In this paper I argue that pluralism at the level of logical systems requires a certain monism at the meta-logical level, and so, in a sense, there cannot be pluralism all the way down. The adequate alternative logical systems bottom out in a shared basic meta-logic, and as such, logical pluralism is limited. I argue that the content of this basic meta-logic must include the analogue of logical rules Modus Ponens and Universal Instantiation. I show this through a (...) detailed analysis of the ‘adoption problem’, which manifests something special about MP and UI. It appears that MP and UI underwrite the very nature of a logical rule of inference, due to all rules of inference being conditional and universal in their structure. As such, all logical rules presuppose MP and UI, making MP and UI self-governing, basic, unadoptable, and required in the meta-logic for the adequacy of any logical system. (shrink)

The more common scheme to explain the classical limit of quantum mechanics includes decoherence, which removes from the state the interference terms classically inadmissible since embodying non-Booleanity. In this work we consider the classical limit from a logical viewpoint, as a quantum-to-Boolean transition. The aim is to open the door to a new study based on dynamical logics, that is, logics that change over time. In particular, we appeal to the notion of hybrid logics to describe semiclassical systems. (...) Moreover, we consider systems with many characteristic decoherence times, whose sublattices of properties become distributive at different times. (shrink)

`This book grew out of the conviction, not in itself strange or startling, that the ordinary person can and should think straight rather than crooked.' Patrick Shaw has written a commonsense introduction to the use of logic in everyday thought and argument. It explains some of the rules of good argument and some of the ways in which arguments can fail, drawing illustrations from a variety of contemporary and international sources, such as the press, radio, and television. Symbols and (...) technicalities are kept to a minimum in this thorough and provocative investigation of the rational approach to thought - and its limitations. Logic and Its Limits emphasizes the use of logic in helping to settle and clarify disputes. It will help the reader to avoid bad arguments, to detect them in others, and so to think and argue more effectively. A wide range of thought-provoking examples and exercises concerned with contemporary social and political issues make this a readable and stimulating guide for the student and general reader alike. (shrink)

Heinrich Rickert was one of the leading neo-Kantian philosophers in Germany and a crucial figure in the discussions of the foundations of the social sciences in the first quarter of the twentieth century. His views were extremely influential, most significantly on Max Weber. The Limits of Concept Formation in Natural Science is Rickert's most important work, and it is here translated into English for the first time. It presents his systematic theory of knowledge and philosophy of science, and deals particularly (...) with historical knowledge and the problem of demarcating the natural from the human sciences. The theory Rickert develops is carefully argued and of great intrinsic interest. It departs from both positivism and neo-Hegelian idealism and is worked out by contrast to the views of others, particularly Dilthey and the early phenomenologists. (shrink)

Arthur Pap’s work played an important role in the development of the analytic tradition. This role goes beyond the merely historical fact that Pap’s views of dispositional and modal concepts were influential. As a sympathetic critic of logical empiricism, Pap, like Quine, saw a deep tension in logical empiricism at its very best in the work of Carnap. But Pap’s critique of Carnap is quite different from Quine’s, and represents the discovery of limits beyond which empiricism cannot go, where there (...) lies nothing other than intuitive knowledge of logic itself. Pap’s arguments for this intuitive knowledge anticipate Etchemendy’s recent critique of the model-theoretic account of logical consequence. Pap’s work also anticipates prominent developments in the contemporary neo-Fregean philosophy of mathematics championed by Wright and Hale. Finally, Pap’s major philosophical preoccupation, the concepts of necessity and possibility, provides distinctive solutions and perspectives on issues of contemporary concern in the metaphysics of modality. In particular, Pap’s account of modality allows us to see the significance of Kripke’s well-known arguments on necessity and apriority in a new light. (shrink)

This is an introduction to set theory and logic that starts completely from scratch. The text is accompanied by many methodological remarks and explanations.

