Showing papers in "The Bulletin of Symbolic Logic in 1999"

The logic of bunched implications.

Abstract: We introduce a logic BI in which a multiplicative (or linear) and an additive (or intuitionistic) implication live side-by-side. The propositional version of BI arises from an analysis of the proof-theoretic relationship between conjunction and implication; it can be viewed as a merging of intuitionistic logic and multiplicative intuitionistic linear logic. The naturality of BI can be seen categorically: models of propositional BI's proofs are given by bicartesian doubly closed categories, i.e., categories which freely combine the semantics of propositional intuitionistic logic and propositional multiplicative intuitionistic linear logic. The predicate version of BI includes, in addition to standard additive quantifiers, multiplicative (or intensional) quantifiers and which arise from observing restrictions on structural rules on the level of terms as well as propositions. We discuss computational interpretations, based on sharing, at both the propositional and predicate levels.

Tarski's System of Geometry

Abstract: This paper is an edited form of a letter written by the two authors (in the name of Tarski) to Wolfram Schwabhauser around 1978. It contains extended remarks about Tarski's system of foundations for Euclidean geometry, in particular its distinctive features, its historical evolution, the history of specific axioms, the questions of independence of axioms and primitive notions, and versions of the system suitable for the development of 1-dimensional geometry.

Hilbert's Programs: 1917-1922

Abstract: Hilbert's finitist program was not created at the beginning of the twenties solely to counteract Brouwer's intuitionism, but rather emerged out of broad philosophical reflections on the foundations of mathematics and out of detailed logical work; that is evident from notes of lecture courses that were given by Hilbert and prepared in collaboration with Bernays during the period from 1917 to 1922. These notes reveal a dialectic progression from a critical logicism through a radical constructivism toward finitism; the progression has to be seen against the background of the stunning presentation of mathematical logic in the lectures given during the winter term 1917/18. In this paper, I sketch the connection of Hilbert's considerations to issues in the foundations of mathematics during the second half of the 19th century, describe the work that laid the basis of modern mathematical logic, and analyze the first steps in the new subject of proof theory. A revision of the standard view of Hilbert's and Bernays's contributions to the foundational discussion in our century has long been overdue. It is almost scandalous that their carefully worked out notes have not been used yet to understand more accurately the evolution of modern logic in general and of Hilbert's Program in particular. One conclusion will be obvious: the dogmatic formalist Hilbert is a figment of historical (de)construction! Indeed, the study and analysis of these lectures reveal a depth of mathematical-logical achievement and of philosophical reflection that is remarkable. In the course of my presentation many questions are raised and many more can be explored; thus, I hope this paper will stimulate interest for new historical and systematic work.

Church's problem revisited

Abstract: Abstract In program synthesis, we transform a specification into a system that is guaranteed to satisfy the specification. When the system is open, then at each moment it reads input signals and writes output signals, which depend on the input signals and the history of the computation so far. The specification considers all possible input sequences. Thus, if the specification is linear, it should hold in every computation generated by the interaction, and if the specification is branching, it should hold in the tree that embodies all possible input sequences. Often, the system cannot read all the input signals generated by its environment. For example, in a distributed setting, it might be that each process can read input signals of only part of the underlying processes. Then, we should transform a specification into a system whose output depends only on the readable parts of the input signals and the history of the computation. This is called synthesis with incomplete information. In this work we solve the problem of synthesis with incomplete information in its full generality. We consider linear and branching settings with complete and incomplete information. We claim that alternation is a suitable and helpful mechanism for coping with incomplete information. Using alternating tree automata, we show that incomplete information does not make the synthesis problem more complex, in both the linear and the branching paradigm. In particular, we prove that independently of the presence of incomplete information, the synthesis problems for CTL and CTL*. are complete for EXPTIME and 2EXPTIME, respectively.

