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Data at the GrammarPragmatics Interface: the case of resumptive pronouns
 in English. In R.D. Borsley (ed.), special volume of Lingua
, 2004
"... This paper explores the relation of grammaticality to acceptability through a discussion of the use of resumptive pronouns in spoken English. It is argued that undergeneration by some grammar of observed linguistic phenomena such as these is as serious a problem for theoretical frameworks as overg ..."
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This paper explores the relation of grammaticality to acceptability through a discussion of the use of resumptive pronouns in spoken English. It is argued that undergeneration by some grammar of observed linguistic phenomena such as these is as serious a problem for theoretical frameworks as overgeneration, and that it has consequences for the way in which grammaticality and acceptability are to be construed. Using the framework of Dynamic Syntax, a theoretical account of relative clauses and anaphora construal is provided from which the use of resumptive pronouns in English emerges as a natural consequence. The fact that examples are considered by native speakers to be unacceptable in neutral contexts is argued to follow from pragmatic effects, explicable from a Relevance Theoretic perspective.
Epsilonsubstitution method for the ramified language and # 1 comprehension rule
 Logic and Foundations of Mathematics
, 1999
"... We extend to Ramified Analysis the definition and termination proof of Hilbert’s ɛsubstitution method. This forms a base for future extensions to predicatively reducible subsystems of analysis. First such system treated here is second order arithmetic with ∆1 1comprehension rule. ..."
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We extend to Ramified Analysis the definition and termination proof of Hilbert’s ɛsubstitution method. This forms a base for future extensions to predicatively reducible subsystems of analysis. First such system treated here is second order arithmetic with ∆1 1comprehension rule.
"Clarifying the Nature of the Infinite": the development of metamathematics and proof theory
, 2001
"... We discuss the development of metamathematics in the Hilbert school, and Hilbert's prooftheoretic program in particular. We place this program in a broader historical and philosophical context, especially with respect to nineteenth century developments in mathematics and logic. Finally, we show how ..."
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We discuss the development of metamathematics in the Hilbert school, and Hilbert's prooftheoretic program in particular. We place this program in a broader historical and philosophical context, especially with respect to nineteenth century developments in mathematics and logic. Finally, we show how these considerations help frame our understanding of metamathematics and proof theory today.
Propositional PSPACE reasoning with Boolean programs versus quantified Boolean formulas
 In ICALP
, 2004
"... Abstract. We present a new propositional proof system based on a somewhat recent characterization of polynomial space (PSPACE) called Boolean programs, due to Cook and Soltys. The Boolean programs are like generalized extension atoms, providing a parallel to extended Frege. We show that this new sys ..."
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Abstract. We present a new propositional proof system based on a somewhat recent characterization of polynomial space (PSPACE) called Boolean programs, due to Cook and Soltys. The Boolean programs are like generalized extension atoms, providing a parallel to extended Frege. We show that this new system, BPLK, is polynomially equivalent to the system G, which is based on the familiar but very different quantified Boolean formula (QBF) characterization of PSPACE due to Stockmeyer and Meyer. This equivalence is proved by way of two translations, one of which uses an idea reminiscent of the ɛterms of Hilbert and Bernays. 1
Mathematical discourse vs. mathematical intuition
 Mathematical reasoning and heuristics, College Publications, London 2005
"... One of the most uninformative statements one could possibly make about mathematics is that the axiomatic method expresses the real nature of mathematics, i.e., that mathematics consists in the deduction of conclusions from given axioms. For the same could be said about several other subjects, for ex ..."
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One of the most uninformative statements one could possibly make about mathematics is that the axiomatic method expresses the real nature of mathematics, i.e., that mathematics consists in the deduction of conclusions from given axioms. For the same could be said about several other subjects, for example, about theology. Think of the first part of Spinoza’sEthica ordine geometrico demonstrata orofGödel’sproofof the existence of God, which are both fine specimens of Theologia ordine geometrico demonstrata. To the objection, ‘Surely theological entities are not mathematicalobjects’,one could answer:How do you know? If mathematics consists in the deduction of conclusions from given axioms, then mathematical objects are given by the axioms. So, if theological entities satisfy the axioms, why should not they be considered mathematical objects? Hilbert says: “Ifinspeakingofmypoints”,linesandplanes“I think of some system of things, e.g. the system: love, law, chimney sweep... and then assume all my axioms as relations between these things,thenmypropositions,e.g.Pythagoras’theorem,arealsovalidfor thesethings”. 1 Similarly he might have said: If in speaking of my points, lines and planes, I think of a suitable triad of theological entities, and assume all my axioms as relations between these things, then my propositions,e.g.Pythagoras’theorem,arealsovalidforthesethings. Indeed, if mathematics consists in the deduction of conclusions from given axioms, then it has no specific content. So it is simply imposibletodistinguishgeometricalobjects,suchas‘points,linesand planes’,from ‘love,law,chimney sweep’,ora suitable triad of theological entities.ThisisvividlyilustratedbyRusel’sstatementthat “mathematicsmaybedefinedasthesubject in which we never know whatwearetalkingabout,norwhetherwhatwearesayingistrue”. 2
Describing proofs by short tautologies
 Annals of Pure and Applied Logic
"... Herbrand’s theorem is one of the most fundamental results about firstorder logic. In the context of proof analysis, Herbranddisjunctions are used for describing the constructive content of cutfree proofs. However, given a proof with cuts, the computation of an Herbranddisjunction is of significa ..."
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Herbrand’s theorem is one of the most fundamental results about firstorder logic. In the context of proof analysis, Herbranddisjunctions are used for describing the constructive content of cutfree proofs. However, given a proof with cuts, the computation of an Herbranddisjunction is of significant computational complexity, as the cuts in the proof have to be eliminated first. In this paper we prove a generalization of Herbrand’s theorem: From a proof with cuts, one can read off a small (linear in the size of the proof) tautology composed of instances of the endsequent and the cut formulas. This tautology describes the proof in the following way: Each cut induces a (propositional) formula stating that a disjunction of instances of the cut formula implies a conjunction of instances of the cut formula. All these cutimplications together then imply the already existing instances of the endsequent. The proof that this formula is a tautology is carried out by transforming the instances in the proof to normal forms and using characteristic clause sets to relate them. These clause sets have first been studied in the context of cutelimination. This extended Herbrand theorem is then applied to cutelimination sequences in order to show that, for the computation of an Herbranddisjunction, the knowledge of only the term substitutions performed during cutelimination is already sufficient.
The Ignorance of Bourbaki
, 1990
"... this article writes: "Which half of his brains did Bourbaki use ? My impression is, the left half. Perhaps I am projecting. The Bourbachistes were uncomfortable with the rightbrain mathematics of the Italian geometers, and for good reason: significant portions were suspect and might, if one takes ` ..."
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this article writes: "Which half of his brains did Bourbaki use ? My impression is, the left half. Perhaps I am projecting. The Bourbachistes were uncomfortable with the rightbrain mathematics of the Italian geometers, and for good reason: significant portions were suspect and might, if one takes `true' and `false' to be leftbrain notions and `right' and `wrong' to be rightbrain ones, be justifiably described as right, but false.
The Square of Individuals
 Proceedings of the Tenth Amsterdam Colloquium
"... this paper. However, readers whose favorite ontology of plurals is different should not be too bothered. In section 6 we will see that the main results of this paper hold also with a Linkian ontology. Getting back to (4), in the set theoretical typing the collective predicate ..."
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this paper. However, readers whose favorite ontology of plurals is different should not be too bothered. In section 6 we will see that the main results of this paper hold also with a Linkian ontology. Getting back to (4), in the set theoretical typing the collective predicate