Francesco Bellucci

Peirce's linear versions of the language of his Existential Graphs (EGs), presented in 1902, are examined. Differences between linear and non-linear languages are explained by permutational invariance and type- vs. occurrence-referentiality: Standard EGs are permutationally invariant with respect to linear EGs, while the Beta part of the system, which corresponds to first-order quantificational theory with identity, is occurrence-referential. However, occurrence-referentiality of Beta graphs constitutes a defect of expressivity: Since the meaning of a quantifier is inextricably connected to the meaning of the sign of identity, certain complex assertions cannot be expressed in the language of Beta graphs without a new extension of its standard notation. © 2021 Elsevier B.V., All rights reserved.

This article offers a critical examination of some central components of Richard Atkins’ Peirce on Inference. First, I discuss Atkins’ reading of Peirce’s notion of a “leading principle” as a conditional whose antecedent is simply the conjunction of an argument’s premises (LPP). Drawing on Peirce’s Baldwin Dictionary entry, I suggest instead that a leading principle is a general description of possible premises and conclusions (LLP), and that the alleged Peircean “confusion” between LPP and LLP is unsubstantiated. I then emphasize Peirce’s distinction between logical and material principles and show how his account of the redundancy of logical leading principles both blocks Carroll’s regress and supports his claim that every mental action is, in form, a valid inference. In the second part, I endorse Atkins’ arguments against identifying Peirce’s late abduction with his early hypothesis, but propose an alternative clarification of the distinction based on the different roles of explanandum and predictions. Finally, I question Atkins’ typology of deduction and his identification of deduction with logical calculus, arguing instead that Peirce takes deductive logic to analyze mathematical reasoning and that deduction proceeds in stages—logical analysis being followed by demonstration, which may be corollarial or theorematic.

In Die Grundgesetze der Arithmetik (I, §48), Frege introduces his rule of the fusion of horizontals, according to which if an occurrence of the horizontal stroke is followed by another occurrence of the same stroke, either in isolation or “contained” in a propositional connective, the two occurrences can be fused with each other. However, the role of this rule, and of the horizontal sign more generally, is controversial; Michael Dummett notoriously claimed, for instance, that the horizontal is “wholly superfluous” in Frege's logical system. In this paper, we challenge Dummett's view by providing a comprehensive analysis of the significance of the horizontal stroke. After some preliminary remarks, we argue that even if Frege's connectives in some sense “contain” the horizontal, yet they are total functions. Then, we take up the question of the sense expressed by the horizontal, and we claim that, unlike other sentential operators, the horizontal is not sense‐compositional. Finally, we consider the semantic and pragmatic aspects of Frege's horizontal in connection to his judgment stroke and the double judgment stroke. Contra Dummett, we argue that the horizontal is a special and indispensable element of Frege's logic.

The paper examines Peirce’s interpretation of Kant’s philosophy of mathematics, with special reference to Kant’s notion of “construction”, and its transformation into a theory of diagrammatic reasoning. For Peirce, all deductive or mathematical reasoning proceeds by the construction and manipulation of diagrams. Peirce does not accept Kant’s idea that only concepts of quanta and quantitas can be constructed. Logical concepts and relations, although they are not quantitative, can be diagrammatized, namely “constructed” in Kant’s sense. What is quantitative in logical construction is not the object that is constructed, but the means of constructing it. This shift from the Kantian idea that the object is quantitative to the idea that the means to construct it is quantitative is operated by Peirce’s notion of a “scale of quantity”, that is, the idea that in mathematical reasoning an intermediate, quantitative construction is introduced and removed afterwards. The paper considers two examples of “construction” of syllogistic logic, Euler diagrams and Boolean algebra. Both are “scales of quantity”, that is, quantitative objects having quantitative relations that serve to the representation of non-quantitative objects having non-quantitative relations.

The paper argues against what I call the “Fregean interpretation” of Peirce’s distinction between the immediate and the dynamic object of a sign, according to which Peirce’s dynamic object is akin to Frege’s Bedeutung, while Peirce’s immediate object is akin to Frege’s Sinn. After having exposed the Fregean interpretation, I briefly reconstruct the genesis of Peirce’s notion of immediate object in his semiotic writings of the years 1904–1909 and defend the view that, according to Peirce, only propositions have immediate objects.

The author argues that “sign” is most often used by Peirce in the sense of “proposition,” or more precisely, that only propositions adequately satisfy Peirce’s definition of the sign. The argument has the form of an exegetical syllogism in Barbara: complete signs are propositions (major premise), Peirce’s definition of the sign is actually a definition of complete signs (minor premise); therefore, the definition of the sign is actually a definition of propositions (conclusion). The author provides textual evidence in support of both the major and the minor premise of my exegetical syllogism, especially in Peirce’s mature production (1902–1912). He also evidences how this view helps making sense of several things that Peirce says about signs and their classification, including his claim that “logic has to be considered as semiotics.”

