The predominant role which the two principles of identity and of sufficient reason currently play, is a tragic symptom of impoverishment and disarray. Whereas the Scholastics very often foray into hostile territory and bring home bits of booty, frequently of highly questionable worth, modern philosophy does not concern itself at all about the Scholastics. No one is any longer in the mood to read the Scholastics and to think through their jargon, sometimes now become a foreign language. An instructive instance of this policy of absolute isolation is the rediscovery of the equivalences here discussed (pp. 145-146) .

George Boole, The Mathematical Analysis of Logic (1847). Oxford: Basil Blackwell, 1948. A classic of modern logic, reprinted in attractive format and available now for those who missed its original first edition. No logic library should be without it.
George Boole, Collected Logical Works. Volume 2: The Laws of Thought (1854). Chicago: The Open Court Publishing Company, 1940. Another classic in the field and indispensable to catch the mood of the man who helped to make modern logic revive from its slumbers.
Nicholas Bourbaki, "The Architecture of Mathematics," The American Mathematical Monthly 57 (1950) 221-232. An expert exposition of what is meant by a "hypothetico-deductive system." Indispensable preamble for those who want to know what it is that they do not like in modern mathematics and logic, and a good source for those who want to know more about it.
Robert S. Brumbaugh, "An Aristotelian Defense of 'Non Aristotelian' Logics," The Journal of Philosophy 48 (1951) 582-585. An irenic attempt on the basis of insights, reminiscent of Veatch, to harmonize both styles of logic. The harmony herein proclaimed is unreal, mainly because an abstractionist doctrine blocks the correct conception of the science of mathematics. Liaison is possible but not by this avenue of reconciliation.
Rudolf Carnap, "Die alte und die neue Logik," Erkenntnis 1 (1930-1931) 12-26. Carnap is a name to conjure with in the field. A prodigious worker and researcher. This article was written in the hey-day of Logical Empiricism, when it was assumed that the techniques of modern logic were the preserve and prerogative and monopoly of that school of Positivism. It is not to be presumed that these views coincide with the contemporary Carnap. Indispensable reading for one who desires to know what has really happened in the area of modern logic.
Rudolf Carnap, The Logical Syntax of Language. London: Routledge and Kegan Paul, 1949. A reissue of a translation of a standard work in the field. Somewhat complicated and technical and impregnated with Positivist assumptions. But indispensable in any survey.
Rudolf Carnap, "Foundations of Logic and Mathematics," International Encyclopedia of Unified Science, Volume 1, Number 3, Chicago: The University of Chicago Press, 1946. A contemporary exposition that will provide an excellent conspectus of the type of problems involved and the methods currently being employed in an intelligent attempt to solve them.
Roderick M. Chisholm, "Sextus Empiricus and Modern Empiricism," Philosophy of Science 8 (1941) 371-384. Especially pp. 380-384 on Stoic logic.  
Roderick M. Chisholm, "The Contrary-to-Fact Conditional," Readings in Philosophical Analysis (New York. Appleton-Century-Crofts, Inc., 1949), pp. 482-497. A most informative piece, to be read in conjunction with Goodman on the same topic and preferably later than Goodman. This is a major probleme de scandals in modern logical analysis. It may just be that philosophers who are more familiar with laws of nature may be able to contribute something significant to the solution of this problem, once they become conversant with its setting and the tools necessary for its formal solution.
Alonzo Church, "A Set of Postulates for the Foundation of Logic," Annals of Mathematics 33 (1932) 346-366; 34 (1933) 839-864. Herein a master hand applies itself to a fundamental task. It is most instructive to watch Church at work. More profitable to read this after one is acquainted with the general outlines of axiomatic theory and hypothetico-deductive systems.
Alonzo Church, "Logic: formal, symbolic, traditional," Dictionary of Philosophy ( New York: Philosophical Library, 1942), pp. 170-182. The contents of this ambitious Dictionary are most uneven. Random reference to its pages is dangerous. But this contribution is among its best. It is condensed. But not dense. A patient and attentive study will pay big dividends in comprehension. Church knows the field and knows how to depict it. A most valuable reference.
Leon Chwistek, The Limits of Science: Outline of Logic and the Methodology of the Exact Sciences. New York: Harcourt, Brace and Company, 1949. A minor classic. Indicates convincingly that there are problems and differences of opinion in modern logical schools. A good sign of health and vigor. The work is a bit diffuse and disorganized. But a topical use of its contents can be rewarding
M. R. Cohen and E. Nagel, An Introduction to Logic and Scientific Method. New York Columbia University Press, 1934 A standard undergraduate text that has just enough fundamentals of modern logic to make it overreach the conventional type of text. Certain basic ideas are given a clear expression for neophytes, and can be still useful where it is not well known.
John C. Cooley, A Primer Of Formal Logic. New York: The Macmillan Company, 1942. Everything that such a book should contain is here handled and handled well. But some have not been attracted by the order of treatment and disposition of the material. Others have noted a superfluity of cumbersome symbolism. Techniques have been successfully streamlined since this work was originally prepared. But it still remains accurate and instructive.
Irving M. Copi, "The 'Intentionality' of Formal Logic," Philosophy and Phenomenological Research 11 (1951) 366-372. A capital and crucial critique of the position of Veatch. See infra.
Louis Couturat, The Algebra of Logic. Chicago: The Open Court Publishing Company, 1914. A minor classic in translation. A period piece but necessary for every relatively complete collection on modern logic.
A.C. Crombie, "Scholastic Logic and the Experimental Method," Archives internationales d'histoire des sciences (1947) 280-285. Written by an expert in the history of science, this article may help to convince some that there were other than vain sectarian or polemical reasons for the general dissatisfaction with Aristotelian logic, characteristic of Renaissance and post-Renaissance man. But note also that modern natural science has altered its logical character also along axiomatic and hypothetico-deductive lines.
Frank C. Dillhoff, "How Is Scholastic Logic Facing Modern Logic?" University of Pittsburgh Bulletin 48 (1952), Number 10. This Abstract of a Doctoral Dissertation was supplied in preprint form by the author. It is a most interesting and informative document, calculated to fill a serious gap in the corporate consciousness of a global but only loosely coordinated contemporary Scholasticism. The author states his problem for inquiry thus (p. 1): "In view of the recent advances, particularly in the fields of symbolic or mathematical logic, semantics, and scientific method, Scholastic logic is now confronted with certain unprecedented and provocative issues. What shall be the attitude of traditional Scholastic logicians and teachers of logic toward these new developments?" An adequate answer was sought by conducting a worldwide questionnaire poll of all known Catholic institutions of higher learning. Of the 413 centers with which contact was made, 196 replied, but 12 of these stated that they do not have philosophy courses as part of their curriculum. Further analysis of replies disclosed 80 foreign institutions and 333 located in the United States. The latter group comprised 31 universities, 46 colleges for men, 110 colleges for women, and 146 major seminaries. Although not complete, the coverage is sufficiently representative to provide a sound cross-section diagnosis of current attitudes and opinions.

I summarize four of the author's conclusions: (1) there is no visible authoritarian and rigid uniformity imposed by administrators, (2) there is a considerable amount of open-mindedness toward newer developments, most of all in colleges for women conducted by Sisters, less so in male institutions, and least of all in the major seminaries, (3) there is mounting evidence for a forthcoming change in the attitude of thinkers within the field of Scholastic logic, and (4) there is a seriously felt need for a complete text for use in the task of teaching Scholastic logic.

