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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.
A History of Natural Deduction and Elementary Logic Textbooks
"... this article is to give a history of the development of this method of doing logic and to characterize what sort of thing is meant nowadays by the name. My view is that the current connotation of the term functions rather like a prototype: there is some exemplar that the term most clearly applies to ..."
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this article is to give a history of the development of this method of doing logic and to characterize what sort of thing is meant nowadays by the name. My view is that the current connotation of the term functions rather like a prototype: there is some exemplar that the term most clearly applies to and which manifests a number of characteristics. But there are other proof systems that differ from this prototypical natural deduction system and are nevertheless correctly characterized as being natural deduction. It is not clear to me just how many of the properties that the prototype exemplifies can be omitted and still have a system that is correctly characterized as a natural deduction system, and I will not try to give an answer. Instead I will focus on a number of features that are manifested to different degrees by the various natural deduction systems. My picture is that if a system ranks `low' on one of these features, it can `make up for it' by ranking high on different features. And it is somehow an overall rating of the total amount of conformity to the entire range of these different features that determines whether any specific logical system will be called a natural deduction system. Some of these features stem from the initial introduction of natural deduction in 1934; but even more strongly, in my opinion, is the effect that elementary textbooks from the 1950s had. There were of course some more technical works that brought the notion of natural deduction into the consciousness of the logical world of the 1950s and 1960s, but I will not consider them in this shortened article. In any case the `ordinary philosopher' of the time would have been little influenced by these works because the huge sway that natural deduction holds over current philosophy is most...
Logical Aspects of Quantum (Non)Individuality
, 2008
"... In this paper I consider some logical and mathematical aspects of the discussion of the identity and individuality of quantum entities. I shall point out that for some aspects of the discussion, the logical basis cannot be put aside; on the contrary, it leads us to unavoidable conclusions which may ..."
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In this paper I consider some logical and mathematical aspects of the discussion of the identity and individuality of quantum entities. I shall point out that for some aspects of the discussion, the logical basis cannot be put aside; on the contrary, it leads us to unavoidable conclusions which may have consequences in how we articulate certain concepts related to quantum theory. Behind the discussion, there is a general argument which suggests the possibility of a metaphysics of nonindividuals, based on a reasonable interpretation of quantum basic entities. I close the paper with a suggestion that consists in emphasizing that quanta should be referred to by the cardinalities of the collections to which they belong, for which an adequate mathematical framework seems to be possible. “The subjectpredicate logic to which we are accustomed depends for its convenience upon the fact that at the usual temperature of the earth there are approximately permanent ‘things’. This would not be true at the temperature of the sun, and is only roughly true at the temperature to which we are accustomed. ” (Russell [1957]) 1