We propose a reconstruction of the constellation of problems and philosophical positions on the nature and number of the primitives of logic in four authors of the nineteenth century logical scene: Peano, Padoa, Frege and Peirce. We argue that the proposed reconstruction forces us to recognize that it is in at least four different senses that a notation can be said to be simpler than another, and we trace the origins of these four senses in the writings of these authors. (...) We conclude that Frege, and even more so Peirce, developed new notations not to make drawing logical conclusions easier but in order to answer the needs of logical analysis. (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)

Omitting experimental data is often considered a violation of scientific integrity. If we consider experimental inquiry as a questioning process, omitting data is seen to be merely an example of tentatively rejecting (‘bracketing’) some of nature’s answers. Such bracketing is not only occasionally permissible; sometimes it is mandated by optimal interrogative strategies. When to omit data is therefore a strategic rather than ethical question. These points are illustrated by reference to Millikan’s oil drop experiment.

The following two articles comprise two sets of Charles Peirce’s manuscripts, “Recent Developments of Existential Graphs and their Consequences for Logic” (MS 498, MS 499, MS 490 & S-36, 1906) and “Assurance through Reasoning” (MS 669 & MS 670, 1911), written for the National Academy of Sciences meetings in 1906 and 1911. The papers are deposited at Houghton Library, Harvard University. Only some parts of MS 470 have been published before, and in somewhat defective form. Although “Assurance” follows “Recent Developments” (...) chronologically, given the expository style of the former it is recommended to be read before “Recent Developments”. As the title indicates, in the latter Peirce goes on to describe his latest discoveries concerning the method and the logic of existential graphs. The transcription reproduces all significant deletions that appear in the original sheets. Editorial comments and additions are given in brackets. [Alt.:] means the beginning of an alternative sequence. [Del. (shrink)

In order to be able to express all possible patterns of dependence and independence between variables, we have to replace the traditional first-order logic by independence-friendly (IF) logic. Our natural concept of truth for a quantificational sentence S says that all the Skolem functions for S exist. This conception of truth for a sufficiently rich IF first-order language can be expressed in the same language. In a first-order axiomatic set theory, one can apparently express this same concept in set-theoretical terms, (...) since the existence of functions can be expressed there. Because of Tarski's theorem, this is impossible. Hence there must exist set-theoretical statements, even provable ones, which are said to be true in first-order models of axiomatic set theory but whose Skolem functions do not all exist. Hence there are provable sentences in axiomatic set theory that are false in accordance with our ordinary conceptions of set-theoretical truth. Such counter-intuitive propositions have been known to exist, but they have been blamed on the peculiarities of very large sets. It is argued here that this explanation is not correct and that there are intuitively false theorems not involving very large sets. Hence the provability or unprovability of a set-theoretical statement, e.g. of the continuum hypothesis (CH) in axiomatic set theory is not necessarily relevant to the truth of CH. (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)

This paper explores the intertwining of uncertainty and values. We consider an important but underexplored field of fundamental uncertainty and values in decision-making. Some proposed methodologies to deal with fundamental uncertainty have included potential surprise theory, scenario planning and hypothetical retrospection. We focus on the principle of uncertainty transduction in hypothetical retrospection as an illustrative case of how values interact with fundamental uncertainty. We show that while uncertainty transduction appears intuitive in decision contexts it nevertheless fails in important ranges of (...) strategic game-theoretic cases. The methodological reasons behind the failure are then examined. (shrink)