It seems to be clear that the preparation and publication of such a qualified text, designed to effect a smooth transition, is the urgent need of the moment. And it is the lesson of the present essay that such a text, richly endowed with accurate historical details, would meet this demand excellently.

The Dissertation seems to be a sound and sober study. It is to be hoped that the author may be able to publish in adequate detail the more significant parts, if not the whole, of his dissertation.

J. Dopp, "L' objet et les methodes de la logique," Revue des Questions Scientifiques 10 (1949) 361-407. A man who knows both goes out of his way to exhibit the structure of both classical and modern logic by correlating it with the different ideals of science that have animated both developments. A capital piece of work, especially helpful for those who know about classical logic but never comprehended what it is all about.
Karl Durr, "Aussagenlogik im Mittelalter," Erkenntnis 7 (1937) 160-169, and supplementary note on p. 356. Alludes to Boethius: de syllogismo hypothetico and Abelard's use of the same. To be correlated with the more general views of Bochenski and Salamucha.
K. Durr, "Les diagrammes logiques de Leonhard Euler et de John Venn," Proceedings of the Tenth International Congress of Philosophy (Amsterdam: North-Holland Publishing Company, 1949), Volume 1, Part 2, pp. 720-721. A brief introduction to diagrammatic techniques and their sponsors in the historical development of modern logic.
Karl Durr, The Propositional Logic of Boethius. Amsterdam: North-Holland Publishing Company, 1951. Prepared in 1939, delayed by political events, and finally published in 1951 when this present essay was already written, Durr's competent and capital study is a most significant contribution to the field of historical logic. The section on Boethius herein would have been considerably improved and pointed by its use. Durr establishes effectively (1) that each of the two forms of propositional logic constructed by Boethius was influential in the early Scholastic period, (2) that the propositional logic of the commentary on Cicero's Topics had its greatest effect on the logic of St. Gallen, and (3) that the propositional logic of the De Syllogismo Hypothetico is continued in the fourth part of Peter Abelard's Dialectics. Moreover the many manuscript commentaries on Boethius' work prove that there was a lively and abiding interest in the subject. Durr again shows that a solid knowledge of the elements of modern logic is indispensable preliminary to an intelligent comprehension and sound interpretation of ancient and medieval logical productions. The book is ideal for the beginner. For Durr first patiently expounds these elementary logical insights and then puts them to use on familiar grounds.
Ralph M. Eaton, "Formal Deduction," Symbolism and Truth (Cambridge: Harvard University Press, 1925), pp. 222-265. There are still some who imagine that use of symbolic techniques is tantamount to an abdication of philosophical realism. This must indeed be a most uncomfortable feeling for those who indulge in it. It may be that a careful reading of these pages will show that abstraction to the level of: "if so and so, then so and so" by no means precludes the truth that there are things which are so and so. But it is not the business of a formal logic to assert what there is and what there is not. It is the business of logic to ascertain "validating forms of inference," independently of the fact whether or not there are ontological structures in the nature of reality. Systematic abstraction is not equal to positivistic denial.
Ralph M. Eaton, General Logic. New York: Charles Scribner's Sons, 1931. A standard text book. Presents both classical and modern logic one after the other. There are many excellent things in this text. Perhaps too many to serve as an efficient class manual. If one desires, he may use the latter sections as a layman's introduction to Principia Mathematica.
B. Einarson, "On Certain Mathematical Terms in Aristotle's Logic," The American Journal of Philology 57 (1936) 33-54; 151-172. Informative and suggestive, even if somewhat inconclusive. As Bochenski has wisely warned: " . . . maxime insistendum est. ut logici Aristotelem graece legant." See Angelicum 13 (1936) 109-123. p. 111. note 1.
Herbert Feigl and Wilfrid Sellars [Editors], Readings in Philosophical Analysis. New York: Appleton-Century-Crofts, Inc., 1949. An excellent anthology, selected cooperatively by representative authorities in the field. Everything in it is good. Some pieces are however only distantly related to the specific concerns of logic. Should be in every library. Particular items are listed separately in this bibliography, whenever occasion calls for it.
R. Feys, "La transcription logistique du raisonnement," Revue neoscolastique de philosophic 26 (1924) 299-324; 27 (1925) 61-86. A very excellent piece of work. Principally concerned with the algebra of logic and the calculus of classes. Capital reference for those who still doubt about the possibility of rapprochement between classical logic and modern logic, or about Scholastic philosophy and modern logic.
R. Feys, "La raisonnement en termes de faits dans la logique russellienne," Revue neoscolastique de philosophic 29 (1927) 393-421; 30 (1928) 154-192; 257-274. A competent essay, dedicated mainly to an exposition for the uninitiated of the objectives and character of Principia Mathematica. Less attempt at formal correlation with Scholasticism than in the preceding article. But at least some attempts are made to do so. There is also some attempt to disengage Russell's project from his philosophical suppositions. Very good work.
R. Feys, "Directions nouvelles de la logistique aux Etats-Unis," Revue neoscolastique de philosophic 40 (1937) 398-411. A report by a foreign observer of the prodigious strides made in logic by the inventive genius of young American logicians. Somewhat technical in character by necessity. But an excellent way to find out what has been done in America and to get an idea of the stature of the Americans in the field
R. Feys, "Les logiques nouvelles des modalites," Revue neoscolastique de philosophie 40 (1937) 517-553; 41 (1938) 217-252. Very technical, of course. But important. A valuable exposition and comparison of several independent lines of development. For a revised version in current perspectives see idem, "Les systemes formalises des modalites aristoteliciennes," Revue philosophique de Louvain 48 (1950) 478-509.
R. Feys, "La technique de la logique combinatoire," Revue philosophique de Louvain 44 (1946) 74-103; 237-270. Sure to discourage those who are symbolically illiterate. Very technical. But for those who grow to its level extremely important. Expounds a metalogic of high elegance.
R. Feys, "Les methodes recentes de deduction naturelle," Revue philosophique de Louvain 44 (1946) 370-400. An exposition and critique of some innovations by Gentzen. Of profound psychological and philosophical interest to those who have been led to believe that the categorical syllogism represents the uniquely natural deductive process of the human mind. For a modern and improved version of the same techniques of deduction see Willard Van Orman Quine, Methods of Logic (New York: Henry Holt and Company, 1950), where it is suggested that one read first the "Historical remarks" on pp. 166-167, and then explore the relevant pages in the text. Or see for more ample discussion Quine, "On Natural Deduction," The Journal of Symbolic Logic 15 (1950) 93-102.
Robert Feys, "Logique formalisee et philosophie," Synthese 6 (1947-1948) 283-298. Good, even if not altogether complete and the last word on the subject.
Adolf Frankel, Einleitung in die Mengenlehre. New York: Dover Publications, Inc., 1949. A recent reprint of a classic on the subject. Belongs ill every representative library.
Gottlob Frege, "On Sense and Nominatum," Readings in Philosophical Analysis (New York: Appleton-Century-Crofts, Inc., 1949), pp. 85-102. An excellent translation of a monumental article by the man without whom there would not be what modern logic there is. The problem of the solitary Venus which is called "Morning Star" and "Evening Star." A profound discussion.
Joseph Frobes, S. I., Tractatus Logicae Formalis. Romae: apud Aedes Pontificiae Universitatis Gregorianae, 1940. For a critical evaluation of this work see Heinrich Scholz' review of it and the author's rejoinder in Philosophisches Jahrbuch 54 (1941) 517-522.
Joseph Frobes, S. J., "Ist die Verwendung der logistischen Formeln in den Lehrbuchern der Logik zu empfehlen?", Travauz du IXe Congres international de Philosophie (Paris: 1937), 6: Logique et mathematiques, pp. 58-63. Interesting to note that the author contributes an enthusiastic affirmative answer to his own question.
A. M. Goichon, "Une logique moderne a l' epoque medievale: la logique d' Avicenne," Archives d' histoire doctrinale et litteraire du moyen-age 16 (1947-1948) 53-68. Those who appreciate the influence of Muslim Scholasticism upon Latin Scholasticism will desire to know the vicissitudes of logic in the Arab world. The record is almost unknown, but what is known is interesting indeed. Goichon reports on p. 56:

