What is an inference rule? This question does not have a unique answer. One usually nds two distinct standard answers in the literature: validity inference ( ` v ' if for every substitution , the validity of [] entails the validity of [']), and truth inference ( ` t ' if for every substitution , the truth of [] entails the truth of [']). In this paper we introduce a general semantic framework that allows us to investigate the notion of inference more carefully. Validity inference and truth inference are in some sense the extremal points in our framework. We investigate the relationship between various types of inference in our general framework, and consider the complexity of deciding if an inference rule is sound, in the context of a number of logics of interest: classical propositional logic, a nonstandard propositional logic, various propositional modal logics, and rst-order logic.

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

this paper gives a full account of all relevant issues, but argue that starting from this generalised viewpoint and working down towards actual learning problems (e.g. decision tree learning, regression, ILP, etc) makes it easier to nd the essential contrasts and similarities between dierent learning problems. Our primary goal (not achieved here) is to abstract away from supercial issues, such as the concrete syntactic representation of a problem or worse the sociological origin of an approach.

ui n'est plus ambigu + @ @ @ @ @ @ @         a + > > > > > > > > ~ ~ ~ ~ ~ ~ ~ ~ b c ## + > > > > > > > > ~ ~ ~ ~ ~ ~ ~ + @ @ @ @ @ @ @ @         c a b Probleme: definir la syntaxe de facon que toutes les ambigu tes syntaxiques soient resolubles simplement par l'analyse syntaxique. Ce n'est pas les cas des vecteurs et fonctions FORTRAN, comme on verra plus avant. Semantique denotationnelle DEA'96 Roberto Di Cosmo 4 Semantique . semantique: Une fois bien defini quels sont les programmes syntaxiquement correctes, on doit pouvoir dire de facon precise et univoque ce que chaque programme "fait" a l'execution (ce qu'il calcule, ou comment il se conduit), y compris si l'execution donne lieu a des erreurs: cela est indispensable pour -- l'utilisateur -- l'implementateur

n'est plus ambigu + @ @ @ @ @ @ @         a + > > > > > > > > ~ ~ ~ ~ ~ ~ ~ ~ b c ## + > > > > > > > > ~ ~ ~ ~ ~ ~ ~ + @ @ @ @ @ @ @ @         c a b Probleme: definir la syntaxe de facon que toutes les ambigu tes syntaxiques soient resolubles simplement par l'analyse syntaxique. Ce n'est pas les cas des vecteurs et fonctions FORTRAN, comme on verra plus avant. Semantique denotationnelle DEA'96 Roberto Di Cosmo 4 Semantique . semantique: Une fois bien defini quels sont les programmes syntaxiquement correctes, on doit pouvoir dire de facon precise et univoque ce que chaque programme "fait" a l'execution (ce qu'il calcule, ou comment il se conduit), y compris si l'execution donne lieu a des erreurs: cela est indispensable pour -- l'utilisateu

Mathematical Services . . . . . . . . . . . . . . . . . . . 10 2.3.2 Autonomy and Decentralization . . . . . . . . . . . . . . . . . . . 11 2.3.3 Coordination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 3 Distributed Articial Intelligence 12 3.1 Agent-Oriented Programming . . . . . . . . . . . . . . . . . . . . . . . . 12 3.2 The Knowledge Query and Manipulation Language . . . . . . . . . . . . 13 3.3 Coordination in Multi-Agent Systems . . . . . . . . . . . . . . . . . . . 13 4 Agent Technology for Distributed Mathematical Reasoning 15 4.1 MathWeb Agents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 4.2 Communication between MathWeb agents . . . . . . . . . . . . . . . . . 18 4.2.1 Technical Issues . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 4.2.2 Characterization of Reasoning Capabilities . . . . . . . . . . . . 18 4.2.3 Context in Mathematical Communication . . . . . . . . . . . . . 19 4.3 Coordination of MathWeb Agents . . . . . . . . . . . . . . . . . . . . . 20 5 Summary and Work Plan 22 5.1 Work Plan . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 1