Charles Peirce's development of his diagrammatic logic, his entitative and existential graphs, was significantly influenced by his affinity for tree graphs that were being used in chemistry. In his development of systems of natural deduction and sequent calculi, Gerhard Gentzen made use of the tree (tableau) method. In presenting the historical sources of both these tools, we draw on unpublished manuscripts from the Peirce Edition Project at the University of Indianapolis, where for many years the first author was a member of the research staff.

The contributions to logic of MacColl and Charles Sanders Peirce (1839-1914) were the two most profound influences upon the work of Ernst Schröder (1841-1902) in algebraic logic. In his Vorlesungen über die Algebra der Logik, Schröder referred to MacColl as one of his most important precursors. Schröder compared Peirce’s considerations with the early parts of MacColl’s series of papers “The calculus of equivalent statements” (published between 1877 and 1880), and he attributed to MacColl priority for having anticipated Peirce’s results. For Schröder, MacColl’s calculus was a preliminary stage of Peirce’s algebra of logic.

In this contribution I recount the history of the development of modern logic in the late 19th century and early 20th century in order to explicate the role played in it of Charles S. Peirce and his semeiotic.

This chapter discusses central issues of Peirce’s conception of logic, comparing his work with the works of other logicians of the time, in particular Boole, Venn, Schröder, and Frege. It presents a detailed analysis of Peirce’s approach to notation, including pasigraphy and Peirce’s claw.

I use van Heijenoort’s published writings and manuscript materials to provide a comprehensive overview of his conception of modern logic as a first-order functional calculus and of the historical developments which led to this conception of mathematical logic, its defining characteristics, and in particular to provide an integral account, from his most important publications as well as his unpublished notes and scattered shorter historico-philosophical articles, of how and why the mathematical logic, whose he traced to Frege and the culmination of its formative period in the incompleteness results of Gödel, became modern logic, as distinct from the traditional logic of Aristotle, and why and how the logistic tradition that led from Frege through Russell, rather than the algebraic tradition that led from De Morgan and Boole through Peirce and Schröder, came, in his view, to define modern logic.

Van Heijenoort’s account of the historical development of modern logic was composed in 1974 and first published in 1992 with an introduction by his former student. What follows is a new edition with a revised and expanded introduction and additional notes.

The historiography of logic conceives of a Fregean revolution in which modern mathematical logic (also called symbolic logic) has replaced Aristotelian logic. The preeminent expositors of this conception are Jean van Heijenoort (1912-1986) and Donald Angus Gillies. The innovations and characteristics that comprise mathematical logic and distinguish it from Aristotelian logic, according to this conception, created ex nihlo by Gottlob Frege (1848-1925) in his Begriffsschrift of 1879, and with Bertrand Russell (1872-1970) as its chief This position likewise understands the algebraic logic of Augustus De Morgan (1806-1871), George Boole (1815-1864), Charles Sanders Peirce (1838-1914), and Ernst Schr\"oder (1841-1902) as belonging to the Aristotelian tradition. The "Booleans" are understood, from this vantage point, to merely have rewritten Aristotelian syllogistic in algebraic guise. The most detailed listing and elaboration of Frege's innovations, and the characteristics that distinguish mathematical logic from Aristotelian logic, were set forth by van Heijenoort. I consider each of the elements of van Heijenoort's list and note the extent to which Peirce had also developed each of these aspects of logic. I also consider the extent to which Peirce and Frege were aware of, and may have influenced, one another's logical writings.

Jean van Heijenoort was best known for his editorial work in the history of mathematical logic. I survey his contributions to model-theoretic proof theory, and in particular to the falsifiability tree method. This work of van Heijenoort's is not widely known, and much of it remains unpublished. A complete list of van Heijenoort's unpublished writings on tableaux methods and related work in proof theory is appended.

We explore the technical details and historical evolution of Charles Peirce's articulation of a truth table in 1893, against the background of his investigation into the truth-functional analysis of propositions involving implication. In 1997, John Shosky discovered, on the verso of a page of the typed transcript of Bertrand Russell's 1912 lecture on "The Philosophy of Logical Atomism" truth table matrices. The matrix for negation is Russell's, alongside of which is the matrix for material implication in the hand of Ludwig Wittgenstein. It is shown that an unpublished manuscript identified as composed by Peirce in 1893 includes a truth table matrix that is equivalent to the matrix for material implication discovered by John Shosky. An unpublished manuscript by Peirce identified as having been composed in 1883-84 in connection with the composition of Peirce's "On the Algebra of Logic: A Contribution to the Philosophy of Notation" that appeared in the American Journal of Mathematics in 1885 includes an example of an indirect truth table for the conditional.

