Important Issues in Foundational Formalisms

This article discusses my work in the last few years on logical formalisms which have been shown to be useful to various aspects of Natural and Programming Languages and for foundational formalisms. In this period, I have been involved in two extensive programs: 1. The rst program concerns languages which exhibit various ways of combining expressiveness with logic. While I do not propose that any of these languages is ideal, I believe that they illustrate the fruitfulness of bringing together ideas from distinct disciplines. Central to the program will be Logic,-calculus and Type Theory, which have played an important role not only in foundational discussions, but also in applied formal semantics; speciically, the semantics of natural language (nl) and of programming languages (pl). The general goal here has been to nd expressive and unifying theories which keep the earlier advantages but bring about new dimensions. This goal moreover extends to nding a general framework which can be used to compare earlier theories and to carry results from one theory to another without duplication of work. Issues that play a great role in the general framework include full expressiveness and logic, intensionality versus extensionality, polymorphism, internal deenability of determiners and quantiiers, xed point operators and self-application, avoidance of Russell's and Curry's paradoxes, and property and truth theories. This program comes from my ongoing work at and collaboration with Edinburgh where the stimulating environment of logic, foundation and language has been invaluable. 2. From the point of view of notation and language, I aimed at studying typed and type-free formalisms and at investigating a-notation which can be used to generalise the various existing type theories, and to improve the computational power of the-calculus by making substitution explicit, by reening reduction, by introducing deenitions to the syntax and by rewriting terms so that their reduction can be made more eecient. This work comes from my ongoing collaboration with Eindhoven, the centre of AUTOMATH, the theorem prover which have inspired much research on language and formalisms. I hope that by studying general formalisms from the outside and by going inside and unpacking the notation, one can gain even much further insights than achieved so far. 1 Full expressiveness and logic As is well-known, combining full type-free-calculus with logic leads to contradiction. The reason is that one can use R x ::xx to derive RR = :(RR). This is known as Russell's paradox. Even when …