I propose that logic may be seen as a science of patterns—however, not in the sense in which mathematics is a science of patterns, but rather in the sense in which physics is. The proposal is that logic identifies, explores, and fixes the inferential patterns which de facto govern our argumentative practices. It can be seen, I argue, as picking up the patterns and working from them toward the state of reflective equilibrium, where the laws it (...) aims at are explicitly articulated. Due to the normativity of logic, this process is not quite the same as that in which a natural science works its way from data to its laws; however, there is no reason why it should be seen as proceeding via a mysterious "a priori analysis.” Instead, it should proceed in the intersubjective space in which the sciences are practiced. (shrink)

Since its publication in 1967, van Heijenoort’s paper, “Logic as Calculus and Logic as Language” has become a classic in the historiography of modern logic. According to van Heijenoort, the contrast between the two conceptions of logic provides the key to many philosophical issues underlying the entire classical period of modern logic, the period from Frege’s Begriffsschrift (1879) to the work of Herbrand, Gödel and Tarski in the late 1920s and early 1930s. The present paper (...) is a critical reflection on some aspects of van Heijenoort’s thesis. I concentrate on the case of Frege and Russell and the claim that their philosophies of logic are marked through and through by acceptance of the universalist conception of logic, which is an integral part of the view of logic as language. Using the so-called “Logocentric Predicament” (Henry M. Sheffer) as an illustration, I shall argue that the universalist conception does not have the consequences drawn from it by the van Heijenoort tradition. The crucial element here is that we draw a distinction between logic as a universal science and logic as a theory. According to both Frege and Russell, logic is first and foremost a universal science, which is concerned with the principles governing inferential transitions between propositions; but this in no way excludes the possibility of studying logic also as a theory, i.e., as an explicit formulation of (some) of these principles. Some aspects of this distinction will be discussed. (shrink)

This introduction to the basic ideas of structural proof theory contains a thorough discussion and comparison of various types of formalization of first-order logic. Examples are given of several areas of application, namely: the metamathematics of pure first-order logic (intuitionistic as well as classical); the theory of logic programming; category theory; modal logic; linear logic; first-order arithmetic and second-order logic. In each case the aim is to illustrate the methods in relatively simple situations and (...) then apply them elsewhere in much more complex settings. There are numerous exercises throughout the text. In general, the only prerequisite is a standard course in first-order logic, making the book ideal for graduate students and beginning researchers in mathematical logic, theoretical computer science and artificial intelligence. For the new edition, many sections have been rewritten to improve clarity, new sections have been added on cut elimination, and solutions to selected exercises have been included. (shrink)

