8,265 research outputs found
Connecting the generalized robustness and the geometric measure of entanglement
The main goal of this paper is to provide a connection between the
generalized robustness of entanglement () and the geometric measure of
entanglement (). First, we show that the generalized robustness is
always higher than or equal to the geometric measure. Then we find a tighter
lower bound to based only on the purity of and its maximal
overlap to a separable state. As we will see it is also possible to express
this lower bound in terms of .Comment: 4 pages, 2 figures. Comments welcome. v2: text improved - some
completely symmetric states were used to illustrate the results. Comments are
always welcome! v3: minor changes. Accepted by Phys. Rev. A. v4: results on
symmetric states fixe
Changing a semantics: opportunism or courage?
The generalized models for higher-order logics introduced by Leon Henkin, and
their multiple offspring over the years, have become a standard tool in many
areas of logic. Even so, discussion has persisted about their technical status,
and perhaps even their conceptual legitimacy. This paper gives a systematic
view of generalized model techniques, discusses what they mean in mathematical
and philosophical terms, and presents a few technical themes and results about
their role in algebraic representation, calibrating provability, lowering
complexity, understanding fixed-point logics, and achieving set-theoretic
absoluteness. We also show how thinking about Henkin's approach to semantics of
logical systems in this generality can yield new results, dispelling the
impression of adhocness. This paper is dedicated to Leon Henkin, a deep
logician who has changed the way we all work, while also being an always open,
modest, and encouraging colleague and friend.Comment: 27 pages. To appear in: The life and work of Leon Henkin: Essays on
his contributions (Studies in Universal Logic) eds: Manzano, M., Sain, I. and
Alonso, E., 201
Turing jumps through provability
Fixing some computably enumerable theory , the
Friedman-Goldfarb-Harrington (FGH) theorem says that over elementary
arithmetic, each formula is equivalent to some formula of the form
provided that is consistent. In this paper we give various
generalizations of the FGH theorem. In particular, for we relate
formulas to provability statements which
are a formalization of "provable in together with all true
sentences". As a corollary we conclude that each is
-complete. This observation yields us to consider a recursively
defined hierarchy of provability predicates which look a lot
like except that where calls upon the
oracle of all true sentences, the recursively
calls upon the oracle of all true sentences of the form . As such we obtain a `syntax-light' characterization of
definability whence of Turing jumps which is readily extended
beyond the finite. Moreover, we observe that the corresponding provability
predicates are well behaved in that together they provide a
sound interpretation of the polymodal provability logic
Variable types for meaning assembly: a logical syntax for generic noun phrases introduced by most
This paper proposes a way to compute the meanings associated with sentences
with generic noun phrases corresponding to the generalized quantifier most. We
call these generics specimens and they resemble stereotypes or prototypes in
lexical semantics. The meanings are viewed as logical formulae that can
thereafter be interpreted in your favourite models. To do so, we depart
significantly from the dominant Fregean view with a single untyped universe.
Indeed, our proposal adopts type theory with some hints from Hilbert
\epsilon-calculus (Hilbert, 1922; Avigad and Zach, 2008) and from medieval
philosophy, see e.g. de Libera (1993, 1996). Our type theoretic analysis bears
some resemblance with ongoing work in lexical semantics (Asher 2011; Bassac et
al. 2010; Moot, Pr\'evot and Retor\'e 2011). Our model also applies to
classical examples involving a class, or a generic element of this class, which
is not uttered but provided by the context. An outcome of this study is that,
in the minimalism-contextualism debate, see Conrad (2011), if one adopts a type
theoretical view, terms encode the purely semantic meaning component while
their typing is pragmatically determined
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