According to the received view formal semantics applies to natural language to some extent only. It is so because natural language is inherently indefinite, in particular, its expressions are ambiguous, vague and admits departures from syntactic rule. Moreover, intensional contexts occur in ordinary language—it results in limitations of the principle of compositionality. The ordinary conversation appeals to various principles, for instance, Grice’s maxims which exceed logical formalism. Thus, ordinary language cannot be fully formalized. On the other hand, if L is (...) a formal language, its metalanguage ML, must be partially informal—for instance, it contains, terms of ordinary mathematics, especially set theory. Even, if, for instance, due to the technique of aritmetization, ML can be represented in L, such a representation is only local. In fact, this view can be derived from some Tarski’s remarks on the role played by natural language. It is usually assumed that the universality of natural language, is the source of troubles associated with antinomies. It is so this circumstance requires a solution, for example by distinguishing levels of language. However, even if antinomies are excluded, what is informal is prior with respect to what is formal. It shows that formal semantics has limitations even with respect to formalized languages. (shrink)

I argue against abductivism about logic, which is the view that rational theory choice in logic happens by abduction. Abduction cannot serve as a neutral arbiter in many foundational disputes in logic because, in order to use abduction, one must first identify the relevant data. Which data one deems relevant depends on what I call one's conception of logic. One's conception of logic is, however, not independent of one's views regarding many of the foundational disputes (...) that one may hope to solve by abduction. (shrink)

Heinrich Rickert (1863-1936) was One of the leading neo-Kantian philosophers in Germany and a crucial figure in the discussions of the foundations of the social sciences in the first quarter of the twentieth century. His views were extremely influential, most significantly on Max Weber. The Limits of Concept Formation in Natural Science is Rickert's most important work, and it is here translated into English for the first time. It presents his systematic theory of knowledge and philosophy of science, and deals (...) particularly with historical knowledge and the problem of demarcating the natural from the human sciences. The theory Rickert develops is carefully argued and of great intrinsic interest. It departs from both positivism and neo-Hegelian idealism and is worked out by contrast to the views of others, particularly Dilthey and the early phenomenologists. (shrink)

Though my ultimate concern is with issues in epistemology and metaphysics, let me phrase the central question I will pursue in terms evocative of philosophy of religion: What are the implications of our logic-in particular, of Cantor and G6del-for the possibility of omniscience?

ABSTRACT:Giving without the expectation of reward is difficult to understand in organizational contexts. In opposition to a logic based on self-interest or a sense of duty, a “logic of gift” has been proposed as a way to understand the phenomenon of free, unconditional giving. However, the rationale behind, and effects of, this logic have been under-explored. This paper responds by first clarifying the three logics of action—the logic of exchange, the logic of duty, and the (...) logic of gift—and then explains how their balanced integration promises to enhance organizational life and outcomes. Having explicated the unique character and contributions of the logic of gift, the paper further suggests practical implications for management. Encouraging the logic of gift fosters more humane relationships within organizations and to enable individuals to be generous in ways that inspire trust and promote creativity. (shrink)

We associate with any abstract logic L a family F(L) consisting, intuitively, of the limit ultrapowers which are complete extensions in the sense of L. For every countably generated [ω, ω]-compact logic L, our main applications are: (i) Elementary classes of L can be characterized in terms of $\equiv_L$ only. (ii) If U and B are countable models of a countable superstable theory without the finite cover property, then $\mathfrak{U} \equiv_L \mathfrak{B}$ . (iii) There exists the "largest" (...) logic M such that complete extensions in the sense of M and L are the same; moreover M is still [ω,ω]-compact and satisfies an interpolation property stronger than unrelativized ▵-closure. (iv) If L = L ωω(Q α ) , then $\operatorname{cf}(\omega_\alpha) > \omega$ and $\lambda^\omega for all $\lambda . We also prove that no proper extension of L ωω generated by monadic quantifiers is compact. This strengthens a theorem of Makowsky and Shelah. We solve a problem of Makowsky concerning L κλ -compact cardinals. We partially solve a problem of Makowsky and Shelah concerning the union of compact logics. (shrink)