In this book, I will discuss three main topics: the roots and aims of scientific knowledge, scientific knowledge in society, and science and values I understand scientific knowledge as being a planned and continuous production of the general and common knowledge of scientific communities. I begin my discussion with a brief analysis of the main differences between sciences, on the one hand, and everyday experience, philosophies, religions, and ideologies, on the other. I define the concept of science as (...) a set of family resemblances (in Wittgenstein's sense). The sciences can be distinguished along three lines, in terms of the difference between empirical and non-empirical sciences, differences in degrees of theoretical and methodological structure among individual sciences and scientific disciplines. Scientific knowledge is based upon some grounds, paradigmatic knowledge bases. Scientific knowledge is founded on a unified paradigmatic knowledge base. The latter includes fundamental theories, basic facts, generally accepted empirical regularities (laws), scientific research methods, scientific explanations, and established relations to other sciences. Scientific knowledge in general is defined well in classical terms by the definition that knowledge is a true and rationally-justified belief. In this connection, scientific truth is relative, depending on the given paradigmatic knowledge base and on the precision and accessibility of the measurements and observations involved. In the book's first part, I consider the production and testing of scientific hypotheses, scientific explanations, the structure of scientific theories, and the problem of scientific realism. I consider explanation an integral part of a wider nexus of heuristic and theoretical activities in science like giving reasons, understanding, clarification, and founding. Continuous production of scientific explanations is a necessary condition for the systematic reproduction of scientific knowledge. Accordingly, it does not suffice to describe them in terms of the logical structure of the arguments on which they are based, but rather, scientific explanations ought to be given complete with an epistemic and pragmatic context of the explanation. I consider especially deductive-nomological, inductive-statistical, dispositional, teleological-functional, and genetic explanations. I emphasize the ever-increasing importance of explanations by analogy and by model. Orthogonal to these distinctions lies the distinction between causal and epistemic explanation. Epistemic explanations relate scientific discoveries to epistemic progress, and causal explanations are these epistemic explanations that relate scientific discoveries to reality too. Causal explanations inform us that certain changes occur in the world as agents of other changes, or rather, that certain changes are real consequences of other changes. Thus, by having explanations of causes, we can intervene in the world to induce by design certain changes in some course of phenomena. I consider scientific theories as logically and epistemically ordered wholeness of scientific knowledge in a realm of research that enables us to construct useful models of reality. After the discussion of different 'statement' views on theories and different interpretations of the theory-experience relation, I introduce the 'model' theoretical (non-statement) views on theories, especially the structural theory of science. The model views on theories transcend the realism-antirealism dualism by introducing models between theory and reality. Therefore, the question of scientific realism is not merely whether and, if so, which element(s), of scientific theories represent reality, but rather which conditions suffice to infer from a theory that one of its models truly expresses reality. In other words, under which conditions do models of a theory represent models of reality? Generally, models are just as much images of reality in a theory as they are images of a theory in reality. The actual problem of scientific realism doesn't originate in the observable-unobservable distinction (even if this distinction is otherwise theoretically loaded), but rather in ontological quandaries, of the sort especially encountered in modern physical theories such as quantum mechanics, relativity theory, and cosmological theories, although they are known to arise in other sciences as well. Scientific theories may be real, even if they describe nothing about the world. However, such theories can be given this status only if their formal, mathematical postulates exhibit some formal homology between reality and mathematical forms. The very fact that it is possible to describe reality with the help of mathematics even in cases, where every (representational, conceptual) language and thought fall short, would indicate that even mathematics itself is, in some sense, a part of reality, although 'outside' of the empirical world. It follows we have to distinguish the non-empirical reality and the empirical world. Formal objectivity and non-arbitrariness of theories do not emerge from either our language or thoughts, or from the world of facts, causes, and effects, but rather from non-empirical reality, which frames the world as some kind of objective transcendental condition for the possibility of the existence of the world, thoughts, and language. In the second part of the book, I consider the incorporation of scientific work and scientific knowledge into modern society and its relations to values and ethics. Production of scientific knowledge goes on as a parallel distributed process of a special kind of social (shared) cognition. The central part of this cognition is the presentation of parts of the world by scientific (theoretical) models which are warranted by the shared and public practice of justification, and in reflective acting in the world as it would. I consider various formal concepts of collective propositional knowledge (distributed, mutual, common knowledge). Scientific knowledge is an achievement to be ascribed to a scientific community or, indirectly, to all those who apply or expand it, rather than to individuals per se. It presents the continuous and partially planned production of non-trivial common knowledge of the research communities in modern societies. However, besides common knowledge, shared dispositional knowledge (knowledge how of scientific teams) and shared knowledge by acquiring (knowledge of fundamental similarities and differences in scientific teams) are parts of the real body of scientific knowledge. Scientific common knowledge implies the projection into idealized universal knowledge of all real or potentially rational and well-informed individuals. The respect of scientific knowledge thus presents the norm of rationality. Science as the intensively shared and socially distributed kind of cognition enters into modern society as a special kind of work (general work), as a productive force (social invention drive), and as a form of capital (human, social, cultural, informational, etc.). However any of these categories can adequately represent the socio-economical role of science in modern societies, and we are still waiting for a more adequate general concept that would theoretically cover the 'scientification' of modern societies. My thesis is that the most important general effect of scientification is (and will be) the growth of social processes that structurally resemble parallel distributed shared social cognition of science, and not so much the rise of the use of scientific knowledge and scientifically based information in all kinds of production. Scientification of modern societies inevitably includes a much stronger need for moral and social responsibility of scientists for their work and its effects in their local society and the global world. This need comes in contradiction with the quite large accepted idea (and even the 'norm') on value-free research. I discuss to some degree different arguments for the absence of non-epistemic values and ethical reasoning from science and find them non-convincing. The sound idea of value-free research lies in the need for the absence of categorical value statements from scientific explanations and theoretical deductions. Values and ethical judgment have, and sometimes they even have to play an important role in another part of research and its application in science or outside it. Not the value-loudness of research is a danger for science but its blind, non-reflective use. It leads to the ideologization and politicization of science. A similar danger lies in the purely value-neutrality of research. It leads to the instrumentalization of science to the technical or bureaucratic means for 'any' aims. (shrink)

Classical logic has proved inadequate in various areas of computer science, artificial intelligence, mathematics, philosopy and linguistics. This is an introduction to extensions of first-order logic, based on the principle that many-sorted logic (MSL) provides a unifying framework in which to place, for example, second-order logic, type theory, modal and dynamic logics and MSL itself. The aim is two fold: only one theorem-prover is needed; proofs of the metaproperties of the different existing calculi can be (...) avoided by borrowing them from MSL. To make the book accessible to readers from different disciplines, whilst maintaining precision, the author has supplied detailed step-by-step proofs, avoiding difficult arguments, and continually motivating the material with examples. Consequently this can be used as a reference, for self-teaching or for first-year graduate courses. (shrink)

