{"product_id":"witness-theory-notes-on-955-calculus-and-logic-paperback","title":"Witness Theory: Notes on \u0026#955;-calculus and Logic - Paperback","description":"\u003cdiv\u003e\u003cp style=\"text-align: right;\"\u003e\u003ca href=\"https:\/\/reportcopyrightinfringement.com\/\" target=\"_blank\" rel=\"nofollow\"\u003e\u003cb\u003eReport copyright infringement\u003c\/b\u003e\u003c\/a\u003e\u003c\/p\u003e\u003c\/div\u003e\u003cp\u003eby \u003cb\u003eAdrian Rezuş\u003c\/b\u003e (Author)\u003c\/p\u003e\u003cp\u003e\u003c\/p\u003e\u003cp\u003eThis book is concerned with the mathematical analysis of the concept of formal proof in classical logic, and records - in substance - a longer exercise in applied λ-calculus.\u003cbr\u003e Following colloquialisms going back to L. E. J. Brouwer, the objects of study in this enterprise are called witnesses. A witness is meant to represent the logical proof of a classically valid formula, in a given proof-context. The formalisms used to express witnesses and their equational behaviour are extensions of the pure  typed' λ-calculus, considered as equational theories.\u003cbr\u003e Formally, a witness is generated from decorated - or  typed' - witness variables, representing assumptions, and witness operators, representing logical rules of inference.\u003cbr\u003e The equational specifications serve to define the witness operators.\u003cbr\u003e In general, this can be done by ignoring the  typing', i.e., the logic formulas themselves.\u003cbr\u003e Model-theoretically, the witnesses are objects of an extensional Scott λ-model.\u003cbr\u003e \u003cbr\u003e The approach - called, generically,  witness theory' - is inspired from work of N. G. de Bruijn, on a mathematical theory of proving, done during the late 1960s and the early 1970s, at the University of Eindhoven (The Netherlands), and is similar to the approach behind the Curry-Howard Correspondence, familiar from intuitionistic logic.\u003cbr\u003e \u003cbr\u003e For the classical case, the decorations - oft called  types' - are classical logic formulas.\u003cbr\u003e At quantifier-free level, the equational theory of concern is the λ-calculus with  surjective pairing' and some subsystens thereof, appropriately decorated.\u003cbr\u003e The extension to propositional, first- and second-order quantifiers is straightforward.\u003cbr\u003e \u003cbr\u003e \u003cbr\u003e The book consists of a collection of notes and papers written and circulated during the last ten years, as a continuation of previous research done by the author during the nineteen eighties.\u003cbr\u003e Among other things, it includes a survey of the origins of modern proof theory - Frege to Gentzen - from a witness-theoretical point of view, as well as a characteristic application of witness theory to a practical logic problem concerning axiomatisability.\u003c\/p\u003e\n            \u003cdiv\u003e\n\u003cstrong\u003eNumber of Pages:\u003c\/strong\u003e 390\u003c\/div\u003e\n            \u003cdiv\u003e\n\u003cstrong\u003eDimensions:\u003c\/strong\u003e 0.8 x 9.21 x 6.14 IN\u003c\/div\u003e\n            \u003cdiv\u003e\n\u003cstrong\u003ePublication Date:\u003c\/strong\u003e March 06, 2020\u003c\/div\u003e\n            ","brand":"BooksCloud","offers":[{"title":"Default Title","offer_id":52705008845107,"sku":"9781848903265","price":40.12,"currency_code":"USD","in_stock":true}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/0300\/5595\/6612\/files\/d2k2QlQzcE1qYmlkcHdzeHZVYTYrQT09.webp?v=1763369957","url":"https:\/\/www.vysn.com\/en-ca\/products\/witness-theory-notes-on-955-calculus-and-logic-paperback","provider":"VYSN","version":"1.0","type":"link"}