A comprehensive one-year graduate (or advanced undergraduate) course in mathematical logic and foundations of mathematics. No previous knowledge of logic is required; the book is suitable for self-study. Many exercises (with hints) are included.

A companion to the logic text by the same authors. -- A. R. A.

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)

An introductory text in linguistic semantics, uniquely balancing empirical coverage and formalism with development of intuition and methodology. -/- This introductory textbook in linguistic semantics for undergraduates features a unique balance between empirical coverage and formalism on the one hand and development of intuition and methodology on the other. It will equip students to form intuitions about a set of data, explain how well an analysis of the data accords with their intuitions, and extend the analysis or seek an alternative. (...) No prior knowledge of linguistics is required. After mastering the material, students will be able to tackle some of the most difficult questions in the field even if they have never taken a linguistics course before. -/- After introducing such concepts as truth conditions and compositionality, the book presents a basic symbolic logic with negation, conjunction, and generalized quantifiers, to serve as the basis for translation throughout the book. It then develops a detailed compositional semantics, covering quantification (scope and binding), adverbial modification, relative clauses, event semantics, tense and aspect, as well as pragmatic phenomena, notably deictic pronouns and narrative progression. -/- A Course in Semantics offers a large and diverse set of exercises, interspersed throughout the text; those labeled “Important practice and looking ahead” prepare students for material to come; those labeled “Thinking about ” invite students to think beyond the content of the book. (shrink)

The ability to reason and think in a logical manner forms the basis of learning for most mathematics, computer science, philosophy and logic students. Based on the author's teaching notes at the University of Maryland and aimed at a broad audience, this text covers the fundamental topics in classical logic in an extremely clear, thorough and accurate style that is accessible to all the above. Covering propositional logic, first-order logic, and second-order logic, as well as (...) proof theory, computability theory, and model theory, the text also contains numerous carefully graded exercises and is ideal for a first or refresher course. (shrink)