A common reaction to Hegel’s suggestion that we collapse Kant’s distinction between form and content is that, since such a move would also deprive us of any way of distinguishing the merely logical from the real possibility of our concepts, it is incoherent and ought to be rejected. It is true that these two distinctions are intimately related in Kant, such that if one goes, the other does as well. But it is less obvious that giving them up as Kant (...) conceives them is as incoherent a proposal as many of Hegel’s critics think. It has been the point of a recent account of Hegel’s idealism to demonstrate that his critique of Kant’s dichotomy between form and content or concept and intuition does not commit him to the view that human cognition is materially creative of its content in the manner of a God-like or intuitive intellect—does not, in other words, signify a return to a pre-Critical metaphysics. This is the interpretative stance I adopt here as well, in hope of giving defenders of Kant reasons for taking Hegel more seriously. (shrink)

The paper discusses in how far medieval logic can appropriately be characterized as a formal science. In this respect, the special mediecal approach to logic as a scientia sermocinalis is examined as well as its main doctrines, namely the theories of supposition and of consequences, and the famous characterization of logic as an ars artium or scientia scientiarum. It is pointed out that medieval logic is not devoted to the setting up of formal systems or (...) any metalogical analysis of formal structures. Logic in the medieval sense of the discipline is necessarily connected with semantical aspects of natural language. Accordingly, we are confronted with a discipline going far beyond the formal structures of discourse. The classification of medieval logic as a formal science is appropriate only under selected perspectives. (shrink)

The connection of Hermann Сohen’s “The Logic of Pure Knowledge” with the revolutionary transformations in physics and mathematics at the end of the 19th century is shown. Сohen criticised Kant’s answer to the question “How is mathematics possible”? If Kant refers to a priori forms of pure intuition, Сohen sees in it a restriction of freedom of mathematical thinking by limits of intuition. It has been shown that Cohen's position is in accordance with the main development of mathematics in (...) the last decades of the 19th century, in particular, with K. Weierstrass’ striving to get rid of geometrical or mechanical images and intuitions in the mathematical analysis. Cohen was also well informed about the latest ideas in physics of his time. They are also discussed in “The Logic of Pure Knowledge.” The revolutionary spirit of physics and mathematics was appealing to Cohen, and he felt a corresponding enthusiasm for it. The ongoing scientific revolution is consonant with Cohen’s assertion that the foundations of science are hypotheses. The purity of pure thinking does not guarantee the correctness of any of its constructions. Each step in the development of science requires a critique of existing notions. The development of knowledge from the naive to the critical one shows a movement from a picture of the world as a set of stable things to a picture of continuous movement and change, where change is more important than what changes. Cohen sees such a development in an evolution of the concept of substance in modern physics and he welcomes the replacement of material substance by energy, seeing this movement as a confirmation of critical idealism. Finally, it is discussed whether we can speak of the actuality of Cohen’s logic of pure knowledge. (shrink)

This work has been selected by scholars as being culturally important and is part of the knowledge base of civilization as we know it. This work is in the public domain in the United States of America, and possibly other nations. Within the United States, you may freely copy and distribute this work, as no entity (individual or corporate) has a copyright on the body of the work. Scholars believe, and we concur, that this work is important enough to be (...) preserved, reproduced, and made generally available to the public. To ensure a quality reading experience, this work has been proofread and republished using a format that seamlessly blends the original graphical elements with text in an easy-to-read typeface. We appreciate your support of the preservation process, and thank you for being an important part of keeping this knowledge alive and relevant. (shrink)

Nowadays the answer to the question “what is logic?” seems very simple and obvious—“logic is a science,” and after that usually one says what is this science about. As for the expressions “logic is an art” or “the art of logic,” then they are only metaphors or some kind of “façon de parler” used in serious scientific discourse. One of my aims here is to trace the line of development of dichotomy “logic as (...) an art—logic as a science” and to demonstrate that both these feat uses of logic have fundamental historical roots and play very important conceptual role in any theorizing about logic. Despite the fact that logic is undoubtedly a science, it can be interpreted as an art, moreover, the analysis of logic from this point of view expands, it seems to me, the researching possibilities in the field of the philosophy of logic at least in better understanding what is logic, what creates its unity independently from the historical period of its development, topics, and methods. (shrink)