The first beginning logic text to employ the tree method--a complete formal system of first-order logic that is remarkably easy to understand and use--this text allows students to take control of the nuts and bolts of formal logic quickly, and to move on to more complex and abstract problems. The tree method is elaborated in manageable steps over five chapters, in each of which its adequacy is reviewed; soundness and completeness proofs are extended at each step, and (...) the decidability proof is extended at the step from truth functions to the logic of nonoverlapping quantifiers with a single variable, after which undecidability is demonstrated by example. The first three chapters are bilingual, with arguments presented twice, in logical notation and in English. The last three chapters consider the discoveries defining the scope and limits of formal methods that marked logic’s coming of age in the 20th century: Godel’s completeness and incompleteness theorems for first and second-order logic, and the Church-Turing theorem on the undecidability of first-order logic. This new edition provides additional problems, solutions to selected problems, and two new Supplements: Truth-Functional Equivalence reinstates material on that topic from the second edition that was omitted in the third, and Variant Methods, in which John Burgess provides a proof regarding the possibility of modifying the tree method so that it will always find a finite model when there is one, and another, which shows that a different modification—once contemplated by Jeffrey--can result in a dramatic speed--up of certain proofs. (shrink)

Dung’s (1995) argumentation framework takes as input two abstract entities: a set of arguments and a binary relation encoding attacks between these arguments. It returns acceptable sets of arguments, called extensions, w.r.t. a given semantics. While the abstract nature of this setting is seen as a great advantage, it induces a big gap with the application that it is used to. This raises some questions about the compatibility of the setting with a logical formalism (i.e., whether it is possible to (...) instantiate it properly from a logical knowledge base), and about the significance of the various semantics in the application context. In this paper we tackle the above questions. We first propose to fill in the previous gap by extending Dung’s (1995) framework. The idea is to consider all the ingredients involved in an argumentation process. We start with the notion of an abstract monotonic logic which consists of a language (defining the formulas) and a consequence operator. We show how to build, in a systematic way, arguments from a knowledge base formalised in such a logic. We then recall some basic postulates that any instantiation should satisfy. We study how to choose an attack relation so that the instantiation satisfies the postulates. We show that symmetric attack relations are generally not suitable. However, we identify at least one ‘appropriate’ attack relation. Next, we investigate under stable, semi-stable, preferred, grounded and ideal semantics the outputs of logic-based instantiations that satisfy the postulates. For each semantics, we delimit the number of extensions an argumentation system may have, characterise the extensions in terms of subsets of the knowledge base, and finally characterise the set of conclusions that are drawn from the knowledge base. The study reveals that stable, semi-stable and preferred semantics either lead to counter-intuitive results or provide no added value w.r.t. naive semantics. Besides, naive semantics either leads to arbitrary results or generalises the coherence-based approach initially developed by Rescher and Manor (1970). Ideal and grounded semantics either coincide and generalise the free consequence relation developed by Benferhat, Dubois, and Prade (1997), or return arbitrary results. Consequently, Dung’s (1995) framework seems problematic when applied over deductive logical formalisms. (shrink)

This paper introduces a logic game which can be used to demonstrate the working of Boolean connectives. The simplicity of the system turns out to lead to some interesting meta-theoretical properties, which themselves carry a philosophical import. After introducing the system, we demonstrate an interesting feature of it—that it, while being an accurate model of propositional logic Booleans, does not contain any tautologies nor contradictions. This result allows us to make explicit a limitation of application of propositional (...) class='Hi'>logic to those sentences with relatively stable truth values. (shrink)

An attempt to show the possibility of penetrating into the ultimate nature of things through the development of what the author calls "buddhi" or "philosophical" reason. Overcoming the limitations of "logical" or "scientific" reason as exemplified in classical physics, philosophical reason reconciles the dualism between knower and known and reveals the unity of the "within" and "without" of nature. This insight into ultimate reality occurs in a state of "dreamless sleep." The world of the "not-self" or of objects is disclosed (...) by philosophical reason to be in its essence spiritual. The external world is a configuration of the unified self.—L. W. (shrink)

The articles in this volume represent a part of the philosophical literature on higher-order logic and the Skolem paradox. They ask the question what is second-order logic? and examine various interpretations of the Lowenheim-Skolem theorem.

