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The Origin of Relation Algebras in the Development and Axiomatization of the Calculus of Relations
, 1991
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What Is Logic?
"... It is far from clear what is meant by logic or what should be meant by it. It is nevertheless reasonable to identify logic as the study of inferences and inferential relations. The obvious practical use of logic is in any case to help us to reason well, to draw good inferences. And the typical form ..."
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It is far from clear what is meant by logic or what should be meant by it. It is nevertheless reasonable to identify logic as the study of inferences and inferential relations. The obvious practical use of logic is in any case to help us to reason well, to draw good inferences. And the typical form the theory of any part of logic seems to be a set of rules of inference. This answer already introduces some structure into a discussion of the nature of logic, for in an inference we can distinguish the input called a premise or premises from the output known as the conclusion. The transition from a premise or a number of premises to the conclusion is governed by a rule of inference. If the inference is in accordance with the appropriate rule, it is called valid. Rules of inference are often thought of as the alpha and omega of logic. Conceiving of logic as the study of inference is nevertheless only the first approximation to the title question, in that it prompts more questions than it answers. It is not clear what counts as an inference or what a theory of such inferences might look like. What are the rules of inference based on? Where do we find them? The ultimate end
Using Features for Automated Problem Solving
, 2006
"... We motivate and present an architecture for problem solving where an abstraction layer of “features ” plays the key role in determining methods to apply. The system is presented in the context of theorem proving with Isabelle, and we demonstrate how this approach to encoding control knowledge is ex ..."
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We motivate and present an architecture for problem solving where an abstraction layer of “features ” plays the key role in determining methods to apply. The system is presented in the context of theorem proving with Isabelle, and we demonstrate how this approach to encoding control knowledge is expressively different to other common techniques. We look closely at two areas where the feature layer may offer benefits to theorem proving — semiautomation and learning — and find strong evidence that in these particular domains, the approach shows compelling promise. The system includes a graphical user interface for Eclipse ProofGeneral and is available from the project web page,
On the logic of Quantum Physics and the concept of the time
 Los Alamos Archives Preprint, quantph/9804040
"... The logic–linguistic structure of quantum physics is analysed. The role of formal systems and interpretations in the representation of nature is investigated. The problems of decidability, completeness, and consistency can affect quantum physics in different ways. Bohr’s complementarity is of great ..."
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The logic–linguistic structure of quantum physics is analysed. The role of formal systems and interpretations in the representation of nature is investigated. The problems of decidability, completeness, and consistency can affect quantum physics in different ways. Bohr’s complementarity is of great interest, because it is a contradictory proposition. We shall see that the flowing of time prevents the birth of contradictions in nature, because it makes a cut between two different, but complementary aspects of the reality. PACS: 03.65.w Quantum mechanics; 03.65.Bz Foundations, theory of measurement, miscellaneous theories; 02.10.By Logic and foundations.
DOI: 10.1017/S000000000000000 Printed in the United Kingdom Towards a diagrammatic classification
"... In this article I present and discuss some criteria to provide a diagrammatic classification. Such a classification is of use for exploring in detail the domain of diagrammatic reasoning. Diagrams can be classified in terms of the use we make of them static or dynamic and of the correspondence bet ..."
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In this article I present and discuss some criteria to provide a diagrammatic classification. Such a classification is of use for exploring in detail the domain of diagrammatic reasoning. Diagrams can be classified in terms of the use we make of them static or dynamic and of the correspondence between their space and the space of the data they are intended to represent. The investigation is not guided by the opposition visual vs. non visual, but by the idea that there is a continuous interaction between diagrams and language. Diagrammatic reasoning is characterized by a duality, since it refers both to an object, the diagram, having its spatial characteristics, and to a subject, the user, who interprets them. A particular place in the classification is occupied by constructional diagrams, which exhibit for the user instructions for the application of some procedures. 1 Introduction: recentering the discussion on visual thinking In the process of describing the world around us, not only language and verbal thought matter. In recent years, an interest has grown around the phenomenon of non verbal thought, such as the use of visualization and visual thinking in reasoning in general and in scientific argumentation in particular. One motivation to pursue this study is the observation that, as Ferguson has suggested,
Some Intuitions Behind Realizability Semantics for Constructive Logic: Tableaux and Läuchli countermodels.