. . . Avicenne dit que l' impossibilite de rendre exactement l' idee exprimee par 'aucun', 'nul', l' oblige en arabe a faire porter la negation sur chaque chose denominee. 'Aucun J' se dira: 'Pas une chose parmi J'. Mais il est clair que la pensee se presente en extension chez Avicenne, comme chez les Stoiciens, car il parle de meme de l' universelle afflrmative: "Si nous disons: 'Tout J est B', nous n' entendons pas par la que l' universalite de J est B. ni que le J universel est B. mais nous voulons dire: 'Chacune des choses qualifices de J . . . est qualifiee de B"'. Classical logicians who assume that the copula is indispensable to the human mind might well consider the fact that in Arabic, for example, a copula is most infrequent; for instance: Allahu kabirun means 'Deus [est] magnus'. See Madkour below.

Nelson Goodman, "The Problem of Counterfactual Conditionals," The Journal of Philosophy 44 (1947) 113-129. A classic piece of exposition and statement of a problem. Goodman does not presume to know the answer. A fertile area for philosophic investigation for those equipped to state and solve the issue in formal terms. Compare Chisholm above on the same topic.
Martin Grabmann, Bearbeitungen und Auslegungen der aristotelischen Logik aus der Zeit von Peter Abaelard bis Petrus Hispanus. Abhandlungen der Preussischen Akademie der Wissenschaften (1937), Phil.-hist. Klasse, Nr. 5 With accurate information and with his customary competence Grabmann here helps significantly to fill some of the serious gaps in the history of medieval logic. The brochure is remarkable for its grasp of the logical issues involved and its scope of research through vast manuscript materials.

The essay is a map to keep a research program of a University Department seriously busy for years.

Martin Grabmann, Kommentare zur aristotelischen Logik aus dem 12 und 13 Jahrhundert. Sitzungsberichte der Preussischen Akademie der Wissenschaften (1938), 18. An amplification of the preceding with new vistas opened into the interesting details of the period.
Thomas Greenwood, Les fondements de la logique symbolique. Paris: Hermann et cie, 1938. Actualites scientifiques et industrielles: 588. 593  
Thomas Greenwood, "The Metaphysical Ground of Logical Operations," The New Scholasticism 15 (1942) 150-166.  
Thomas Greenwood, "The Unity of Logic," The Thomist 8 (1945) 457 470. A very interesting attempt to comprehend modern insights within the familiar apophantic framework:

'S is P' Should strike familiar and comforting chords in the minds of those who assume that it is impossible for the human mind to express itself except it be in the categorical proposition. Greenwood's grasp of modern logic is modestly competent. But it may be said that even if his metaphysical reduction is correct, its techniques are operationally more cumbersome than the conventional ones. But no one should overlook his serious attempt at reconciliation. All three references expound the same insight and program of reduction

O. Helmer, "On the Theory of Axiom-Systems," Analysis 3 In a few brief and lucid pages a master expounds an important topic. A patient and careful reading will be very instructive.
David Hilbert, The Foundations of Geometry. Chicago: The Open Court Publishing Company, 1910. A fundamental work in translation. Should be in every library. Instance of hypothetico-deductive theory in operation.
David Hilbert, "Axiomatisches Denken," Mathematische Annalen 78 (1918) 405-419. With the simplicity characteristic of greatness and the clarity which only an expert can command, Hilbert answers all the newcomer's queries. I know nothing more authoritative, nothing better.
D. Hilbert and W. Ackermann, Grundzage der theoretischen Logik. New York: Dover Publications, 1949. Reprint of a classic, long unavailable. Should be in every library.
D. Hilbert and P. Bernays, Grundlagen der Mathematik (1934). Ann Arbor, Michigan: Edwards Brothers, Inc., 1944. Offset reproduction of an indispensable classic. Quite advanced in sections but also contains a masterly exposition of fundamentals, superior to many more explicit introductory texts. Should be in every library.
Pierre Hoenen, S. J., Recherches de logique formelle. Romae: apud Aedes Universitatis Gregorianae, 1947. Seems to me that the modern logical atmosphere has led a productive mind to develop the theory of the classical syllogism along independent lines. An interesting performance.
J. Homans, "La logique algorithmique," Revue neoscolastique de philosophie 9 (1902) 344-364. To my knowledge one of the first Scholastic notices to follow upon the appearance of Schroder's expansion of Boole's algebra. Accepting the narrow range of its treatment, it is indeed very good, and conciliatory.
J. Hontheim, S.J., Der logische Algorithmus in seinem Wesen, in seiner Anwendung , und in seiner philosophischen Bedeutung. Berlin: Verlag von Felix L. Dames, 1895. A museum piece of rare value. The first Scholastic exposition and critique of the class algebra, as developed by Schroder after Boole. There was no lag in Hontheim's alertness to the new phaenomenon. His critique followed immediately the completion of Schroder's publication. Erant gigantes in diebus illis.
Edward V. Huntington, "The Postulational Method in Mathematics." American Mathematical Monthly 41 (1934) 84-92. A master in the exercise of the method herein expounds in clear and simple language the essentials of its structure. A capital reference.
E. V. Huntington, "The Method of Postulates," Philosophy of Science 4 (1937) 482-495. Here Huntington answers all the questions a newcomer would like to put to an expert. Both this and the preceding item pair nicely.
Martha Hurst, "Implication in the 4th Century B. C.," Mind 44 (1935) 484-495. Accurate in data but inaccurate on some points of interpretation.
[John of Saint Thomas] John J. Glanville, G. Donald Hollenhorst and Yves R. Simon, "John of St. Thomas: Entia Rationis and Second Intentions," The New Scholasticism 23 (1949) 395-413. A translation of relevant sections of John of St. Thomas' Cursus Philosophicus Thomisticus. A competent job. The material is of some importance and has figured largely in the arguments of Veatch for a mild reactionary position against modern logic.
Z. Jordan, "The Development of Mathematical Logic and of Logical Positivism in Poland between Two Wars," Polish Science and Learning (1945) Number 6. New York: Oxford University Press. An invaluable brochure. Almost everything one would want to know, and there is very much to know about the Warsaw School, is here contained accurately and briefly. An indispensable introduction to the recent past for one who decides to get into the field at the current moment. It is remarkable how well Jordan depicts the crucial points involved at the decisive junctures of development.
Jorgen Jorgenson, "Einige Hauptpunkte der Entwicklung der formalen Logik seit Boole," Erkenntnis 5 (1935) 131-142. The content here carries out the promise of its title. A guidebook to the main lines of development that will tell its reader how modern logic progressed to its near contemporary position.
Ernst Kapp, Greek Foundations of Traditional logic. New York: Columbia University Press, 1942. A very interesting, if incomplete, book. Attention is here called to one documented thesis of the author:

. . . we shall certainly not get rid of what we have to get rid of so long as we ourselves remain unconsciously dependent upon a misunderstood and misinterpreted Aristotelian logic. Thus we have to get acquainted with the historical fact that according to Aristotle's concept of a syllogism, the syllogism itself and the preceding mental activity run in opposite directions. See page 74. There is more of the same and it is all interesting and provocative.

