Analytic Renormings of C(K) Spaces
A ladder system is disposed between two spaced storage shelves. The ladder may move longitudinally along the shelves, and may also be moved laterally between the shelves. Further, the ladder may pivot about a lateral axis such that the ladder may be picked up over obstructions in an aisle between the shelves. In this way, one can quickly and easily position the ladder at any position along or between the two storage shelves for accessing items stored on the shelves. This ladder system is particularly useful in the crowded warehouse-type retail outlets which are now gaining popularity.

title: Polynomial algebras on classical banach spaces
abstract: For the classical Banach spacesX = ℓp,C(K) we identify alln such that every polynomial of degreen + 1 onX is uniformly approximable on the unit ball by elements of the algebra generated by all polynomials of degree up ton onX.

title: Remarks on symmetric smooth norms

title: Dual renormings of Banach spaces
abstract: We prove that a Banach space admitting an equivalent WUR norm is an Asplund space. Some related dual renormings are also presented.

title: Smooth norms that depend locally on finitely many coordinates
abstract: We characterize separable normed spaces that admit equivalent COO-smooth norms depending locally on finitely many coordinates. It follows, in particular, that such norms exist on any normed space with countable algebraic basis. We use the method of Talagrand operators developed by Haydon in  and certain integral convolution techniques [2], [13] to characterize separable normed spaces that admit C??-smooth norms depending locally on finitely many coordinates as spaces that admit norms with countable boundaries. As corollaries we obtain improvements on some results of Fonf  and Vanderwerff  and a new simple proof of a result of Haydon on Cw-smooth renormings of spaces of continuous functions on countable compact sets. Our results should be compared to the result of Godun, Lin, and Troyanski in  where it is shown that every separable Banach space can be given an equivalent norm, the unit ball of which contains countably many strongly extreme points. From this result and Theorem 1 below it follows that strongly extreme points need not necessarily form a boundary of Bx* . We will use the notation standard in Banach space theory. In particular, Bx and Sx will denote respectively the unit ball and the unit sphere of a Banach space X. We say that 11 11 depends locally on finitely many coordinates if for each x E Sx there exist an open neighbourhood 0 of x, a finite set {x, ...,Xk*} C X*, and a function f: Rk -+ R such that IIYII =f(X1*(y), ...,xk*(y)) fory EO. If (X, 11) is a normed space, the set B c Sx* is called a boundary of (X, * 11) if for each x E Sx there exists x* E B such that x*(x) = 1. Theorem 1. Let (X, 11 * 11) be a normed space.  (i) X admits an equivalent norm having a countable boundary. (ii) X admits an equivalent norm with a boundary B, such that there is a sequence {Kn}nEN of norm compact sets in X* satisfing B C UnEN KnX Received by the editors October 13, 1993 and, in revised form, June 15, 1994.

title: Combining Functions for Certainty Degrees in Consulting Systems
abstract: Consulting systems are rule-based systems of Artificial Intelligence working with propositions that may be uncertain, i.e. may have a truth-value different from “true” and “false”. The work of a consulting system consists, roughly, in propagation of uncertain knowledge throughout the net of rules according to some combining functions. It is shown that the notion of an ordered Abelian group is central for the definition of these combining functions. Various particular groups, both Archimedean and non-Archimedean, are considered for use in consulting systems. Practical experiments with them are also described.

title: Ten Questions and One Problem on Fuzzy Logic
abstract: Recently, I had a very interesting friendly e-mail discussion with Professor Parikh on vagueness and fuzzy logic. Parikh published several papers concerning the notion of vagueness. They contain critical remarks on fuzzy logic and its ability to formalize reasoning under vagueness [10,11]. On the other hand, for some years I have tried to advocate fuzzy logic (in the narrow sense, as Zadeh says, i.e. as formal logical systems formalizing reasoning under vagueness) and in particular, to show that such systems (of many-valued logic of a certain kind) offer a fully fledged and extremely interesting logic [4, 5]. But this leaves open the question of intuitive adequacy of many-valued logic as a logic of vagueness. Below I shall try to isolate eight questions Parikh asks, add two more and to comment on all of them. Finally, I formulate a problem on truth (in)definability in Łukasiewicz logic which shows, in my opinion, that fuzzy logic is not just “applied logic” but rather belongs to systems commonly called “philosophical logic” like modal logics, etc.

title: Epistemic Entrenchment and Arithmetical Hierarchy
abstract: If the underlying theory is sufficiently rich (e.g. like first-order arithmetic), then no epistemic entrenchment preorder of sentences is recursively enumerable. Consequently, the set of all defeasible proofs (determined by such a fixed preorder) is not recursively enumerable and hence, a fortiori, nonrecursive. On the other hand there is a satisfactorily rich epistemic entrenchment preorder

title: Fuzzy logic as logic
abstract: Formal logical calculi underlying fuzzy logic are investigated, main attention being paid to Lukasiewicz’s logic and (a variant of) its generalization by Pavelka. The paper has an introductory and survey character.