Êtes-vous un étudiant de l'EPFL à la recherche d'un projet de semestre?
Travaillez avec nous sur des projets en science des données et en visualisation, et déployez votre projet sous forme d'application sur Graph Search.
Concept
Branching quantifier
Résumé
In logic a branching quantifier, also called a Henkin quantifier, finite partially ordered quantifier or even nonlinear quantifier, is a partial ordering
of quantifiers for Q ∈ {∀,∃}. It is a special case of generalized quantifier. In classical logic, quantifier prefixes are linearly ordered such that the value of a variable ym bound by a quantifier Qm depends on the value of the variables
y1, ..., ym−1
bound by quantifiers
Qy1, ..., Qym−1
preceding Qm. In a logic with (finite) partially ordered quantification this is not in general the case.
Branching quantification first appeared in a 1959 conference paper of Leon Henkin. Systems of partially ordered quantification are intermediate in strength between first-order logic and second-order logic. They are being used as a basis for Hintikka's and Gabriel Sandu's independence-friendly logic.
The simplest Henkin quantifier is
It (in fact every formula with a Henkin prefix, not just the simplest one) is equivalent to its second-order Skolemization, i.e.
It is also powerful enough to define the quantifier (i.e. "there are infinitely many") defined as
Several things follow from this, including the nonaxiomatizability of first-order logic with (first observed by Ehrenfeucht), and its equivalence to the -fragment of second-order logic (existential second-order logic)—the latter result published independently in 1970 by Herbert Enderton and W. Walkoe.
The following quantifiers are also definable by .
Rescher: "The number of φs is less than or equal to the number of ψs"
Härtig: "The φs are equinumerous with the ψs"
Chang: "The number of φs is equinumerous with the domain of the model"
The Henkin quantifier can itself be expressed as a type (4) Lindström quantifier.
Hintikka in a 1973 paper advanced the hypothesis that some sentences in natural languages are best understood in terms of branching quantifiers, for example: "some relative of each villager and some relative of each townsman hate each other" is supposed to be interpreted, according to Hintikka, as:
which is known to have no first-order logic equivalent.
Source officielle
À propos de ce résultat
Cette page est générée automatiquement et peut contenir des informations qui ne sont pas correctes, complètes, à jour ou pertinentes par rapport à votre recherche. Il en va de même pour toutes les autres pages de ce site. Veillez à vérifier les informations auprès des sources officielles de l'EPFL.