Cut elimination for a simple formulation of epsilon calculus

AbstractA simple cut elimination proof for arithmetic with the epsilon symbol is used to establish the termination of a modified epsilon substitution process. This opens a possibility of extension to much stronger systems

Semantics and Proof Theory of the Epsilon Calculus

The epsilon operator is a term-forming operator which replaces quantifiers in ordinary predicate logic. The application of this undervalued formalism has been hampered by the absence of well-behaved proof systems on the one hand, and accessible presentations of its theory on the other. One significant early result for the original axiomatic proof system for the epsilon-calculus is the first epsilon theorem, for which a proof is sketched. The system itself is discussed, also relative to possible semantic interpretations. The problems facing the development of proof-theoretically well-behaved systems are outlined.Comment: arXiv admin note: substantial text overlap with arXiv:1411.362

The Epsilon Calculus and Herbrand Complexity

Hilbert's epsilon-calculus is based on an extension of the language of predicate logic by a term-forming operator $\epsilon_{x}$. Two fundamental results about the epsilon-calculus, the first and second epsilon theorem, play a role similar to that which the cut-elimination theorem plays in sequent calculus. In particular, Herbrand's Theorem is a consequence of the epsilon theorems. The paper investigates the epsilon theorems and the complexity of the elimination procedure underlying their proof, as well as the length of Herbrand disjunctions of existential theorems obtained by this elimination procedure.Comment: 23 p

Hilbert's "Verunglueckter Beweis," the first epsilon theorem, and consistency proofs

In the 1920s, Ackermann and von Neumann, in pursuit of Hilbert's Programme, were working on consistency proofs for arithmetical systems. One proposed method of giving such proofs is Hilbert's epsilon-substitution method. There was, however, a second approach which was not reflected in the publications of the Hilbert school in the 1920s, and which is a direct precursor of Hilbert's first epsilon theorem and a certain 'general consistency result' due to Bernays. An analysis of the form of this so-called 'failed proof' sheds further light on an interpretation of Hilbert's Programme as an instrumentalist enterprise with the aim of showing that whenever a `real' proposition can be proved by 'ideal' means, it can also be proved by 'real', finitary means.Comment: 18 pages, final versio

Some proof theoretical remarks on quantification in ordinary language

This paper surveys the common approach to quantification and generalised quantification in formal linguistics and philosophy of language. We point out how this general setting departs from empirical linguistic data, and give some hints for a different view based on proof theory, which on many aspects gets closer to the language itself. We stress the importance of Hilbert's oper- ator epsilon and tau for, respectively, existential and universal quantifications. Indeed, these operators help a lot to construct semantic representation close to natural language, in particular with quantified noun phrases as individual terms. We also define guidelines for the design of the proof rules corresponding to generalised quantifiers.Cet article dresse un rapide panorama de l'approche commune de la quantification généralisée ou non en linguistique formelle et en philosophie du langage. Nous montrons que ce cadre général va parfois 'a l'encontre des données linguistiques, et nous donnons quelques indications pour une approche différente basée sur la théorie de la démonstration, qui sur bien des points s'avère plus proche de la langue. Nous soulignons l'importance des opérateurs tau et epsilon de Hilbert, qui rendent respectivement compte de la quantification universelle et existentielle. En effet, ces opérateurs permettent de construire des représentations sémantiques qui suivent la langue avec, en particulier des groupes nominaux quantifiées qui soient des termes individuels. Nous donnons aussi des principes pour définir des règles de déduction qui correspondent aux quantificateurs généralisés

The Montagovian Generative Lexicon ΛT yn: a Type Theoretical Framework for Natural Language Semantics

International audienceWe present a framework, named the Montagovian generative lexicon, for computing the semantics of natural language sentences, expressed in many-sorted higher order logic. Word meaning is described by several lambda terms of second order lambda calculus (Girard’s system F): the principal lambda term encodes the argument structure, while the other lambda terms implement meaning transfers. The base types include a type for propositions and many types for sorts of a many-sorted logic for expressing restriction of selection. This framework is able to integrate a proper treatment of lexical phenomena into a Montagovian compositional semantics, like the (im)possible arguments of a predicate, and the adaptation of a word meaning to some contexts. Among these adaptations of a word meaning to contexts, ontological inclusions are handled by coercive subtyping, an extension of system F introduced in the present paper. The benefits of this framework for lexical semantics and pragmatics are illustrated on meaning transfers and coercions, on possible and impossible copredication over different senses, on deverbal ambiguities, and on “fictive motion”. Next we show that the compositional treatment of determiners, quantifiers, plurals, and other semantic phenomena is richer in our framework. We then conclude with the linguistic, logical and computational perspectives opened by the Montagovian generative lexicon

A termination proof for epsilon substitution using partial derivations

AbstractEpsilon substitution method introduced by Hilbert is a successive approximation process providing numerical realizations from proofs of existential formulas. Most convergence (termination) proofs for it use assignments of decreasing ordinals to stages of the process and work only for predicative systems. We describe a new ordinal assignment for the case of first-order arithmetic admitting extension to impredicative systems. It is based on an interpretation of individual epsilon substitutions forming the substitution process as incomplete finite proofs, each encoding a complete but infinite proof