Reviewed by: The Logic Pamphlets of Charles Lutwidge Dodgson and Related Pieces Irving H. Anellis Francine F. Abeles, editor. The Logic Pamphlets of Charles Lutwidge Dodgson and Related Pieces. The Pamphlets of Lewis Carroll, 4. New York-Charlottesville-London: Lewis Carroll Society of North America-University Press of Virginia, 2010. Pp. xx + 271. Cloth, $75.00. Until William Bartley’s rediscovery and reconstruction of Dodgson’s lost Part II of Symbolic Logic, Lewis Carroll’s reputation in logic, when taken seriously, rested upon his few published articles on logical paradoxes and puzzles, while his published books, The Game of Logic and Symbolic Logic, were regarded as suitable perhaps as pedagogical tools, but dismissed otherwise as amusements, on a par with the Alice works. Richard Braithwaite asserted (“Lewis Carroll as Logician”) that “Carroll regarded formal and symbolic logic not as a corpus of systematic knowledge about valid thought nor yet as an art for teaching a person to think correctly.” He suggested that Dodgson used his pseudonym for his popular and whimsical writings, reserving his legal name only for his serious mathematical products. Philosophers practicing linguistic analysis in their assault on nonsense deriving from misuse of language meanwhile developed an interest in Carroll (e.g. George Pitcher, “Wittgenstein, Nonsense, and Lewis Carroll”). Abeles, on the contrary, stresses the seriousness of Dodgson’s pedagogical and popularizing mission (195–99). Bartley’s discovery opened the door for reevaluating Dodgson as a serious logician, and Abeles has been in the forefront of that reevaluation. Her general introduction and introductions to the main divisions of this collection of Carroll’s publications (and additional material from his Nachlass and a handful of publications of others, composed in response to Carroll’s published articles) place his serious logical work and its significance in historical perspective, providing a deeper, more sophisticated view than found in Bartley’s “Editor’s Introduction” to Lewis Carroll’s Symbolic Logic. Abeles also offers brief introductions to the individual entries, which describe their physical characteristics and circumstances of composition. Dodgson aligned with the majority of those British logicians of his day, starting with George Boole, who understood formal logic to be Aristotelian syllogistic, and symbolic logic to be syllogistic logic expressed algebraically. His notational innovation was employing indices to terms so that, independently of the arrangement of terms, propositions could easily be read; thus, “xy0” can be read as “no x are y” or “no y are x” and “x1y0” as “all x are not y” or “no x are y”. In dealing with the classical elimination problem in the class calculus—to determine the maximum information, without duplications, obtainable from a given set of premises—Dodgson in his later work and unpublished letters anticipated several concepts of automated theorem proving. Central to both geometry and to formal logic are the rules of logical inference that establish the validity of arguments and guarantee that conclusions inferred from true [End Page 506] premises are true. The modern analytic tableaux, or tree method, is a fundamental tool in this regard, both for deriving theorems and for determining the validity or invalidity of the proofs for propositional calculus and first-order predicate calculus. Introducing his reconstruction of Part II of Symbolic Logic, Bartley noted that, along with the logic diagrams that Carroll devised, he also devised a tableau method that strongly resembled Beth’s semantic tableaux. Abeles went more deeply and convincingly into that resemblance in her earlier research. Beth’s tableaux, together with Hintikka’s model sets, were the direct ancestors of Smullyan’s analytic tableaux, which in turn are the theoretical basis for the Robinson resolution method that continues to be central to much work in programming logic. Abeles has also shown that the work in Studies in Logic by Charles Peirce and his logic students at Johns Hopkins University served as a source of inspiration for Dodgson. More critically, she demonstrated that Dodgson devised and employed the tree method for carrying out proofs of syllogisms and chains of syllogisms, or soriteses, in Part II of Symbolic Logic (see her “Lewis Carroll’s Method of Trees: Its Origins in Studies in Logic,” Modern Logic 1 [1990]: 25–34). The tree...

The contributions to logic of MacColl and Charles Sanders Peirce (1839-1914) were the two most profound influences upon the work of Ernst Schröder (1841-1902) in algebraic logic. In his Vorlesungen über die Algebra der Logik, Schröder referred to MacColl as one of his most important precursors. Schröder compared Peirce’s considerations with the early parts of MacColl’s series of papers “The calculus of equivalent statements” (published between 1877 and 1880), and he attributed to MacColl priority for having anticipated Peirce’s results. For Schröder, MacColl’s calculus was a preliminary stage of Peirce’s algebra of logic.