In the paper, a detailed analysis of some new logical aspects of Cantor’s diagonal proof of the uncountability of continuum is presented. For the first time, strict formal, axiomatic, and algorithmic definitions of the notions of potential and actual infinities are presented. It is shown that the actualization of infinite sets and sequences used in Cantor’s proof is a necessary, but hidden, condition of the proof. The explication of the necessary condition and its factual usage within the framework of Cantor’s (...) proof makes, Cantor’s proof invalid. It’s shown that traditional Cantor’s proof has a second necessary, but hidden as well, condition which is teleological by its nature, i.e., is not mathematical. The explication of the second necessary condition makes Cantor’s statement on the uncountability of continuum unprovable from the point of view of classical logic.One of the most dramatic facts in history of science is connected just with the notion of actual infinity and consists in the following. On the one hand, Aristotle, Berkeley, Locke, Descartes, Spinoza, Gauss, Kant, Cauchy, Kronecker, Hermite, Poincaré, Bair, Borel, Brouwer, Quine, Wittgenstein, Weyl, Luzin, and a lot of other outstanding creators of classical logic and classical mathematics, during millenniums, stated categorically and insistently warned about that the actual infinity is a self-contradictory notion and its usage in mathematics is inadmissible (e.g., according to Poincare, “There is no actual infinity. - Cantorians forgot that and fell into contradictions. Later generations will regard Mengenlehre (set theory) as a disease from which one has recovered”). On the other hand, a lot of outstanding scientists of the XX c.c., such as Hilbert, Church, Turing, Gödel, etc., and modern axiomatic set theory as a whole ignored this (intuitive though) opinion of genii and accepted Cantor’s “transfinite paradise”, based on the actual infinity. In the paper some logical, philosophical, and psychological aspects of this historical “favorable opposition” are analyzed, and the fact that the scientific intuition of genii has won the historical battle against the mytho-’logic’ of Cantor’s transfinite ‘paradise’ is explained and logically justified. (shrink)

In the paper, a detailed analysis of some new logical aspects of Cantor’s diagonal proof of the uncountability of continuum is presented. For the first time, strict formal, axiomatic, and algorithmic definitions of the notions of potential and actual infinities are presented. It is shown that the actualization of infinite sets and sequences used in Cantor’s proof is a necessary, but hidden, condition of the proof. The explication of the necessary condition and its factual usage within the framework of Cantor’s (...) proof makes, Cantor’s proof invalid. It’s shown that traditional Cantor’s proof has a second necessary, but hidden as well, condition which is teleological by its nature, i.e., is not mathematical. The explication of the second necessary condition makes Cantor’s statement on the uncountability of continuum unprovable from the point of view of classical logic.One of the most dramatic facts in history of science is connected just with the notion of actual infinity and consists in the following. On the one hand, Aristotle, Berkeley, Locke, Descartes, Spinoza, Gauss, Kant, Cauchy, Kronecker, Hermite, Poincaré, Bair, Borel, Brouwer, Quine, Wittgenstein, Weyl, Luzin, and a lot of other outstanding creators of classical logic and classical mathematics, during millenniums, stated categorically and insistently warned about that the actual infinity is a self-contradictory notion and its usage in mathematics is inadmissible (e.g., according to Poincare, “There is no actual infinity. - Cantorians forgot that and fell into contradictions. Later generations will regard Mengenlehre (set theory) as a disease from which one has recovered”). On the other hand, a lot of outstanding scientists of the XX c.c., such as Hilbert, Church, Turing, Gödel, etc., and modern axiomatic set theory as a whole ignored this (intuitive though) opinion of genii and accepted Cantor’s “transfinite paradise”, based on the actual infinity. In the paper some logical, philosophical, and psychological aspects of this historical “favorable opposition” are analyzed, and the fact that the scientific intuition of genii has won the historical battle against the mytho-’logic’ of Cantor’s transfinite ‘paradise’ is explained and logically justified. (shrink)

Logical monism is the claim that there is a single correct logic, the 'one true logic' of our title. The view has evident appeal, as it reflects assumptions made in ordinary reasoning as well as in mathematics, the sciences, and the law. In all these spheres, we tend to believe that there aredeterminate facts about the validity of arguments. Despite its evident appeal, however, logical monism must meet two challenges. The first is the challenge from logical pluralism, according (...) to which there is more than one correct logic. The second challenge is to determine which form of logicalmonism is the correct one. One True Logic is the first monograph to explicitly articulate a version of logical monism and defend it against the first challenge. It provides a critical overview of the monism vs pluralism debate and argues for the former. It also responds to the second challenge by defending a particularmonism, based on a highly infinitary logic. It breaks new ground on a number of fronts and unifies disparate discussions in the philosophical and logical literature. In particular, it generalises the Tarski-Sher criterion of logicality, provides a novel defence of this generalisation, offers a clearnew argument for the logicality of infinitary logic and replies to recent pluralist arguments. (shrink)

