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)

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)

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)

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)

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)

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)

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)

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)

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.

Jonathan Walkan challenges cognitive science's dominant model of mental representation and proposes a novel, well-devised alternative. The traditional view in the cognitive sciences uses a linguistic model of mental representation. That logic-based model of cognition informs and constrains both the classical tradition of artificial intelligence and modeling in the connectionist tradition. It falls short, however, when confronted by the frame problem---the lack of a principled way to determine which features of a representation must be updated when new information (...) becomes available. So far, proposed alternatives, including the imagistic model, have not resolved the problem. Waskan proposes the Intrinsic Cognitive Models hypothesis, according to which representational states can be concpetualized as the cognitive equivalent of scale models.Waskan argues further that the proposal that humans harbor and manipulate cognitive counterparts to scale models offers the only viable explanation for what most clearly differentiates humans from other creatures: the capacity to engage in truth-preserving manipulation of representations. The ICM hypothesis, he claims, can be distinguished from sentence-based accounts of truth preservation in a way that is fully compatible with what is known about the brain. (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)

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)

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)

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)

In lieu of an abstract, here is a brief excerpt of the content:Reviewed by:Unmodern Philosophy and Modern Philosophy by John DeweyCharles A. HobbsJohn Dewey. Unmodern Philosophy and Modern Philosophy. Carbondale and Edwardsville: Southern Illinois University Press, 2012, 351 pp., index.John Dewey’s latest publication marks a watershed moment for scholarship in American philosophy, and, in addition to Dewey himself, we have editor Phillip Deen to thank for discovering it (among the Dewey papers in Special Collections at Morris Library of Southern Illinois (...) University) and editing it. We see Dewey late in life (early 1940s) engaged mainly in a critique of the modern, ancient, and (fragmentarily) medieval philosophical periods, which Dewey argues, are eras not nearly as distinct as traditional understanding admits. In the Introduction, Deen helpfully explains how we are now bequeathed incomplete but in most [End Page 122] cases quite robust chapter drafts that amount to another substantial work by Dewey. While published by Southern Illinois University Press, this volume is not itself part of The Collected Works of John Dewey. Still, the careful editing (of what are actually a number of manuscripts) is comparable in its excellence. There is also a well-done Index.The scope is twofold: retrospective and prospective. Chapters one through seven trace the development of so-called modern philosophy along with its central epistemological problem, with the aim being to demonstrate how thoroughly “unmodern” such philosophy has been. Then, chapters eight through fourteen constitute a genuinely modern approach that naturalistically and functionally undercuts the epistemic problem of subject/object and points to a revolution in the process of knowing.We turn to the retrospective, beginning in the first chapter with the acknowledgement that Dewey comes to philosophy itself as a kind of anthropologist, something we note as consistent with the influence of Franz Boas on Dewey (with whom Dewey was a colleague and intellectual ally at Columbia for about thirty-five years). This is further underscored by a lengthy quotation of anthropologist Bronislaw Malinowski on the meaning of culture (p. 290). The ongoing history of philosophy is itself a cultural activity. Indeed, citing James, Dewey declares that “The sickliest way in which a student of philosophy can approach his subject-matter is that of a search for ultimate impersonal revelation of truth.” (p. 16) Rather, let the philosopher have the pragmatic attitude of openness “... to truth from whatever source it comes” (ibid).With that, Dewey once again turns to the Greeks, who initiated “The Discovery of Rational Discourse” (Ch. III) as underscored by the Socratic means of cooperative discourse, with the dialogues—not lectures—being evidence of the Athenian philosophical revolution in turning “from discussing nature and giving an account of it to the examination of the nature and procedure of discussion and report” (p. 38). Yet this project, despite its initial promise, failed for lack of a genuine logic of experimentation or inquiry. Then philosophy focused on going beyond the natural, beginning with the medieval “Search for Salvation” (Ch. IV, a significantly less developed and unified chapter draft). “Modern” philosophers then rightly replaced grand cosmic schemes with a focus on human nature (Ch. V), but they went too far with their introspective methods, gradually resulting in the same consequence supernaturalism had produced: a dualism of mind and nature, as if human nature is somehow not a part of nature. As we recall from Human Nature and Conduct (1922), human nature is not independent of but rather interdependent (or, using Dewey’s later term, transactional) with the rest of nature. The key is to avoid hypostatizing mind and better to talk of culture as nature. [End Page 123]With the Renaissance onward, humans have been “Wandering between Two Worlds” (Ch. VI), and endless arguments between “rationalists” and “empiricists” overlook what these camps share, namely a psychology assuming a fundamental split between subjective/objective and assumptions about certainty as founded upon various first principles or substances. The result is the stagnation of philosophy while science flourishes. Dewey cites Bertrand Russell as an example of continued insistence on such a priori principles (p. 106).We find ourselves in an epoch split between scientism and moral and religious fundamentalism, the result of a lingering tenacious investment in bad elements from ancient and medieval... (shrink)