[Wilfrid Hodges] During the last forty or so years it has become popular to offer explanations of logical notions in terms of games. There is no doubt that many people find games helpful for understanding various logical phenomena. But we ask whether anything is really 'explained' by these accounts, and we analyse Paul Lorenzen's dialogue foundations for constructive logic as an example. The conclusion is that the value of games lies in their ability to provide helpful metaphors and representations, rather (...) than in any true conceptual analysis. In fact some of the standard explanations of logical notions in terms of competitive games simply don't work. /// [Erik C. W. Krabbe] In an attempt to redeem the Lorenzen-type dialogues from their detractors, it is perhaps best first to provide a survey of the various benefits these dialogues have been supposed to yield. This will be done in Section I. It will not be possible, within the confines of this paper, to scrutinize them all, but in Section II we shall delve deeper into the capacity of this type of dialogue to yield a model for the immanent criticism of philosophical positions. Section III will extend the concept of a dialogue in such a way as to conform better with our intuitive conceptions of what a rational discussion of a position should contain. This will be followed up by a concept of 'winning a dialogue' that takes position midway between the old conception of 'winning one play' and that of the full-fledged presentation of a winning strategy. Concepts of 'rational discussion' are thus shown to be, plausibly, more fundamental than those of proof. In Section IV, I shall discuss the specific problems about dialogical foundations put forward by Wilfrid Hodges. (shrink)

The world of philosophy can perhaps be seen as a microcosm of the world at large. In the course of the last few decades, the world has seen the collapse of the communist system of Russia, a major crisis of the free market economy in the USA, Europe and Japan, and massive economic changes in China. One perspective on contemporary philosophical research is reached by asking what crises the major philosophical traditions, if not literally “systems”, are likewise undergoing and what (...) can be done to find a road ahead. What might a “stimulus package” for philosophy be like (except for inevitably being controversial)? (shrink)

A century ago, Charles S. Peirce proposed a logical approach to modalities that came close to possible-worlds semantics. This paper investigates his views on modalities through his diagrammatic logic of Existential Graphs. The contribution of the GAMMA part of EGs to the study of modalities is examined. Some ramifications of Peirce's remarks are presented and placed into a contemporary perspective. An appendix is included that provides a transcription with commentary of Peirce's unpublished manuscript on modality from 1901.

The theory of existential graphs, which Peirce ultimately divided into four quadrants , is a rich method of analysis in the philosophy of logic. Its $$\upbeta $$ β -part boasts a diagrammatic theory of quantification, which by 1902 Peirce had used in the logical analysis of natural-language expressions such as complex donkey-type anaphora, quantificational patterns describing new mathematical concepts, and cognitive information processing. In the $$\upbeta $$ β -quadrant, he came close to inventing independence-friendly logic, the idea of which he (...) found indispensable in fulfilling the tasks –. (shrink)

We examine Charles S. Peirce's mature views on the logic of science, especially as contained in his later and still mostly unpublished writings. We focus on two main issues. The first concerns Peirce's late conception of retroduction. Peirce conceived inquiry as performed in three stages, which correspond to three classes of inferences: abduction or retroduction, deduction, and induction. The question of the logical form of retroduction, of its logical justification, and of its methodology stands out as the three major threads (...) in his later writings. The other issue concerns the second stage of scientific inquiry, deduction. According to Peirce's later formulation, deduction is divided not only into two kinds but also into two sub-stages: logical analysis and mathematical reasoning, where the latter is either corollarial or theorematic. Save for the inductive stage, which we do not address here, these points cover the essentials of Peirce's latest thinking on the l.. (shrink)

Like Leibniz, C. S. Peirce drew much of the inspiration for his philosophical work from a close study of logical and mathematical reasoning. Now what insights did this study reveal to Peirce? His own answer is formulated as follows: “My first real discovery about mathematical procedure was that there are two kinds of necessary reasoning, which I call the Corollarial and the Theorematic.…” The import of this discovery was lost on philosophers for a long time. The purpose of the present (...) paper is to show what Peirce's insight amounts to, to point out some of its applications, relate it to certain recent developments in the philosophy of logic, and thereby vindicate Peirce's distinction. This undoubtedly will also help to show its present day relevance. (shrink)

Charles Peirce had significant interests in economics. He reworked the mathematical economic models of Cournot and Jevons in the 1870s. He conceived of the transitive axiom of consumer preferences in 1874. Peirce also developed a thesis of the cognitive efficiency of the human mind, abduction. He criticized Newcomb's economic writings. These forays into economics affected the six essays on pragmatism. These interests in economics are integrated with the meaning of the pragmatic maxim in Peirce's 1903 Harvard Lectures.