We demonstrate that the quantum-mechanical description of composite physical systems of an arbitrary number of similar fermions in all their admissible states, mixed or pure, for all finite-dimensional Hilbert spaces, is not in conflict with Leibniz's Principle of the Identity of Indiscernibles (PII). We discern the fermions by means of physically meaningful, permutation-invariant categorical relations, i.e. relations independent of the quantum-mechanical probabilities. If, indeed, probabilistic relations are permitted as well, we argue that similar bosons can also be discerned in all (...) their admissible states; but their categorical discernibility turns out to be a state-dependent matter. In all demonstrated cases of discernibility, the fermions and the bosons are discerned (i) with only minimal assumptions on the interpretation of quantum mechanics; (ii) without appealing to metaphysical notions, such as Scotusian haecceitas, Lockean substrata, Postian transcendental individuality or Adamsian primitive thisness; and (iii) without revising the general framework of classical elementary predicate logic and standard set theory, thus without revising standard mathematics. This confutes: (a) the currently dominant view that, provided (i) and (ii), the quantum-mechanical description of such composite physical systems always conflicts with PII; and (b) that if PII can be saved at all, the only way to do it is by adopting one or other of the thick metaphysical notions mentioned above. Among the most general and influential arguments for the currently dominant view are those due to Schrodinger, Margenau, Cortes, Dalla Chiara, Di Francia, Redhead, French, Teller, Butterfield, Giuntini, Mittelstaedt, Castellani, Krause and Huggett. We review them succinctly and critically as well as related arguments by van Fraassen and Massimi. (shrink)

When first published twenty years ago, The Logic of Medicine presented a new way of thinking about clinical medicine as a scholarly discipline as well as a profession. Since then, advances in research and technology have revolutionized both the practice and theory of medicine. In this new, extensively rewritten edition, Dr. Murphy includes changes to show how these different areas of scholarship may affect details of "the logic of medicine" without compromising its fundamental coherence. New to this edition (...) are discussions of the challenge of the flood of new empirical data, new ideas in genetics, molecular biology, homeostasis, pathogenesis, cancer, aging, and Alzheimer's disease. Murphy also comments on such new theoretical topics as dynamic systems, chaos, and fractals and their impact on the burgeoning fields of philosophy and practice of medicine. Written with medical students in mind, the book includes a glossary, many new examples, and problems for solutions with comments on each. An entirely new chapter deals with modeling. Clinicians and researchers will also find the principles thought-provoking and illuminating. (shrink)

We introduce Boolean-like algebras of dimension n ($$n{\mathrm {BA}}$$ n BA s) having n constants $${{{\mathsf {e}}}}_1,\ldots,{{{\mathsf {e}}}}_n$$ e 1, …, e n, and an $$(n+1)$$ ( n + 1 ) -ary operation q (a “generalised if-then-else”) that induces a decomposition of the algebra into n factors through the so-called n-central elements. Varieties of $$n{\mathrm {BA}}$$ n BA s share many remarkable properties with the variety of Boolean algebras and with primal varieties. The $$n{\mathrm {BA}}$$ n BA s provide the (...) algebraic framework for generalising the classical propositional calculus to the case of n–perfectly symmetric–truth-values. Every finite-valued tabular logic can be embedded into such a n-valued propositional logic, $$n{\mathrm {CL}}$$ n CL, and this embedding preserves validity. We define a confluent and terminating first-order rewriting system for deciding validity in $$n{\mathrm {CL}}$$ n CL, and, via the embeddings, in all the finite tabular logics. (shrink)

This intriguing and ground-breaking book is the first in-depth study of the development of philosophy of science in the United States during the Cold War. It documents the political vitality of logical empiricism and Otto Neurath's Unity of Science Movement when these projects emigrated to the US in the 1930s and follows their de-politicization by a convergence of intellectual, cultural and political forces in the 1950s. Students of logical empiricism and the Vienna Circle treat these as strictly intellectual (...) non-political projects. In fact, the refugee philosophers of science were highly active politically and debated questions about values inside and outside science, as a result of which their philosophy of science was scrutinized politically both from within and without the profession, by such institutions as J. Edgar Hoover's FBI. It will prove absorbing reading to philosophers and historians of science, intellectual historians, and scholars of Cold War studies. (shrink)

A. J. Ayer was one of the foremost analytical philosophers of the twentieth century, and was known as a brilliant and engaging speaker. In essays based on his influential Dewey Lectures, Ayer addresses some of the most critical and controversial questions in epistemology and the philosophy of science, examining the nature of inductive reasoning and grappling with the issues that most concerned him as a philosopher. This edition contains revised and expanded versions of the lectures and two additional essays. (...) Ayer begins by considering Hume's formulation of the problem of induction and then explores the inferences on which we base our beliefs in factual matters. In other essays, he defines the three kinds of probability that inform inductive reasoning and examines the various criteria for verifiability and falsifiability. In his extensive introduction, Graham Macdonald discusses the arguments in _Probability and Evidence_, how they relate to Ayer's other works, and their influence in contemporary philosophy. He also provides a brief biographical sketch of Ayer, and includes a bibliography of works about and in response to _Probability and Evidence_. (shrink)