Shepard has challenged psychologists to identify nonarbitrary principles of mind upon which to build a more explanatory and general cognitive science. I suggest that such nonarbitrary principles may fruitfully be sought not only in the laws of physics and mathematics, but also in the logical entailments of different categories of representation. In the example offered here, conceptualizing mental events as indexical with respect to the events they represent enables one to account parsimoniously for a wide range of empirical psychological (...) phenomena. [Shepard]. (shrink)

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 article discusses the differences between the classical logic of science (17th-20th centuries) and non-classical logic (20th century). While classical logic is based on the general properties of the objects studied, non-classical logic is based on the special, individual. The classical logic singled out in the studied objects their common properties that united them and ensured their independence of human. The scientist and his social connections are volatile and cannot serve as a stable basis (...) for obtaining the only possible true result during examination of particular object. At the same time, it was not taken into account that scientists themselves endowed nature with this property of independence. It would be more correct to say that scientist distinguished from all properties of reality precisely these properties as the most convenient for constructing a logical system that explains the entire surrounding world. All other properties were ignored. In the 20th century attention was shifted to the properties of objects and events that determine their identity, which means their ability to communicate. After all, for a communication at least two interlocutors are needed who differ from each other. Each of them has its own origin, which is formed by the context. At the origin of the formation of knowledge there is still no division between subject and object. The surgeon works with his hands guided by his theoretical knowledge. When he encou difficulty, he can modify the theory, thus solving the practical problem that has arisen. In the structure of science, knowledge is embedded in this form, without division between man and the surrounding world. However this last division remains only possible but not essential for logic. The main problem that needs to be further seriously considered is to understand how the social is included in the philosophy of science. (shrink)

Marc Lange’s new book on laws offers a restatement and development of the account he proposed in Natural Laws and Scientific Practice (Oxford University Press, 2000), henceforth NLSP, and the new material is helpfully summarized in the preface. Laws and Lawmakers presents the key idea from NLSP in a rather more reader-friendly manner – this idea being roughly that the difference between laws and accidents is that laws, unlike accidents, form a ‘stable’ set, i.e. a logically closed set of truths (...) such that they would all still hold under any counterfactual supposition consistent with the set. So, for example, the natural laws all still hold under counterfactual suppositions such as ‘had this match been struck …’, ‘had Bill Gates wanted to build a gold cube one mile across’ and so on; thus this set is stable. But the set of laws plus the accidental claim ‘there is no gold cube one mile across’ fails to hold under such counterfactual suppositions because had Bill Gates wanted to build a gold cube one mile across, such a cube might well have come into existence; thus this set is not stable. While the basic outline and defence of this idea is provided in Chapter 1, those wishing to delve into the intricate …. (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)

