This volume is dedicated to Prof. Dag Prawitz and his outstanding contributions to philosophical and mathematical logic. Prawitz's eminent contributions to structural proof theory, or general proof theory, as he calls it, and inference-based meaning theories have been extremely influential in the development of modern proof theory and anti-realistic semantics. In particular, Prawitz is the main author on natural deduction in addition to Gerhard Gentzen, who defined natural deduction in his PhD thesis published in 1934. The book opens with an (...) introductory paper that surveys Prawitz's numerous contributions to proof theory and proof-theoretic semantics and puts his work into a somewhat broader perspective, both historically and systematically. Chapters include either in-depth studies of certain aspects of Dag Prawitz's work or address open research problems that are concerned with core issues in structural proof theory and range from philosophical essays to papers of a mathematical nature. Investigations into the necessity of thought and the theory of grounds and computational justifications as well as an examination of Prawitz's conception of the validity of inferences in the light of three “dogmas of proof-theoretic semantics” are included. More formal papers deal with the constructive behaviour of fragments of classical logic and fragments of the modal logic S4 among other topics. In addition, there are chapters about inversion principles, normalization of proofs, and the notion of proof-theoretic harmony and other areas of a more mathematical persuasion. Dag Prawitz also writes a chapter in which he explains his current views on the epistemic dimension of proofs and addresses the question why some inferences succeed in conferring evidence on their conclusions when applied to premises for which one already possesses evidence. (shrink)

The concept of “necessity of thought” plays a central role in Dag Prawitz’s essay “Logical Consequence from a Constructivist Point of View” (Prawitz 2005). The theme is later developed in various articles devoted to the notion of valid inference (Prawitz, 2009, forthcoming a, forthcoming b). In section 1 I explain how the notion of necessity of thought emerges from Prawitz’s analysis of logical consequence. I try to expound Prawitz’s views concerning the necessity of thought in sections 2, 3 and 4. (...) In sections 5 and 6 I discuss some problems arising with regard to Prawitz’s views. (shrink)

We discuss the semantical categories of base and object implicit in the Curry-Howard theory of types and we derive derive logic and, in particular, the comprehension principle in the classical version of the theory. Two results that apply to both the classical and the constructive theory are discussed. First, compositional semantics for the theory does not demand ‘incomplete objects’ in the sense of Frege: bound variables are in principle eliminable. Secondly, the relation of extensional equality for each type is definable (...) in the Curry-Howard theory. (shrink)

This paper is an introduction: in particular, to algebras of relations of various ranks, and in general, to the part of algebraic logic algebraizing quantifier logics. The paper has a survey character, too. The most frequently used algebras like cylindric-, relation-, polyadic-, and quasi-polyadic algebras are carefully introduced and intuitively explained for the nonspecialist. Their variants, connections with logic, abstract model theory, and further algebraic logics are also reviewed. Efforts were made to make the review part relatively comprehensive. In some (...) directions we tried to give an overview of the most recent results and research trends, too. (shrink)

Keith Hossack's thesis is that knowledge is a conceptually primitive and metaphysically fundamental relation between a mind and a fact. He argues that in terms of the simple relation of knowledge we can analyze central notions of epistemology , of semantics , of modality and a priori knowledge , of psychology , and of linguistics . He does so in a framework that includes a fairly rich faculty psychology and that stresses causation: knowledge can be caused by belief, but because (...) knowledge is simple, it is not any kind of belief. He regards his enterprise as metaphysics, and he proceeds in a rather grand manner. The beginning of his Preface might remind one of the opening paragraphs of Leibniz's The Monadology.If knowledge is a simple relation between minds and facts, then we want to know a bit about facts. Hossack takes facts to consist in universals being instantiated among things. He favours a view of universals as respects in which things resemble each other , but his theory is at least officially silent about which universals exist, leaving that as a matter for scientific inquiry . But he takes metaphysics to be a science . Here he is being coy, and it is worth making out how. Hossack uses multigrade relations; these are relations of no fixed polyadicity that for any natural number n may be instantiated by n things. Vectors are bearers of such relations. The relation among working men of uniting to form a union is a multigrade relation, and the founders of the International Workers of the World are a vector bearing this relation. (shrink)