In this text, a variety of modal logics at the sentential, first-order, and second-order levels are developed with clarity, precision and philosophical insight. All of the S1-S5 modal logics of Lewis and Langford, among others, are constructed. A matrix, or many-valued semantics, for sentential modal logic is formalized, and an important result that no finite matrix can characterize any of the standard modal logics is proven. Exercises, some of which show independence results, help to develop logical skills. A separate (...) sentential modal logic of logical necessity in logical atomism is also constructed and shown to be complete and decidable. On the first-order level of the logic of logical necessity, the modal thesis of anti-essentialism is valid and every de re sentence is provably equivalent to a de dicto sentence. An elegant extension of the standard sentential modal logics into several first-order modal logics is developed. Both a first-order modal logic for possibilism containing actualism as a proper part as well as a separate modal logic for actualism alone are constructed for a variety of modal systems. Exercises on this level show the connections between modal laws and quantifier logic regarding generalization into, or out of, modal contexts and the conditions required for the necessity of identity and non-identity. Two types of second-order modal logics, one possibilist and the other actualist, are developed based on a distinction between existence-entailing concepts and concepts in general. The result is a deeper second-order analysis of possibilism and actualism as ontological frameworks. Exercises regarding second-order predicate quantifiers clarify the distinction between existence-entailing concepts and concepts in general. Modal Logic is ideally suited as a core text for graduate and undergraduate courses in modal logic, and as supplementary reading in courses on mathematical logic, formal ontology, and artificial intelligence. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:"And Her Substance Would Be Mine":Envy, Hate, and Ontological Evacuation in Josephine Hart's SinA. Samuel Kimball (bio)Envy involuntarily testifies to a lack of being that puts the envious to shame.—René Girard, A Theatre of EnvySin, offspring of snt-ya, "that which is," in Germanic sun(d)jo, "it is true," "the sin is real," and ultimately from es-, "to be," source of am, is, sooth, soothe; of the Sanskrit roots sat- and (...) sant-, "existing, true, virtuous"; of the Latin, esse, "to be," from which is derived such words as essence, presence, absence, and represent; of the Greek roots einei, "to be" as well as ont-, "being," which provides the name for the philosophical study of being itself, ontology.—Adapted from The American Heritage Dictionaryof Indo-European RootsI: Ruth Garton's Envious NarrationThe etymology of sin suggests that the evil it names reveals something fundamental about being, something ruinous and yet unavoidable, something that endangers or even nullifies being and yet is inseparable from it, even essential to it. In Josephine Hart's second novel, Sin, this something is a species of Girardian desire in the form of a monstrous envy, which impels the first-person narrator, Ruth Garton, to a lifelong hatred of her cousin and stepsister, Elizabeth. The focus of Ruth's confessional narrative, envy, drives Ruth to take covert and increasingly virulent revenge against her quasi-sibling in a self-deluding attempt to appropriate not only the objects of Elizabeth's desire but ultimately Elizabeth's desire itself and the apparent ontological fullness that Ruth attributes to it. Over the course of her life, then, Ruth loses the promise [End Page 239] of her name in her ruthless pursuit of her enemy. As she becomes increasingly frustrated in her efforts to destroy her sister's life, Ruth finally turns her hatred of her nemesis against herself as well as the readers to whom she appeals for understanding. If envy heightens Ruth's consciousness to the point where she can perceive that her desire signals her ontological emptiness, nevertheless, it so captures her, so dominates her emotional and cognitive life, that she is unable to find a way to affirm her own—or anyone else's—being in the world. The result is a spectacular narrative record of the cost of recognizing but being unable to resist the mimeticism of desire.1II. The Girardian Dynamic of SinAs Girard has demonstrated, one of the most difficult existential challenges is to understand that desire is mimetically mediated. The result is that we come into our desire only through someone else's and always want something more and other than we think we want. Behind every move to possess a particular object is a largely unrecognized and in principle unsatisfiable desire for absolute satisfaction in the form of ontological completeness, self-sufficiency, intactness. On the one hand, Girard writes, "by the example of his own desire…the model conveys to the subject the supreme desirability of the object" (1977, 146). On the other hand, since to desire mimetically means to come into desire through the other rather than through oneself, by this same example the model conveys to the subject the contingency, the dependency, the secondariness, hence the emptiness of the subject's desire, even the subject's lack of desire, the lack that, paradoxically, constitutes his or her desire. Therefore, to be subjected to the other's want ultimately has the effect of signaling the seeming priority, fullness, and self-origination of this other person's desire. For this reason Girard asserts that "man... desires being, something he himself lacks and which some other person seems to possess" (146).If at first the other person models the desire that the subject lacks, it is in the logic of mimesis that the other will become a rival and, unless the rivalry is somehow interrupted or abrogated, inevitably an enemy to be destroyed. The subject will become locked into a potentially deadly game of trying to obtain the same object that the model desires, so as to experience the satisfactions—ultimately, the sense of being—that the object in question seems to provide... (shrink)

Crash Course in Logic is a booklet designed to introduce basic principles of logic and critical thinking to students so they can better express their ideas. Many high school and college students have trouble constructing theoretical arguments and writing clearly because they are not acquainted with the forms of reasoning that are presented in this booklet. Intended as a supplement to other instructional material for a variety of courses, this booklet will guide students through a mini-course (...) on logic that includes many examples and exercises. With knowledge of the basic forms of reasoning, students will have the tools necessary to solve problems and evaluate arguments as well as articulate their own ideas and insights clearly. Crash Course in Logic will be of great value to teachers of any subject who are searching for an accessible way to teach critical thinking and reasoning to their students. (shrink)

A First Course in Logic is an introduction to first-order logic suitable for first and second year mathematicians and computer scientists. There are three components to this course: propositional logic; Boolean algebras; and predicate/first-order, logic. Logic is the basis of proofs in mathematics — how do we know what we say is true? — and also of computer science — how do I know this program will do what I think it will? Surprisingly (...) little mathematics is needed to learn and understand logic (this course doesn't involve any calculus). The real mathematical prerequisite is an ability to manipulate symbols: in other words, basic algebra. Anyone who can write programs should have this ability. (shrink)

