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VII Zusammenfassung IX Extended Abstract XI Acknowledgements XIII I

Consistency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 3.2 Hintikka Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 3.3 Primitive Equality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 3.4 Model Existence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 4 Extensional Higher-Order Resolution: ER 42 4.1 A Review of HORES and ER . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 4.2 Basic Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 4.3 Lifting Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 4.4 Completeness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 4.5 Theorem Equivalence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 5 Extensional Higher-Order Paramodulation: EP 57 5.1 A Naive and Incomplete Adaptation...

Abstract. In this article, we study the prehistory of type theory up to 1910 and its development between Russell and Whitehead’s Principia Mathematica ([71], 1910–1912) and Church’s simply typed λ-calculus of 1940. We first argue that the concept of types has always been present in mathematics, though nobody was incorporating them explicitly as such, before the end of the 19th century. Then we proceed by describing how the logical paradoxes entered the formal systems of Frege, Cantor and Peano concentrating on Frege’s Grundgesetze der Arithmetik for which Russell applied his famous paradox 1 and this led him to introduce the first theory of types, the Ramified Type Theory (rtt). We present rtt formally using the modern notation for type theory and we discuss how Ramsey, Hilbert and Ackermann removed the orders from rtt leading to the simple theory of types stt. We present stt and Church’s own simply typed λ-calculus (λ→C 2) and we finish by comparing rtt, stt and λ→C. §1. Introduction. Nowadays, type theory has many applications and is used in many different disciplines. Even within logic and mathematics, there are many different type systems. They serve several purposes, and are formulated in various ways. But, before 1903 when Russell first introduced

Functions play a central role in type theory, logic and computation. We describe how the notions of functionalisation (the way in which functions can be constructed) and instantiation (the process of applying a function to an argument) have been developed in the last century. We explain how both processes were implemented in Frege's Begriffschrift [17], Russell's Ramified Type Theory [42] and the lambda-calculus (originally introduced by Church [12, 13]) showing that the lambda-calculus misses a crucial aspect of functionalisation. We then pay attention to some special forms of function abstraction that do not exist in the lambda-calculus and we show that various logical constructs (e.g., let expressions and definitions and the use of parameters in mathematics), can be seen as forms of the missing part of functionalisation. Our study of the function concept leads...

This article is a brief review of the type free λ-calculus and its basic rewriting notions, and of the pure type system framework which generalises many type systems. Both the type free λ-calculus and the pure type systems are presented using variable names and de Bruijn indices. Using the presentation of the λ-calculus with de Bruijn indices, we illustrate how a calculus of explicit substitutions can be obtained. In addition, de Bruijn's notation for the λ-calculus is introduced and some of its advantages are outlined.

This paper provides an abstract syntax for a formal language of mathematics. We call our language Weak Type Theory (abbreviated WTT ). WTT will be as faithful as possible to the mathematician 's language yet will be formal and will not allow ambiguities. WTT can be used as an intermediary between the natural language of the mathematician and the formal language of the logician. As far as we know, this is the rst extensive formalization of an abstract syntax of a formal language of mathematics. We compare our work with existing formalizations of languages of mathematics. 1

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In the following we describe the basic ideas underlying\Omega\Gamma mkrp, an interactive proof development environment [6]. The requirements for this system were derived from our experiences in proving an interrelated collection of theorems of a typical textbook on semi-groups and automata [3] with the first-order theorem prover mkrp [11]. An important finding was that although current automated theorem provers have evidently reached the power to solve non-trivial problems, they do not provide sufficient assistance for proving the theorems contained in such a textbook. On account of this, we believe that significantly more support for proof development can be provided by a system with the following two features: -- The system must provide a comfortable human-oriented problem-solving environment. In particular, a human user should be able to specify the problem to be solved in a natural way and communicate on proof