Engaging with Kant’s transcendental logic seems to be a question of mere scholarly historical interest today. It is most commonly regarded a mixture between logic and psychology or epistemology, and by that, not a serious form of logic. Transcendental logic seems to be of no systematical impact on the concept of logic. My paper aims to disclose a different account on the endeavour of Kant’s transcendental logic in particular and of the “Critique of Pure (...) Reason” (CPR) in general. Kant’s fundamental question is in a revolutionary way aiming to ground the character of necessity of knowledge, which means to justify the claim that thinking in accordance with the forms and principles of formal logic does not lead to sheer tautologies or an unsolved contradiction, but to knowledge that is objectively valid. In a first part, I shall demonstrate the necessity and the significance of this new fundamental question of the CPR with respect to its genesis out of pre-Kantian metaphysics. A brief outline of Kant’s answer to this question, with special emphasis on his revolutionary new comprehension of logical form, will be given as well. A second part shall open up a perspective that lies beyond Kant’s standpoint with reference to Nietzsche and eventually to Hegel. I will answer the question: What knowledge do we achieve about being or actuality by means of formal logic? I will argue that Kant shows that formal logic is the logic of all technical-practical conduct but also, at least indirectly, the limitation of the technical-practical knowledge and its legitimate sphere of application. (shrink)

For Tarski logics, there are simple criteria that enable one to conclude that two premise sets are equivalent. We shall show that the very same criteria hold for adaptive logics, which is a major advantage in comparison to other approaches to defeasible reasoning forms. A related property of Tarski logics is that the extensions of equivalent premise sets with the same set of formulas are equivalent premise sets. This does not hold for adaptive logics. However a very similar criterion does. (...) We also shall show that every monotonic logic weaker than an adaptive logic is weaker than the lower limit logic of the adaptive logic or identical to it. This highlights the role of the lower limit for settling the adaptive equivalence of extensions of equivalent premise sets. (shrink)

Badiou and Wittgenstein focus in their works on potentialities of innovation in the realm of thought as well as in the realm of politics. These innovations manifest themselves especially when two seemingly contrasting jurisdictions of thought – present in politics and logic – meet and merge. For Badiou a set-theoretical process of enforcement may change pre-established templates of a political doxa. For Wittgen-stein it is the spontaneity of concept-formation that crisscross referential relations within the “space of reason” and through (...) performative enactments makes visible unexpected places of unprompted innovation. For both Wittgenstein and Badiou, the subject is of vital importance in this union of politics and logic. It is both a “limit of the world” as well as a “supernumerary agency”. Characterized as such, it can provoke new worlds to appear with the aid of what I will call in the following paragraphs self-proclaimed logics of scission. (shrink)

Contributed articles presented at the National Seminar on the Limits of Thought and Beyond, organized by the Department of Philosophy, Assam University, Silchar, India, during 28-29 November 2011.