Completeness before Post: Bernays, Hilbert, and the Development of Propositional Logic

Abstract: Some of the most important developments of symbolic logic took place in the 1920s. Foremost among them are the distinction between syntax and semantics and the formulation of questions of completeness and decidability of logical systems. David Hilbert and his students played a very important part in these developments. Their contributions can be traced to unpublished lecture notes and other manuscripts by Hilbert and Bernays dating to the period 1917–1923. The aim of this paper is to describe these results, focussing primarily on propositional logic, and to put them in their historical context. It is argued that truth-value semantics, syntactic (“Post-”) and semantic completeness, decidability, and other results were first obtained by Hilbert and Bernays in 1918, and that Bernays's role in their discovery and the subsequent development of mathematical logic is much greater than has so far been acknowledged.

Gap Forcing: Generalizing the Lévy-Solovay Theorem

Abstract: The Levy-Solovay Theorem[8] limits the kind of large cardinal embeddings that can exist in a small forcing extension. Here I announce a generalization of this theorem to a broad new class of forcing notions. One consequence is that many of the forcing iterations most commonly found in the large cardinal literature create no new weakly compact cardinals, measurable cardinals, strong cardinals, Woodin cardinals, strongly compact cardinals, supercompact cardinals, almost huge cardinals, huge cardinals, and so on.

New Directions in Descriptive Set Theory

Abstract: I will start with a quick definition of descriptive set theory: It is the study of the structure of definable sets and functions in separable completely metrizable spaces. Such spaces are usually called Polish spaces. Typical examples are R^n, C^n, (separable) Hilbert space and more generally all separable Banach spaces, the Cantor space 2^N, the Baire space N^N, the infinite symmetric group S_∞, the unitary group (of the Hilbert space), the group of measure preserving transformations of the unit interval, etc.

Between Russell and Hilbert: Behmann on the Foundations of Mathematics

Abstract: After giving a brief overview of the renewal of interest in logic and the foundations of mathematics in Gottingen in the period 1914-1921, I give a detailed presentation of the approach to the foundations of mathematics found in Behmann's doctoral dissertation of 1918, Die Antinomie der transfiniten Zahl und ihre Auflosung durch die Theorie von Russell und Whitehead. The dissertation was written under the guidance of David Hilbert and was primarily intended to give a clear exposition of the solution to the antinomies as found in Principia Mathematica. In the process of explaining the theory of Principia, Behmann also presented an original approach to the foundations of mathematics which saw in sense perception of concrete individuals the Archimedean point for a secure foundation of mathematical knowledge. The last part of the paper points out an important numbers of connections between Behmann's work and Hilbert's foundational thought.

Models of Second-Order Zermelo Set Theory

Abstract: In [12], Ernst Zermelo described a succession of models for the axioms of set theory as initial segments of a cumulative hierarchy of levels Uα Vα . The recursive definition of the Vα's is: Thus, a little reflection on the axioms of Zermelo-Fraenkel set theory (ZF) shows that Vω, the first transfinite level of the hierarchy, is a model of all the axioms of ZF with the exception of the axiom of infinity. And, in general, one finds that if κ is a strongly inaccessible ordinal, then Vκ is a model of all of the axioms of ZF. (For all these models, we take ∈ to be the standard element-set relation restricted to the members of the domain.) Doubtless, when cast as a first-order theory, ZF does not characterize the structures 〈Vκ,∈∩(Vκ×Vκ )〉 for κ a strongly inaccessible ordinal, by the Löwenheim-Skolem theorem. Still, one of the main achievements of [12] consisted in establishing that a characterization of these models can be attained when one ventures into second-order logic. For let second-order ZF be, as usual, the theory that results from ZF when the axiom schema of replacement is replaced by its second-order universal closure. Then, it is a remarkable result due to Zermelo that second-order ZF can only be satisfied in models of the form 〈Vκ,∈∩(Vκ×Vκ )〉 for κ a strongly inaccessible ordinal.