Alexander Koyre, "'The Liar'," Philosophy and Phenomenological Research 6 (1946) 344-362. Herein a philosopher of high competence in general, but not over-trained in modern logic, handles rather cavalierly a genuine logical problem of major proportions. Those who think that just general ability is enough to dispel similar sophisms without recourse to metalanguages should see ibidem 8 (1947) 245-253 for a vigorous and rigorous reply.
L. Lachance, O. P., "Saint Thomas dans l' histoire de la logique," Etudes d' histoire litteraire et doctrinale du XIlIe siecle (Paris: J. Vrin, 1932). Volume 1, pp. 61-103. Kapp, opere citato, p. 30, remarks that: "Traditional logic has its own traditional view of its own history." Lachance appears to adopt that view, and to know no other. But Lachance does know his Thomas texts very well and selects them expertly. He puts Thomas in a-well defined niche in his sketch of the "traditional" history of logic. But it is highly questionable whether the "traditional" history of logic can stand in the face of modern evidence.
Phillip de Lacy, "The Stoic Categories as Methodological Principles," Transactions and Proceedings of the American Philological Association 76 (1945) 246-263. Will help a newcomer to orient himself concerning the background to Stoic logic. And without a clear comprehension of Stoic logic one cannot hope to acquire an accurate history of logic.
Jean Ladriere, "Le role du theoreme de Godel dans le developpement de la theorie de la demonstration," Revue philosophique de Louvain 47 (1949) 469-492. Kurt Godel is a name to conjure with in the field. The problem he faces is delicate, the method of working towards its solution subtle and complex, the solution abstruse in form but providing common sense with a sudden and startling shock. One may find the theorem in Kurt Godel, "Uber formal unentscheidbare Satze der Principia Mathematica und verwandter Systeme I," Monatshefte fur Mathematik and Physik 38 (1931) 173-198. The theorem turns upon the essential incompletability of logic and mathematics, in particular arithmetic and geometry. This is the report of Tarski, Introduction to Logic and the Methodology of the Deductive Sciences ( New York: Oxford University Press, 1946), p. 137:

. . . it turns out that arithmetic and advanced geometry are incomplete; for it has been possible to set up problems of a purely arithmetical or geometrical character that can be neither positively nor negatively decided within these disciplines. It might be supposed that this fact is merely an outcome of the imperfection of the axiom systems and methods of proof at our disposal up to date, and that a suitable modification (for instance, an extension of the axiom system) may, in the future, yield complete systems. Deeper investigations, however, have shown this conjecture to be erroneous, never will it be possible to build up a consistent and complete deductive theory containing as its theorems all the sentences of arithmetic or of advanced geometry....

This is perhaps just enough introduction to render Ladriere intelligible and instructive. See too the invaluable article of W. V. Quine, "On Decidability and Completeness," Synthese 7 (1948-49) 441-446. A masterful piece of integration.

Susanne K. Langer, An Introduction to Symbolic Logic. Boston: Houghton, Mifflin Co., 1937. Another introductory text, with some individual excellences. But now a bit out of date.
H. D. P. Lee, "Geometrical Method in Aristotle's Account of First Principles," Classical Quarterly 29 (1935) 113-124. While perhaps more dependent upon the orientations of Heath than is either necessary or good, Lee succeeds quite well in marshalling the texts and correlating them in detail. I quote his conclusions (p. 117): "Euclid's Common Notions and Aristotle's Axioms, and the Definitions of both, are exactly parallel. The common notions and axioms are principles of reasoning whose scope extends further than that of a single science: the definitions are statements of the meaning of terms. To Aristotle's hypotheses answer Euclid's postulates. Both are a minimum of further assumptions necessary besides the axioms or common notions and the definitions. The hypotheses assume existence, the postulates the possibility of constructions. And these assumptions are in effect the same at any rate in the case of the first three postulates; and there was nothing answering to the last two postulates till Euclid. The term postulate was adopted by him in preference to Hypothesis because of the associations which would naturally cling to the latter and which he would wish to avoid."
Clarence I. Lewis, A Survey of Symbolic Logic. Berkeley: University of California Press, 1918. An American classic. Almost unobtainable. The Boole-Schroder algebra is expounded and developed. There are some interesting appendices concerning Leibniz and symbolic logic.
Clarence I. Lewis, "Notes on the Logic of Intension," Structure, Method, and Meaning: Essays in Honor of Henry M. Sheffer (New York: Liberal Arts Press, 1951), pp. 25-34. A remarkable essay that opens up avenues of fruitful investigation and may contain the formula for reconciling two antagonistic logical temperaments. It is, for example, remarked:

The difference between an extensional and an intensional system is not one with respect to the analytic character of the asserted functions; no valid calculus of logic includes any nonanalytic assertion. The difference is that, in an extensional system, the analytic character of postulates and theorems is not asserted, and not distinguished from the character of nonanalytic truth, because "A is analytic" cannot be symbolized or defined in terms of the logical constants of the system without addition. In an intensional system, the modalities of function"analytic," "nonanalytic"; "consistent," "inconsistent"; "deducible," "nondeducible"; and so on are symbolized and distinguished from nonmodal truth. As a consequence, in any application of an extensional calculus, the peculiarly logical questions of analyticity, consistency, and deducibility have to be discussed in terms of some adjoined metalogic or metalanguage; whereas the laws governing such properties are, for an intensional calculus, statable within the system itself (pp. 25-26). These significant points are well taken. Nor are they all. For the author continues:

. . . Logic can assert no expression to be true unless it is true by virtue of its intensional meaning, and it can assert no expression to be false except one which is contravened by virtue of its intension. That there is such a thing as extensional logic is due to the fact that certain truth-functions are analytic by reason of the intensional meaning of the logical constants which figure in them and of their syntax (p. 28).

And although currently "there exists no calculus symbolizing both extensional and intensional functions of terms" (p. 28), nor "even any calculus which is recognized as an intentional logic of terms" (p. 28), yet the Boolean algebra exists and is "interpretable as a logic of terms in intension" (p. 29). But not without difficulty. Hence a "convenient basis for an intentional logic of terms can be developed much more simply . . . An outline of this system . . . the 'calculus of predicates' . . . follows" (p. 30). And after depicting the anatomy of this system, Lewis significantly remarks that "the whole logic of propositions could be regarded as a special case of this calculus of predicates [as] . . . propositional terms signifying states of affairs. Thus the calculus of predicates could be taken as the basic branch of logic in general" (p. 34). This is a lucid and profound essay, bristling with constructive suggestions for those who desire the devices of modern logistical analysis but are constitutionally opposed to the orientations of an exclusively extensional development. See my review of this Festschrift volume in The Modern Schoolman (1951), pp. {;3-74

