187 research outputs found
Some remarks concerning modal propositional logic of questions
Recently, it has become a custom to treat questions (or, better, questioning) as a game between two subjects. Unfortunately, one rarely goes beyond the scheme of Questioner-Scientist and Answerer-Nature, although the Interlocutor so conceived displays some undesirable features. This paper argues for the idea that logic of questions can be build as a logic of the game between “knowledge resources” persons or theories, rather than errant Scientist and omniscient Nature. To this end the concept of epistemically-possible worlds is discussed, which is conceived as analogous to that of possible worlds in modal logic. And, furthermore, the concepts of relation of epistemic alternativeness and of epistemically-alternative worlds are introduced. On this basis a version of semantics for propositional, three-valued logic of questions is offered and semantic proofs of some theses are given
Logic and cognition. Two faces of psychologism
In this paper two concepts of psychologism in logic are outlined: the one which Frege and Husserl fought against and the new psychologism, or cognitivism, which underlies a cognitive turn in contemporary logic. Four issues such cognitively oriented logic should be interested in are indicated. They concern: new fields opened for logical analysis, new methods and tools needed to address these fields, neural basis of logical reasoning, and an educational problem: how to teach such logic? Several challenging questions, which arise in the context of these issues, are listed
The Dynamics and Geometry of Semi-Hyperbolic Rational Semigroups
We deal with various classes of finitely generated semi-hyperbolic rational
semigroups acting on the Riemann sphere. Our primary tool is the associated
skew product map . For every real number we define the
topological pressure ascribed to the (possibly) unbounded potential
. We show that for some non-degenerate open (in
) interval containing and the Hausdorff dimension of the Julia
set , the function is real-analytic.
We further show that for all in such an interval there exist a unique
-conformal measure and a unique Borel probability -invariant
measure absolutely continuous with respect to . We then show that
and are equivalent measures and that the dynamical system
is metrically exact (thus ergodic), and that for H\"older
continuous observables satisfies the CLT, LIL, and the
Exponential Decay of Correlations. We prove a Variational Principle for the
potentials , and we characterize the measures as
their unique equilibrium states.
Concerning geometry, we first introduce the Nice Open Set Condition, and we
prove that each *semi-hyperbolic rational semigroup satisfying this condition
is of finite type. We then introduce the class of non-exceptional semigroups
and perform a full multifractal analysis of the equilibrium states . In
particular, we show that the corresponding multifractal spectrum is non-trivial
for every non-exceptional totally non-recurrent rational semigroup satisfying
the Nice Open Set Condition.
Finally, we settle a long standing problem in the theory of rational
semigroups by proving that for our class of semigroups the Hausdorff dimension
of each fiber Julia set is strictly smaller than the Hausdorff dimension of the
global Julia set of the semigroup.Comment: 151 pages, 2 figure
Geometry and dynamics in Gromov hyperbolic metric spaces: With an emphasis on non-proper settings
Our monograph presents the foundations of the theory of groups and semigroups
acting isometrically on Gromov hyperbolic metric spaces. Our work unifies and
extends a long list of results by many authors. We make it a point to avoid any
assumption of properness/compactness, keeping in mind the motivating example of
, the infinite-dimensional rank-one symmetric space of
noncompact type over the reals. The monograph provides a number of examples of
groups acting on which exhibit a wide range of phenomena not
to be found in the finite-dimensional theory. Such examples often demonstrate
the optimality of our theorems. We introduce a modification of the Poincar\'e
exponent, an invariant of a group which gives more information than the usual
Poincar\'e exponent, which we then use to vastly generalize the Bishop--Jones
theorem relating the Hausdorff dimension of the radial limit set to the
Poincar\'e exponent of the underlying semigroup. We give some examples based on
our results which illustrate the connection between Hausdorff dimension and
various notions of discreteness which show up in non-proper settings. We
construct Patterson--Sullivan measures for groups of divergence type without
any compactness assumption. This is carried out by first constructing such
measures on the Samuel--Smirnov compactification of the bordification of the
underlying hyperbolic space, and then showing that the measures are supported
on the bordification. We study quasiconformal measures of geometrically finite
groups in terms of (a) doubling and (b) exact dimensionality. Our analysis
characterizes exact dimensionality in terms of Diophantine approximation on the
boundary. We demonstrate that some Patterson--Sullivan measures are neither
doubling nor exact dimensional, and some are exact dimensional but not
doubling, but all doubling measures are exact dimensional.Comment: A previous version of this document included Section 12.5 (Tukia's
isomorphism theorem). The results of that subsection have been split off into
a new document which is available at arXiv:1508.0696
Diophantine approximation in Banach spaces
In this paper, we extend the theory of simultaneous Diophantine approximation
to infinite dimensions. Moreover, we discuss Dirichlet-type theorems in a very
general framework and define what it means for such a theorem to be optimal. We
show that optimality is implied by but does not imply the existence of badly
approximable points
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