Automated Deduction – CADE 28

This open access book constitutes the proceeding of the 28th International Conference on Automated Deduction, CADE 28, held virtually in July 2021. The 29 full papers and 7 system descriptions presented together with 2 invited papers were carefully reviewed and selected from 76 submissions. CADE is the major forum for the presentation of research in all aspects of automated deduction, including foundations, applications, implementations, and practical experience. The papers are organized in the following topics: Logical foundations; theory and principles; implementation and application; ATP and AI; and system descriptions

Metalogic and the psychology of reasoning.

The central topic of the thesis is the relationship between logic and the cognitive psychology of reasoning. This topic is treated in large part through a detailed examination of the recent work of P. N. Johnson-Laird, who has elaborated a widely-read and influential theory in the field. The thesis is divided into two parts, of which the first is a more general and philosophical coverage of some of the most central issues to be faced in relating psychology to logic, while the second draws upon this as introductory material for a critique of Johnson-Laird's `Mental Model' theory, particularly as it applies to syllogistic reasoning. An approach similar to Johnson-Laird's is taken to cognitive psychology, which centrally involves the notion of computation. On this view, a cognitive model presupposes an algorithm which can be seen as specifying the behaviour of a system in ideal conditions. Such behaviour is closely related to the notion of `competence' in reasoning, and this in turn is often described in terms of logic. Insofar as a logic is taken to specify the competence of reasoners in some domain, it forms a set of conditions on the 'input-output' behaviour of the system, to be accounted for by the algorithm. Cognitive models, however, must also be subjected to empirical test, and indeed are commonly built in a highly empirical manner. A strain can therefore develop between the empirical and the logical pressures on a theory of reasoning. Cognitive theories thus become entangled in a web of recently much-discussed issues concerning the rationality of human reasoners and the justification of a logic as a normative system. There has been an increased interest in the view that logic is subject to revision and development, in which there is a recognised place for the influence of psychological investigation. It is held, in this thesis, that logic and psychology are revealed by these considerations to be interdetermining in interesting ways, under the general a priori requirement that people are in an important and particular sense rational. Johnson-Laird's theory is a paradigm case of the sort of cognitive theory dealt with here. It is especially significant in view of the strong claims he makes about its relation to logic, and the role the latter plays in its justification and in its interpretation. The theory is claimed to be revealing about fundamental issues in semantics, and the nature of rationality. These claims are examined in detail, and several crucial ones refuted. Johnson- Laird's models are found to be wanting in the level of empirical support provided, and in their ability to found the considerable structure of explanation they are required to bear. They fail, most importantly, to be distinguishable from certain other kinds of models, at a level of theory where the putative differences are critical. The conclusion to be drawn is that the difficulties in this field are not yet properly appreciated. Psychological explantion requires a complexity which is hard to reconcile with the clarity and simplicity required for logical insights

Formal methods in the theories of rings and domains

In recent years, Hilbert's Programme has been resumed within the framework of constructive mathematics. This undertaking has already shown its feasability for a considerable part of commutative algebra. In particular, point-free methods have been playing a primary role, emerging as the appropriate language for expressing the interplay between real and ideal in mathematics. This dissertation is written within this tradition and has Sambin's notion of formal topology at its core. We start by developing general tools, in order to make this notion more immediate for algebraic application. We revise the Zariski spectrum as an inductively generated basic topology, and we analyse the constructive status of the corresponding principles of spatiality and reducibility. Through a series of examples, we show how the principle of spatiality is recurrent in the mathematical practice. The tools developed before are applied to specific problems in constructive algebra. In particular, we find an elementary characterization of the notion of codimension for ideals of a commutative ring, by means of which a constructive version of Krull's principal ideal theorem can be stated and proved. We prove a formal version of the projective Eisenbud-Evans-Storch theorem. Finally, guided by the algebraic intuition, we present an application in constructive domain theory, by proving a finite version of Kleene-Kreisel density theorem for non-flat information systems.In den vergangenen Jahren wurde das Hilbertsche Programm im Rahmen der konstruktiven Mathematik wiederaufgenommen. Diese Unternehmung hat sich vor allem in der kommutativen Algebra als praktikabel erwiesen. Insbesondere spielen punktfreie Methoden eine wesentliche Rolle: sie haben sich als die angemessene Sprache herausgestellt, um das Zwischenspiel von "real'" und "ideal" in der Mathematik auszudrücken. Die vorliegende Dissertation steht in dieser Tradition; zentral ist Sambins Begriff der formalen Topologie. Zunächst entwickeln wir ein allgemeines Instrumentarium, das geeignet ist, diesen Begriff seinen algebraischen Anwendungen näherzubringen. Sodann arbeiten wir das Zariski-Spektrum in eine induktiv erzeugte "basic topology" um und analysieren den konstruktiven Status der einschlägigen Varianten von Spatialität und Reduzibilität. Durch Angabe einer Reihe von Instanzen zeigen wir, wie häufig das Prinzip der Spatialität in der mathematischen Praxis vorkommt. Die eigens entwickelten Werkzeuge werden schließlich auf spezifische Probleme aus der konstruktiven Algebra angewandt. Insbesondere geben wir eine elementare Charakterisierung der Kodimension eines Ideals in einem kommutativen Ring an, mit der eine konstruktive Fassung des Krullschen Hauptidealsatzes formuliert und bewiesen werden kann. Ferner beweisen wir eine formale Fassung des Satzes von Eisenbud-Evans-Storch im projektiven Fall. Geleitet von der algebraischen Intuition stellen wir zuletzt eine Anwendung in der konstruktiven Bereichstheorie vor, indem wir eine finite Variante des Dichtheitssatzes von Kleene und Kreisel für nicht-flache Informationssysteme beweisen

Chains of life: Turing, Lebensform, and the emergence of Wittgenstein’s later style

This essay accounts for the notion of Lebensform by assigning it a logical role in Wittgenstein’s later philosophy. Wittgenstein’s additions of the notion to his manuscripts of the PI occurred during the initial drafting of the book 1936-7, after he abandoned his effort to revise The Brown Book. It is argued that this constituted a substantive step forward in his attitude toward the notion of simplicity as it figures within the notion of logical analysis. Next, a reconstruction of his later remarks on Lebensformen is offered which factors in his reading of Alan Turing’s “On computable numbers, with an application to the Entscheidungsproblem“ (1936/7), as well as his discussions with Turing 1937-1939. An interpretation of the five occurrences of Lebensform in the PI is then given in terms of a logical “regression” to Lebensform as a fundamental notion. This regression characterizes Wittgenstein’s mature answer to the question, “What is the nature of the logical?