[Wilfrid Hodges] During the last forty or so years it has become popular to offer explanations of logical notions in terms of games. There is no doubt that many people find games helpful for understanding various logical phenomena. But we ask whether anything is really 'explained' by these accounts, and we analyse Paul Lorenzen's dialogue foundations for constructive logic as an example. The conclusion is that the value of games lies in their ability to provide helpful metaphors and representations, rather (...) than in any true conceptual analysis. In fact some of the standard explanations of logical notions in terms of competitive games simply don't work. /// [ Erik C. W. Krabbe] In an attempt to redeem the Lorenzen-type dialogues from their detractors, it is perhaps best first to provide a survey of the various benefits these dialogues have been supposed to yield. This will be done in Section I. It will not be possible, within the confines of this paper, to scrutinize them all, but in Section II we shall delve deeper into the capacity of this type of dialogue to yield a model for the immanent criticism of philosophical positions. Section III will extend the concept of a dialogue in such a way as to conform better with our intuitive conceptions of what a rational discussion of a position should contain. This will be followed up by a concept of 'winning a dialogue' that takes position midway between the old conception of 'winning one play' and that of the full-fledged presentation of a winning strategy. Concepts of 'rational discussion' are thus shown to be, plausibly, more fundamental than those of proof. In Section IV, I shall discuss the specific problems about dialogical foundations put forward by Wilfrid Hodges. (shrink)

Like Leibniz, C. S. Peirce drew much of the inspiration for his philosophical work from a close study of logical and mathematical reasoning. Now what insights did this study reveal to Peirce? His own answer is formulated as follows: “My first real discovery about mathematical procedure was that there are two kinds of necessary reasoning, which I call the Corollarial and the Theorematic.…” The import of this discovery was lost on philosophers for a long time. The purpose of the present (...) paper is to show what Peirce's insight amounts to, to point out some of its applications, relate it to certain recent developments in the philosophy of logic, and thereby vindicate Peirce's distinction. This undoubtedly will also help to show its present day relevance. (shrink)

During the last forty or so years it has become popular to offer explanations of logical notions in terms of games. There is no doubt that many people find games helpful for understanding various logical phenomena. But we ask whether anything is really 'explained' by these accounts, and we analyse Paul Lorenzen's dialogue foundations for constructive logic as an example. The conclusion is that the value of games lies in their ability to provide helpful metaphors and representations, rather than in (...) any true conceptual analysis. In fact some of the standard explanations of logical notions in terms of competitive games simply don't work. /// [Erik C. W. Krabbe] In an attempt to redeem the Lorenzen-type dialogues from their detractors, it is perhaps best first to provide a survey of the various benefits these dialogues have been supposed to yield. This will be done in Section I. It will not be possible, within the confines of this paper, to scrutinize them all, but in Section II we shall delve deeper into the capacity of this type of dialogue to yield a model for the immanent criticism of philosophical positions. Section III will extend the concept of a dialogue in such a way as to conform better with our intuitive conceptions of what a rational discussion of a position should contain. This will be followed up by a concept of 'winning a dialogue' that takes position midway between the old conception of 'winning one play' and that of the full-fledged presentation of a winning strategy . Concepts of 'rational discussion' are thus shown to be, plausibly, more fundamental than those of proof. In Section IV, I shall discuss the specific problems about dialogical foundations put forward by Wilfrid Hodges. (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)

The theory of the economics of research played a central role in the analysis of scientific method of Charles Sanders Peirce. The present paper describes Peirce's project as he saw it and then puts its machinery to work in an analysis of current issues in the philosophy of science. The aim is to show that, even apart from their historical interest, Peirce's ideas on this subject have a substantial systematic interest.