1,197 research outputs found
Correspondences between Classical, Intuitionistic and Uniform Provability
Based on an analysis of the inference rules used, we provide a
characterization of the situations in which classical provability entails
intuitionistic provability. We then examine the relationship of these
derivability notions to uniform provability, a restriction of intuitionistic
provability that embodies a special form of goal-directedness. We determine,
first, the circumstances in which the former relations imply the latter. Using
this result, we identify the richest versions of the so-called abstract logic
programming languages in classical and intuitionistic logic. We then study the
reduction of classical and, derivatively, intuitionistic provability to uniform
provability via the addition to the assumption set of the negation of the
formula to be proved. Our focus here is on understanding the situations in
which this reduction is achieved. However, our discussions indicate the
structure of a proof procedure based on the reduction, a matter also considered
explicitly elsewhere.Comment: 31 page
On the existence of Stone-Cech compactification
In [G. Curi, "Exact approximations to Stone-Cech compactification'', Ann.
Pure Appl. Logic, 146, 2-3, 2007, pp. 103-123] a characterization is obtained
of the locales of which the Stone-Cech compactification can be defined in
constructive type theory CTT, and in the formal system CZF+uREA+DC, a natural
extension of Aczel's system for constructive set theory CZF by a strengthening
of the Regular Extension Axiom REA and the principle of dependent choice. In
this paper I show that this characterization continues to hold over the
standard system CZF plus REA, thus removing in particular any dependency from a
choice principle. This will follow by a result of independent interest, namely
the proof that the class of continuous mappings from a compact regular locale X
to a regular a set-presented locale Y is a set in CZF, even without REA. It is
then shown that the existence of Stone-Cech compactification of a
non-degenerate Boolean locale is independent of the axioms of CZF (+REA), so
that the obtained characterization characterizes a proper subcollection of the
collection of all locales. The same also holds for several, even impredicative,
extensions of CZF+REA, as well as for CTT. This is in contrast with what
happens in the context of Higher-order Heyting arithmetic HHA - and thus in any
topos-theoretic universe: by constructions of Johnstone, Banaschewski and
Mulvey, within HHA Stone-Cech compactification can be defined for every locale
Perspectives for proof unwinding by programming languages techniques
In this chapter, we propose some future directions of work, potentially
beneficial to Mathematics and its foundations, based on the recent import of
methodology from the theory of programming languages into proof theory. This
scientific essay, written for the audience of proof theorists as well as the
working mathematician, is not a survey of the field, but rather a personal view
of the author who hopes that it may inspire future and fellow researchers
Multiple Conclusion Rules in Logics with the Disjunction Property
We prove that for the intermediate logics with the disjunction property any
basis of admissible rules can be reduced to a basis of admissible m-rules
(multiple-conclusion rules), and every basis of admissible m-rules can be
reduced to a basis of admissible rules. These results can be generalized to a
broad class of logics including positive logic and its extensions, Johansson
logic, normal extensions of S4, n-transitive logics and intuitionistic modal
logics
- …