The summary is devoted to the meetings of the Presidium of the Russian Federal Educational and Methodological Association held in September 2018 in Astrakhan and in February 2019 in Moscow. The authors review main educational and methodological issues discussed at the meetings. The summary covers the issues of online courses in philosophy. It also touches upon key issues of religious studies, the specifics of its teaching in Russia and foreign countries, current problems facing religious studies in Russia. The article contains (...) a review of the current state of logic education in Russia, its current problems and possible ways to solve them. Also it outlines issues related to the implementation of the Russian national project “Science” as well as philosophy education at Astrakhan State University and the issues of realization of new educational standards. (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 paper we conduct an analysis of the critical narratives of Stardew Valley and compare them to other relevant videogames in order to develop new possibilities for an ecological critique of capitalist extractive econo­mies. Critical narratives of this game are aimed primarily at the alienating conditions of labour and deeply devastating modes of production under capitalism that impact and severely damage the environment. Analysing these narratives, we superimpose the immediate messages of the game with the procedural rhetoric and material (...) conditions of their existence. Over the course of our analysis, we highlight a material-narrative disso­nance which, in the case of Stardew Valley, fails to function as a commu­nicative strategy of the game and remains its mere external contradiction. Although the game’s critical narrative may seem overly utopian and its political imaginary a bit underdeveloped, the game elaborates on concrete ways to tackle the alienation of labour and resolve the ecological crisis. In addition to this, the paper covers the history of the interplay between the video game industry and the field of global ecological crisis research. We compare the attempts to raise awareness of the videogames’ own material­ity that preceded Stardew Valley. We conclude that Stardew Valley utilises the language of sustainable co-existence and wasteless local production, expanding this logic both to the sphere of labour and the spectrum of en­vironmental problems. In the case of ecological critique, some gameplay decisions in Stardew Valley enable us to come up with new strategies aimed at creating critical narratives about the environment in videogames. Thus supplementing Stardew Valley’s findings with critical tropes derived from other games (mainly, Rain World), we were able to gather a set of theoretical instruments that could facilitate the creation of games about ecosystems. Following Donna Haraway’s emphasis on the crucial role of narrative fram­ing (“it matters which stories tell stories”), we highlight new opportunities for the entire medium of videogames. (shrink)

This is a free book, 165 pages. It is for anyone who has had a solid introductory logic course and wants more. Topics covered include soundness and completeness for first-order logic, Tarski's theorem on the undefinability of truth, Gödel's incompleteness theorems, the undecidability of first-order logic, a smattering of second-order logic, and modal logic (both propositional and quantificational). I wrote it for use in my own course, because I thought I could present the (...) most important results and concepts more clearly than the available textbooks. (shrink)

The author's motivation for constructing the calculi of this paper\nis so that time and tense can be "discussed together in the same\nlanguage" (p. 282). Two types of enriched propositional caluli for\ntense logic are considered, both containing ordinary propositional\nvariables for which any proposition may be substituted. One type\nalso contains "clock-propositional" variables, a,b,c, etc., for\nwhich only clock-propositional variables may be substituted and that\ncorrespond to instants or moments in the semantics. The other type\nalso contains "history-propositional" variables, u,v,w, etc., for\nwhich only history-propositional variables (...) may be substituted and\nthat correspond to maximally linearly ordered subsets of instants.\nThis second type is for tense logics in which time is not linear\nand where at any moment there may exist alternative future courses\nof events. Quantifiers are included in both types but only for clock-\nand history-propositional variables. However, these quantifiers are\ngiven an essentially substitutional interpretation. In addition,\nbesides each clock-propositional variable being true at exactly one\ninstant, the semantics requires that for each instant there be at\nleast one clock-propositional variable true at that instant, which\nin effect amounts to having each instant of the model "denoted" in\nthe formalism by some clock-propositional variable. Thus it is not\nsurprising that quantification over clock-propositional variables\nturns out to have a "well behaved first-order semantics" (p. 284).\n{One axiom relating history- with clock-propositions seems to need\nrevision: CKTauTbuAUabUba. Since the clock-propositional variables\nmight be true at the same instant the consequent should include an\nalternative, namely, LCab.} Besides the standard truth-functional\nand tense operators G,H,F,P, the author includes L for "it is\nalways the case that". The model-theoretic earlier-than relation\nordering instants and the semantic truth-at (an instant) relation\nare formalized by Uab and Taα (for a wff α) which\nare defined respectively as LCbPa (whenever the clock-proposition\nb is the case the clock-proposition a was the case) and LCaα\n(α is the case whenever the clock-proposition a is the case).\nBecause the future tense operators are definable in terms of U\nand T and the latter are definable in terms of L and P, the\ndual of H, the author notes that his completeness proofs can be\nrestricted to L and H as primitive and with the future tense\noperators used for other interpretations, especially Prior, A.'s\nindeterminist "it will be the case that", where time is not linear.\nThe author concludes with some rather technical observations showing\nthat "in ordinary tense logics the instants are nonstandard elements\nof the models. This provides", according to the author, "a semantic\npressure to build the instants into tense logic, so that they will\nappear as standard elements in the models" (p. 283). (shrink)