Clarence Irving Lewis and Cooper Harold Langford, Symbolic Logic. New York: The Century Company, 1932. Reprinted New York: Dover Publications, Inc., 1951. Lewis' theory of "strict implication" is a sine qua non in the complete inventory of the ideas of modern logic. In general it is a remarkable attempt to link and adapt logical implication to more natural notions of deducibilityso far, at least, as this is possible. Those who find the notion of material implication unpalatable may be attracted to this line of development. Before adventuring on their own lines, they should acquaint themselves in detail with the origins and the fate of Lewis' system of "strict implication." Langford has many good things to say in a separate second section of the book on "formal structure" in general. Compare W. V. Quine, "The Problem of Interpreting Modal Logic," The Journal of Symbolic Logic 12 (1947) 43-48.
Jan Lukasiewicz, Aristotle's Syllogistic from the Standpoint of Modern Formal Logic. Oxford: Clarendon Press, 1951. With a combination of philological competence and logical acumen unique in the long annals of Aristotelian criticism, Lukasiewicz here presents to English readers the first trustworthy account of Aristotle's theory of assertoric syllogisms (Prior Analytics 1. 1, 2, 4-7). Technical finesse commends the book to the contemporary expert. But simplicity and clarity of exposition render it suitable alike for the intelligent layman in logic. The volume abounds with significant disclosures, some central, others incidental, some new, others less recent, but all of them important and most of them established beyond further question. The Aristotle section of this essay would have benefitted immensely had the material been available at the time of writing. This is indeed an invaluable book. There is no other to be compared with it. It may at long last ''persuade living philosophers that they should cease to write about logic or its history before having acquired a solid knowledge of what is called 'mathematical logic.' It would otherwise be a waste of time for them as well as for their readers" (p. 47). See my review of this book in a forthcoming issue of The Philosophical Review (1952).
J. Lukasiewicz, "Philosophische Bemerkungen zu mehrwertigen Systemen des Aussagenkalkuls," Comptes rendus des seances de la society des sciences et des lettres de Varsovie 23 (1930) Third Section, pp. 51-77.  
Jan Lukasiewicz, "Zur Geschichte der Aussagenlogik," Erkenntnis 5 (1935) 111-131. Lukasiewicz is a master in the field. Recovered the true sense of the logic of the Stoics, which had been hopelessly entombed in the confusions of Prantl. Knew his Aristotle and was sensitive to his provocative passages where vistas are opened and not explored. His style is simple and clear nine accurate. The founder of modern historical logic.
Ibrahim Madkour, L' Organon d' Aristote dans le monde arabe. Paris: J. Vrin, 1934. A study principally grounded on the works of Avicenna [Ibn Sina] in the original Arabic. The topic is of extreme importance for a knowledge of later Latin Scholasticism. It is unfortunate that the author's competence in logic does not equal his mastery of Arabic. Would seem that this work will have to be repeated and revised. The information here included is significant rather than definitive.
Benson Mates, "Diodorean Implication," The Philosophical Review 58 (1949) 234-242. Alternative to Philo.  
Benson Mates, "Stoic Logic and the Text of Sextus Empiricus," American Journal of Philology 70 (1949) 290-298. With an impressive mastery of the textual materials and a deft logical analysis of their contents, Mates here shows by definitive examples how a modern logical competence allows one to cut like a razor edge through the Gordian knots of classic philological puzzles in the manuscript tradition of Sextus Empiricus. Mates is brilliantly clear on the Stoic conditional and distinguishes it neatly from the Stoic argument with which it has often been confused. An excellent study.
Karl Menger, "The New Logic," Philosophy of Science 4 (1937) 299-336. Good indeed.
James W. Miller, The Structure of Aristotelian Logic. London: Kegan Paul, Trench, Trench, Co., Ltd., 1938. A critical examination by a man well versed in the techniques and discoveries of modern logic. Indispensable for all who care to see how and why Aristotle measures up to modern analysis:

". . . Traditional logic and modern logic are in perfect agreement with each other. The apparent disagreements between them reflect merely a difference of vocabulary. Either 'class' or No s is p, is given a different meaning in one system from that which it has in the other. But this difference is one of convention only. All of the principles of traditional logic can be expressed in the language of modern logic; when so expressed, they are principles of modern logic.

Modern logic is of course a much more extensive system than traditional logic. Nothing in traditional logic corresponds to the modern logic of relations, nor to the modern logic of unanalyzed propositions. Traditional logic is concerned simply with the logic of classes. But it does not even treat that branch of logic in its full generality. On the one hand it confines itself to a restricted conception of 'class' (or No s is p). On the other hand it does not deal with logical sums and products of classes.

Traditional logic thus coincides with a part of modern logicbut with a part only. It is genuinely a part of logicbut only a part. It is a special case." (p. 95)

E. A. Moody, The Logic of William of Ockham. New York: 1935.  
Augustus de Morgan, Formal Logic. London: The Open Court Company, 1926. A reprint of a classic, first published in 1847, revival date of logic.
Joseph P. Mullalley, The Summulae Logicales of Peter of Spain. Notre Dame: The University of Notre Dame, 1945. Publications in Medieval Studies: 8. Long introductory essay that survives its own deficiencies. Will orient well but should be read critically. See Bochenski's estimate in his own complete edition of Peter of Spain's Summulae Logicales.
Jean Nicod, Foundations of Geomety and Induction. London. Routledge and Kegan Paul, Ltd., 1950. Reprint of a classic of minor dimensions. But important. Should be in any library of the history of logic.
Caspar Nink, S. J., "Die mathematisch-logistische Symbolsprache in philosophischer Sicht," Scholastik 15 (1940) 57-62. Those who still feel queasy and uneasy about symbolic techniques and a loyalty to Scholastic realism may find their fears allayed by Nink's use of several principles of Scholastic psychology in their favor. Brief but good.
[Charles Saunders Pierce] Charles Hartshorne and Paul Weiss [Editors], Collected Papers of Charles Saunders Pierce. Cambridge: Harvard University Press, 1933. Pierce was one of early American modern logicians. His papers are replete with profound, if systematically unorganized, insights. See especially Volume 3. Indispensable if one is to understand what happened in the development of American thought about logic.
Ch. Perelman and L. Albrechts-Tyteca, "Logique et rhetorique," Revue philosophique 75 (1950) 1-35. Aristotle attempted an effective adaptation of his logic to the more humanistic needs of rhetoric. The above article suggests similar lines in the contemporary context of modern formal logic and human needs of persuasion and influence. Humanists who are sad over the fact that modern logic is so formal and human needs so real, may find these gropings suggestive.
Richard H. Popkin, "An Examination of Two Inconsistencies in Aristotelian Logic," The Philosophical Review 56 (1947) 670-681. A most interesting study. The alleged inconsistencies are exposed, analysed, and criticised. There are two conclusions: (1) if certain assumptions are made, and Aristotelians are free to make them, then the inconsistencies are avoided, but (2) at the cost of restricting the range of the logic to the point of insignificance in contemporary contexts. The author concludes on p. 680: ''Thus, although these two inconsistencies do not invalidate classical logic, we must restrict its field in order to solve them. This restriction makes traditional logic inadequate for present day needs." The entire study deserves close attention by contemporary Aristotelians.
P. S. Popov, "The Logic of Aristotle and Formal Logic," Philosophy and Phenomenological Research 8 (1947) 1-22. Pure formalism tries to consider the elements of thought separately from the object. that is, without taking into account the fact that thought with its forms is primarily the reflection of the surrounding material world. Instead, the formal elements come to be treated purely subjectively without [being brought into] relation with external nature, the material world, whereas a genuine logic should be an objective logic. Actually, the subjective or what appertains to the subject (as thought necessarily does), in order to manifest its cognitive functions must be in uninterrupted relation with the object of knowledge, the external world. Only then will judgments and arguments (proofs, demonstrations, etc.) have cognitive meaning and perform a cognitive function and express the real connections of nature, the material world. "In customary logic thought is formalistically detached from objectivity," says Lenin [Philosophical Notebooks (Moscow: 1938), p. 177], and further on, ''The laws of logic are the reflection of the objective in man's subjective knowledge."

