264 research outputs found
Unifying Functional Interpretations: Past and Future
This article surveys work done in the last six years on the unification of
various functional interpretations including G\"odel's dialectica
interpretation, its Diller-Nahm variant, Kreisel modified realizability,
Stein's family of functional interpretations, functional interpretations "with
truth", and bounded functional interpretations. Our goal in the present paper
is twofold: (1) to look back and single out the main lessons learnt so far, and
(2) to look forward and list several open questions and possible directions for
further research.Comment: 18 page
On Various Negative Translations
Several proof translations of classical mathematics into intuitionistic
mathematics have been proposed in the literature over the past century. These
are normally referred to as negative translations or double-negation
translations. Among those, the most commonly cited are translations due to
Kolmogorov, Godel, Gentzen, Kuroda and Krivine (in chronological order). In
this paper we propose a framework for explaining how these different
translations are related to each other. More precisely, we define a notion of a
(modular) simplification starting from Kolmogorov translation, which leads to a
partial order between different negative translations. In this derived
ordering, Kuroda and Krivine are minimal elements. Two new minimal translations
are introduced, with Godel and Gentzen translations sitting in between
Kolmogorov and one of these new translations.Comment: In Proceedings CL&C 2010, arXiv:1101.520
On Affine Logic and {\L}ukasiewicz Logic
The multi-valued logic of {\L}ukasiewicz is a substructural logic that has
been widely studied and has many interesting properties. It is classical, in
the sense that it admits the axiom schema of double negation, [DNE]. However,
our understanding of {\L}ukasiewicz logic can be improved by separating its
classical and intuitionistic aspects. The intuitionistic aspect of
{\L}ukasiewicz logic is captured in an axiom schema, [CWC], which asserts the
commutativity of a weak form of conjunction. This is equivalent to a very
restricted form of contraction. We show how {\L}ukasiewicz Logic can be viewed
both as an extension of classical affine logic with [CWC], or as an extension
of what we call \emph{intuitionistic} {\L}ukasiewicz logic with [DNE],
intuitionistic {\L}ukasiewicz logic being the extension of intuitionistic
affine logic by the schema [CWC]. At first glance, intuitionistic affine logic
seems very weak, but, in fact, [CWC] is surprisingly powerful, implying results
such as intuitionistic analogues of De Morgan's laws. However the proofs can be
very intricate. We present these results using derived connectives to clarify
and motivate the proofs and give several applications. We give an analysis of
the applicability to these logics of the well-known methods that use negation
to translate classical logic into intuitionistic logic. The usual proofs of
correctness for these translations make much use of contraction. Nonetheless,
we show that all the usual negative translations are already correct for
intuitionistic {\L}ukasiewicz logic, where only the limited amount of
contraction given by [CWC] is allowed. This is in contrast with affine logic
for which we show, by appeal to results on semantics proved in a companion
paper, that both the Gentzen and the Glivenko translations fail.Comment: 28 page
- …