1,751 research outputs found
The modal logic of forcing
What are the most general principles in set theory relating forceability and
truth? As with Solovay's celebrated analysis of provability, both this question
and its answer are naturally formulated with modal logic. We aim to do for
forceability what Solovay did for provability. A set theoretical assertion psi
is forceable or possible, if psi holds in some forcing extension, and
necessary, if psi holds in all forcing extensions. In this forcing
interpretation of modal logic, we establish that if ZFC is consistent, then the
ZFC-provable principles of forcing are exactly those in the modal theory known
as S4.2.Comment: 31 page
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
Topic-Sensitive Epistemic 2D Truthmaker ZFC and Absolute Decidability
This paper aims to contribute to the analysis of the nature of mathematical modality, and to the applications of the latter to unrestricted quantification and absolute decidability. Rather than countenancing the interpretational type of mathematical modality as a primitive, I argue that the interpretational type of mathematical modality is a species of epistemic modality. I argue, then, that the framework of two-dimensional semantics ought to be applied to the mathematical setting. The framework permits of a formally precise account of the priority and relation between epistemic mathematical modality and metaphysical mathematical modality. The discrepancy between the modal systems governing the parameters in the two-dimensional intensional setting provides an explanation of the difference between the metaphysical possibility of absolute decidability and our knowledge thereof. I also advance an epistemic two-dimensional truthmaker semantics, if hyperintenisonal approaches are to be preferred to possible worlds semantics. I examine the relation between epistemic truthmakers and epistemic set theory
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
The First-Order Hypothetical Logic of Proofs
The Propositional Logic of Proofs (LP) is a modal logic in which the modality â–¡A is revisited as [​[t]​]​A , t being an expression that bears witness to the validity of A . It enjoys arithmetical soundness and completeness, can realize all S4 theorems and is capable of reflecting its own proofs ( ⊢A implies ⊢[​[t]​]A , for some t ). A presentation of first-order LP has recently been proposed, FOLP, which enjoys arithmetical soundness and has an exact provability semantics. A key notion in this presentation is how free variables are dealt with in a formula of the form [​[t]​]​A(i) . We revisit this notion in the setting of a Natural Deduction presentation and propose a Curry–Howard correspondence for FOLP. A term assignment is provided and a proof of strong normalization is given.Fil: Steren, Gabriela. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Departamento de Computación; ArgentinaFil: Bonelli, Eduardo Augusto. Universidad Nacional de Quilmes. Departamento de Ciencia y TecnologÃa; Argentina. Consejo Nacional de Investigaciones CientÃficas y Técnicas; Argentin
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