The above quotation (pp. 1-2) from Popov suggests the unhappy thought that too much realistic ardor, animating an opposition to the formal symbolic methods of modern logic, may be construed as giving comfort and support to the "party line" of Soviet logic. Stalin and the Stagirite make strange bed-fellows. And so, too, would their disciples.

E. L. Post, "Introduction to a General Theory of Elementary Propositions," American Journal of Mathematics 43 (1921) 163-185. Indispensable material for anyone who takes the calculus of propositions as seriously interesting and logically important. In its field, a classic. See also W. V. Quine, "Completeness of the Propositional Calculus," The Journal of Symbolic Logic 3 (1938) 37-40.
Carl von Prantl, Geschichte der Logik im Abendlande. Leipzig: S. Hirzel, 1855-1870. The erudition is vast, the lore is immense, but the interpretation is erratic and therefore generally unreliable. But this monumental work still remains useful as a documentary source.

When an acceptably definitive history of logic has been written, I suspect that its total structure will somehow disclose two major epochs: (1) the ancient and medieval Aristotelian system, and (2) the modern mathematical or symbolic systems en bloc.

I further suspect that the first era will somehow describe a system of logic that (a) began with a study of the conventional propositions and deductions of scientific discourse, (b) probed behind the written or spoken proposition to the psychological origins of the judgment which the sentence or statement purported to exhibit, (c) and there became involved (i) in an epistemological critique of the objective validity of its constituent terms, (ii) in a metaphysical investigation of the ontological nature of the objects to which the terms referred, and (iii) in a psychological inquiry into the machinery of manufacture of its terms from the raw experience of natural objects. To the apparent satisfaction of the researchers such investigations delivered a homogeneous Weltanschauung in which a metaphysics of nature complemented a psychology of mind and both together determined the character of its epistemology and the syllogistic structure of its logic. Thereafter ancient and medieval Aristotelian logic rotated serenely around its own axis in a universe that was established and stabilized by simultaneous reference to the triple coordinates of a complementary epistemology, psychology, and metaphysics. Such a logic did not, and could not, exist or function independently of the props of its general philosophical context. Such a logic could not but be affected by contact with its supporting philosophical sciences. Such a logic was not, nor could it be, pure or formal logic.

I further suspect that the second era will begin when the old epistemology was abandoned in favor of a scientific empiricism, the old metaphysics discarded in the antinomies of a phenomenological criticism, and the old psychology dissolved in the failure of experimental techniques to disclose a psyche. For in the new philosophical universe the old logic was sure to become unacceptable and embarrassing. It would seem from the record that the search for a new logic, which was in fact the search for a pure logic, has followed a descending spiral path through three successive stages of de-emphasis: (1) a de-emphasis on the logical relevance of the psychological processes of judgment and a converse emphasis on the proposition, as containing for public analysis the pith of the judgment; (2) a de-emphasis on the logical import of the intelligible content of the proposition and an increased focus of attention on the structure of the sentence that purported to express it; (3) a growing de-emphasis on the logical significance of the sentence as vehicle for the proposition and a corresponding increase of interest in the skeletal structure of statements and their anatomical -- not organic  -- interrelationships. At every step one discerns an instinctive urge to liberate logic from epistemological, metaphysical, or psychological entanglements of the prior stage. Beneath the vicissitudes of modern logic lies the persistent purpose to render the science pure and neat, innocent and formal, neutral and symbolic, unencumbered with irrelevant commitments.

W. V. Quine, "Ontological Remarks on the Propositional Calculus," Mind 43 (1934) 472-476. I quote pp. 473-474 and 476:

. . . the signs 'p', 'q', etc. are customarily construed as proposition variables, i.e. as signs ambiguously denotative of propositions, i. e. as signs ambiguously denotative of the things which sentences denote. We now cancel this circuit through denoted entities, and explain the signs: 'p', 'q', etc., directly as ambiguously abbreviated sentences -- which comes to the same thing as before except that the existence of denoted entities, propositions, is no longer presupposed . . . [And further] we can reconstrue the theory of deduction as a branch of semantics, whose elements are shapes, signs, specifically sentences. The signs: 'p', 'q', etc., thus become sentence variables; neither signs ambiguously denotative of propositions, nor signs ambiguously abbreviative of sentences, but signs ambiguously denotative of sentences....

The flight from a farrago of logically irrelevant commitments is clear. In similar vein see idem, "On Universals," The Journal of Symbolic Logic 12 (1947) 74-84.

W. V. Quine, "Notes on Existence and Necessity," The Journal of Philosophy 4 (1943) 113-127. Sharp, cautious and wise guidance for those interested in the development of modal logics.
W. V. Quine, "Designation and Existence," Readings in Philosophical Analysis (New York: Appleton-Century-Crofts, Inc., 1949), pp. 44-51. A most important piece of keen analysis that dissolves a semantic confusion which has plagued philosophers throughout history. I quote from p. 49 and p. 50:

. . . Perhaps we can reach no absolute decision as to which words have designata and which have none, but at least we can say whether or not a given pattern of linguistic behavior construes a word W as having a designatum. This is decided by judging whether existential generalization with respect to W is accepted as a valid form of inference. A name -- not in the sense of a mere noun, but in the semantic sense of an expression designating something -- becomes describable as an expression with respect to which existential generalization is valid. . . . The universe of entities is the range of values of variables. To be is to be the value of a variable.

Willard Van Orman Quine, Elementary Logic. New York: Ginn and Company, 1941. Now surpassed by Quine's latest text, cited below. But if one has trouble in grasping the notion of a "schema," this book will successfully impart the idea.
W. V. Quine, "On the Logic of Quantification," Journal of Symbolic Logic 10 (1945) 1-12. Excellent, authentic, accurate.
Willard Van Orman Quine, Mathematical Logic. Cambridge: Harvard University Press, 1947. A master work. From the basic notions throughout the development and up to an original reworking of Godel's theorem. Elegant in systematic treatment. A classic in the field. Not for beginners, of course. But the beginning of the book will deepen a beginner's knowledge of the fundamentals, imparted in more elementary fashion elsewhere. Important revised edition, 1951.
Willard Van Orman Quine, "On What There Is," The Review of Metaphysics 1 (1948) 21-38. A very significant expression of opinion that clears the atmosphere. To be read after and in connection with Quine's "Designation and Existence," as above. I quote Quine's summary in part, p. 38:

. . . In earlier pages I undertook to show that some common arguments in favor of certain ontologies are fallacious. Further, I advanced an explicit standard whereby to decide what the ontological commitments of a theory are. But the question what ontology actually to adopt still stands open, and the obvious counsel is tolerance and an experimental spirit....

