This article is a theoretical study of the relations between logic and Anglo-American cognitive science. It uses temporal, historical, and intentional evidence, and it is based on two consequential assumptions: (a) between the late 19th century and the first decades of the 20th century, some logical systems attempted to explain the foundations of mathematics and (b) since the 1940s and 1950s, these same systems became implicit sources of contents and methods of the rising Cognitivism. This study produces an (...) important conclusion: Anglo-American cognitive science partly derives from logic and partly shares many similarities with this science. (shrink)

Written before the impressive collapse of the socialist system in Eastern Europe, this book offers a quite objective and serious systematic analysis of the Marxian labor theory of value, Marx's main scientific legacy. After reconstructing the 'prototype' of this theory - which is the theory as it was left by Marx himself in Capital - the author proceeds to a careful and detailed analysis of its foundational problems, taking into account Bohm-Bawerk's important criticisms. After introducing advanced contemporary formal tools, the (...) author proceeds to a thorough discussion of the dialectical method, just in order to tackle the foundational problems of the theory. He provides a formally precise and well motivated definition of abstract labor, and then proceeds to prove the existence of a measurement of abstract labor - i.e. the existence of numerical labor-values. Using this result, the author provides rigorous axiomatic foundations for the theory of value and then proves the existence of a Marxian competitive equilibrium, which is tantamount to the proof of the possibility of reproduction for a capitalist economy. The author finishes the book by showing in detail how the problems of the prototype are solved, by reconstructing the Leontief model of the labor theory of value on the new logical bases. Written in a very clear style, in the language of contemporary philosophy of science, the book is of interest to philosophers of science and economists, applied logicians and all those interested in the scientific legacy of Karl Marx. (shrink)

APA PsycNET abstract: This is the first volume of a two-volume work on Probability and Induction. Because the writer holds that probability logic is identical with inductive logic, this work is devoted to philosophical problems concerning the nature of probability and inductive reasoning. The author rejects a statistical frequency basis for probability in favor of a logical relation between two statements or propositions. Probability "is the degree of confirmation of a hypothesis (or conclusion) on the basis of some given (...) evidence (or premises)." Furthermore, all principles and theorems of inductive logic are analytic, and the entire system is to be constructed by means of symbolic logic and semantic methods. This means that the author confines himself to the formalistic procedures of word and symbol systems. The resulting sentence or language structures are presumed to separate off logic from all subjectivist or psychological elements. Despite the abstractionism, the claim is made that if an inductive probability system of logic can be constructed it will have its practical application in mathematical statistics, and in various sciences. 16-page bibliography. (PsycINFO Database Record (c) 2016 APA, all rights reserved). (shrink)

The book is about logical analysis of natural language. Since we humans communicate by means of natural language, we need a tool that helps us to understand in a precise manner how the logical and formal mechanisms of natural language work. Moreover, in the age of computers, we need to communicate both with and through computers as well. Transparent Intensional Logic is a tool that is helpful in making our communication and reasoning smooth and precise. It deals with (...) all kinds of linguistic context in a fully compositional and anti-contextual way. (shrink)

In many ways set theory lies at the heart of modern mathematics, and it does powerful work both philosophical and mathematical – as a foundation for the subject. However, certain philosophical problems raise serious doubts about our acceptance of the axioms of set theory. In a detailed and original reassessment of these axioms, Sharon Berry uses a potentialist approach to develop a unified determinate conception of set-theoretic truth that vindicates many of our intuitive expectations regarding set theory. Berry further defends (...) her approach against a number of possible objections, and she shows how a notion of logical possibility that is useful in formulating Potentialist set theory connects in important ways with philosophy of language, metametaphysics and philosophy of science. Her book will appeal to readers with interests in the philosophy of set theory, modal logic, and the role of mathematics in the sciences. (shrink)

This chapter contains sections titled: Foundations and Logical Foundations Foundations for Mathematics Mathematics and Set Theory Sets, Classes, and Logic.

This book addresses the logical aspects of the foundations of scientific theories. Even though the relevance of formal methods in the study of scientific theories is now widely recognized and regaining prominence, the issues covered here are still not generally discussed in philosophy of science. The authors focus mainly on the role played by the underlying formal apparatuses employed in the construction of the models of scientific theories, relating the discussion with the so-called semantic approach to scientific theories. (...) The book describes the role played by this metamathematical framework in three main aspects: considerations of formal languages employed to axiomatize scientific theories, the role of the axiomatic method itself, and the way set-theoretical structures, which play the role of the models of theories, are developed. The authors also discuss the differences and philosophical relevance of the two basic ways of aximoatizing a scientific theory, namely Patrick Suppes’ set theoretical predicates and the "da Costa and Chuaqui" approach. This book engages with important discussions of the nature of scientific theories and will be a useful resource for researchers and upper-level students working in philosophy of science. (shrink)

First-order logic. The origin of modern foundational studies. Frege's system and the paradoxes. The teory of types. Zermelo-Fraenkel set theory. Hilbert's program and Godel's incompleteness theorems. The foundational systems of W.V. Quine. Categorical algebra.

