590 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
Deduction modulo theory
This paper is a survey on Deduction modulo theor
Provability Logic and the Completeness Principle
In this paper, we study the provability logic of intuitionistic theories of
arithmetic that prove their own completeness. We prove a completeness theorem
for theories equipped with two provability predicates and
that prove the schemes and for
. Using this theorem, we determine the logic of fast provability
for a number of intuitionistic theories. Furthermore, we reprove a theorem
previously obtained by M. Ardeshir and S. Mojtaba Mojtahedi determining the
-provability logic of Heyting Arithmetic
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
Towards a Paraconsistent Quantum Set Theory
In this paper, we will attempt to establish a connection between quantum set
theory, as developed by Ozawa, Takeuti and Titani, and topos quantum theory, as
developed by Isham, Butterfield and Doring, amongst others. Towards this end,
we will study algebraic valued set-theoretic structures whose truth values
correspond to the clopen subobjects of the spectral presheaf of an orthomodular
lattice of projections onto a given Hilbert space. In particular, we will
attempt to recreate, in these new structures, Takeuti's original isomorphism
between the set of all Dedekind real numbers in a suitably constructed model of
set theory and the set of all self adjoint operators on a chosen Hilbert space.Comment: In Proceedings QPL 2015, arXiv:1511.0118
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