19th Century Logic Between Philosophy and Mathematics

Abstract: The history of modern logic is usually written as the history of mathematical or, more general, symbolic logic. As such it was created by mathematicians. Not regarding its anticipations in Scholastic logic and in the rationalistic era, its continuous development began with George Boole’s The Mathematical Analysis of Logic of 1847, and it became a mathematical subdiscipline in the early 20th century. This style of presentation cuts off one eminent line of development, the philosophical development of logic, although logic is evidently one of the basic disciplines of philosophy. One needs only to recall some of the standard 19th century definitions of logic as, e.g., the art and science of reasoning (Whateley) or as giving the normative rules of correct reasoning (Herbart). In the paper the relationship between the philosophical and the mathematical development of logic will be discussed. Answers to the following questions will be provided: 1. What were the reasons for the philosophers’ lack of interest in formal logic? 2. What were the reasons for the mathematicians’ interest in logic? 3. What did “logic reform” mean in the 19th century? Were the systems of mathematical logic initially regarded as contributions to a reform of logic? 4. Was mathematical logic regarded as art, as science or as both?

The Syllogism's Final Solution

Abstract: In 1883, while a student of C. S. Peirce at Johns Hopkins University, Christine Ladd-Franklin published a paper titled On the Algebra of Logic, in which she develops an elegant and powerful test for the validity of syllogisms that constitutes the most significant advance in syllogistic logic in two thousand years. Sadly, her work has been all but forgotten by logicians and historians of logic. Ladd-Franklin's achievement has been overlooked, partly because it has been overshadowed by the work of other logicians of the nineteenth century renaissance in logic, but probably also because she was a woman. Though neglected, the significance of her contribution to the field of symbolic logic has not been diminished by subsequent achievements of others. In this paper, I bring to light the important work of Ladd-Franklin so that she is justly credited with having solved a problem over two millennia old. First, I give a brief survey of the history of syllogistic logic. In the second section, I discuss the logical systems called “algebras of logic”. I then outline Ladd-Franklin's algebra of logic, discussing how it differs from others, and explain her test for the validity of the syllogism, both in her symbolic language and the more familiar language of modern logic. Finally I present a rigorous proof of her theorem. Ladd-Franklin developed her algebra of logic before the methods necessary for a rigorous proof were available to her. Thus, I do now what she could not have done then.

Degree spectra of relations on computable structures

Abstract: The study of additional relations on computable structures began with the work of Ash and Nerode [2]. The concept of degree spectra of relations was later introduced by Harizanov [18]. In this dissertation, several new examples of possible degree spectra of relations on computable structures are given. In particular, it is shown that, for every c.e. degree a , the set {0, a} can be realized as the degree spectrum of an intrinsically c.e. relation on a structure of computable dimension two, thus answering a question of Goncharov and Khoussainov [16]. Some extensions of this result are given, and the methods used in proving it are employed to construct a computably categorical structure whose expansion by a single constant has computable dimension w . Degree spectra of relations on computable models of particular algebraic theories are also investigated. For example, it is shown that, for every n > 0, there is a computable integral domain with a subring whose degree spectrum consists of exactly n c.e. degrees, including 0. In contrast to this result, it is shown, for instance, that the degree spectrum of a computable relation on a computable linear ordering is either a singleton or infinite. In both cases, sufficient criteria for similar results to hold of a given class of structures are provided.

Accessible Recursive Functions

Abstract: The class of all recursive functions fails to possess a natural hierarchical structure, generated predicatively from “within”. On the other hand, many (proof-theoretically significant) sub-recursive classes do. This paper attempts to measure the limit of predicative generation in this context, by classifying and characterizing those (predictably terminating) recursive functions which can be successively defined according to an autonomy condition of the form: allow recursions only over well-orderings which have already been “coded” at previous levels. The question is: how can a recursion code a well-ordering? The answer lies in Girard's theory of dilators, but is reworked here in a quite different and simplified framework specific to our purpose. The “accessible” recursive functions thus generated turn out to be those provably recursive in ( –CA) 0 .