, 1996
"... We use formal semantic analysis based on new, modeltheoretic constructions to generate intuitive confidence that the Heyting Calculus is an appropriate system of deduction for constructive reasoning. Wellknown modal semantic formalisms have been defined by Kripke and Beth, but these have no fo ..."
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We use formal semantic analysis based on new, modeltheoretic constructions to generate intuitive confidence that the Heyting Calculus is an appropriate system of deduction for constructive reasoning. Wellknown modal semantic formalisms have been defined by Kripke and Beth, but these have no formal concepts corresponding to constructions, and shed little intuitive light on the meanings of formulae. In particular, the wellknown completeness proofs for these semantics do not generate confidence in the sufficiency of the Heyting Calculus, since we have no reason to believe that every intuitively constructive truth is valid in the formal semantics. Lauchli has proved completeness for a realizability semantics with formal concepts analogous to constructions, but the analogy is, in our view, inherently inexact. We argue in some detail that, in spite of this inexactness, every intuitively constructive truth is valid in Lauchli semantics, and therefore the Heyting Calculus is p...
c○University of Notre Dame BOOK REVIEW Geraldine Brady. From Peirce to Skolem: A Neglected Chapter in
"... The thesis of this book is that Löwenheim’s and Skolem’s work on what is now known as the downward Löwenheim–Skolem theorem developed directly from Schröder’s Algebra der Logik, which was itself an avowed elaboration of the ..."
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The thesis of this book is that Löwenheim’s and Skolem’s work on what is now known as the downward Löwenheim–Skolem theorem developed directly from Schröder’s Algebra der Logik, which was itself an avowed elaboration of the
JACQUES HERBRAND: LIFE, LOGIC, AND AUTOMATED DEDUCTION
"... The lives of mathematical prodigies who passed away very early after groundbreaking work invoke a fascination for later generations: The early death of Niels Henrik Abel (1802–1829) from ill health after a sled trip to visit his fiancé for Christmas; the obscure circumstances of Evariste Galois ’ (1 ..."
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The lives of mathematical prodigies who passed away very early after groundbreaking work invoke a fascination for later generations: The early death of Niels Henrik Abel (1802–1829) from ill health after a sled trip to visit his fiancé for Christmas; the obscure circumstances of Evariste Galois ’ (1811–1832) duel; the deaths of consumption of Gotthold Eisenstein (1823–1852) (who sometimes lectured his few students from his bedside) and of Gustav Roch (1839–1866) in Venice; the drowning of the topologist Pavel Samuilovich Urysohn (1898–1924) on vacation; the burial of Raymond Paley (1907–1933) in an avalanche at Deception Pass in the Rocky Mountains; as well as the fatal imprisonment of Gerhard Gentzen (1909–1945) in Prague1 — these are tales most scholars of logic and mathematics have heard in their student days. Jacques Herbrand, a young prodigy admitted to the École Normale Supérieure as the best student of the year1925, when he was17, died only six years later in a mountaineering accident in La Bérarde (Isère) in France. He left a legacy in logic and mathematics that is outstanding.
The birth of analytic philosophy ∗
"... Analytic philosophy was, at its birth, an attempt to escape from an earlier tradition, that of Kant, and the first battleground was mathematics. Kant had claimed that mathematics is grounded neither in experience nor in logic but in the spatiotemporal structure which we ourselves impose on experien ..."
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Analytic philosophy was, at its birth, an attempt to escape from an earlier tradition, that of Kant, and the first battleground was mathematics. Kant had claimed that mathematics is grounded neither in experience nor in logic but in the spatiotemporal structure which we ourselves impose on experience. First Frege tried to refute Kant’s account in the case of arithmetic by showing that it could be derived from logic; then Russell extended the project to the whole of mathematics. Both failed, but in addressing the problems which the project generated they founded what is nowadays known as analytic philosophy or, perhaps more appropriately, as the analytic method in philosophy. What this brief summary masks, however, is that it is far from easy to say what the analytic method in philosophy amounts to. By tracing the outlines of the moment when it was born we shall here try to identify some of its distinctive features.