These two small works are a good supplement to Rescher’s recent trilogy. Whereas the systems-theoretic approach is employed in Methodological Pragmatism in dealing with the problem of the legitimation of claims to factual knowledge or cognitive rationality, Dialectics deals with the argumentation aspect of thesis-introduction rather than the logical aspect of thesis-derivation. Although some key notions such as the idea of burden of proof and presumption have been stated in the former work, what is offered here is a systematic discussion (...) of a disputational model of inquiry. The principal aims are "to exhibit the sociocommunal roots of the foundations of rationality, to provide an instrument for the critique of scepticism implicit in the cognitive solipsism of the Cartesian approach, and to illuminate the communal and controversy-oriented aspects of rational argumentation and inquiry—scientific inquiry in particular." Again, Rescher limits his discussion to "the probative mechanism in the factual domain." This book, if it were not for its price, should be a good supplement to a course in "informal" logic which focuses on argumentation rather than on mere evaluation of arguments. Plausible Reasoning provides the mechanism for evaluation of plausibility claims as distinct from probabilistic ones. The basic differences between these two types of claims are clearly discussed. "Plausibility is essentially a classificatory concept which ranks theses in terms of the standing and solidity of their cognitive basis. Plausibility grades theses by an external or extrinsic standard of the hierarchical nature of their supporting bases. It classifies propositions by the status of the evidential sources or validating principles that vouch for them. Probability weighs alternatives and evaluates theses by this relative contentual weight of the supporting considerations."—A.S.C. (shrink)

A selection of readings on the philosophy of logic, intended for use in introductory logic courses. Areas covered are: the nature of logic, the syllogism, the laws of thought, symbolic logic, and induction. The selections are well diversified and, for the most part, substantial.—A. E. J.

A number of authors have argued that the fact that certain indexicals depend for their reference-determination on the speaker’s referential intentions demonstrates the inadequacy of associating such expressions with functions from contexts to referents (characters). By distinguishing between different uses to which the notion of context is put in these argument, I show that this line of argument fails. In the course of doing so, I develop a way of incorporating the role played by intentions into a character-based semantics (...) for indexicals and I argue that the framework I prefer is superior to an alternative which has been proposed by others. (shrink)

In this first paper of a series of works on the foundations of science, we examine the significance of logical and mathematical frameworks used in foundational studies. In particular, we emphasize the distinction between the order of a language and the order of a structure to prevent confusing models of scientific theories with first-order structures, and which are studied in standard model theory. All of us are, of course, bound to make abuses of language even in putatively precise contexts. (...) This is not a problem—in fact, it is part of scientific and philosophical practice. But it is important to be sensitive to the dierent uses that structure, model, and language have. In this paper, we examine these topics in the context of classical logic; only in the last section we touch upon briefly on non-classical ones. (shrink)

Originally prepared as a doctoral thesis which was presented in 1940, the present work ranges over the major figures in British idealism, and in the Angloamerican schools of neorealism and logical atomism. What is understood here as the problem of relations is, of course, the controversy regarding the internality or externality of relations. This controversy begins with some issues involved in the definition and classification of relations, issues which affect the definition and classification of other categories such as individual, (...) quality, and even quantity. Soon enough, however, this set of issues is augmented by the problems involved in the definition of the relations emerging with each different kind of knowing, and by the problems concerning the bearing that knowledge has on the reality of relations. The main body of Datta's work is the succinct exposition of the doctrine of relations in a great number of authors such as Lotze, Green, Bradley, Bosanquet, James, Perry, Moore, Alexander, Russell, Broad, Cook Wilson, all of which complement each other in an exhaustive analysis of the above issues. Of great interest is the exposition of the Indian doctrine of the categories, including relation, in the Nyaya-Vaisesika system which protested against the Buddhistic conception of reality as an eternal flux. Datta concludes that "relations are always inter-objective, so that whenever cognition is concerned the question of relation does not arise. For cognition is not a relational fact but the ground of all relational facts." In backing these conclusions, the author sketches a brief outline of an ontology in which the main postulate proclaims that the spatiotemporal world is not the whole of reality: reality must include the meaning world as well.--A. M. (shrink)

