158 research outputs found
Coherence in Modal Logic
A variety is said to be coherent if the finitely generated subalgebras of its
finitely presented members are also finitely presented. In a recent paper by
the authors it was shown that coherence forms a key ingredient of the uniform
deductive interpolation property for equational consequence in a variety, and a
general criterion was given for the failure of coherence (and hence uniform
deductive interpolation) in varieties of algebras with a term-definable
semilattice reduct. In this paper, a more general criterion is obtained and
used to prove the failure of coherence and uniform deductive interpolation for
a broad family of modal logics, including K, KT, K4, and S4
Quantified intuitionistic propositional logic and Cantor space
We consider propositional quantification in intuitionistic logic. We prove that, under topological interpretation over the Cantor space, it enjoys surprising and interesting properties such as the maximum property and a kind of distribution of existential quantifier over conjunction. Moreover, by pointing to appropriate examples, we show that the set of quantified formulas valid in the Cantor space strictly contains the set of formulas provable in the minimal system of intuitionistic logic with propositional quantification
Semantics out of context: nominal absolute denotations for first-order logic and computation
Call a semantics for a language with variables absolute when variables map to
fixed entities in the denotation. That is, a semantics is absolute when the
denotation of a variable a is a copy of itself in the denotation. We give a
trio of lattice-based, sets-based, and algebraic absolute semantics to
first-order logic. Possibly open predicates are directly interpreted as lattice
elements / sets / algebra elements, subject to suitable interpretations of the
connectives and quantifiers. In particular, universal quantification "forall
a.phi" is interpreted using a new notion of "fresh-finite" limit and using a
novel dual to substitution.
The interest of this semantics is partly in the non-trivial and beautiful
technical details, which also offer certain advantages over existing
semantics---but also the fact that such semantics exist at all suggests a new
way of looking at variables and the foundations of logic and computation, which
may be well-suited to the demands of modern computer science
Categories for Dynamic Epistemic Logic
The primary goal of this paper is to recast the semantics of modal logic, and
dynamic epistemic logic (DEL) in particular, in category-theoretic terms. We
first review the category of relations and categories of Kripke frames, with
particular emphasis on the duality between relations and adjoint homomorphisms.
Using these categories, we then reformulate the semantics of DEL in a more
categorical and algebraic form. Several virtues of the new formulation will be
demonstrated: The DEL idea of updating a model into another is captured
naturally by the categorical perspective -- which emphasizes a family of
objects and structural relationships among them, as opposed to a single object
and structure on it. Also, the categorical semantics of DEL can be merged
straightforwardly with a standard categorical semantics for first-order logic,
providing a semantics for first-order DEL.Comment: In Proceedings TARK 2017, arXiv:1707.0825
Representation and duality of the untyped lambda-calculus in nominal lattice and topological semantics, with a proof of topological completeness
We give a semantics for the lambda-calculus based on a topological duality
theorem in nominal sets. A novel interpretation of lambda is given in terms of
adjoints, and lambda-terms are interpreted absolutely as sets (no valuation is
necessary)
From types to sets by local type definitions in higher-order logic
Types in Higher-Order Logic (HOL) are naturally interpreted as nonempty sets—this intuition is reflected in the type definition rule for the HOL-based systems (including Isabelle/HOL), where a new type can be defined whenever a nonempty set is exhibited. However, in HOL this definition mechanism cannot be applied inside proof contexts. We propose a more expressive type definition rule that addresses the limitation and we prove its soundness. This higher expressive power opens the opportunity for a HOL tool that relativizes type-based statements to more flexible set-based variants in a principled way. We also address particularities of Isabelle/HOL and show how to perform the relativization in the presence of type classes
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Tripos models of Internal Set Theory
This thesis provides a framework to make sense of models of E. Nelson’s Internal Set Theory (and hence of nonstandard analysis) in elementary toposes by exploiting the technology of tripos theory and Lawvere’s hyperdoctrines. A new doctrinal account of nonstandard phenomena is described, which avoids a few key restrictions in Nelson’s approach: chiefly, the dependence on Set Theory (which is done by replacing a model of set theory with a topos as the starting point) and reliance on an internally defined notion of standard element. From the new perspective, validity of the schemes of Idealisation, Standardisation, and Transfer correspond to the existence of certain relationships between hyperdoctrines, leading to the new notion of a tripos model of IST. After discussing the properties of such models that make them a suitable abstraction of classic IST we explore situations in which such structures arise, leading to two distinct main classes of models: what we refer to as Nelson models, which are those for which there is a well-behaved `predicate of standard elements’ (providing thus a close approximation of classic IST valid for toposes), and models obtained by elaborating on the work started by A. Kock and C. J. Mikkelsen on the categorification of Transfer. We then elaborate upon one particular kind of model of each type: the ultrapower models, which are Nelson models obtained from a choice of adequate ultrafilter and that mirror the constructions of classical Robinson nonstandard analysis, and the localic models for Transfer and Standardisation, which are obtained from any given open surjection of locales (e.g universal covers of topological spaces) via the
Kock-Mikkelsen construction at the level of sheaf toposes.This study was funded by the Coordenação de Aperfeiçoamento
de Pessoal de NÃvel Superior - Brasil (CAPES) (process n∘ 8881.128278/2016-01)
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