Since the world is full of indeterminacy, the Neutrosophics found their place into contemporary research. We now introduce for the first time the notions of Neutrosophic Crisp Sets and Neutrosophic Topology on Crisp Sets. We develop the 2012 notion of Neutrosophic Topological Spaces and give many practical examples. Neutrosophic Science means development and applications of Neutrosophic Logic, Set, Measure, Integral, Probability etc., and their applications in any field. It is possible to define the neutrosophic measure and consequently the (...) neutrosophic integral and neutrosophic probability in many ways, because there are various types of indeterminacies, depending on the problem we need to solve. Indeterminacy is different from randomness. Indeterminacy can be caused by physical space, materials and type of construction, by items involved in the space, or by other factors. In 1965 [51], Zadeh generalized the concept of crisp set by introducing the concept of fuzzy set, corresponding to the situation in which there is no precisely defined set;there are increasing applications in various fields, including probability, artificial intelligence, control systems, biology and economics. Thus, developments in abstract mathematics using the idea of fuzzy sets possess sound footing. In accordance, fuzzy topological spaces were introduced by Chang [12] and Lowen [33]. After the development of fuzzy sets, much attention has been paid to the generalization of basic concepts of classical topology to fuzzy sets and accordingly developing a theory of fuzzy topology [1-58]. In 1983, the intuitionistic fuzzy set was introduced by K. Atanassov [55, 56, 57] as a generalization of the fuzzy set, beyond the degree of membership and the degree of non-membership of each element. In 1999 and 2002, Smarandache [71, 72, 73, 74] defined the notion of Neutrosophic Sets, which is a generalization of Zadeh’s fuzzy set and Atanassov's intuitionistic fuzzy set. Some neutrosophic concepts have been investigated by Salama et al. [61-70]. Forwarding the study of neutrosophic sets, this book consists of seven chapters, targeting to:  generalize the previous studies in [1-59], and[91-94] so to define the neutrosopic crisp set and neutrosophic set concepts;  discuss their main properties; A. A. Salama & Florentin Smarandache  introduce and study some concepts of neutrosophic crisp and neutrosophic topological spaces and deduce their properties;  deduce many types of functions and give the relationships between different neutrosophic topological spaces, which helps to build new properties of neutrosophic topological spaces;  stress once more the importance of Neutrosophic Ideal as a nontrivial extension of neutrosophic set and neutrosophic logic [71, 72, 73, 74];  propose applications on computer sciences by using neutrosophic sets. (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.

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.

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)

Intensional logic is the technical study of such intensional phenomena in human reasoning as modality, knowledge, or flow of time. These all require a richer semantic picture than standard truth values in one static environment. Such a picture is provided by so-called possible worlds semantics, a paradigm which is surveyed in this book, both as to its external sources of motivation and as to the internal dynamics of the resulting program. In particular, Manual of Intensional Logic presents the (...) major classical topics, including modal logic, tense logic, and conditional logic, all of which illustrate motivations coming from philosophy and linguistics. The Book also discusses recent computational applications in computer science and AI. Finally, Manual of Intensional Logic takes up recent developments in the study of language and information making themselves felt in the area. The Book examines the role of partial information--with illustrations drawn from different branches of Intensional Logic--and various influences stemming from current theories of the semantics of natural language, involving generalized quantifiers and theories of types. (shrink)

It is often claimed that there can be no such thing as a logic of scientific discovery, but only a logic of verification. By 'logic of discovery' is usually meant a normative theory of discovery processes. The claim that such a normative theory is impossible is shown to be incorrect; and two examples are provided of domains where formal processes of varying efficacy for discovering lawfulness can be constructed and compared. The analysis shows how one can treat (...) operationally and formally phenomena that have usually been dismissed with fuzzy labels like 'intuition' and 'creativity'. (shrink)

Context: Philosophy of science is the branch of philosophy that deals with methods, foundations, and implications of science. It is a theory of how to create scientific knowledge. Presently, there is widespread agreement on how to do science, namely conjectures, ideally in the form of a mathematical model, and refutations, testing the model using empirical evidence. Problem: Many social scientists are using a conception of science created for the physical sciences. Expanding philosophy of science so (...) that it more successfully encompasses social systems would create new avenues of inquiry. Two dimensions could be added to philosophy of science: the amount of attention paid to the observer and the amount of impact of a theory on the system described. Method: My approach is to illuminate underlying assumptions. I claim that there are at least three epistemologies and that they can be combined to form a more robust conception of knowledge and of how to do research. There are at least four models and four basic elements being used by scientists. Results: The article identifies the logical propositions underlying second-order science. It suggests strategies for developing second-order science. And it describes several methods that can be used to practice second-order science, including how past theories have not only described but also changed the phenomenon being studied. Implications: The task for members of the scientific community, particularly social scientists, is to practice second-order science and to develop further its theories and methods. A practical implication is to accept methods for acting as well as theories as a contribution to science, since methods explicitly define the role of an observer/ participant. Constructivist content: The paper is an extension of the work of Heinz von Foerster and other second-order cyberneticians. (shrink)

