Conservativity of embeddings in the lambda Pi calculus modulo rewriting (long version)

The lambda Pi calculus can be extended with rewrite rules to embed any functional pure type system. In this paper, we show that the embedding is conservative by proving a relative form of normalization, thus justifying the use of the lambda Pi calculus modulo rewriting as a logical framework for logics based on pure type systems. This result was previously only proved under the condition that the target system is normalizing. Our approach does not depend on this condition and therefore also works when the source system is not normalizing.Comment: Long version of TLCA 2015 pape

Extending the Extensional Lambda Calculus with Surjective Pairing is Conservative

We answer Klop and de Vrijer's question whether adding surjective-pairing axioms to the extensional lambda calculus yields a conservative extension. The answer is positive. As a byproduct we obtain a "syntactic" proof that the extensional lambda calculus with surjective pairing is consistent.Comment: To appear in Logical Methods in Computer Scienc

Truth-value semantics and functional extensions for classical logic of partial terms based on equality

We develop a bottom-up approach to truth-value semantics for classical logic of partial terms based on equality and apply it to prove the conservativity of the addition of partial description and partial selection functions, independently of any strictness assumption.Comment: 15 pages, to appear in the Notre Dame Journal of Formal Logi

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

Lewis meets Brouwer: constructive strict implication

C. I. Lewis invented modern modal logic as a theory of "strict implication". Over the classical propositional calculus one can as well work with the unary box connective. Intuitionistically, however, the strict implication has greater expressive power than the box and allows to make distinctions invisible in the ordinary syntax. In particular, the logic determined by the most popular semantics of intuitionistic K becomes a proper extension of the minimal normal logic of the binary connective. Even an extension of this minimal logic with the "strength" axiom, classically near-trivial, preserves the distinction between the binary and the unary setting. In fact, this distinction and the strong constructive strict implication itself has been also discovered by the functional programming community in their study of "arrows" as contrasted with "idioms". Our particular focus is on arithmetical interpretations of the intuitionistic strict implication in terms of preservativity in extensions of Heyting's Arithmetic.Comment: Our invited contribution to the collection "L.E.J. Brouwer, 50 years later