In this essay I focus on the idea of the univocity of being, championed by Duns Scotus and given prominence more recently by Deleuze. Although I am interested in how this idea can be established, my primary concern is with something more basic: how the idea can even be properly thought. In the course of exploring this issue, which I do partly by borrowing some ideas about logical syntax from Wittgenstein's Tractatus, I try to show how there can be (...) dialogue between analytic philosophers and those of other traditions. (shrink)

In the past three decades there has been a rapid development of the formal machinery for modal logic. Quantified modal logic has developed along with a semantics and model theory that is appropriate to it. With this technical development there has been relatively little discussion of what modality is all about. There are two fundamental questions that have gone unanswered. First, to what does necessity amount? Is this a new logical notion, or is it something that can be (...) further analyzed in terms of other notions that we already have at our disposal? The second question is what makes truths involving necessity true? What is their ontological grounding in the world? This essay is directed at the first of these questions. ;There are three possible reductions of metaphysical modality. First, there is the reduction suggested by the semantics for modal logic: a reduction in terms of possible worlds. Second, there is the reduction in terms of cognition. What is possible is what is conceivable. Third, there is the reduction in terms of language. Necessities arise from the structures of language and/or language use. ;I argue that none of these is successful. All of them suffer either from presupposing some form of metaphysical necessity in the reductive base, rendering them circular and not truly reductive, or they do not do justice to the modality being reduced. ;Since necessity is not reducible to any non-modal features of the world, we need to determine whether we ought to reject it as unacceptable on this basis, or whether we ought to retain it, but treat is as a new primitive that is introduced into a philosophical or logical theory. I urge the latter course is the one that we ought to take because without it too many things will likewise be deemed unacceptable. Causal necessity and explanation, normally accepted theories or knowledge and perception, logic and logical validity, and one common manner of distinguishing sets from properties will be deemed unacceptable if de re necessity is unacceptable. All of these presuppose de re metaphysical necessity in some way. I conclude that de re metaphysical necessity ought to be retained in our theory as primitive, unanalyzable notion. (shrink)

This book is a selection of papers from a conference which took place at the University of Keele in July 1982. The conference was an extraordinarily enjoyable one, and we would like to take this opportunity of thanking all participants for helping to make it so. The conference was intended to allow scholars working on different aspects of symbolic behaviour to compare findings, to look for common ground, and to identify differences between the various areas. We hope that it was (...) successful in these aims: the assiduous reader may judge for himself. Several themes emerged during the course of the conference. Some of these were: 1. There is a distinction to be made between those symbol systems which attempt, more or less directly, to represent a state of affairs in the world (e. g. language, drawing, map and navigational skill) and those in which the representational function is complemented, if not overshadowed, by properties of the symbol system itself, and the systematic inter-relations that symbols can have to one another (e. g. music, mathematics). The distinction is not absolute, for the nature of all symbolic skills is, in part, a function of the structure of the symbolic system employed. Nonetheless, this distinction helps us to understand some common acquisition difficulties, such as that experienced in mathematics, where mental manipulation of symbols can go awry if a child assumes too close a correspondence between mathematical symbols and the world they represent. 2. (shrink)

In 1929 Carnap gave a paper in Prague on Investigations in General Axiomatics; a briefsummary was published soon after. Its subject lookssomething like early model theory, and the mainresult, called the Gabelbarkeitssatz, appears toclaim that a consistent set of axioms is complete justif it is categorical. This of course casts doubt onthe entire project. Though there is no furthermention of this theorem in Carnap''s publishedwritings, his Nachlass includes a largetypescript on the subject, Investigations inGeneral Axiomatics. We examine this work (...) here,showing that it provides important insights intoCarnap''s development during this critical period, thetransition from Aufbau to Syntax,especially regarding the nature and motivation ofCarnap''s logicism. Moreover, we show how theAxiomatics influenced Carnap''s student Gödel inreaching the fundamental logical results that soonafterwards undermined Carnap''s project. (shrink)

