Metamathematics of Fuzzy Logic
Preface. 1. Preliminaries. 2. Many-Valued Propositional Calculi. 3. Lukasiewicz Propositional Logic. 4. Product Logic, Godel Logic. 5. Many-Valued Predicate Logics. 6. Complexity and Undecidability. 7. On Approximate Inference. 8. Generalized Quantifiers and Modalities. 9. Miscellanea. 10. Historical Remarks.

title: Fuzzy logic and arithmetical hierarchy
abstract: Fuzzy logic is understood as a logic with a comparative and truth-functional notion of truth. Arithmetical complexity of sets of tautologies (identically true sentences) and satisfiable sentences (sentences true in at least one interpretation) as well of sets of provable formulas of the most important systems of fuzzy predicate logic is determined or at least estimated.

title: Arithmetical Interpretations of Dynamic Logic
abstract: An arithmetical interpretation of dynamic propositional logic (DPL) is a mapping f satisfying the following: (1) fassociates with each formula A of DPL a sentencef(A) of Peano arithmetic (PA) and with each program a a formula f(a) of PA with one free variable describing formally a supertheory of PA; (2) f commutes with logical connectives; (3) f([aK]A) is the sentence saying that f(A) is provable in the theory f(a); (4) for each axiom A of DPL, f(A) is provable in PA (and consequently, for each A provable in DPL, f(A) is provable in PA). The arithmetical completeness theorem is proved saying that a formula A of DPL is provable in DPL iff for each arithmetical interpretationf,f(A) is provable in PA. Various modifications of this result are considered.  The aim of the present paper is to apply techniques of arithmetical interpretations (or: provability interpretations) of modal propositional logic (as developed and studied by deJongh, Sambin, Solovay, Smoryn'ski, Guaspari and others, see ,  ) to dynamic logic. Dynamic logic (or: logic of programs) is a generalization of modal logic extensively studied in computer science (Pratt, Fischer, Ladner, Segerbergh, Parikh, see , , ). Both various systems of modal logic and systems of dynamic logic have well-behaving Kripke semantics (necessity means validity in all alternative states-worlds) and complete axiomatizations (for modal logic see [6] and , for dynamic logic see ). The idea behind arithmetical interpretations of modal logic is to interpret necessity as formal provability (in arithmetic). Now, dynamic logic is a modal logic with many modalities of necessity (corresponding to programs) so that the question arises whether one could interpret these modalities by formal provability in various theories (and obtain some completeness results). This question will be answered positively. What is the meaning of our results? First, the question just formulated seems to be a very natural question and it seems not to be trivial to answer it. Second, our result could elucidate from an unexpected point of view the saying of Kowalski : "algorithm logic + control". We shall see that programs (algorithms)i.e. their logical components-may be modelled by some formal theories. Third, it has proved to be useful to investigate modal calculi having various modalities, not just the necessity, when studying self-reference in Peano arithmetic ((see [10], ). And we have in dynamic logic an example of such a calculus having both Received October 15, 1981. ? 1983, Association for Symbolic Logic 0022-4812/83/4803-0021 /$02.00

title: The New Version of the GUHA Procedure ASSOC (Generating Hypotheses on Associations) — Mathematical Foundations
abstract: Design choices for a new version of the GUHA procedure ASSOC are formulated and mathematical foundations are 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: 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.