821 research outputs found
Reasoning about Knowledge in Linear Logic: Modalities and Complexity
In a recent paper, Jean-Yves Girard commented that âit has been a long time since philosophy has stopped intereacting with logicâ[17]. Actually, it has no
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
Non-normal modalities in variants of Linear Logic
This article presents modal versions of resource-conscious logics. We
concentrate on extensions of variants of Linear Logic with one minimal
non-normal modality. In earlier work, where we investigated agency in
multi-agent systems, we have shown that the results scale up to logics with
multiple non-minimal modalities. Here, we start with the language of
propositional intuitionistic Linear Logic without the additive disjunction, to
which we add a modality. We provide an interpretation of this language on a
class of Kripke resource models extended with a neighbourhood function: modal
Kripke resource models. We propose a Hilbert-style axiomatization and a
Gentzen-style sequent calculus. We show that the proof theories are sound and
complete with respect to the class of modal Kripke resource models. We show
that the sequent calculus admits cut elimination and that proof-search is in
PSPACE. We then show how to extend the results when non-commutative connectives
are added to the language. Finally, we put the logical framework to use by
instantiating it as logics of agency. In particular, we propose a logic to
reason about the resource-sensitive use of artefacts and illustrate it with a
variety of examples
From IF to BI: a tale of dependence and separation
We take a fresh look at the logics of informational dependence and
independence of Hintikka and Sandu and Vaananen, and their compositional
semantics due to Hodges. We show how Hodges' semantics can be seen as a special
case of a general construction, which provides a context for a useful
completeness theorem with respect to a wider class of models. We shed some new
light on each aspect of the logic. We show that the natural propositional logic
carried by the semantics is the logic of Bunched Implications due to Pym and
O'Hearn, which combines intuitionistic and multiplicative connectives. This
introduces several new connectives not previously considered in logics of
informational dependence, but which we show play a very natural role, most
notably intuitionistic implication. As regards the quantifiers, we show that
their interpretation in the Hodges semantics is forced, in that they are the
image under the general construction of the usual Tarski semantics; this
implies that they are adjoints to substitution, and hence uniquely determined.
As for the dependence predicate, we show that this is definable from a simpler
predicate, of constancy or dependence on nothing. This makes essential use of
the intuitionistic implication. The Armstrong axioms for functional dependence
are then recovered as a standard set of axioms for intuitionistic implication.
We also prove a full abstraction result in the style of Hodges, in which the
intuitionistic implication plays a very natural r\^ole.Comment: 28 pages, journal versio
Constructive Provability Logic
We present constructive provability logic, an intuitionstic modal logic that
validates the L\"ob rule of G\"odel and L\"ob's provability logic by permitting
logical reflection over provability. Two distinct variants of this logic, CPL
and CPL*, are presented in natural deduction and sequent calculus forms which
are then shown to be equivalent. In addition, we discuss the use of
constructive provability logic to justify stratified negation in logic
programming within an intuitionstic and structural proof theory.Comment: Extended version of IMLA 2011 submission of the same titl
Logical operators for ontological modeling
We show that logic has more to offer to ontologists than standard first order
and modal operators. We first describe some operators of linear logic which we
believe are particularly suitable for ontological modeling, and suggest how to interpret
them within an ontological framework. After showing how they can coexist
with those of classical logic, we analyze three notions of artifact from the literature
to conclude that these linear operators allow for reducing the ontological commitment
needed for their formalization, and even simplify their logical formulation
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