Peirce algebras combine sets, relations and various operations linking the two in a unifying setting. This paper offers a modal perspective on Peirce algebras. Using modal logic a characterization of the full Peirce algebras is given, as well as a finite axiomatization of their equational theory that uses so-called unorthodox derivation rules. In addition, the expressive power of Peirce algebras is analyzed through their connection with first-order logic, and the fragment of first-order (...) logic corresponding to Peirce algebras is described in terms of bisimulations. (shrink)

Peirce algebras combine sets, relations and various operations linking the two in a unifying setting. This paper offers a modal perspective on Peirce algebras. Using modal logic as a characterization of the full Peirce algebras is given, as well as a finite axiomatization of their equational theory that uses so-called unorthodox derivation rules. In addition, the expressive power of Peirce algebras is analyzed through their connection with first-order logic and the fragment of (...) first-order logic corresponding to Peirce algebras is described in terms of bisimulations. (shrink)

Peirce introduced Existential Graphs in late 1896, and they were systematically investigated in his 1903 Lowell Lectures. Alpha graphs for classical propositional logic constitute the first part of EGs. The second and the third parts are the beta graphs for first-order logic and the gamma graphs for modal and higher-order logics, among others. As a logical syntax, EGs are two-dimensional graphs, or diagrams, in contrast to the linear algebraic notations. Peirce's theory of EGs is not only a theory (...) of logical syntax but also a proof theory for many logics. A set of transformation rules of graphs defines a proof system for a graphical logic. In this... (shrink)

Robert Burch describes Peircean Algebraic Logic as a language to express Peirce's “unitary logical vision” , which Peirce tried to formulate using different logical systems. A “correct” formulation of Peirce's vision then should allow a mathematical proof of Peirce's Reduction Thesis, that all relations can be generated from the ensemble of unary, binary, and ternary relations, but that at least some ternary relations cannot be reduced to relations of lower arity.Based on Burch's algebraization, the authors further (...) simplify the mathematical structure of PAL and remove a restriction imposed by Burch, making the resulting system in its expressiveness more similar to Peirce's system of existential graphs. The drawback, however, is that the proof of the Reduction Thesis from Burch no longer holds. A new proof was introduced in Hereth Correia, and Pöschel and was published in full detail in Hereth .In this paper, we provide proof of Peirce's Reduction Thesis using a graph notation similar to Peirce's existential graphs. (shrink)

This article investigates Charles Peirce’s development of logical calculi for classical propositional logic in 1880–1896. Peirce’s 1880 work on the algebra of logic resulted in a successful calculus for Boolean algebra. This calculus, denoted byPC, is here presented as a sequent calculus and not as a natural deduction system. It is shown that Peirce’s aim was to presentPCas a sequent calculus. The law of distributivity, which Peirce states in 1880, is proved using Peirce’s Rule, which (...) is a residuation, inPC. The transitional systems of the algebra of the copula that Peirce develops since 1880 paved the way to the 1896 graphical system of the alpha graphs. It is shown how the rules of the alpha system reinterpret Boolean algebras, answering Peirce’s statement that logical graphs supply a new system of fundamental assumptions to logical algebra. A proof-theoretic analysis is given for the connection betweenPCand the alpha system. (shrink)

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?1884 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. (shrink)

Charles S. Peirce’s pragmatist theory of logic teaches us to take the context of utterances as an indispensable logical notion without which there is no meaning. This is not a spat against compositionality per se , since it is possible to posit extra arguments to the meaning function that composes complex meaning. However, that method would be inappropriate for a realistic notion of the meaning of assertions. To accomplish a realistic notion of meaning (as opposed e.g. to algebraic meaning), (...) Sperber and Wilson’s Relevance Theory (RT) may be applied in the spirit of Peirce’s Pragmatic Maxim (PM): the weighing of information depends on (i) the practical consequences of accommodating the chosen piece of information introduced in communication, and (ii) what will ensue in actually using that piece in further cycles of discourse. Peirce’s unpublished papers suggest a relevance-like approach to meaning. Contextual features influenced his logic of Existential Graphs (EG). Arguments are presented pro and con the view in which EGs endorse non-compositionality of meaning. (shrink)

We study logical reduction (factorization) of relations into relations of lower arity by Boolean or relative products that come from applying conjunctions and existential quantifiers to predicates, i.e. by primitive positive formulas of predicate calculus. Our algebraic framework unifies natural joins and data dependencies of database theory and relational algebra of clone theory with the bond algebra of C.S. Peirce. We also offer new constructions of reductions, systematically study irreducible relations and reductions to them and introduce a new characteristic (...) of relations, ternarity, that measures their ‘complexity of relating’ and allows to refine reduction results. In particular, we refine Peirce’s controversial reduction thesis, and show that reducibility behaviour is dramatically different on finite and infinite domains. (shrink)