In Memoriam: Albert G. Dragalin 1941–1998

Abstract: Albert G. Dragalin, an important figure in the development of mathematical logic in the former Soviet Union, passed away from a sudden heart attack in Debrecen (Hungary) on December 18, 1998, at the age of 57. He was born on April 10, 1941 in Morzchevec (in the Arkhangelsk region in northern Russia). A. Dragalin received his Master’s Degree in 1963 and his Ph.D. in 1968 under A. A. Markov Jr. from the Department of Mathematics and Mechanics of Moscow State University. After joining the Chair of Mathematical Logic in the same Department in 1968, he established several graduate courses in mathematical logic, especially in proof theory, intuitionistic logic, and axiomatic set theory. Almost every year he introduced a new course. His seminar on proof theory was for many years one of the most important centers of activity in mathematical logic in Moscow. It attracted many young mathematicians. He was a very lively and enthusiastic teacher with broad views on logic andmathematics. Practically every undergraduate or graduate student interested in mathematical logic was influenced by him, even if he was not always a formal advisor. Dragalin supervised a large number of Master students and more than a dozen Ph.D. disserations in mathematical logic. He played an important part in establishing a course in mathematical logic for all mathematicsmajors atMoscow StateUniversity. This course was first taught by A. N. Kolmogorov. Dragalin coauthored with A. N. Kolmogorov two textbooks [7, 8] for this course. Dragalin’s independent personality did not get along well with the Soviet realities in general and the depressing situation in Soviet mathematical logic in particular. In 1983 in the middle of his academic career he left Moscow for Debrecen, Hungary, with his second wife Svetlana Buzàsi, who was a Hungarian mathematician of Russian origin. In Hungary, Albert struggled to establish himself in a new cultural and scientific environment and eventually succeeded. He joined the Computer Center of the Lajos Kossuth University at Debrecen and moved to the Department of Computer Science of the same university in 1990. Dragalin received his Doctor of Sciences Degree in mathematics from the Hungarian Academy of Sciences in 1988. In 1993, he became chair of the Department of Computer Science of the Lajos Kossuth University.

In Memoriam: Nelson Goodman, 1906-1998

Abstract: Nelson Goodman, analytic philosopher of science and of the arts who made seminal contributions in a striking variety of areas, died November 25, 1998, in Needham,Massachusetts. He was 92. Goodman was a frequent early contributor to the Journal, and was Vice President of the Association from 1950 to 1952. In later years he and the Journal were tomove in different directions, though his work continued to involve applications of logic. Goodmanpublished eight papers in the Journal, three of them coauthored, and a like number of reviews. Four of the papers sought to develop methods for determining the relative simplicity of sets of extralogical primitives. Constructional systems, like that of Carnap’s Aufbau, were a major focus for Goodman; he offered a detailed study of them in his first book, Structure of appearance [1]. As he put it, “The motives for seeking economy in the basis of a system are much the same as the motives for constructing the system itself.” The paper Elimination of extra-logical postulates [9], written with Quine, showed how such postulates might be replaced by “mere definition” in a wide range of cases. A ready illustration is elimination of the postulate of transitivity for the “part of” relation: rather than take ‘Pt’ as primitive, define it in terms of ‘O’, overlaps, by

1998–1999 Winter Meeting of the Association for Symbolic Logic

Abstract: s of invited talks and contributed talks given (in person or by title) by members of the Association for Symbolic Logic follow. For the Program Committee Peter Cholak Abstracts of invited talkss of invited talks MICHAEL BENEDIKT,What you can and can’t define in a model. Bell Laboratories, Room 2f-329, 263 Shuman Blvd., Naperville, Illinois 60566, USA. E-mail: benedikt@research.bell-labs-com. We overview some work in the definability of second order properties with first-order sentences. It works like this: You fix a modelM (e.g., the real field), and a class of subsets C of the model (e.g., the definable subsets), and some property P(S) of elements S from C , e.g., “S is of even cardinality”, and you ask: Is there a first-order formula in the language of M plus a predicate for S that defines within C exactly the sets that have P(S)? In the case where C is the collection of finite subsets of M , you get a generalization of finite model theory parameterized by the model M . In the case where M has a natural topology—as with models based on the real line—you get a kind of topological model theory parameterized byM . In both cases there are connections between the resulting theory and questions in stability theory and real algebraic geometry. c © 1999, Association for Symbolic Logic 1079-8986/99/0502-0006/$2.10

