The language of the basic logic of proofs extends the usual propositional language by forming sentences of the sort x is a proof of F for any sentence F. In this paper a complete axiomatization for the basic logic of proofs in Heyting Arithmetic HA was found.

This highly acclaimed book is a major contribution to the philosophy of language as well as a systematic interpretation of Frege, indisputably the father of ...

The present volume is intended as an all-round introduction to constructivism. Here constructivism is to be understood in the wide sense, and covers in particular Brouwer's intuitionism, Bishop's constructivism and A.A. Markov's constructive recursive mathematics. The ending "-ism" has ideological overtones: "constructive mathematics is the (only) right mathematics"; we hasten, however, to declare that we do not subscribe to this ideology, and that we do not intend to present our material on such a basis.

Following the speech act theory, we take hypotheses and assertions as linguistic acts with different illocutionary forces. We assume that a hypothesis is justified if there is at least a scintilla of evidence for the truth of its propositional content, while an assertion is justified when there is conclusive evidence that its propositional content is true. Here we extend the logical treatment for assertions given by Dalla Pozza and Garola by outlining a pragmatic logic for assertions and hypotheses. On the (...) basis of this extension we analyse the standard logical opposition relations for assertions and hypotheses. We formulate a pragmatic square of oppositions for assertions and a hexagon of oppositions for hypotheses. Finally, we give a mixed hexagon of oppositions to point out the opposition relations for assertions and hypotheses. (shrink)

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)

C.S. Peirce held what is nowadays called a “commitment view” of assertion. According to this type of view, assertion is a kind of act that is determined by its “normative effects”: by asserting a proposition one undertakes certain commitments, typically to be able to provide reason to believe what one is asserting, or, in Peirce’s words, one “takes responsibility” for the truth of the proposition one asserts. Despite being an early adopter of the view, if Peirce’s commitment view of assertion (...) is mentioned at all in contemporary discussions it is only in passing. His view is, however, far more complex and nuanced than he has been given credit for. My primary goal here, then, is to get a better understanding of Peirce’s version of a commitment view of assertion. I also argue that figuring out the details of Peirce’s theory of assertion can also provide us with a viable way to respond to problems that contemporary commitment views of assertion face. (shrink)

1 INTRODUCTION Above the other titles he might justly have claimed, Charles S. Peirce prized the title 'logician'. He expressed in several places his ...

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)

Charles Peirce’s alpha system \ is reformulated into a deep inference system where the rules are given in terms of deep graphical structures and each rule has its symmetrical rule in the system. The proof analysis of \ is given in terms of two embedding theorems: the system \ and Brünnler’s deep inference system for classical propositional logic can be embedded into each other; and the system \ and Gentzen sequent calculus \ can be embedded into each other.

Admissible rules of a logic are those rules under which the set of theorems of the logic is closed. In this paper, a Gentzen-style framework is introduced for analytic proof systems that derive admissible rules of non-classical logics. While Gentzen systems for derivability treat sequents as basic objects, for admissibility, the basic objects are sequent rules. Proof systems are defined here for admissible rules of classes of modal logics, including K4, S4, and GL, and also Intuitionistic Logic IPC. With minor (...) restrictions, proof search in these systems terminates, giving decision procedures for admissibility in the logics. (shrink)

In the present paper I wish to regard constructivelogic as a self-contained system for the treatment ofepistemological issues; the explanations of theconstructivist logical notions are cast in anepistemological mold already from the outset. Thediscussion offered here intends to make explicit thisimplicit epistemic character of constructivism.Particular attention will be given to the intendedinterpretation laid down by Heyting. This interpretation, especially as refined in the type-theoretical work of Per Martin-Löf, puts thesystem on par with the early efforts of Frege andWhitehead-Russell. This quite (...) recent work, however,has proved valuable not only in the philosophy andfoundations of mathematics, but has also foundpractical application in computer science, where thelanguage of constructivism serves as an implementableprogramming language, and within the philosophy oflanguage.\footnote{Nordstr\"{o}m et al. give an overview of the work in computerscience, whereas Ranta provides an impressiveconstructivist alternative to Montague Grammar usingthe richer type structure of Martin-L\"{o}f in placeof the simple classical type theory of Church.} Mypresentation will be carried out through a contrastwith standard metamathematical work.\footnote{Troelstra and van Dalen give an encyclopedictreatment of the metamathematics of constructivism.}In the course of the development I have occasion tooffer some novel considerations on thenature of proof and inference. (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)

Following the speech act theory, we take hypotheses and assertions as linguistic acts with different illocutionary forces. We assume that a hypothesis is justified if there is at least a scintilla of evidence for the truth of its propositional content, while an assertion is justified when there is conclusive evidence that its propositional content is true. Here we extend the logical treatment for assertions given by Dalla Pozza and Garola (1995, Erkenntnis, 43, 81–109) by outlining a pragmatic logic for assertions (...) and hypotheses. On the basis of this extension we analyse the standard logical opposition relations for assertions and hypotheses. We formulate a pragmatic square of oppositions for assertions and a hexagon of oppositions for hypotheses. Finally, we give a mixed hexagon of oppositions to point out the opposition relations for assertions and hypotheses. (shrink)

Peirce and Frege both distinguished between the propositional content of an assertion and the assertion of a propositional content, but with different notational means. We present a modification of Peirce’s graphical method of logic that can be used to reason about assertions in a manner similar to Peirce’s original method. We propose a new system of Assertive Graphs, which unlike the tradition that follows Frege involves no ad hoc sign of assertion. We show that axioms of intuitionistic logic can be (...) derived from AGs, and argue that AGs analyse and represent assertions and illocutionary content in a way which is motivated both by its logical properties and its historical connection with the ideas that led to the development of the graphical method. (shrink)