Two algebraic structures, the contrapositionally complemented Heyting algebra (ccHa) and the contrapositionally |$\vee $| complemented Heyting algebra (c|$\vee $|cHa), are studied. The salient feature of these algebras is that there are two negations, one intuitionistic and another minimal in nature, along with a condition connecting the two operators. Properties of these algebras are discussed, examples are given and comparisons are made with relevant algebras. Intuitionistic Logic with Minimal Negation (ILM) corresponding to ccHas and its extension |${\textrm {ILM}}$|-|${\vee (...) }$| for c|$\vee $|cHas are then investigated. Besides its relations with intuitionistic and minimal logics, ILM is observed to be related to Peirce’s logic and Vakarelov’s logic MIN. With a focus on properties of the two negations, relational semantics for ILM and |${\textrm {ILM}}$|-|${\vee }$| are obtained with respect to four classes of frames, and inter-translations between the classes preserving truth and validity are provided. ILM and |${\textrm {ILM}}$|-|${\vee }$| are shown to have the finite model property with respect to these classes of frames and proved to be decidable. Extracting features of the two negations in the algebras, a further investigation is made, following logical studies of negations that define the operators independently of the binary operator of implication. Using Dunn’s logical framework for the purpose, two logics |$K_{im}$| and |$K_{im-{\vee }}$| are discussed, where the language does not include implication. The |$K_{im}$|-algebras are reducts of ccHas and are different from relevant algebraic structures having two negations. The negations in the |$K_{im}$|-algebras and |$K_{im-{\vee }}$|-algebras are shown to occupy distinct positions in an enhanced form of Dunn’s kite of negations. Relational semantics for |$K_{im}$| and |$K_{im-{\vee }}$| is provided by a class of frames that are based on Dunn’s compatibility frames. It is observed that this class coincides with one of the four classes giving the relational semantics for ILM and |${\textrm {ILM}}$|-|${\vee }$|⁠. (shrink)

In his formal papers on existential graphs , Peirce tended to obscure the simplicity of EGs with distracting digressions. In MS 514, however, he presented his simplest introduction to the EG syntax, semantics, and rules of inference. This article reproduces Peirce's original words and diagrams with further commentary, explanations, and examples. Unlike the syntax-based approach of most current textbooks, Peirce's method addresses the semantic issues of logic in a way that can be transferred to any notation. The (...) concluding section shows that his rules of inference can clarify the foundations of proof theory and relate diverse methods, such as resolution and natural deduction. To relate EGs to other notations for logic, this article uses the Existential Graph Interchange Format , which is a subset of the CGIF dialect of Common Logic. EGIF is a linear notation that can be mapped to and from the Alpha, Beta, and Gamma variants of EGs. It can also be translated to or from other formalisms, algebraic or geometrical. (shrink)

Peirce's classifications of signs started to be developed in 1865 and it extends up to 1909. I will present on the period that begins in 1865, and that has two moments of intense production - "On a New List of Categories"and "On the Algebra of Logic: a contribution to the philosophy of notation". It is an introductory approach whose intention is to make the reader be familiar with the Peircean complex classifications of signs.

Below are reflections on Peirce’s conception of abductive methods and Russell’s conception of regressive methods. Along the way, it will be necessary to examine the marked differences between Russell and Frege on the ins and outs of logicism, from which latter the regressivist ideas first emerged. Russell was aware of Peirce’s contributions to the algebraization of logic and Peirce was aware of Russell’s writings on logicism. However, in framing his thoughts about regressive methods, Russell showed no familiarity (...) with Peirce’s treatment of abductive methods. In 1907, Russell read to the Cambridge Mathematics Club an essay entitled “The regressive method of discovering the premises of mathematics.” Since that paper didn’t see the published light of day until three years after Russell’s death in 1970, Peirce couldn’t have taken notice of it in developing his ideas about abduction. Even so, it has been suggested that there exists a noteworthy similarity between Russell’s regressivism and Peirce’s abductivism. The principal purpose of this essay is to show the resemblance to have been misjudged, in which case, my title would have to be corrected. (shrink)

The aim of this article is Schröder’s treatment of the so called solution problem [Auflösungsproblem]. First, I will introduce Schröder’s ideas; then I will discuss them taking into account Peirce’s considerations in The Logic of Relatives ([13, pp. 161–217] now republished in [9, pp. 288–345]).

The calculus of relations was created and developed in the second half of the nineteenth century by Augustus De Morgan, Charles Sanders Peirce, and Ernst Schröder. In 1940 Alfred Tarski proposed an axiomatization for a large part of the calculus of relations. In the next decade Tarski's axiomatization led to the creation of the theory of relation algebras, and was shown to be incomplete by Roger Lyndon's discovery of nonrepresentable relation algebras. This paper introduces the calculus of (...) relations and the theory of relation algebras through a review of these historical developments. (shrink)