W. V. Quine, "Ontology and Ideology," Philosophical Studies 2 (1951) 11-15. A very precise study of fine points involved in a theory of reference and a theory of meaning.
Willard Van Orman Quine, Methods of Logic. New York: Henry Holt and Company, 1950. The best available text in English. What it sets out to do, it does with elegance and perfection. Bochenski's Precis and Quine's Methods make an excellent tandem. If one had to choose two books and only two books, these are my choice. Quine supplies exercises. Bochenski supplies relations. And they enrich each other throughout the remainder of their texts.
Frank Plumpton Ramsey, The Foundations of Mathematics and other Logical Essays. London: Routledge, and Kegan Paul, Ltd., 1950. A minor classic, recently reprinted. Should be in every library. Essential to understand what happened after Principia Mathematica appeared.
Hans Reichenbach, Elements of Symbolic Logic. New York: The Macmillan Company, 1947. All the usual material is here. And much else besides. In this sense it is more than an ordinary text. Newer issues that Quine and Bochenski do not treat explicitly are included. Reichenbach lapses into symbolic formulation on the slightest provocation. It helps the problem. But sometimes it hinders the beginner. But a sound introduction to the periphery problems of strictly contemporary modern logic.
Hans Reichenbach, "The Syllogism Revised," Philosophy of Science 19 (1952) 1-16. Conventional logic purports to be a logic of 'notions' and hence is inclined to minimize modern logic as a logic of 'notations.' Many therefore find it difficult to grasp just precisely what is the fruitful role of symbolization for the improvement of thinking. Reichenbach's present paper excellently illustrates how simple notational advances can improve notional comprehension. Persons familiar with the conventional syllogism should benefit much from a careful perusal of this study. From this point of view it is highly recommended.
Paul Rosenbloom, The Elements of Mathematical Logic. New York: Dover Publications, Inc., 1951. A comprehensive and critical survey of classic and contemporary systematic formulations. Invaluable.
W. D. Ross, "The Discovery of the Syllogism," The Philosophic cal Review 48 (1939) 251-272. Interesting discussion to which I am indebted in this paper. See Solmsen's comments, as noted below. Note the points of agreement. They are significant.
Josiah Royce, "Axioms," Hasting's Encyclopedia of Religion and Ethics. (New York: Charles Scribner's Sons, 1928), Volume 2, pp. 279-282. A patient and persistent inquiry wherein the author unravels confusions and disengages definitively several distinct meanings of this ambiguous and troublesome expression.
Louis Rougier, "La Scolastique et la Logique," Erkenntnis 5 (1935) 100-109. One need not agree altogether with the author but one cannot fail to learn much from his comparisons and contrasts.
Louis Rougier, "La relativite de la logique," Erkenntnis 8 (1939-1940) 193-217. Same caliber as above and same comments. Brilliant and incisive in parts.
Bertrand Russell, Introduction to Mathematical Philosophy. New York: The Macmillan Company, 1930. A minor classic and layman's handbook to the ideas of Principia Mathematica. Russell's style is interesting to follow. His metaphors and turns of phrase are calculated to impart clarity to subtle problems.
Bertrand Russell, "On Denoting," Readings in Philosophical Analysis (New York: Appleton-Century-Crofts, Inc., 1949), pp. 103-115. Anyone who has ever been uneasy about the conventional practice of regarding singular propositions in syllogistic theory as if they were universals," will be interested in the way in which Russell transforms; singular terms out of the furniture of logic. Pure logic need never concern itself with singular terms. A tremendous systematic advance.
Bertrand Russell, The Principles of Mathematics. New York: W. W. Norton and Co., 1938. A classic. Should be in every library. The matrix whence came the Principia Mathematica. The discussion ranges far and wide. But most of the basic notions of symbolic logic are included and clarified.
Paul Arthur Schlipp [Editor], "The Philosophy of Bertrand Russell," The Library of Living Philosophers 5 (Evanston, Illinois: The Library of Living Philosophers, 1946). A collection of essays: expository, critical, evaluative. Indispensable if one is to know how others take Russell and take to him. Contributes perspective to the reader.
Joseph Salamucha, "Die Aussagenlogik bei Wilhelm Ockham," Franziskanische Studien 32 (1950) 97-134. Excellent German translation of a monumental article, long concealed in the original Polish. The translator transforms Salamucha's symbolism into a form more usual in Western Europe and the Anglo-American school. Its appearance in a world-language is as much an event as its original publication. The latter sections of this study lean heavily upon its contents.
Heinrich Scholz, "Die Axiomatik der Alten," Blatter f Or devutsche Philosophic 4 (1930) 259-278. A model piece of textual analysis and philological interpretation of Aristotle's theory of deductive science, embedded in a rich context of modern logical insights and precisions. An exemplary essay where critical acumen does not dull the edge of an intelligent sympathy for the real stature of Aristotle's genius.
Heinrich Scholz, "Die mathematische Logik und Metaphysik," Philosophisches Jahrbuch 51 (1938) 1-35. An excellent piece. Information and insight.
Heinrich Scholz, "Die klassische und die moderne Logik," Blatter fur deutsche Philosophie 10 (1937) 254-281. With an immense patience and the simplicity that only experts can command Scholz here introduces the layman philosopher to the basic intricacies of modern logic. The symbolism employed is that of the Warsaw School, but it is clearly explained. Scholz notes that Leibniz, Bolzano, Frege, and Hilbert were all professional mathematicians and then remarks significantly (pp. 256-257): "This intimate association with mathematics has had a most deplorable result. It has frightened the philosophers away from contact with modern logic and has given them the impression that this new mathematical logic is a specialized interest of mathematicians only, and has no relevance at all to the pursuits of qualified philosophers. This impression is false."
Heinrich Scholz, Gesohichte der Logik. Berlin: Junker and Dunnhaupt, 1931. The only reliable history of logic in the field. Incomplete. But the right orientation. On this further monographs can build constructively. It should be in every library, if available. If enough requests are made, it may be reprinted.
E. Schroder, Vorlesungen uber die Algebra den Logik. Leipzig: Teubner, 1890-1895. A classic. Develops and improves systematically the logic of Boole. Necessary item in every library of logic. Modern logic has gone a long way since Schroder. But he marks a milestone in the Dassaze.
H. M. Sheffer, "A Set of Five Independent Postulates for Boolean Algebras," Transactions of the American Mathematical Society 14 (1913) 481-488. An illustration of the method of construction of a hypothetico-deductive theory. Analysis of this contribution assists one in grasping the relevance of abstract explanations of the method. The hand that alzanges this set of postulates is sharp and sure.
Yves R. Simon and Karl Menger, "Atistotelian Demonstration and Postulational Method," The Modern Schoolman 25 (1948) 183-192. A most important paper for people on both sides.
Edward O. Sisson, "The Copula in Aristotle and Afterwards~^ The Philosophical Review 48 (1939) 57-64. One need not agree with the author completely in order to gain a new perspective from his contentions and to reexamine a fundamental question.
Friedrich Solmsen, "Boethius and the history of the Organon," American Journal of Philology 65 (1944) 69-74 From a rich background of philological lore Solmsen here estimates the significance of Boethius' testimony for the problem of establishing when, wbereR how, and by whom Aristotle's Organon came into standardized existence. The systematic and internal reasons here disclosed for the conventional sequence of the works are interesting and instructive. For we are told that the term precedes the sentence and the sentence precedesthe syllogism. Solmsen concludes (p. 73): 'If we are cautious we shall say that the edition on which Boethius and the Byzantinesand, except for the inclusion of the Eisagoge, all modern editors and almost all writers on Aristotledepend came into existence between the end of the third and the end of the fifth century."