This a general account of the school of Mo-tzu, its social basis as a movement of craftsmen, its isolated place in the Chinese tradition, and the nature of its later contributions to logic, ethics, and science. It assesses the relation of Mohist thinking to the structure of the Chinese language, and grapples with the textual dynamics of later Mohist writings, particularly in regard to grammar and style, technical terminology, the use and significance of stock examples, and overall organization. (...) Includes edited and annotated Chinese text with an English translation and commentary, a glossary, and a photographic reproduction of the unemended text from the Taoist Patrology. (shrink)

Moving beyond identity politics while continuing to respect diverse entities and concerns, Whitney A. Bauman builds a planetary politics that better responds to the realities of a pluralistic world. Calling attention to the historical, political, and ecological influences shaping our understanding of nature, religion, humanity, and identity, Bauman collapses the boundaries separating male from female, biology from machine, human from more than human, and religion from science, encouraging readers to embrace hybridity and the inherent fluctuations of an open, evolving (...) global community. As he outlines his planetary ethic, Bauman concurrently develops an environmental ethic of movement that relies not on place but on the daily connections we make across the planet. He shows how both identity politics and environmental ethics fail to realize planetary politics and action, limited as they are by foundational modes of thought that create entire worlds out of their own logic. Introducing a postfoundational vision not rooted in the formal principles of "nature" or "God" and not based in the idea of human exceptionalism, Bauman draws on cutting-edge insights from queer, poststructural, and deconstructive theory and makes a major contribution to the study of religion, science, politics, and ecology. (shrink)

Logic as Universal Science offers a detailed reconstruction of the underlying philosophy in The Principles of Mathematics showing how Russell sought to deliver a death blow to the dominant Kantian view that formal logic is a concise and dry science and unable to enlarge our understanding.

The paper proposes an idea of explicating the invariant universal structure of dialogue through the mathematics of knots and braids, which is relevant, both for the development of particular models of communication and/or dialogue, and for constructing a general theory of dialogue, or the theory of utterances. The possibility of modeling dialogue with the help of the mathematics of braids and knots—categories, entities and their attributes—is shown by use of some well-known examples such as parts of the sentence in grammar. (...) Entities and their attributes can be considered as knots, either right or false. The idea is to visualize the chains of these entities: to formalize them not to keep in mind, neither in the text, but to work with them graphically.Nodes, as final structures, and braids that generate them, allow to reveal paradigmatic anomies, conflicts, logical and ethical disagreements, etc., and vice versa: “synonimies,” mutual understanding, unanimity, in a fundamentally new format of perception and understanding of the context of genesis (braids) and results (knots) of dialogue.The visualization of verbal utterance on the basis of the knot and braid mathematics is a significant step towards formalizing the theories of dialogue, both in general theoretical and purely practical plans. The latter is analyzed by the example of the person’s inner dialogue with herself/himself in making decisions: rational and/or irrational (emotional), which allows conducting a specific work with clients of psychotherapy, coaching, management consulting, mediation, etc. (shrink)

The assessment of Soviet logic and philosophy of logic development in the 20th and even the dawn of the twenty-first century shows the tight correlation between state policy towards higher education and the official attitude towards these fields of research. Progressive stages of Russia’s/Soviet State evolvement are marked with positive treatment of logic and philosophy of logic. Reactionary stages may be described in the context of the so-called ideologized science phenomenon, and they are marked by (...) negative treatment of philosophy of logic and even logic as a field of research, stagnation, or, at worst, total abandonment of logical education. This correlation has not been studied well enough. We will fulfill this lacuna from the social studies of science perspective. (shrink)

This paper describes the topological aspect of a logic-based, artificial intelligence approach to formalising the qualitative description of spatial properties and relations, and reasoning about those properties and relations. This approach, known as RCC theory, has been under development for several years at the University of Leeds. The main rationale for this project is that qualitative descriptions of spatial properties and relationships, and qualitative spatial reasoning, are of fundamental importance in human thinking about the world: even where quantitative spatial (...) data are most important, they must be attached to the components of a perceived spatial structure if we are to make use of them. RCC theory covers other qualitative aspects of spatial description and reasoning, but the topological properties and relations of spatially extended entities are fundamental to our work. The topological formalisms used by mathematicians are, in general, not well suited to the task of formalising the kinds of ‘common-sense’ or ‘everyday’ qualitative spatial description and reasoning which are our primary interest. Nevertheless, we must come to grips with the concepts of topology as practised by mathematicians if we are not to risk constantly ‘reinventing wheels’. (shrink)