Built on the foundations laid by Peirce, Schröder, and others in the 19th century, the modern development of relation algebras started with the work of Tarski and his colleagues [21, 22]. They showed that relation algebras can capture strong first‐order theories like ZFC, and so their equational theory is undecidable. The less expressive class WA of weakly associative relation algebras was introduced by Maddux [7]. Németi [16] showed that WA's have a decidable universal theory. There has (...) been extensive research on increasing the expressive power of WA by adding new operations [1, 4, 11, 13, 20]. Extensions of this kind usually also have decidable universal theories. Here we give an example – extending WA's with set‐theoretic projection elements – where this is not the case. These “logical” connectives are set‐theoretic counterparts of the axiomatic quasi‐projections that have been investigated in the representation theory of relation algebras [22, 6, 19]. We prove that the quasi‐equational theory of the extended class PWA is not recursively enumerable. By adding the difference operator D one can turn WA and PWA to discriminator classes where each universal formula is equivalent to some equation. Hence our result implies that the projections turn the decidable equational theory of “WA + D ” to non‐recursively enumerable (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim). (shrink)

The process of ‘logical differentiation’ was introduced by Peirce in 1870. Directly analogous to mathematical differentiation, it uses logical terms instead of mathematical variables. Here, this mysterious process receives new interpretations which serve to clarify Peirce’s use of logical terms. I introduce the logical terms, the operation of multiplication, the logical analogy to the binomial theorem, infinitesimal relatives, the concepts of numerical coefficients and the number associated with each term. I also analyse the algebraic development of ‘logical differentiation’ (...) and consider in depth one application of the process. (shrink)

We study algebras whose elements are relations, and the operations are natural “manipulations” of relations. This area goes back to 140 years ago to works of De Morgan, Peirce, Schröder . Well known examples of algebras of relations are the varieties RCAn of cylindric algebras of n-ary relations, RPEAn of polyadic equality algebras of n-ary relations, and RRA of binary relations with composition. We prove that any axiomatization, say E, of RCAn has to be very (...) complex in the following sense: for every natural number k there is an equation in E containing more than k distinct variables and all the operation symbols, if 2 < n < ω. Completely analogous statement holds for the case n ω. This improves Monk's famous non-finitizability theorem for which we give here a simple proof. We prove analogous nonfinitizability properties of the larger varieties SNrnCAn + k. We prove that the complementation-free subreducts of RCAn do not form a variety. We also investigate the reason for the above “non-finite axiomatizability” behaviour of RCAn. We look at all the possible reducts of RCAn and investigate which are finitely axiomatizable. We obtain several positive results in this direction. Finally, we summarize the results and remaining questions in a figure. We carry through the same programme for RPEAn and for RRA. By looking into the reducts we also investigate what other kinds of natural algebras of relations are possible with more positive behaviour than that of the well known ones. Our investigations have direct consequences for the logical properties of the n-variable fragment Ln of first order logic. The reason for this is that RCAn and RPEAn are the natural algebraic counterparts of Ln while the varieties SNrnCAn + k are in connection with the proof theory of Ln. This paper appears in two parts. The first part contains the non-finite axiomatizability results. The present second part contains finite axiomatizations of some fragments together with a figure summarizing the finite and non-finite axiomatizability results in this area and the problems left open. (shrink)

We study algebras whose elements are relations, and the operations are natural “manipulations” of relations. This area goes back to 140 years ago to works of De Morgan, Peirce, Schröder . Well known examples of algebras of relations are the varieties RCAn of cylindric algebras of n-ary relations, RPEAn of polyadic equality algebras of n-ary relations, and RRA of binary relations with composition. We prove that any axiomatization, say E, of RCAn has to be very (...) complex in the following sense: for every natural number k there is an equation in E containing more than k distinct variables and all the operation symbols, if 2 < n < ω. Completely analogous statement holds for the case n ω. This improves Monk's famous non-finitizability theorem for which we give here a simple proof. We prove analogous nonfinitizability properties of the larger varieties SNrnCAn + k. We prove that the complementation-free subreducts of RCAn do not form a variety. We also investigate the reason for the above “non-finite axiomatizability” behaviour of RCAn. We look at all the possible reducts of RCAn and investigate which are finitely axiomatizable. We obtain several positive results in this direction. Finally, we summarize the results and remaining questions in a figure. We carry through the same programme for RPEAn and for RRA. By looking into the reducts we also investigate what other kinds of natural algebras of relations are possible with more positive behaviour than that of the well known ones. Our investigations have direct consequences for the logical properties of the n-variable fragment Ln of first order logic. The reason for this is that RCAn and RPEAn are the natural algebraic counterparts of Ln while the varieties SNrnCAn + k are in connection with the proof theory of Ln. This paper appears in two parts. The first part contains the non-finite axiomatizability results. The present second part contains finite axiomatizations of some fragments together with a figure summarizing the finite and non-finite axiomatizability results in this area and the problems left open. (shrink)

When one tries to determine what the iconic dimension of thought consists in for Peirce and what its range is, one might have the impression that his remarks on this matter are inconsistent. For instance, on the one hand he writes the following: Remember it is by icons only that we really reason, and abstract statements are valueless in reasoning except so far as they aid us to construct diagrams. The sectaries of the opinion I am combating seem, on (...) the contrary, to suppose that reasoning is performed with abstract "judgments," and that an icon is of use only as enabling me to frame abstract statements as premisses. In an apparently similar fashion, Peirce argues that the predicate of a... (shrink)