I begin by asking whether there was a Fregean revolution in logic, and, if so, in what did it consist. I then ask whether, and if so, to what extent, Russell played a decisive role in carrying through the Fregean revolution, and, if so, how. A subsidiary question is whether it was primarily the influence of The Principles of Mathematics or Principia Mathematica, or perhaps both, that stimulated and helped consummate the Fregean revolution. Finally, I examine cases in which logicians sought, in the years immediately following publication of the Principles and Principia, to integrate traditional logic into the Fregean paradigm, focusing on the case of Henry Bradford Smith. My proposed conclusion is that there were different means adopted for rewriting the syllogism, in terms of the logic of relations, in terms of the propositional calculus, or as formulas of the monadic predicate calculus. This suggests that the changes implemented as a result of the adoption of the Russello-Fregean conception of logic could more accurately be called by Grattan-Guinness's term convolution, rather than revolution.

Journal Article Joong Fang (1923–2010) Get access Irving H. Anellis Irving H. Anellis *Peirce Edition, Institute for American Thought, Indiana University-Purdue University at Indianapolis, Indianapolis, Indiana 46202-5159, U.S.A.ianellis@iupui.edu Search for other works by this author on: Oxford Academic Google Scholar Philosophia Mathematica, Volume 18, Issue 2, June 2010, Pages 137–143, https://doi.org/10.1093/philmat/nkq008 Published: 01 June 2010

Reviewed by: Handbook of the History of Logic, volume 3: The Rise of Modern Logic from Leibniz to Frege Irving H. Anellis Dov M. Gabbay and John Woods (Editors). Handbook of the History of Logic, volume 3: The Rise of Modern Logic from Leibniz to Frege. Amsterdam, etc.: Elsevier, 2004. 750pp. plus index. The Handbook of the History of Logic, under the general editorship of Dov Gabbay and John Woods, is intended as an extensive and encyclopedic survey of the entire history of logic from earliest times to the present. Every contribution to the Handbook is written by a specialist in a particular era or topic, and most of the articles run well over fifty pages, in many cases twice that much. Because of the range of inclusion and the diversity of contributors, the styles vary greatly from essay to essay. So also does the perspective from which the contributors work. Thus, some essays are largely philosophical, others are mathematically technical, ranging from exposition to critical analysis to rational reconstruction, and even to unadulterated speculation. (For example, in an earlier volume, covering ancient Greek, Indian, and Buddhist logics, there is an effort to rewrite Buddhist logic in Aristotelian terms.) This kind of diversity does not necessarily depend upon the specialized topic or upon the logician whose work is being considered, but rather upon the interests and focus of the authors of the essays. Additionally, the editors have allowed "logic" to be defined quite broadly, to include what many would doubtlessly consider more properly to be philosophical logic, or even philosophy of logic; and while there is little, in the volumes published thus far, to satisfy readers who would be interested in non-deductive logics—Hilpinen alone gives any attention to abduction or induction, and only in a most cursory manner—nonclassical deductive logics are represented, both within the context of specific logicians and chronological frameworks, or in [End Page 456] distinct volumes of the series devoted specifically to the history of non-classical logics. This expansive definition of logic coincides in many respects with Peirce's own, which takes within its compass the methodology of the sciences and critical thinking, and includes abduction and induction along with deduction. There are three essays in the volume under review that impinge to a significant extent upon the work in logic of Charles Peirce, each by well-known contributors. They are: • Theodore Hailperin, "Algebraical Logic 1685–1900," pp. 323–88; • Victor Sánchez Valencia, "The Algebra of Logic," pp. 389–544; • Risto Hilpinen, "Peirce's Logic," 611–58. The first two essays include significant discussions of Peirce's work in logic against the broader historical context in which that work was carried out. The third essay is not a complete account of Peirce's work in logic, but carefully avoids duplicating those aspects of the Peircean corpus that have been discussed by the other two authors. Thus it is absolutely essential to work through all three essays in order to gain a comprehensive overview of the full extent of Peirce's work in deductive logic. There is a wealth of technical detail in the surveys of Hailperin, Sánchez Valencia, and Hilpinen, and much of their treatment may be daunting for readers without a solid background in algebraic logic. The incorporation of criticisms along with exposition makes the task the more difficult for those not already acquainted with the technical aspects of the work being discussed. Because many of the readers of this review will not necessarily be either historians of logic or trained as logicians, as well as constraints imposed by limitation of space in this journal as compared with the length of the contributions being reviewed, I will provide merely a sketch of a survey of the scope of the contents of the contributions and a general overview of the character of the essays. Hailperin (p. 323) defines "algebraic logic" as "a style of doing logic, a style in which concepts and relations are expressed by mathematical symbols." He provides a general survey of the evolution of algebraic logic, beginning with Leibniz and ending with Whitehead's Treatise. The most prominent figures in his account are Leibniz, De Morgan, Boole, and...