In this paper, we provide a logical formalization of the emotion triggering process and of its relationship with mental attitudes, as described in Ortony, Clore, and Collins’s theory. We argue that modal logics are particularly adapted to represent agents’ mental attitudes and to reason about them, and use a specific modal logic that we call Logic of Emotions in order to provide logical definitions of all but two of their 22 emotions. While these definitions may be subject to (...) debate, we show that they allow to reason about emotions and to draw interesting conclusions from the theory. (shrink)

Mathematics as a Science of Patterns is the definitive exposition of a system of ideas about the nature of mathematics which Michael Resnik has been elaborating for a number of years. In calling mathematics a science he implies that it has a factual subject-matter and that mathematical knowledge is on a par with other scientific knowledge; in calling it a science of patterns he expresses his commitment to a structuralist philosophy of mathematics. He links this to a (...) defence of realism about the metaphysics of mathematics--the view that mathematics is about things that really exist. Resnik's distinctive philosophy of mathematics is here presented in an accessible and systematic form: it will be of value not only to specialists in this area, but to philosophers, mathematicians, and logicians interested in the relationship between these three disciplines, or in truth, realism, and epistemology. (shrink)

In order that a degree-of-belief function be coherent it is necessary and sufficient that it satisfy the axioms of probability theory. This theorem relies heavily for its proof on the two-valued sentential calculus, which emerges as a limiting case of a continuous scale of truth-values. In this "continuum of certainty" a theorem analogous to that instanced above is proved.

The Joint International Conference on Logic Programming, sponsored by the Association for Logic Programming, is a major forum for presentations of research, applications, and implementations in this important area of computer science. Logic programming is one of the most promising steps toward declarative programming and forms the theoretical basis of the programming language Prolog and its various extensions. Logic programming is also fundamental to work in artificial intelligence, where it has been used for nonmonotonic and (...) commonsense reasoning, expert systems implementation, deductive databases, and applications such as computer-aided manufacturing. Krzysztof R. Apt is project leader at the Centre for Mathematics and Computer Science in Amsterdam and part-time Professor in Computer Science at the University of Amsterdam. Topics covered: Theory Foundations. Programming Languages. Implementation. Programming Methodologies and Tools. Applications. Deductive Databases. Artificial Intelligence. Parallelism. (shrink)

The appearance of consciousness in the universe remains one of the major mysteries unsolved by science or philosophy. Absent an agreed-upon definition of consciousness or even a convenient system to test theories of consciousness, a confusing heterogeneity of theories proliferate. In pursuit of clarifying this complicated discourse, we here interpret various frameworks for the scientific and philosophical study of consciousness through the lens of social insect evolutionary biology. To do so, we first discuss the notion of a forward test (...) versus a reverse test, analogous to the normal and revolutionary phases of the scientific process. Contemporary theories of consciousness are forward tests for consciousness, in that they strive to become a means to classify the level of consciousness of arbitrary states and systems. Yet no such theory of consciousness has earned sufficient confidence such that it might be actually used as a forward test in ambiguous settings. What is needed now is thus a legitimate reverse test for theories of consciousness, to provide internal and external calibration of different frameworks. A reverse test for consciousness would ideally look like a method for referencing theories of consciousness to a tractable (and non-human) model system. We introduce the Ant Colony Test (ACT) as a rigorous reverse test for consciousness. We show that social insect colonies, though disaggregated collectives, fulfill many of the prerequisites for conscious awareness met by humans and honey bee workers. A long lineage of philosophically-neutral neurobehavioral, evolutionary, and ecological studies on social insect colonies can thus be redeployed for the study of consciousness in general. We suggest that the ACT can provide insight into the nature of consciousness, and highlight the ant colony as a model system for ethically performing clarifying experiments about consciousness. (shrink)

This essay offers a conception of logic by which logic may be considered to be exceptional among the sciences on the backdrop of a naturalistic outlook. The conception of logic focused on emphasises the traditional role of logic as a methodology for the sciences, which distinguishes it from other sciences that are not methodological. On the proposed conception, the methodological aims of logic drive its definitions and principles, rather than the description of scientific phenomena. The (...) notion of a methodological discipline is explained as a relation between disciplines or practices. Logic serves as a methodological discipline with respect to any theoretical practice, and this generality, as well as logic’s reflexive nature, distinguish it from other methodological disciplines. Finally, the evolution of model theory is taken as a case study, with a focus on its methodological role. Following recent work by John Baldwin and Juliette Kennedy, we look at model theory from its inception in the mid-twentieth century as a foundational endeavour until developments at the end of the century, where the classification of theories has taken centre-stage. (shrink)

This book looks at the ways in which conditionals, an integral part of philosophy and logic, can be of practical use in computer programming. It analyzes the different types of conditionals, including their applications and potential problems. Other topics include defeasible logics, the Ramsey test, and a unified view of consequence relation and belief revision. Its implications will be of interest to researchers in logic, philosophy, and computer science, particularly artificial intelligence.