The standard account of the history of logic, in its crudest outline, runs as follows. Logic as a discipline was invented by Aristotle with his creation of syllogistic theory, which, despite the emergence of propositional logic in the work of the Stoics, refinements by mediæval logicians, and Leibniz's sketches of a characteristica universalis, dominated philosophy until the middle of the nineteenth century, when Boole's work on the "algebra of logic" and Frege's Begriffsschrift finally absorbed it into a larger framework. Boolean (...) and Fregean logic were rivals until the beginning of the twentieth century, when the superior power of the Fregean system, as developed by Peano and Russell, won the day, and modern logic as we know it began to exert its hegemony. However, as with any crude account, there are glaring omissions and anomalies that threaten its usefulness even as a first approximation, and Peirce's role in the story has perhaps been the most seriously underplayed of all. Whilst rooted in the Boolean tradition, Peirce also—admittedly slightly later, but nevertheless independently of Frege—invented a notation for quantifiers: a notation, indeed, that influenced modern notation to a far greater extent than Frege's own cumbersome two-dimensional script. Furthermore, Peirce's logic of relations was a significant development in its own right, and both this and the stirrings of the model-theoretic approach in his work was to feed into the mainstream tradition through the work of Schröder, Löwenheim, Skolem, and Tarski, in particular. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Reviewed by:The Correspondence of Charles S. Peirce and the Open Court Publishing Company, 1890–1913 ed. by Stetson J. RobinsonCornelis de WaalEdited by Stetson J. RobinsonThe Correspondence of Charles S. Peirce and the Open Court Publishing Company, 1890–1913 Berlin: De Gruyter, 2022. 666pp., incl. indexThe fifth volume in the Peirceana series brings us the extensive correspondence between Peirce and the Open Court Publishing Company (abbreviated to OCP (...) by Robinson). The double-barreled opening shot consists of two letters (one by Francis Russell and one by Open Court’s Editor Paul Carus) aimed at soliciting a contribution from [End Page 109] Peirce for the inaugural issue of a new quarterly journal to be published by the Open Court, called The Monist. Carus had been particularly impressed by Peirce’s Illustrations of the Logic of Science papers, which had appeared over a decade earlier in Popular Science Monthly, and to which Russell had drawn Carus’s attention.1 Peirce quickly responded, suggesting not one but an entire series of articles, the first of which to be titled “The Architecture of Theories.” It proved the beginning of a productive but also turbulent relationship between Peirce and the publishing company. The final letter in the exchange comes again from Carus, written September 10, 1913, about seven months before Peirce died.The Open Court Collection at Southern Illinois University Carbondale, from which much of the correspondence is drawn, is a truly magnificent collection. It contains not only business correspondence and financial records, but also the various stages of the publishing process. It includes the original texts that Peirce sent to the publisher, as well as the often heavily corrected galleys and proofs—material that is used appreciatively by the Peirce Edition Project for its critical edition of Peirce’s Writings. When we combine this record with the Nachlass of a philosopher who lived in a continuously expanding mansion with a penchant for keeping each and every scrap of paper, we get a remarkably full picture of everything that went into the publication of Peirce’s work with the Open Court. In contrast, we know painfully little of what went on at the Century Company, which published the Century Dictionary to which Peirce so heavily contributed, or of James Mark Baldwin’s treatment of Peirce’s submissions when he compiled his Dictionary of Philosophy and Psychology. Neither venture seems to have left any records, apart from what survived within Peirce’s own papers— at least no such records have so far been found. We know a bit more about Peirce’s dealings with The Nation, but again, almost exclusively because of what has been preserved in Peirce’s papers.Robinson’s hefty tome makes the correspondence between Peirce and the Open Court widely available for the first time. The book opens with a short explanation of the editorial method, a helpful chronology, and a brief historical introduction describing Peirce’s relationship with the Open Court. The work also contains fourteen illustrations, mostly of letters. The introduction to the volume is brief and largely piggybacks on a handful of sources, such as the Introduction to Volume 8 of the Writings, rather than that it was inspired by the collection of letters that follows it, or by a closer familiarity with the archival holdings of the Open Court.2 In his introduction, Robinson divides Peirce’s relationship with the publishing company in four periods: A very active 1890–1894, which includes Peirce’s familiar Monist Metaphysical papers, his contributions to the fortnightly The Open Court, and his [End Page 110] attempts at writing an elementary arithmetic. This period ends with a falling out between Peirce and Hegeler. It is followed by a far more subdued 1894–1897, where the relationship with the publisher is largely restored by Russell, and which results in an extensive discussion by Peirce of the third volume of Ernst Schröder’s Vorlesungen über die Algebra der Logik, which appeared in two parts in The Monist. This period too ends with a falling out. The activity picks up again in 1904, resulting in Peirce’s famous pragmatism papers as well as the “Some Amazing Mazes” series of 1908... (shrink)