1999 Spring Meeting of the Association for Symbolic Logic

Abstract: s of invited and contributed talks (given in person or by title) follow. For the Program Committee Charles Parsons Abstracts of contributed talkss of contributed talks G. ALDO ANTONELLI, Free set algebras satisfying systems of equations. Logic and Philosophy of Science, University of California, Irvine, California 92697-5100, USA. E-mail: aldo@uci.edu. In this talk we introduce the notion of a set algebra S satisfying a system E of equations. After defining a notion of freeness for such algebras, we show that, for any system E of equations, set algebras that are free in the class of structures satisfying E exist and are unique up to a bisimulation. Along the way, we mention analogues of classical set-theoretic and algebraic properties as well as connections between the algebraic and set-theoretic viewpoints. PHILIP EHRLICH, The reals and the surreals. Department of Philosophy, Ohio University, Athens, Ohio 45701, USA. E-mail: ehrlich@ohiou.edu. In his monograph On numbers and games [1], J. H. Conway introduced an ordered field No which contains (in a suitable sense that can be made precise) “all numbers great and small.” However, in addition to its distinguished structure as an ordered field No has a rich c © 1999, Association for Symbolic Logic 1079-8986/99/0504-0004/$1.60

XI Latin American Symposium on Mathematical Logic

Abstract: s of invited and contributed talks, and contributed communications presented by title follow. For the Program Committee Carlos Augusto Di Prisco Abstracts of invited and contributed talkss of invited and contributed talks MANUELABAD, Separating sets for maximal subalgebras of pseudocomplemented distributive lattices. Departamento de Matemática, Universidad Nacional del Sur, Av. Alem 1253, (8000) Bahía Blanca, Argentina. E-mail: imabad@criba.edu.ar. The Frattini subalgebra of an algebra A, denoted Φ(A), is defined to be the intersection c © 1999, Association for Symbolic Logic 1079-8986/99/0504-0006/$4.00

1998–99 Annual Meeting of the Association for Symbolic Logic

Abstract: s of talks given in person or by title by members of the Association for Symbolic Logic follow. For the Program Committee Sam Buss c © 1999, Association for Symbolic Logic 1079-8986/99/0503-0007/$3.70

Australasian Association for Logic 29th Annual Conference

Abstract: s of contributed talks SEIKI AKAMA, Negative facts and constructible falsity. Computational Logic Laboratory, Department of Information Systems, Teikyo Heisei University, 2289 Uruido, Ichihara-shi, Chiba 290-0193, Japan. E-mail: akama@cn.thu.ac.jp. We formalize the notion of negative facts due to Russell (1956) in terms of Nelson’s (1949) logic of constructible falsity. We generalize van Fraassen’s (1969) theory of facts so that we can deal with a positive fact as a true proposition and a negative fact as a false proposition in a constructive manner. We extend the previous work in Akama and Sylvan (1994). We provide some adequacy results for the revised van Fraassen semantics with respect to several versions of Kripke semantics for strong negation (cf. Routley (1974), Akama (1988, 1990)). The advantage of van Fraassen semantics is to avoid the unnecessary commitment of semantic entities like possible worlds or set-ups in that we allow for various kinds of facts. The proposed semantics can thus be expanded for other non-classical logics. JC BEALL, The justification of deduction one last time. School of Philosophy, Faculty ofArts, TheUniversity of Tasmania,GPOBox 252-41Hobart, Tasmania 7001, Australia. E-mail: j.c.beall@utas.edu.au. David Hume told us that induction faces a dilemma: Its alleged justification will be c © 1999, Association for Symbolic Logic 1079-8986/99/0504-0005/$2.00