Abstract:"Issues of Pragmaticism" (1905) contains the only published version of Peirce's doctrine of "critical common-sensism." One of the claims of that doctrine is that common sense beliefs are invariably vague. In this paper, we seek to explain this claim. We begin by providing a philological reconstruction of the drafts of "Issues of Pragmaticism" and a comparison of Peirce's several, successive expositions of critical common-sensism across those drafts. Then we examine Peirce's theory of vagueness; we show that there is both a "subjectal" and a "predicative" variety of vagueness, and that Peirce construes predicative vagueness according to three distinct models. Finally, we assess in what sense, i.e., according to which of the three models, common sense beliefs may be said to be invariably vague.

The problem of the unity of the proposition—what distinguishes a proposition from a mere list of constituents, so that the former is able to say something while the latter is not?—is as old as philosophy. It is evoked at the end of Plato’s Sophist, where the Stranger affirms that when one makes a statement “he does not merely give names, but he reaches a conclusion by combining verbs with nouns” ; and it is discussed by Aristotle in De Interpretatione, where it is said that since “falsity and truth have to do with combination and separation”, then “the first single statementmaking sentence is the affirmation, next is the negation” ; but “every statement-making..

In his later years, Peirce devoted much energy to the project of a book on logic, whose intended title was “Logic, considered as Semeiotic.” That the science of logic is better considered as semeiotic is indeed one of the most fundamental tenets of Peirce’s mature philosophy of logic. But what is the primary motivation for considering logic as semeiotic and what advantages did Peirce see in doing so? If logic is to be considered as semeiotic, this can only mean that its objects and their functioning are to be described in purely semeiotical terms. But did Peirce succeed in providing such a description? This paper focuses on the semeiotical functioning of the fundamental triad of logic: terms (rhemes), propositions (dici-signs), and arguments; it also discusses the idea of an extension of the science of logic and offers examples of the kind of extension that Peirce had in mind.

The article offers an analysis of Peirce's notion of “rhema.” It examines and explains Peirce's definition of the rhema; it identifies and solves two problems that are direct consequences of the definition. The first problem is that proper names, while classified as rhemata, do not satisfy Peirce's definition of the rhema. The second problem is that Peirce also calls “rhemata” the results of propositional analysis that however do not satisfy his own definition of the rhema. Peirce himself solves the first problem by generalizing the notion of rhema into that of “seme.” I argue that we can solve the second problem if, following M. Dummett, we distinguish propositional analysis from propositional decomposition.

The paper seeks to explain in what sense pragmatism was for Peirce a doctrine of logic. It is argued that pragmatism is a doctrine of logic for Peirce because its maxim, the pragmatic maxim, is a maxim of the methodeutic of abduction, i.e., concerns the method of selecting hypotheses for experimental testing. The paper also connects this idea to Peirce’s 1913 thesis according to which pragmatism contributes to the security but not to the uberty of reasoning. The connection consists in that by excluding hypotheses that cannot be tested, the maxim renders the whole process of reasoning more secure, while any explanatory hypotheses is “uberous” in Peirce’s sense as any other.

Peirce’s claims that methodeutic “concerns abduction alone” and that “pragmatism contributes to the security of reasoning but hardly to its uberty” are explained. They match as soon as a third claim is taken into account, namely that “pragmatism is the logic of abduction,” not of deduction or induction. Since methodeutic concerns abduction and not deduction or induction, it follows that pragmatism is a maxim of methodeutic. Then, since pragmatism contributes to the security of reasoning but not to its uberty, it follows that methodeutic contributes to the security of the only reasoning it is concerned with, namely abduction. We then explain two related issues of methodeutic of abduction. First, in addition to the maxim of pragmatism, which suggests how to choose among experimental hypotheses contributing to the security of reasoning, there is the maxim of simplicity, which suggests hypotheses that are preferable for investment and which contributes to uberty of reasoning. Second, a third maxim of abduction is economy, which suggests adopting hypotheses that contribute to the advantageousness of reasoning even when pragmatism and simplicity cease to apply. These three maxims—experientiality for security, simplicity for uberty, and economy for advantageousness—are the bedrocks of Peirce’s methodeutic of abduction.