This clearly written book is designed to introduce the student to the content and methods of formal logic up to and including the lower predicate calculus with identity. There is also a final chapter on undecidability and incompleteness in which the theorems of Church and a version of Gödel's incompleteness theorem are discussed. Primary attention is paid throughout to the development of a system of natural deduction, and the author makes recourse at numerous points to examples drawn from arguments (...) in ordinary natural language. Aside from the material in the last chapter, the principal novelty of the book lies in its use of a method of trees to test for validity. While this device is often informative and pedagogically helpful, it occasionally takes on some of the hallmarks of gimmickry, and sometimes encourages the student to apply a predesigned program, rather than his mind. On the other hand, Jeffrey does provide a good treatment of topics which often baffle beginning logic students, such as tensed verbs, the use of multiple quantifiers, selection of a universe of discourse and relation to elementary set-theoretical notions. There is also a brief section on restricted quantifiers. Each chapter contains numerous exercises for the student.—H. P. K. (shrink)

Robert Hanna (Rationality and logic. MIT Press, Cambridge, 2006) articulates and defends the thesis of logical cognitivism, the claim that human logical competence is grounded in a cognitive faculty (in Chomsky’s sense) that is not naturalistically explicable. This position is intended to steer us between the Scylla of logical Platonism and the Charybdis of logical naturalism (/psychologism). The paper argues that Hanna’s interpretation of Chomsky is mistaken. Read aright, Chomsky’s position offers a defensible version of naturalism, one Hanna may (...) accept as far as his version of naturalism goes, although not one that supports the claim that cognitive science offers a place for logic that is somehow outside the natural, contingent order. (shrink)

First published in 2000. Routledge is an imprint of Taylor & Francis, an informa company.

Logical abductivism is the method which purports to use Inference to the Best Explantion (IBE) to determine the best logical theory. The present essay argues that this is not the case, since the method fails to meet the criteria requisite for the fruitful application of IBE. This occurs due to an intrinsic difficulty in choosing the appropriate evidence and theoretical virtues which guide theory revision in logic: one’s previous conception of logic influences both these choices. Logical abductivism fails, (...) moreover, to select the best logical theory, exactly because a lack of agreement on theory and virtues for Logic. Rather than direct comparison between two options, a more suitable approach to theory revision in logic is piecemeal, because this method neither assumes nor needs a neutral ground from which to start revising theories. (shrink)

Cet article maintient que l’intérêt de Husserl pour le développement d’une logique pure en tant que théorie de la science limite sa conception de l’ontologie. L’ontologie formelle est, pour Husserl, une théorie formelle des objets de connaissance, dont les catégories fondamentales sont celles de substance, propriété et relation. En outre, les ontologies régionales évoluent au sein des limites catégorielles définies par l’ontologie formelle. Mais une telle ontologie laisse de côté les activités et les processus de tout genre, parmi lesquels le (...) plus important est celui de l’ « avoir à l’esprit ». La phénoménologie transcendantale, en revanche, s’occupe de l’être-conscient-du-monde, un être-conscient qui est cependant inséparable de l’être-dans-le-monde. Cette phénoménologie se situe sur le même terrain ontologique que l’ontologie fondamentale de Heidegger. Mais alors, bien que la notion husserlienne d’ontologie soit limitée par le rôle dominant attribué à la logique en tant que théorie de la science et à la connaissance vraie qui lui est propre, il y a plus d’ontologie chez Husserl que dans son « ontologie ». (shrink)

Candrakīrti is known for his commentaries on the major works of Nāgārjuna and Āryadeva. This chapter examines how Candrakīrti uses language and logic to undermine people's confidence in cherished beliefs about a self and point them towards the Buddha's path and its goal the peace of nirvana that transcends the limitations of language and logic. Candrakīrti first sets out his view on the two truths in the Madhyamakāvatāra. He associates both truths with the soteriological goal of Nāgārjuna's path: (...) the peaceful state of nirvana. Madhyamaka teaching amounts to nihilism. Candrakīrti responds that the charge of nihilism is wrong because there is no repudiation of moral actions. Emptiness is also related to the concept of intrinsic nature, and Candrakīrti uses this term in more than one sense. (shrink)

Journal of Chinese Philosophy, EarlyView.