In his Locke Lectures Brandom proposes to extend what he calls the project of analysis to encompass various relationships between meaning and use. As the traditional project of analysis sought to clarify various logical relations between vocabularies so Brandom’s extended project seeks to clarify various pragmatically mediated semantic relations between vocabularies. The point of the exercise in both cases is to achieve what Brandom thinks of as algebraic understanding. Because the pragmatist critique of the traditional project of analysis was precisely (...) to deny that such understanding is appropriate to the case of natural language, the very idea of an analytic pragmatism is called into question by that critique. My aim is to clarify the prospects for Brandom’s project, or at least something in the vicinity of that project, through a comparison of it with what I will suggest we can think of as Kant’s analytic pragmatism as developed by Peirce. (shrink)

Peirce considered the principal business of logic to be the analysis of reasoning. He argued that the diagrammatic system of Existential Graphs, which he had invented in 1896, carries the logical analysis of reasoning to the furthest point possible. The present paper investigates the analytic virtues of the Alpha part of the system, which corresponds to the sentential calculus. We examine Peirce’s proposal that the relation of illation is the primitive relation of logic and defend the view that (...) this idea constitutes the fundamental motive of philosophy of notation both in algebraic and graphical logic. We explain how in his algebras and graphs Peirce arrived at a unifying notation for logical constants that represent both truth-function and scope. Finally, we show that Shin’s argument for multiple readings of Alpha graphs is circular. (shrink)

We describe Peirce’s 1903 system of modal gamma graphs, its transformation rules of inference, and the interpretation of the broken-cut modal operator. We show that Peirce proposed the normality rule in his gamma system. We then show how various normal modal logics arise from Peirce’s assumptions concerning the broken-cut notation. By developing an algebraic semantics we establish the completeness of fifteen modal logics of gamma graphs. We show that, besides logical necessity and possibility, Peirce proposed an (...) epistemic interpretation of the broken-cut modality, and that he was led to analyze constructions of knowledge in the style of epistemic logic. (shrink)

We give a new and elementary construction of primitive positive decomposition of higher arity relations into binary relations on finite domains. Such decompositions come up in applications to constraint satisfaction problems, clone theory and relational databases. The construction exploits functional completeness of 2-input functions in many-valued logic by interpreting relations as graphs of partially defined multivalued ‘functions’. The ‘functions’ are then composed from ordinary functions in the usual sense. The construction is computationally effective and relies on well-developed methods of functional (...) decomposition, but reduces relations only to ternary relations. An additional construction then decomposes ternary into binary relations, also effectively, by converting certain disjunctions into existential quantifications. The result gives a uniform proof of Peirce’s reduction thesis on finite domains, and shows that the graph of any Sheffer function composes all relations there. (shrink)

The pragmatic notion of assertion has an important inferential role in logic. There are also many notational forms to express assertions in logical systems. This paper reviews, compares and analyses languages with signs for assertions, including explicit signs such as Frege’s and Dalla Pozza’s logical systems and implicit signs with no specific sign for assertion, such as Peirce’s algebraic and graphical logics and the recent modification of the latter termed Assertive Graphs. We identify and discuss the main ‘points’ of (...) these notations on the logical representation of assertions, and evaluate their systems from the perspective of the philosophy of logical notations. Pragmatic assertions turn out to be useful in providing intended interpretations of a variety of logical systems. (shrink)

It is well-known that by 1882, Peirce, influenced by Cayley’s, Clifford’s and Sylvester’s works on algebraic invariants and by the chemical analogy, had already achieved something like a diagrammatic treatment of quantificational logic of relatives. The details of that discovery and its implications to some wider issues in logical theory merit further investigation, however. This paper provides a reconstruction of the genesis of Peirce’s logical graphs from the early 1880s until 1896, covering the period of time during which (...) he already was acquainted with the works of his Johns Hopkins colleagues on the mathematical theory of graphs and was reaching the very first forms of his theory and method of... (shrink)

In 1883, while a student of C. S. Peirce at Johns Hopkins University, Christine Ladd-Franklin published a paper titled On the Algebra of Logic, in which she develops an elegant and powerful test for the validity of syllogisms that constitutes the most significant advance in syllogistic logic in two thousand years. Sadly, her work has been all but forgotten by logicians and historians of logic. Ladd-Franklin's achievement has been overlooked, partly because it has been overshadowed by the work of (...) other logicians of the nineteenth century renaissance in logic, but probably also because she was a woman. Though neglected, the significance of her contribution to the field of symbolic logic has not been diminished by subsequent achievements of others.In this paper, I bring to light the important work of Ladd-Franklin so that she is justly credited with having solved a problem over two millennia old. First, I give a brief survey of the history of syllogistic logic. In the second section, I discuss the logical systems called “algebras of logic”. I then outline Ladd-Franklin's algebra of logic, discussing how it differs from others, and explain her test for the validity of the syllogism, both in her symbolic language and the more familiar language of modern logic. Finally I present a rigorous proof of her theorem. Ladd-Franklin developed her algebra of logic before the methods necessary for a rigorous proof were available to her. Thus, I do now what she could not have done then. (shrink)