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)

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)

Cognitive science is, more than anything else, a pursuit of cognitive mechanisms. To make headway towards a mechanistic account of any particular cognitive phenomenon, a researcher must choose among the many architectures available to guide and constrain the account. It is thus fitting that this volume on contemporary debates in cognitive science includes two issues of architecture, each articulated in the 1980s but still unresolved: " • Just how modular is the mind? – a debate initially pitting encapsulated (...) mechanisms against highly interactive ones. • Does the mind process language-like representations according to formal rules? – a debate initially pitting symbolic architectures against less language-like architectures. " Our project here is to consider the second issue within the broader context of where cognitive science has been and where it is headed. The notion that cognition in general—not just language processing—involves rules operating on language-like representations actually predates cognitive science. In traditional philosophy of mind, mental life is construed as involving propositional attitudes—that is, such attitudes towards propositions as believing, fearing, and desiring that they be true—and logical inferences from them. On this view, if a person desires that a proposition be true and believes that if she performs a certain action it will become true, she will make the inference and perform the action. (shrink)

The subject of this thesis is physicalism, understood not as some particular doctrine pertaining narrowly to the philosophy of mind, but rather as a quite general metaphysical claim to the effect that everything is, or is fundamentally, physical. Thus physicalism explicates the thought that in some sense physics is the basic science. The aim of the thesis is to defend a particular brand of physicalism, which I call eliminative type physicalism. It claims, roughly, that every property is a physical (...) property, a property mentioned in the laws of physics, and hence that any putative property not identifiable with a physical property must be eliminated from our ontology. Eliminative type physicalism is apt to face three objections, and so my thesis, like Caesar's Gaul, falls into three parts. In the first, I argue against the idea that there are tenable positions, both physicalist and non-physicalist, alternative to eliminative type physicalism. I argue that each of these positions token physicalism (Fodor, middle Putnam), supervenience physicalism (Lewis, Horgan) and and a non-physicalist view I call pluralism (Goodman, late Putnam) is defective. In the second part, responding to the objection that there is just no reason to be a physicalist, I develop a positive argument for eliminative type physicalism, an argument resting upon a strong version of the explanatory test for reality according to which only explanatorily indispensable properties can justifiably be said to exist. In the third and final part, I argue, against the charge that eliminative type physicalism cannot accommodate what I call phenomenal properties (qualia, raw feels etc.), that there is no good reason to deny, and one good reason to affirm, that phenomenal properties just are physical properties. (shrink)

Some modern thinkers have supposed that “cause” is an outworn notion, or at least that it is one of which modern science has no need. This is due mainly to the discovery that, while the scientist can give us general laws as to what in fact happens, he cannot help us to discern the reason for the laws or the inward nature of the forces on which they depend. He can tell us the “that” but not the “why”; he (...) cannot show us in a single case that the effect follows necessarily a priori from the nature of the cause, that any other effect than the one which actually takes place would be logically impossible. He has studied the law of gravitation, but this law does not enable him to see why material bodies should attract each other in this fashion; it is only a generalized statement of the fact that they do. He knows that certain substances, if absorbed by eating, will nourish and others destroy our tissues; but he cannot say why they should do so. He can no doubt analyse them further and discover that, for example, meat is nourishing because it contains a large proportion of nitrogenous matter, but he could not tell a priori whether this nitrogenous matter would be likely to nourish or to poison us. Only where mathematics can be applied do we see necessity in such a way that any alternative becomes inconceivable to us; but mathematics alone can never establish from a quantity present here and now what quantity there will be at a later time or in another part of space. Mathematics can show, e.g. , that, if there is 2 + 2 here and now, there must be 4 here and now, not that, if there is 2 + 2 here and now, there will be 4 in an hour's time or a mile away; and therefore it cannot be made the sole basis of any causal law whatever. (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)

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)