Friedrich Solmsen, "The Discovery of the Syllogism," The Philosophical Review 50 (1941) 410-421. Compare Ross above. Their Points of agreement are most important.
H. B. Smith, "On the Relation of Aristotelian Logic to that of Boole-Schroder," The Monist 42 (1932) 275-282. Some find Smith very dense and obscure. Otherwise an excellent and stimulating, if challenging, reference.
H. B. Smith, "The Algebra of Propositions," Philosophy of Science 3 (1936) 551-578. Do not read this unless you feel that you can follow a symbolic discussion with some ease. Otherwise by all means do.
L. S. Stebbing, A Modern Introduction to Logic. London: Methuen and Co., Ltd., 1933. An excellent text from many points of view. The material is derivative. But the exposition is clear. Some students "see" a point in Stebbing that escapes them in other texts. Worth an examination.
L. Susan Stebbing, A Modern Elementary Logic. London: Methuen and Company, Ltd., 1949. A junior edition, more or less, of the above. But good so far as it goes. Same excellences as above.
Alfred Tarski, "The Semantic Conception of Truth," Readings in Philosophical Analysis ( New York: Appleton-CenturyCrofts, Inc., 1949), pp. 52-84. A contemporary classic. Indispensable. Slightly labored hilt elrrr
Alfred Tarski, Introduction to Logic and the Methodology of the Deductive Sciences. New York: Oxford University Press, 1946. One of the best. For basic logic as well as for hypothetico-deductive theories. Many illustrations are mathematical in character. It is good to study them, too. The problems and exercises are thought-provoking. Excellent practice in techniques and thinking. If one is limited to three books, choose Bochenski's Precis, Quine's Method's, and Tarski's Introduction.
Ivo Thomas, O. P., "Material Implication in John of St. Thomas," Dominican Studies 3 (1950), p. 180. A brief report by the author. But check the chapter in John of St. Thomas yourself.
J. Ullmo, "Physique et axiomatique," Revue d e metaphysique et de morale 54 (1949) 126-138. If one desires to glimpse the range and possible fruitfulness of the hypothetico-deductive system technique outside the areas of pure logic or mathematics, let him study the instances of isomorphism in physics which the author introduces and explains. A fascinating contribution.
Rene van den Driessche, "Sur le 'De syllogismo hypothetico' de Boece," Methodos 1 (1949) 293-307. A capital study, unavailable when this essay was written. Complements the section on Boethius in chapter on Scholastic logic.
B. L. van der Waerden, Einfuhrung in die algebraische Geometrie. New York: Dover Publications, Inc., 1949. A modern classic. Rigorous and therefore most helpful. Get it for the library, so that it will be there when one is ready to look at it.96 Conventional Logic and Modern Logic
Henry Veatch, "Concerning the Ontological Status of Logical Forms." The Review of Metaphysics 2 (1948) 40-64.  
Henry Veatch, "Aristotelian and Mathematical Lgic," The Thomist 13 (1950) 50-96.  
Henry Veatch, "In Defense of the Syllogism," The Modern Schoolman 27 (1950) 1S4-202.  
Henry Veatch, "Basic Confusions in Current Notions of Propositional Calculi," The Thomist 14 (1951) 238-258.  
Henry Veatch, "Formalisrn and/or Intentionality in Logic," Philosophy and Phenomenological Research 11 (1951) 348364. I find it fascinating to follow Veatch, difficult to evaluate his connected contributions. Is he a center for a reactionary, rather than a conciliatory movement? He has his points and he makes them well. But their relevance does not seem as decisive as the author seems to think. Veatch demonstrates that the objects of modern logic are not "second intentions," as such. Veatch defends the classical doctrine that science is a knowledge through causes. But how crucial is the doctrine of "second intentions?" And is there not a spectacular world of science which is not per causas 2 Veatch should be read and studied. Nothing would help the situation more than if he were to found a reactionary and articulate school, dedicated to clarifying the issues and bringing them to a head. Confer Copi supra.
A. Virieux-Reymond, "La logique stoicienne," Proceedings of the Tenth Intezrnational Congress of Philosophy (Amsterdam: North-Holland Publishing Company, 1949), Yolume 1, Part 2, pp. 718-719. Brief but historically accurate abstract. Excellent summary.
Antoinette Virieux-Reymond, La logique et 1' etpiste'mologie des Stoiciens. Chambery: editions "Lire," n.d. (1950). Completely derivative from secondary sources. But accurate in the main and richly informative to the tyro.
John J. Wellmuth, "Some Comments on the Nature of Mathematical Logic," The New Scholasticism 15 (1942) 9-15. Brief and hesitant presentation of an excellent but hitherto unpublished idea of D. A. Steele, S.J. The value of the idea survives its derivative and second hand presentation An important fragment until the original is published. If isomorphism interests anyone, this is an indispensable reference.
A. N. Whitehead, An Introduction to Mathematics. New YorkOxford University Press, 1948. A reprint of a classic. The less one knows and likes mathematics, the more one should read and study this book. If this fails to impart to a reader the genuine genius of mathematics, nothing will.
Alfred North Whitehead and Bertrand Russell, Principia Mathematica. Cambridge: at the University Press, Volume 12 (1936), Volume 22 (1927), Volume 3 (1927). The classic par excellence. Nothing ean take its place. Do not even look at Volumes 2 and 3 until one has made solid progress through at least the first 10 sections of Volume 1. Should be in every library. In these volumes all that went before is distilled and organized, and all that has happened afterwards lies in germ. Indispensable. Reprinted by publishers in 1951.
Warner Arms Wick, Metaphysics and the New Logic. Chicago: the University of Chicago Press, 1942. I found Wick disappointing. But read him yourself. The problem is real and he sees it. But the approach toward solution seems very distant. Like tracing current Democracy-Totalitarianism sociological conflicts back to original sin. No one who wants to see and perhaps effect 3 correlation can afford to miss the book, even though it may not help him significantly.
Philip Paul Wiener, 'iNotes on Leibniz' Conception of Logic and Its Historical Context," The Philosophical Review 48 (1939) 567-586. Will serve as an introduction to the subject of Leibniz and logic and acquaint the reader with the best studies on the subject. Profoundly interesting.
John Wild, "An Introduction to the Phenomenology of Signs," Philosophy and Phenomenological Research 8 (1947-1948) 217-223. A vigorous and gifted philosopher controverts a behavioristic semiotic with materials drawn from the heritage of Scholasticism. Provocative.
Ludwig Wittgenstein, Tractatus Logico-Philosophicus. London: Routledge and Kegan Paul, Ltd., 1949. Wittgenstein's Tractatus is to modern logic what Thomas a Kempis is to traditional asceticism. There are even stylistic resemblances. Both are the type of book to serve as origins of meditations. An indispensable classic.
I. H. Woodger, "The Technique of Theory Construction," International Encyclopedia of Unified Science, Volume 2, Number 5. Chicago: the University of Chicago Press, 1947. Reminds me of Leonardo da Vinci's "anticipations" of aeroplanes and submarines: excellent theoretical ideas but far in advance of technical possibility of execution. But most instructive and suggestive. Important for all who think that formal logic is irrelevant to concrete experience and its organization along scientific lines. Will also introzil7P one ta the literature on the subject.
Kllan B. Wolter, O. F. M., "Ockham and the Textbooks: On the Origin of Possibility," Franziskanische Studied 32 (1950) 70-96. The coupe-de-grace to the moribund "black legend" about Ockham. Victims of this legend will be disaffected toward Ockham's logical competence, and on altogether unhistorical grounds. One must first get Ockham straight and only then undertake to evaluate his contributions to logic. All adverse criticism, constructed on the basis of the legend, is unacceptable evidence.