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 $G$ acting on the Riemann sphere. Our primary tool is the associated skew product map $\tilde f$. For every real number $t\ge 0$ we define the topological pressure $P(t)$ ascribed to the (possibly) unbounded potential $-t\log|\tilde f'|$. We show that for some non-degenerate open (in $[0,+\infty)$) interval containing $0$ and the Hausdorff dimension of the Julia set $J(G)$, the function $t\mapsto P(t)$ is real-analytic. We further show that for all $t$ in such an interval there exist a unique $t$-conformal measure $m_t$ and a unique Borel probability $\tilde f$-invariant measure $\mu_t$ absolutely continuous with respect to $m_t$. We then show that $\mu_t$ and $m_t$ are equivalent measures and that the dynamical system $(\tilde f,\mu_t)$ is metrically exact (thus ergodic), and that for H\"older continuous observables $(\tilde f,\mu_t)$ satisfies the CLT, LIL, and the Exponential Decay of Correlations. We prove a Variational Principle for the potentials $-t\log|\tilde f'|$, and we characterize the measures $\mu_t$ 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 $\mu_t$. 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 $\mathbb H^\infty$, the infinite-dimensional rank-one symmetric space of noncompact type over the reals. The monograph provides a number of examples of groups acting on $\mathbb H^\infty$ 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