Homotopy Type Theory is a proposed new language and foundation for mathematics, combining algebraic topology with logic. An important rule for the treatment of identity in HoTT is path induction, which is commonly explained by appeal to the homotopy interpretation of the theory's types, tokens, and identities as spaces, points, and paths. However, if HoTT is to be an autonomous foundation then such an interpretation cannot play a fundamental role. In this paper we give a derivation of path induction, motivated (...) from pre-mathematical considerations, without recourse to homotopy theory. (shrink)

Reviewed Works:Gaisi Takeuti, Proof Theory.Georg Kreisel, Proof Theory: Some Personal Recollections.Wolfram Pohlers, Contributions of the Schutte School in Munich to Proof Theory.Stephen G. Simpson, Subsystems of $\mathbf{Z}_2$ and Reverse Mathematics.Solomon Feferman, Proof Theory: A Personal Report.

In this 1992 book, Professor Koslow advances an account of the basic concepts of logic. A central feature of the theory is that it does not require the elements of logic to be based on a formal language. Rather, it uses a general notion of implication as a way of organizing the formal results of various systems of logic in a simple, but insightful way. The study has four parts. In the first two parts the various sources of the general (...) concept of an implication structure and its forms are illustrated and explained. Part 3 defines the various logical operations and systematically explores their properties. A generalized account of extensionality and dual implication is given, and the extensionality of each of the operators, as well as the relation of negation and its dual, are given substantial treatment because of the novel results they yield. Part 4 considers modal operators and studies their interaction with logical operators. By obtaining the usual results without the usual assumptions this new approach allows one to give a very simple account of modal logic minus the excess baggage of possible world semantics. (shrink)

According to Michael Dummett, we may think of the meaning of an expression as given by the principles governing the use we make of it. The principles regulating our linguistic practices can then be grouped into two broad categories (Dummett 1973: 396, 1991: 211). We might state them as follows: I-principles: state the circumstances under which an assertion of a sentence containing the expression in question is warranted. E-principles: state the consequences of asserting a sentence containing the expression. In the (...) case of the logical constants, we may associate a constant's I-principles with the set of its introduction rules and its E-principles with the set of its elimination rules (Dummett 1973: 454).1. (shrink)

The business of mathematics is definition and proof, and its foundations comprise the principles which govern them. Modern mathematics is founded upon set theory. In particular, both the axiomatic method and mathematical logic belong, by their very natures, to the theory of sets. Accordingly, foundational set theory is not, and cannot logically be, an axiomatic theory. Failure to grasp this point leads obly to confusion. The idea of a set is that of an extensional plurality, limited and definite in size, (...) composed of well defined objects.It is the extension of Greek notion of 'number' (arithmos) into Cantor's 'transfinite'. (shrink)

. The term inversion principle goes back to Lorenzen who coined it in the early 1950s. It was later used by Prawitz and others to describe the symmetric relationship between introduction and elimination inferences in natural deduction, sometimes also called harmony. In dealing with the invertibility of rules of an arbitrary atomic production system, Lorenzen’s inversion principle has a much wider range than Prawitz’s adaptation to natural deduction. It is closely related to definitional reflection, which is a principle for reasoning (...) on the basis of rule-based atomic definitions, proposed by Hallnäs and Schroeder-Heister. After presenting definitional reflection and the inversion principle, it is shown that the inversion principle can be formally derived from definitional reflection, when the latter is viewed as a principle to establish admissibility. Furthermore, the relationship between definitional reflection and the inversion principle is investigated on the background of a universalization principle, called the ω- principle, which allows one to pass from the set of all defined substitution instances of a sequent to the sequent itself. (shrink)

La prima vedere pare surprinzătoare corelaţia semnalată în titlul de mai sus, cu atât mai mult cu cât investigarea ei este rodul unei personalităţi de prim rang din domeniul filosofiei antice. Vom găsi aici menţionate, fireşte, teme majore ale reflecţiei tradiţionale, cum sunt cele privitoare la liberul arbitru, problema corp-suflet sau existenţa divinităţii, teme deloc specifice filosofiei analitice.

This is a revised and expanded version of Dummett's 1976 William James Lectures at Harvard. Dummett aims to construct a "base camp," in the theory of linguistic meaning, for an "assault on metaphysical peaks." The book begins with a brief discussion of metaphysical disputes over realism, and ends, fifteen chapters later, with a brief treatment of realism and the theory of meaning. The intervening chapters take up such semantical/logical topics as the following: inference and truth; meaning, knowledge, and understanding; truth (...) and meaning-theories; the justification of deduction; and semantical holism. Dummett frequently interacts with the relevant views of Frege, Tarski, Quine, and Donald Davidson. Several of the chapters are rather technical, focusing on topics in the philosophy of logic. (shrink)