361 research outputs found
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
Temporal Justification Logic
Justification logics are modal-like logics with the additional capability of recording the reason, or justification, for modalities in syntactic structures, called justification terms. Justification logics can be seen as explicit counterparts to modal logics. The behavior and interaction of agents in distributed system is often modeled using logics of knowledge and time. In this paper, we sketch some preliminary ideas on how the modal knowledge part of such logics of knowledge and time could be replaced with an appropriate justification logic
Subset models for justification logic
We introduce a new semantics for justification logic based on subset
relations. Instead of using the established and more symbolic interpretation of
justifications, we model justifications as sets of possible worlds. We
introduce a new justification logic that is sound and complete with respect to
our semantics. Moreover, we present another variant of our semantics that
corresponds to traditional justification logic.
These types of models offer us a versatile tool to work with justifications,
e.g.~by extending them with a probability measure to capture uncertain
justifications. Following this strategy we will show that they subsume
Artemov's approach to aggregating probabilistic evidence
Computational reverse mathematics and foundational analysis
Reverse mathematics studies which subsystems of second order arithmetic are
equivalent to key theorems of ordinary, non-set-theoretic mathematics. The main
philosophical application of reverse mathematics proposed thus far is
foundational analysis, which explores the limits of different foundations for
mathematics in a formally precise manner. This paper gives a detailed account
of the motivations and methodology of foundational analysis, which have
heretofore been largely left implicit in the practice. It then shows how this
account can be fruitfully applied in the evaluation of major foundational
approaches by a careful examination of two case studies: a partial realization
of Hilbert's program due to Simpson [1988], and predicativism in the extended
form due to Feferman and Sch\"{u}tte.
Shore [2010, 2013] proposes that equivalences in reverse mathematics be
proved in the same way as inequivalences, namely by considering only
-models of the systems in question. Shore refers to this approach as
computational reverse mathematics. This paper shows that despite some
attractive features, computational reverse mathematics is inappropriate for
foundational analysis, for two major reasons. Firstly, the computable
entailment relation employed in computational reverse mathematics does not
preserve justification for the foundational programs above. Secondly,
computable entailment is a complete relation, and hence employing it
commits one to theoretical resources which outstrip those available within any
foundational approach that is proof-theoretically weaker than
.Comment: Submitted. 41 page
Detecting and Correcting Conservativity Principle Violations in Ontology-to-Ontology Mappings
In order to enable interoperability between ontology-based systems, ontology matching techniques have been proposed. However, when the generated mappings suffer from logical flaws, their usefulness may be diminished. In this paper we present an approximate method to detect and correct violations to the so-called conservativity principle where novel subsumption entailments between named concepts in one of the input ontologies are considered as unwanted. We show that this is indeed the case in our application domain based on the EU Optique project. Additionally, our extensive evaluation conducted with both the Optique use case and the data sets from the Ontology Alignment Evaluation Initiative (OAEI) suggests that our method is both useful and feasible in practice.Copyright 2014 Springer International Publishing Switzerland. The final publication is available at http://link.springer.com/chapter/10.1007%2F978-3-319-11915-1_
- …