Philosophy

The technical and aesthetic foundations of György Ligeti's concept of micropolyphony, which he employed most prominently in his 1961 orchestral work, Atmosphères, can be credited, in part, to his post-emigration experiments with electronic composition at the studios of the NWDR in Cologne in the late 1950s. Although Ligeti had already theorized general concepts of musical texture and space prior to his emigration to the West in 1956, the nature of the micropolyphony he employs in his later work is characteristically distinct, exhibiting a greater sensitivity to density and timbre in addition to processes of aural integration and interaction. To this end, this analysis examines the way, and more importantly the extent to which, Ligeti’s often overlooked work in electronic music directly influenced his approach to the implementation of sound-mass in his later methodology. This is done through a comparison of Pièce Électronique No. 3, an electronic work begun in 1957, and Atmosphèr...

Gödel's incompleteness results are two of the most fundamental and important contributions to logic and the foundations of mathematics. He showed that no axiomatizable formal system strong enough to capture elementary number theory can prove every true sentence in its language. This theorem is an important limiting result regarding the power of formal axiomatics, but has also been of immense importance in other areas, such as the theory of computability.

A general class of labeled sequent calculi is investigated, and necessary and sufficient conditions are given for when such a calculus is sound and complete for a finite-valued logic if the labels are interpreted as sets of truth values (sets-as-signs). Furthermore, it is shown that any finite-valued logic can be given an axiomatization by such a labeled calculus using arbitrary" systems of signs," ie, of sets of truth values, as labels. The number of labels needed is logarithmic in the number of truth values, and it is shown that this bound is tight.

Abstract. Although Kurt Gödel does not figure prominently in the history of computabilty theory, he exerted a significant influence on some of the founders of the field, both through his published work and through personal interaction. In particular, Gödel's 1931 paper on incompleteness and the methods developed therein were important for the early development of recursive function theory and the lambda calculus at the hands of Church, Kleene, and Rosser.

Many-valued logic is not much younger than the whole field of symbolic logic. It was introduced in the early twenties of this century by Lukasiewicz [1920] and Post [1921] and has since developed into a very large area of research. Most of the early work done has concentrated on problems of axiomatizability on the one hand, and algebraical/model theoretic investigations on the other. The proof theory of many-valued systems has not been investigated to any comparable extent.

Abstract: A uniform construction for sequent calculi for finite-valued first-order logics with distribution quantifiers is exhibited. Completeness, cut-elimination and midsequent theorems are established. As an application, an analog of Herbrand's theorem for the four-valued knowledgerepresentation logic of Belnap and Ginsberg is presented. It is indicated how this theorem can be used for reasoning about knowledge bases with incomplete and inconsistent information.

Abstract: Although arguments for and against competing theories of vagueness often appeal to claims about the use of vague predicates by ordinary speakers, such claims are rarely tested. An exception is Bonini et al.(1999), who report empirical results on the use of vague predicates by Italian speakers, and take the results to count in favor of epistemicism. Yet several methodological difficulties mar their experiments; we outline these problems and devise revised experiments that do not show the same results.

Jan Kraj����ek posed the following problem: Is there is a generalization result in the theory of real closed fields of the form: If A (1+���+ 1)(n occurrences of 1) is provable in length k for all n����, then (��� x) A (x) is provable? It is argued that the answer to this question depends on the particular formulation of the ���theory of real closed fields.��� Four distinct formulations are investigated with respect to their generalization behavior. It is shown that there is a positive answer to Kraj����ek's question for

(A, B, C arbitrary formulas) is the propositional pendant of the schema of identity. It can be argued that, apart form the usual propositional tautologies and inference schemas which are given as axiomatizations of propositional logic (eg, modus pohens, modus tollens, case distinction, chain rule), the schema of equivalence is also used extensively in mathematical reasoning. However, it seems that Eq has not been used or investigated in the proof theory of propositional logic to any significant extent.

A simple model of dynamic databases is studied from a modal logic perspecitve. A state �� of a database is an atomic update of a state �� if at most one atomic statement is evaluated differently in �� compared to ��. The corresponding restriction on Kripke-like structures yields so-called update logics. These logics are studied also in a many-valued context. Adequate tableau calculi are given.

The problem of algorithmic structuring of proofs in the sequent calculi LK and LK B (LK where blocks of quantifiers can be introduced in one step) is investigated, where a distinction is made between linear proofs and proofs in tree form. In this framework, structuring coincides with the introduction of cuts into a proof. The algorithmic solvability of this problem can be reduced to the question of k/l-compressibility: Given a proof of of length k, and lk: Is there is a proof of of length l?

Abstract Entailment in propositional Godel logics can be defined in a natural way. While all infinite sets of truth values yield the same sets of tautologies, the entailment relations differ. It is shown that there is a rich structure of infinite-valued Godel logics, only one of which is compact. It is also shown that the compact infinite-valued Godel logic is the only one which interpolates, and the only one with an re entailment relation

Next to the logicist project of Frege and Russell of reducing mathematics (or at least arithmetic) wholesale to logic, Hilbert's program is the main contribution to the foundation of mathematics of the last century. Taking as his motivation the paradoxes of set theory and the revisionist attacks on classical mathematics by such predicativists as Poincaréand such intuitionists as Brouwer and the later Weyl, Hilbert formulated a new program for the grounding of arithmetic.

Abstract Carnap's doctrine of linguistic pluralism, enshrined in the "Principle of Tolerance" of his Logical Syntax of Language, became a cornerstone of Carnap's mature philosophy in general and conception of scientific method in particular. Not least because of widespread misunderstandings of Carnap's project on the basis of insufficient attention to this doctrine, it has become the subject of significant attention in recent years.

Abstract. The generalization properties of algebraically closed fields of characteristic and of characteristic 0 are investigated in the sequent calculus with blocks of quantifiers. It is shown that admits finite term bases, and admits term bases with primality constraints. From these results the analogs of Kreisel's Conjecture for these theories follow: If for some,(1's) is provable in steps, then is provable.

G��del's incompleteness results are two of the most fundamental and important contributuions to logic and the foundations of mathematics. G��del showed that no axiomatizable formal system strong enough to capture elementary number theory can prove every true sentence in its language. This theorem is an important limiting result regarding the power of formal axiomatics, but has also been of immense importance in other areas, eg, the theory of computability.