This paper focuses on the normative analysis—in the sense of the classic decision-theoretic formulation—of decision problems that arise in connection with forensic expert reporting. We distinguish this analytical account from other common types of decision analyses, such as descriptive approaches. While decision theory is, since several decades, an extensively discussed topic in legal literature, its use in forensic science is more recent, and with an emphasis on goals such as the analysis of the logical structure of forensic expert conclusions (...) regarding, for example, propositions of common source of evidential and known materials. Typical examples are so-called identification decisions, especially categorical conclusions according to which fingermarks come from a particular a person of interest. We will present and compare ways of stating forensic identification decisions in decision-theoretic terms and explain their underlying rationale. In particular, we will emphasize the importance of viewing this analysis as normative in the sense of providing a reflective rather than a prescriptive reference point against which people in charge of forensic identification decisions may compare their otherwise intuitive and informal reasoning, before acting. Normative decision analysis in forensic science thus provides a vector through which current practice can be articulated, scrutinized and rethought. (shrink)

In order that a degree-of-belief function be coherent it is necessary and sufficient that it satisfy the axioms of probability theory. This theorem relies heavily for its proof on the two-valued sentential calculus, which emerges as a limiting case of a continuous scale of truth-values. In this "continuum of certainty" a theorem analogous to that instanced above is proved.

In order that a degree-of-belief function be coherent it is necessary and sufficient that it satisfy the axioms of probability theory. This theorem relies heavily for its proof on the two-valued sentential calculus, which emerges as a limiting case of a continuous scale of truth-values. In this “continuum of certainty” a theorem analogous to that instanced above is proved.

Predicates are supposed to slice reality neatly in two halves, one for which the predicate holds, the other for which it fails. Yet far from being razors, predicates tend to be dull knives that mangle reality. If reality is a tomato and predicates are knives, then when these knives divide the tomato, plenty of mush remains unaccounted for. Of course some knives are sharper than others, just as some predicates are less vague than others. “x is water” is certainly sharper (...) than “x is beautiful.” But perfect sharpness, perfect boundaries, and perfect separation seem only to obtain in mathematics. The vagueness inherent in many predicates became particularly evident in the twentieth century. Quantum mechanics, the revival of certain ancient paradoxes, and the philosophy of science all contributed to a growing awareness that vagueness was ineliminable from many predicates. Quantum mechanical superposition seems to allow mutually exclusive simultaneous states. In searching for plausible interpretations of quantum mechanics, some researchers attempted to do away with classical bivalent logic. In its place they substituted multivalent quantum logics. Any logic with more than two values forces its predicates to slice reality into more than two parts. Ancient paradoxes involving heaps and baldness also pointed up the 1 Random Predicate Logic 2.. (shrink)

The impact of social factors upon the philosophical investigations in a broad sense is quite evident. Nevertheless their impact upon epistemology as a branch of philosophy, logic, and history of science as fields of research with noticeable philosophical content is not evident enough. We are keen to claim that this impact exists within some limits, although it is not so overtly evident. Moreover in the case of Marxism it is of a paradoxical nature. Marxism always puts the accent (...) on the role of social and economic factors in the development, development of science included. To a large extent due to Marxism, externalism emerged; the key idea of externalism may be expressed through the statement that social and economic reasons are the main sources of development of science. B.M. Hessen declared and did his best in 1931 to justify this statement through the example of the emergence of classical mechanics. Meanwhile the social milieu of Marxist countries placed a taboo on the externalist approach towards epistemology, the interpretation of logic, and history of science. All these fields of knowledge were evolved in the Marxist era in the USSR and Eastern Europe - despite the spirit of Marxism - within strict internalist boarders. We offer the explanation of this contradiction. (shrink)

1. Lemmon’s stratification. By a “tense logic” I mean a system with the following features: it contains sentential variables which stand for sentences which in some cases are true at some times and false at others; it contains the usual truth-functions, whose truth-conditions are given the obvious modifications, e.g. Np is true when and only when p is false, Kpq is true when and only when both its conjuncts are; and it contains two additional functions which may be interpreted (...) as ‘It will be the case that p’ and ‘It has been the case that p’, the former being true when and only when the plain p will be true later on and the latter when and only when the plain p has been true at some earlier time. (shrink)