Christine Ladd-Franklin is often hailed as a guiding star in the history of women in logic—not only did she study under C. S. Peirce and was one of the first women to receive a PhD from Johns Hopkins, she also, according to many modern commentators, solved a logical problem which had plagued the field of syllogisms since Aristotle. In this paper, we revisit this claim, posing and answering two distinct questions: Which logical problem did Ladd-Franklin solve in her thesis, (...) and which problem did she think she solved? We show that in neither case is the answer “a long-standing problem due to Aristotle.” Instead, what Ladd-Franklin solved was a problem due to Jevons that was first articulated in the nineteenth century. (shrink)

It has been suggested that Russell and or Wittgenstein arrived at a truth-table device in or around 1912 [Shosky 1997], and that, since the history of its development is so complex, the best one can claim is that theirs may be the first identifiably ascribable example. However, Charles Peirce had, unbeknownst to most logicians of the time, already developed a truth table for binary connectives of his algebra of logic in 1902.

This is a companion article to the translation of ‘Zasada sprzeczności a logika symboliczna’, the appendix on symbolic logic of Jan Łukasiewicz's 1910 book O zasadzie sprzeczności u Arytotelesa (On the Principle of Contradiction in Aristotle). While the appendix closely follows Couturat's 1905 book L'algebra de la logique (The Algebra of Logic), footnotes show that Łukasiewicz was aware of the work of Peirce, Huntington and Russell (before Principia Mathematica). This appendix was influential in the development of the Polish school (...) of logic, directly inspiring Stanisław Leśniewski and Leon Chwistek and more widely by serving as a text of the new symbolic logic. This appendix was an important source of the dominant algebraic logic in Poland, but also indicates that Łukasiewicz appreciated Russell's axiomatic approach to logic. (shrink)

The paper concentrates on the problem of adequate reflection of fragments of reality via expressions of language and inter-subjective knowledge about these fragments, called here, in brief, language adequacy. This problem is formulated in several aspects, the most being: the compatibility of language syntax with its bi-level semantics: intensional and extensional. In this paper, various aspects of language adequacy find their logical explication on the ground of the formal-logical theory T of any categorial language L generated by the so-called classical (...) categorial grammar, and also on the ground of its extension to the bi-level, intensional and extensional semantic-pragmatic theory ST for L. In T, according to the token-type distinction of Ch.S. Peirce, L is characterized first as a language of well-formed expression-tokens (wfe-tokens) - material, concrete objects - and then as a language of wfe-types - abstract objects, classes of wfe-tokens. In ST the semantic-pragmatic notions of meaning and interpretation for wfe-types of L of intensional semantics and the notion of denotation of extensional semanics for wfe-types and constituents of knowledge are formalized. These notions allow formulating a postulate (an axiom of categorial adequacy) from which follow all the most important conditions of the language adequacy, including the above, and a structural one connected with three principles of compositionality. (shrink)

The paper offers a survey of four key moments in which symbolisms for quantification were first introduced: §§11–2 of Frege’s Begriffsschrift ; Peirce’s ‘Algebra of Logic’ ; Peano’s ‘St...