Reference and Existence, Saul Kripke's John Locke Lectures for 1973, can be read as a sequel to his classic Naming and Necessity. It confronts important issues left open in that work -- among them, the semantics of proper names and natural kind terms as they occur in fiction and in myth; negative existential statements; the ontology of fiction and myth. In treating these questions, he makes a number of methodological observations that go beyond the framework of his earlier book -- (...) including the striking claim that fiction cannot provide a test for theories of reference and naming. In addition, these lectures provide a glimpse into the transition to the pragmatics of singular reference that dominated his influential paper, " Speaker's Reference and Semantic Reference " -- a paper that helped reorient linguistic and philosophical semantics. Some of the themes have been worked out in later writings by other philosophers -- many influenced by typescripts of the lectures in circulation -- but none have approached the careful, systematic treatment provided here. The virtuosity of Naming and Necessity -- the colloquial ease of the tone, the dazzling, on-the-spot formulations, the logical structure of the overall view gradually emerging over the course of the lectures -- is on display here as well. (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 problem of Galileo's logical methodology has long interested scholars. In this volume William A. Wallace offers a solution that is completely unexpected, yet backed by convincing documentary evidence. His analysis starts with an early notebook Galileo wrote at Pisa, appropriating a Jesuit professor's exposition of the Posterior Analystics of Aristotle, and ends with one of the last letters Galileo wrote, stating that in logic he has been a Peripatetic all his life. Wallace's detective work unearths the complete (...) class='Hi'>logic course from which the notebook was excerpted, then proceeds to show how its terminology and methodology continue to surface in Galileo's later writings in which he founds his new sciences of the heavens and of local motion. The result is a tour de force that commends itself not only to Galileo's scholars and to logicians, philosophers, and historians, but to anyone interested in the epistemic roots of modern science. (shrink)

This collection of 17 articles offers an overview of the philosophical activities of a group of philosophers (who have been) working at the Groningen University. The meta-methodological assumption which unifies the research of this group, holds that there is a way to do philosophy which is a middle course between abstract normative philosophy of science and descriptive social studies of science. On the one hand it is argued with social studies of science that philosophy should take notice of what (...) scientists actually do. On the other hand, however, it is claimed that philosophy can and should aim to reveal cognitive patterns in the processes and products of scientific and common sense knowledge. Since it is thought that those patterns can function as guidelines in new research and/or in research in other disciplines, philosophy can nevertheless hold on to the normative aim which is characteristic of 'classical' philosophy of science. Compared to this common assumption, there is a diversity of subjects. Some papers deal with general problems of science, knowledge, cognition and argumentation, others with topics relating to foundational problems of particular sciences. Therefore this volume is of interest to philosophers of science, to philosophers of knowledge and argumentation in general, to philosophers of mind, as well as for scientists working in the physical and applied sciences, biology, psychology and economy who are interested in the foundations of their disciplines. After a foreword by Leszek Nowak and a general introduction by the editors, the book is divided into four parts, with special introductions. - I: Conceptual Analysis in Service of Various Research Programmes (Henk Zandvoort, Rein Vos, Rick Looijen, Gerben Stavenga, Renée Dalitz); - II: The Logic of the Evaluation of Arguments, Hypotheses, Default Rules, and Interesting Theorems (Erik Krabbe, Theo Kuipers, Alfons Keupink, Maarten Janssen/Yao-Hua Tan, Bert Hamminga); - III: Three Challenges to the Truth Approximation Programme (Sjoerd Zwart, Hinne Hettema/Theo Kuipers, Roberto Festa); - IV: Explicating Psychological Intuitions (Anne-Ruth Mackor, Jeanne Peijnenburg, Lex Guichard, Michel ter Hark). The Groningen research group was recently qualified, by an official international assessment committee, as one of the best philosophy research groups in the Netherlands. (shrink)