• The transitive closure of R is the smallest relation S for which: –R⊆S, – S is transitive. • To express ^r one would need an ‘infinite formula’: {(x, y) | R(x, y) ∨ ∃z(R(x, z) ∧ R(z, y)) ∨∃z, v(R(x, z) ∧ R(z, v) ∧ R(v, y)) ∨∃z, v, w(R(x, z) ∧ R(z, v) ∧ R(v, w) ∧ R(w, y)) ∨ · · ·.

Many books explain what is known about the universe. This book investigates what cannot be known. Rather than exploring the amazing facts that science, mathematics, and reason have revealed to us, this work studies what science, mathematics, and reason tell us cannot be revealed. In The Outer Limits of Reason, Noson Yanofsky considers what cannot be predicted, described, or known, and what will never be understood. He discusses the limitations of computers, physics, logic, and our own thought processes. Yanofsky (...) describes simple tasks that would take computers trillions of centuries to complete and other problems that computers can never solve; perfectly formed English sentences that make no sense; different levels of infinity; the bizarre world of the quantum; the relevance of relativity theory; the causes of chaos theory; math problems that cannot be solved by normal means; and statements that are true but cannot be proven. He explains the limitations of our intuitions about the world -- our ideas about space, time, and motion, and the complex relationship between the knower and the known. Moving from the concrete to the abstract, from problems of everyday language to straightforward philosophical questions to the formalities of physics and mathematics, Yanofsky demonstrates a myriad of unsolvable problems and paradoxes. Exploring the various limitations of our knowledge, he shows that many of these limitations have a similar pattern and that by investigating these patterns, we can better understand the structure and limitations of reason itself. Yanofsky even attempts to look beyond the borders of reason to see what, if anything, is out there. (shrink)

This work investigates some of the most important logical limits of scientific knowledge. We argue that scientific knowledge is based on different logicalforms and paradigms. The logical forms, which represent the rational structure of scientific knowledge, show their limits through logical antinomies. The paradigms, which represent the scientific points of view on the world, show their limits through the theoretical anomalies. When these limits arise in science and when scientists become fully and deeply aware of them, they can determine logical (...) or paradigmatic revolutions. These are different in their respective courses, although the logical forms and the paradigms are parts of the same type of knowledge. In the end, science can avoid or can integrate its different limits. In fact, the limits of science can become new opportunities for its growth and development. (shrink)

What is the relation of logic to thinking? My dissertation offers a new argument for the claim that logic is constitutive of thinking in the following sense: representational activity counts as thinking only if it manifests sensitivity to logical rules. In short, thinking has to be minimally logical. An account of thinking has to allow for our freedom to question or revise our commitments – even seemingly obvious conceptual connections – without loss of understanding. This freedom, I argue, (...) requires that thinkers have general abilities to respond to support and tension among their thoughts. And these abilities are constituted by following logical rules. So thinkers have to follow logical rules. But there isn’t just one correct logic for thinking. I show that my view is consistent with logical pluralism: there are a range of correct logics, any one of which a thinker might follow. A logic for thinking does, however, have to contain certain minimal principles: Modus Ponens and Non-Contradiction, and perhaps others. We follow logical rules by exercising logical capacities, which display a distinctive first-person/third-person asymmetry: a subject can find the instances of a rule compelling without seeing them as instances of a rule. As a result, there are two limits on illogical thinking. First, thinkers have to tend to find instances of logical rules compelling. Second, thinkers can’t think in obviously illogical ways. So thinking has to be logical – but not perfectly so. When we try to think, but fail, we produce nonsense. But our failures to think are often subjectively indistinguishable from thinking. To explain how this occurs, I offer an account of nonsense. To be under the illusion that some nonsense makes sense is to enter a pretence that the nonsense is meaningful. Our use of nonsense within the pretence relies on the role of logical form in understanding. Finally, while the normativity of logic doesn’t fall directly out of logical constitutivism, it’s possible to build an attractive account of logical normativity which has logical constitutivism as an integral part. I argue that thinking is necessary for human flourishing, and that this is the source of logical normativity. (shrink)