While many books have been written about Bertrand Russell's philosophy and some on his logic, I. Grattan-Guinness has written the first comprehensive history of the mathematical background, content, and impact of the mathematical logic and philosophy of mathematics that Russell developed with A. N. Whitehead in their Principia mathematica (1910-1913).? This definitive history of a critical period in mathematics includes detailed accounts of the two principal influences upon Russell around 1900: the set theory of Cantor and the mathematical logic of (...) Peano and his followers. Substantial surveys are provided of many related topics and figures of the late nineteenth century: the foundations of mathematical analysis under Weierstrass; the creation of algebraic logic by De Morgan, Boole, Peirce, Schröder, and Jevons; the contributions of Dedekind and Frege; the phenomenology of Husserl; and the proof theory of Hilbert. The many-sided story of the reception is recorded up to 1940, including the rise of logic in Poland and the impact on Vienna Circle philosophers Carnap and Gödel. A strong American theme runs though the story, beginning with the mathematician E. H. Moore and the philosopher Josiah Royce, and stretching through the emergence of Church and Quine, and the 1930s immigration of Carnap and GödeI. Grattan-Guinness draws on around fifty manuscript collections, including the Russell Archives, as well as many original reviews. The bibliography comprises around 1,900 items, bringing to light a wealth of primary materials. Written for mathematicians, logicians, historians, and philosophers--especially those interested in the historical interaction between these disciplines--this authoritative account tells an important story from its most neglected point of view. Whitehead and Russell hoped to show that (much of) mathematics was expressible within their logic; they failed in various ways, but no definitive alternative position emerged then or since. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Reviews 91 THE ROOTS OF MODERN LOGIC ALASDAIR URQUHART Philosophy/ U. ofToronto Toronro, ON, Canada M5S IAI URQUHART@CS.TORONTO.EDU I. Grattan-Guinness. The Searchfor Mathematical Roots,r870--r940: logics, Set Theoriesand the Foundations of Mathematicsfrom Cantor through Russellto Godel Princeron: Princeton U. P.,2000. Pp. xiv,690. us$45.oo. Grattan-Guinness's new hisrory of logic is a welcome addition to the literature. The title does not quite do justice ro the book, since it begins with the (...) prehistory of English work in algebraic logic, including the work of the French analycicalschool, and extends to just after Godel's great incompleteness paper of 1931.The core of the book, though, is the philosophical and mathematical developmems leading up to and immediately following from chework ofWhicehead and Russell.Russell is ac chehearcof che book, and as a whole che hisrory forms an imporranr conrribucion ro Russell scudies. Commemarors on Russell have often confined themselves to a rather narrow hisrorical perspeccivein which Russell is seen as the (problemacic) heir of Gottlob Frege, and few ocher hisrorical figures (ocher than Peano) emer imo che picture. Graccan-Guinness corrects chis hisrorica.limbalance by placing Russell in che mucl1 wider context of the development of machemacics on che continent, and in parricu.laremphasizing strongly che key influence of Camor on 92 Reviews Russell's work. Cantor has usually received short shrift from philosophers, as unlike Frege,he appears rarher naive from rhe philosophical point of view. The story proper begins in Chaprer 2 wirh L-igrange'sversion of analysis in which rhe basic concepts were to be defined in rerms of algebraic manipularion of power series. This lead to rhe founding of rhe Analyrical Socierywhere Babbage, Herschel and Peacock were active. It is in chis English tradicion of algebraic analysis char rhe pioneering work of De Morgan and Boole found irs roors. Grarran-Guinness givesa derailed account of the work of both logicians, although his discussion of Boole's merhods does not seem entirely adequate. The question is:what are we to make of Boole'spuzzling insistence chat expressions like x + y are "uninterpretable", while he nevertheless manipulated them freely in his mathematical derivations? It is not correct to say that the addition sign can only link disjoint class symbols, since ir is belied by Boole's formal pracrice. A possible solution has been suggested by Hailperin, in his book on Boole's logic, in which he proposes interpreting the "uninterpretable" expressions as denoting signed mulrisets. Oddly, Grattan-Guinness refers to Hailperin 's work,' bur elsewhere (p. 42) adopts rhe view that Boole's addition sign could only link disjoint classes.The chapter concludes with brief accounts of the work of Cauchy, Weierstrass and Bolz.'lno. The next chapter is a detailed account of rhe work of Cantor and his creation of Mengenlehre. The origins of set theory in the theory of trigonometrical series, and rhe ensuing discoveryof rransfinire ordinal and cardinal numbers, are described clearly and succinctly. In addition, the chapter contains an account of Dedekind's philosophy of arithmetic and Cantor's philosophy of mathematics. Cantor's philosophy is an uneasy blend of formalism, platonism and idealism, and has tmderstandably aroused little enthusiasm among philosophers of matl1ematics, alrhough Michael Hallerr has recently smdied it in detail. GrammGuinness emphasizes, rhough, Cantor's magnificent marhematica.l achievements, in defining and clarifying basic conceprs such as measure, dimension and cardinality of sets. Chaprer 4 is a rather miscellaneous chapter, in which six partly independent, partly interrwined stories are told. It begins with developments in sectheory in Germany and France up to the turn of rhe century, goes on to discuss American logic in rhe work of C. S. Peirce and his students, and continues the rheme of algebraic logic with Schroder and his logic of relatives. The remainder of the chapter is given over to Frege, Husserl and Hilbert. The section on Frege is one of rhe more idiosyncratic parts of the book. Grattan-Guinness distinguishes berween Frege, "a mathematician who wrore in ' T Hailperin, Book's logic and Probability, 2nd ed. (Amsterdam: North-Holland, 1986). Reviews 93 German, in a markedly Platonic spirit", and Frege, "a philosopher of language and founder of... (shrink)

