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What is an Inference Rule?
 Journal of Symbolic Logic
, 1992
"... 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 ..."
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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 rstorder logic.
Types in logic and mathematics before 1940
 Bulletin of Symbolic Logic
, 2002
"... 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 λcalculus of 1940. We first argue that the concept of types has always been present in mathematics, thou ..."
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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 λcalculus 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 λcalculus (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
Revisiting the Notion of Function
"... 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 pro ..."
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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 lambdacalculus (originally introduced by Church [12, 13]) showing that the lambdacalculus misses a crucial aspect of functionalisation. We then pay attention to some special forms of function abstraction that do not exist in the lambdacalculus 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...
Reviewing the classical and the de Bruijn notation for λcalculus and pure type systems
 Logic and Computation
, 2001
"... This article is a brief review of the type free λcalculus and its basic rewriting notions, and of the pure type system framework which generalises many type systems. Both the type free λcalculus and the pure type systems are presented using variable names and de Bruijn indices. Using the presentat ..."
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This article is a brief review of the type free λcalculus and its basic rewriting notions, and of the pure type system framework which generalises many type systems. Both the type free λcalculus and the pure type systems are presented using variable names and de Bruijn indices. Using the presentation of the λcalculus with de Bruijn indices, we illustrate how a calculus of explicit substitutions can be obtained. In addition, de Bruijn's notation for the λcalculus is introduced and some of its advantages are outlined.
AttributeValue and Relational Learning: A Statistical Viewpoint
 PROCEEDINGS OF THE ICML2000 WORKSHOP ON ATTRIBUTEVALUE AND RELATIONAL LEARNING: CROSSING THE BOUNDARIES
, 2000
"... ... 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 lea ..."
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... 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.
An Abstract Syntax for a Formal Language of Mathematics
, 2001
"... 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 be ..."
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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
DEA Programmation 1996 Semantique denotationnelle
"... n'est plus ambigu + @ @ @ @ @ @ @ a + > > > > > > > > ~ ~ ~ ~ ~ ~ ~ ~ b c ## + > > > > > > > > ~ ~ ~ ~ ~ ~ ~ + @ @ @ @ @ @ @ @ c a b Probleme: d ..."
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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
Bayesian Learning of Bayesian Networks Bayesian Learning of Bayesian Networks with Informative
"... This paper presents and evaluates an approach to Bayesian model averaging where the models are Bayesian nets (BNs). Prior distributions are dened using stochastic logic programs and the MCMC MetropolisHastings algorithm is used to (approximately) sample from the posterior. Experiments using data g ..."
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This paper presents and evaluates an approach to Bayesian model averaging where the models are Bayesian nets (BNs). Prior distributions are dened using stochastic logic programs and the MCMC MetropolisHastings algorithm is used to (approximately) sample from the posterior. Experiments using data generated from known BNs have been conducted to evaluate the method. The experiments used 6 dierent BNs and varied: the structural prior, the parameter prior, the MetropolisHasting proposal and the data size. Each experiment was repeated three times with dierent random seeds to test the robustness of the MCMCproduced results. Our results show that with eective priors (i) robust results are produced and (ii) informative priors improve results signicantly.