This paper provides an analysis of the notational difference between Beta Existential Graphs, the graphical notation for quantificational logic invented by Charles S. Peirce at the end of the 19th century, and the ordinary notation of first-order logic. Peirce thought his graphs to be “more diagrammatic” than equivalently expressive languages for quantificational logic. The reason of this, he claimed, is that less room is afforded in Existential Graphs than in equivalently expressive languages for different ways of representing the same fact. The reason of this, in turn, is that Existential Graphs are a non-linear, occurrence-referential notation. As a non-linear notation, each graph corresponds to a class of logically equivalent but syntactically distinct sentences of the ordinary notation of first-order logic that are obtained by permuting those elements that in the graphs lie in the same area. As an occurrence-referential notation, each Beta graph corresponds to a class of logically equivalent but syntactically distinct sentences of the ordinary notation of first-order logic in which the identity of reference of two or more variables is asserted. In brief, Peirce’s graphs are more diagrammatic than the linear, type-referential notation of first-order logic because the function that translates the latter to the graphs does not define isomorphism between the two notations.

The Syllabus for Certain Topics of Logic is a long treatise that Peirce wrote in October and November to complement the material of his 1903 Lowell Lectures. The last of the eight lectures was on abduction, first entitled “How to Theorize” and then “Abduction.” Of abduction, the Syllabus states that its “conclusion is drawn in the interrogative mood ”.1 This is not the first time that Peirce associates abduction to interrogations,2 but the statement is significant because it is the first time that the “interrogative mood” is ascribed to speculative grammar. At the same time, such...

Expressively equivalent logical languages can enunciate logical notions in notationally diversified ways. Frege’s Begriffsschrift, Peirce’s Existential Graphs, and the notations presented by Wittgenstein in the Tractatus all express the sentential fragment of classical logic, each in its own way. In what sense do expressively equivalent notations differ? According to recent interpretations, Begriffsschrift and Existential Graphs differ from other logical notations because they are capable of “multiple readings.” We refute this interpretation by showing that there are at least three different kinds of such multiple readings. While readings of the first kind do not capture any essential difference among notations but only among vocabularies, corresponding to readings of the second and the third kind two general parameters according to which notations may differ are defined: linearity vs. non-linearity, and tabularity vs. non-tabularity. This answers the question of how there can be substantially different but expressively equivalent logical notations.

A rose is a rose is a rose is a rose.As is well known, according to Charles S. Peirce one of the principal tasks of logic is the analysis of reasoning. This was indeed the explicit purpose of his logical algebras and graphical logic, and Peirce often credits himself with possessing a special gift for logical analysis. Yet he surprisingly also holds that “absolute completeness of logical analysis is no less unattainable [than] is omniscience. Carry it as far as you please, and something will always remain unanalyzed.” 1 The question thus arises as to what Peirce could have meant by saying that, in logical analysis, something remains unanalyzed. This question, important..

Cet article soutient que s’il existe une philosophie du langage chez Peirce, il faut la chercher dans sa conception de la grammaire spéculative. Je reconstitue l’évolution de la grammaire spéculative de Peirce dans la période 1894- 1906, et je montre que, tandis que dans les années 1890 la grammaire spéculative est considérée comme une théorie de la proposition, Peirce la conçoit dès 1903 comme une classification générale des signes, incluant une théorie des actes de langage tout à fait pionnière.

In a brilliant article published in a past issue of the Transactions, Jorge A. Flórez examines Peirce’s theory of the origin of abduction in Aristotle. In the article Flórez makes two substantial points. In the first place, he argues that Peirce’s theory of the origin of abduction in the 25th chapter of the second book of the Prior Analytics is mistaken, because in that chapter Aristotle discusses first-figure syllogisms with a dialectic or contingent minor premise, and not, as Peirce thought, second-figure syllogisms. The second substantial point that Flórez makes is that something having the form of a Peircean abduction is rather to be found in the 13th chapter of the first book of the Posterior Analytics...

This paper focuses on two important notational devices that were embedded in Husserl’s and Peirce’s notations, the sign of inclusion and the sign of assertion. Husserl first criticizes, then follows Schröder in taking inclusion to be a simpler concept than equality, and endows his logical notation with a sign of inclusion. This was due to Peirce’s notational innovations and arguments, by which Husserl is indirectly influenced through of Schröder. Further, like Frege Husserl endows his notation with a sign of assertion, claiming that without some such sign it would be impossible to distinguish the assertion of a propositional content from its occurrence unasserted in some larger propositional context. Against this idea of Frege and Husserl, Peirce had a powerful argument—the compositional structure of a formula is de facto the sign of its assertion—, and Peirce’s algebraical and graphical notations can be seen as notational realizations of this principle.