Sweet describes the pragmatic foundations of standard logic and applies these foundations to the task of developing a theory of intended models as an extension of standard model theory in which the relevant "intending" is represented pragmatically. Methods of formal logic are used to investigate the structure of the relation between language and the world. The truism which holds that this relation includes the speaker as well as the object spoken about is formally explicated and applied to the problem of (...) illuminating one of the deepest phenomena in standard model theory: the existence of non-isomorphic models of complete theories. To this end it is shown that standard logic admits pragmatic foundations upon which a theory of intended models can be built as an extension of standard model theory. The relevant "intending" is represented by the very forms of verbal behavior which determine the grammatical and logical structure of the sentences whose referential meaning is in question. The uniqueness properties of the class of intended models may then be described. The first section of the book states the immediate goal of standard pragmatics as that of recovering the algebraic structure first-order logic by means of a purely pragmatic construction. The second section, the major portion of the work, then provides the foundation for a semiotic theory of intended models and referential meaning. The theory is then applied to the problem of referential indeterminancy, which has been associated with the phenomenon of scientific revolutions. The theory is also applied to the problem of the apparent synonymy of observationally equivalent theories. Sweet concludes that such theories are not referentially synonymous in any natural sense which is analogous to the paradigm sense in which theories of alternative scales of measurement are referentially synonymous. A novel feature of this book is the formal explication of the idea that the factors, pragmatical in nature, which distinguish the actual meaning of a sentence from among its possible meanings, whose range is defined by the manner in which the sentence is parsed, determine that very parsing. Applicability to natural language of the model-theoretic semantics thereby obtained is made possible by another feature of the book: the development of a theory of locally standard grammar which provides the foundation for representing the structure of natural language as that of standard first-order logic, in a local, as distinguished from a global, sense. This book is intended for scholars in logic, semiotics, and the philosophies of language and of science. Those concerned specifically with such philosophers as Peirce, Martin, and Davidson will also find the study valuable. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Reviewed by:Handbook of Cognitive Mathematics ed. by Marcel DanesiNathan HaydonMarcel Danesi (Ed) Handbook of Cognitive Mathematics Cham, Switzerland: Springer International, 2022, vii + 1383, including indexFor one acquainted with C.S. Peirce, it is hard to see Springer's recent Handbook of Cognitive Mathematics (editor: Marcel Danesi) through none other than a Peircean lens. Short for the cognitive science of mathematics, such a modern, scientific pursuit into the nature and (...) study of mathematical practice would no doubt be found agreeable to Peirce. The fact that references to Peirce appear often throughout the Handbook is a welcome find, with Peirce's ideas being a key subject of half a dozen chapters, and where a reader of any other chapter may well find further connections to Peirce's ideas. After spending time with the Handbook, it is clear that cognitive mathematics has not only embraced some of Pierce's ideas but may be at an important forefront of Peirce studies. In the end, the field may well be an area a Peircean should pay attention to.The book itself is pitched as a reference volume to the field (p. v) with the necessary background to familiarize oneself with the aims and results. The connection to cognitive science may call to mind detailed cognitive models [Ch. 10–14], theories on the origins of numeracy and other theories behind the biological and evolutionary requirements of mathematical thought [Ch.15–18], and the like. These are all present. But a key theme of the Handbook is to situate mathematics not just within more traditional 'cognitive' faculties and the more formal, i.e. algebraic, presentations of mathematics, but also to place mathematics within other human faculties and practices, from the arts to language [Ch. 19–22], within education and learning more broadly [Ch. 23–26], and in relation to the significant, though at first perhaps less quantitative parts of mathematics, like the use of metaphor, gesture, analogy, [End Page 243] abstraction, as well as further cultural and ethnographic considerations [Ch. 5–8]. The Handbook has explicitly taken a broader, more interdisciplinary approach (p. vi–vii) towards the scientific aspects of mathematical practice—choosing to regulate the study not by antecedently drawn opinions about what mathematics is (or has traditionally been taken to be) but by what future quantifiable and diverse study may come to bear on the practice and have to say about those engaging in it.This broad interdisciplinarity has a pragmatist ring, where theory cannot so easily be separated from the normative, social, and, more generally, the more thoroughly human aspects that we encounter and employ when we engage in it. Two further commitments of cognitive mathematics steer us even closer towards Peirce's views. The first—going back, for example, to Lakoff and Núñez's Where Mathematics Comes From (2000), which is taken to be a key early work in shaping the field—is that mathematics is taken to be a language like any other and as such must be learned and situated within our other cognitive faculties (p. vi, Ch. 4). The second—and largely, one imagines, following from a concern for seeing how mathematics is actually practiced—is that mathematics is essentially diagrammatic, involving as it does experimenting in diagrams and employing other signs (one may think here even of the mathematician sketching equations or geometric figures on a sheet of paper). What is perhaps initially most noteworthy about Peirce's philosophy of mathematics is his early concern for both these aspects of the practice.The book (more like a tome of over 1300 pages) is too large to be covered here in all its aspects, hence the review will focus on those aspects a Peircean or Pragmatist may find of interest. In particular, this review focuses on several of the themes above—situating mathematics more broadly within our everyday (cognitive) lives, the necessarily diagrammatic, fallible, and creative aspects of the practice, and, finally, what these aspects may have to say about Peirce's philosophy more generally, particularly with respect to (scientific) inquiry.A Peircean reader may well begin with Pietarinen's "Pragmaticism as Philosophy of Cognitive Mathematics" [Ch. 39] and "Peirce on Mathematical... (shrink)

Mathematics is a struggle for many. To make it more accessible, behavioral and educational scientists are redesigning how it is taught. To a similar end, a few rogue mathematicians and computer scientists are doing something more radical: they are redesigning mathematics itself, improving its ergonomic features. Charles Peirce, an important contributor to ordinary symbolic logic, also introduced a rigorous but non-symbolic, graphical alternative to it that is easier to picture. In the spirit of this iconic logic, George Spencer-Brown founded (...) iconic mathematics. Performing iconic arithmetic, algebra, and even trigonometry, resembles doing calculations on an abacus, which is still popular in education today, has aided humanity for millennia, helps even when it is merely imagined, and ameliorates severe disability in basic computation. Interestingly, whereas some intellectually disabled individuals excel in very complex numerical tasks, others of normal intelligence fail even in very simple ones. A comparison of their wider psychological profiles suggests that iconic mathematics ought to suit the very people traditional mathematics leaves behind. (shrink)