A translation of an important and neglected work of Averroes, with an excellent introduction, placing the book in its historical context and evaluating it philosophically. In the course of his close and meticulous refutation of Al Ghazali, who had denied the possibility of a rational philosophy and advanced arguments for the priority of mystical revelation, Averroes discusses the eternality of the world, the logical relation between cause and effect, and the relation between potentiality and actuality. There is also a (...) separate volume of notes, indices and an interesting list of contradictions to be found in Aristotle. A significant scholarly work. --A. R. (shrink)

Topics in the standard introductory logic course are all covered, with emphasis on the understanding of concepts rather than the development of skills in symbolic manipulation. Exposition is informal and rich in illustrations from reasoning in ordinary life. Exercises are also drawn from ordinary discourse, but they often call for the application of symbolic techniques. In combining accuracy and sophistication with simplicity and liveliness of style, the book is indeed a pedagogic success. --A. P. D. M.

This paper is divided into two parts. In the first one I distinguish between weak and strong Anti-Archimedeanisms, the latter being the view that metaethics, just as any other discipline attempting to work out a second-order conceptual, metaphysical non-committed discourse about the first-order discourse composing normative practices, is conceptually impossible or otherwise incoherent. I deal in particular with Ronald Dworkin’s famous exposition of the view. I argue that strong Anti-Archimedeanism constitutes an untenable philosophical stance, therefore making logical space for the (...) practice of a discipline such as metaethics—conceived as ethically neutral. This makes space, concurrently, for neutral conceptual jurisprudence. In the second part of the article, I attempt to show two things. On the one hand, that Dworkin’s widely discussed ‘challenge of disagreements’ to legal positivism is founded upon strong Anti-Archimedeanism. On the other hand, that having rejected strong Anti-Archimedeanism we should consequently reject the challenge as a serious challenge to positivism. This move, of course, does not thereby imply that accounting for legal disagreements is not an important jurisprudential task. But it marks—contra Dworkin—that there is no principled or a priori impossibility of doing so within a positivist framework. (shrink)

This is a translation of the chapter on perception by Kumarilabhatta's magnum opus, the Slokavarttika , which is one of the central texts of the Hindu response to the criticism of the logical-epistemological school of Buddhist thought. It is crucial for understanding the debates between Hindus and Buddhists about metaphysical, epistemological and linguistic questions during the classical period. In an extensive commentary, the author explains the course of the argument from verse to verse and alludes to other theories of (...) classical Indian philosophy and numerous other technical matters. Notes to the translation and commentary go further into the historical and philosophical background of Kumarila's ideas. The book provides an introduction to the history and the development of Indian epistemology, a synopsis of Kumarila's work and an analysis of its argument. It is a valuable contribution to the field of Indian philosophical studies. (shrink)

This is a most helpful survey of the rise and development of a new attitude towards philosophical investigation among a number of influential thinkers in England. The prime movers in this “revolution in philosophy” have been mathematicians, like Russell, logicians, like Ramsey and Wisdom, professors of philosophy, such as Ayer and Ryle; they have all felt the massive influence of Wittgenstein, and they make use of symbolic logic as an instrument. A movement which, in origin, was concerned especially with (...) mathematics and logic, and soon became an attempt to work out a perfect language, and, from an assumed empiricism, to use the structure of language to reveal the structure of reality, was bound to be highly technical and to give rise to many divergent theories as various obstacles to this ambitious programme were met with. Moreover, many of the more notable contributions of these thinkers are found in books that are intelligible only to one acquainted with symbolic logic, or in articles in specialized reviews that are not easily accessible to all. The movement has already passed through two well-defined phases, and has now entered on another. The time is thus ripe for such a book as this which traces, in a language which all can follow, the course of development, and the reasons for it, within this tendency, and introduces us to the present state of investigation and the main lines along which such investigation is proceeding. (shrink)

In the modal literature various notions of completeness have been studied for normal modal logics. Four of these are defined here, viz. (plain) completeness, first-order completeness, canonicity and possession of the finite model property — and their connections are studied. Up to one important exception, all possible inclusion relations are either proved or disproved. Hopefully, this helps to establish some order in the jungle of concepts concerning modal logics. In the course of the exposition, the interesting properties of first-order (...) definability and preservation under ultrafilter extensions are introduced and studied as well. (shrink)