In order to represent legal knowledge adequately, it is vital to create a formal device that can freely construct an individual concept directly from a predicate expression. For this purpose, a Compound Predicate Formula (CPF) is formulated for use in legal expert systems. In this paper, we willattempt to explain the nature of CPFs by rigorous logical foundation, i.e., establishing their syntax and semantics precisely through the use of appropriate examples. We note the advantages of our system over other (...) such systems and discuss the significance of CPFs with regard to the formalization of legal reasonings using examples from the United Nations Convention for the International Sale of Goods. (shrink)

This paper addresses the logical foundations of goal-regression planning in autonomous rational agents. It focuses mainly on three problems. The first is that goals and subgoals will often be conjunctions, and to apply goal-regression planning to a conjunction we usually have to plan separately for the conjuncts and then combine the resulting subplans. A logical problem arises from the fact that the subplans may destructively interfere with each other. This problem has been partially solved in the AI (...) literature (e.g., in SNLP and UCPOP), but the solutions proposed there work only when a restrictive assumption is satisfied. This assumption pertains to the computability of threats. It is argued that this assumption may fail for an autonomous rational agent operating in a complex environment. Relaxing this assumption leads to a theory of defeasible planning. The theory is formulated precisely and an implementation in the OSCAR architecture is discussed. (shrink)

The Logical Foundations of Social Theory describes Gert Mueller’s argument that physical, biological, social, moral, and cultural reality form an asymmetrical hierarchy of founding and controlling relationships that condition social reality rather than mechanically determining it. This book analyzes social stratification, the moral order, and culture systems.

The Logical Foundations of Social Theory describes Gert Mueller’s argument that physical, biological, social, moral, and cultural reality form an asymmetrical hierarchy of founding and controlling relationships that condition social reality rather than mechanically determining it. This book analyzes social stratification, the moral order, and culture systems.

Internal logic is the logic of content. The content is here arithmetic and the emphasis is on a constructive logic of arithmetic (arithmetical logic). Kronecker's general arithmetic of forms (polynomials) together with Fermat's infinite descent is put to use in an internal consistency proof. The view is developed in the context of a radical arithmetization of mathematics and logic and covers the many-faceted heritage of Kronecker's work, which includes not only Hilbert, but also Frege, Cantor, Dedekind, Husserl and Brouwer. The (...) book will be of primary interest to logicians, philosophers and mathematicians interested in the foundations of mathematics and the philosophical implications of constructivist mathematics. It may also be of interest to historians, since it covers a fifty-year period, from 1880 to 1930, which has been crucial in the foundational debates and their repercussions on the contemporary scene. (shrink)

Carnap's inductive logic (or confirmation) project is revisited from an "increase in firmness" (or probabilistic relevance) point of view. It is argued that Carnap's main desiderata can be satisfied in this setting, without the need for a theory of "logical probability." The emphasis here will be on explaining how Carnap's epistemological desiderata for inductive logic will need to be modified in this new setting. The key move is to abandon Carnap's goal of bridging confirmation and credence, in favor of (...) bridging confirmation and evidential support. (shrink)

This volume, the fourth in the Vancouver Studies in Cognitive Science published by OUP, examines the role of logic in cognitive psychology in light of recent developments. Gonzalo Reyes's new semantic theory has brought the fields of cognitive psychology and logic closer together, and has shed light on how children may master proper names and count nouns, and thus acquire knowledge. The chapters highlight the inadequacies of classical logic in its handling of ordinary language and reveals the prospects of applying (...) the new theory of kinds to cognitive psychology, cognitive science, linguistics, the philosophy of language and logic. (shrink)

The aim of this paper is to shed new light on the logical roots of arithmetic by presenting a logical framework that takes seriously ordinary locutions like ‘at least n Fs’, ‘n more Fs than Gs’ and ‘n times as many Fs as Gs’, instead of paraphrasing them away in terms of expressions of the form ‘the number of Fs’. It will be shown that the basic concepts of arithmetic can be intuitively defined in the language of ALA, (...) and the Dedekind–Peano axioms can be derived from those definitions by logical means alone. It will also be shown that some fundamental facts about cardinal numbers expressed using singular terms of the form ‘the number of Fs’, including Hume’s Principle, can be derived solely from definitions. (shrink)

This book is a major contribution to the study of the philosopher F. H. Bradley, the most influential member of the nineteenth-century school of British Idealists. It offers a sustained interpretation of Bradley's Principles of Logic, explaining the problem of how it is possible for inferences to be both valid and yet have conclusions that contain new information. The author then describes how this solution provides a basis for Bradley's metaphysical view that reality is one interconnected experience and how this (...) gives rise to a new problem of truth. (shrink)

The abstract status of Kant's account of his ‘general logic’ is explained in comparison with Gödel's general definition of a formal logical system and reflections on ‘abstract’ (‘absolute’) concepts. Thereafter, an informal reconstruction of Kant's general logic is given from the aspect of the principles of contradiction, of sufficient reason, and of excluded middle. It is shown that Kant's composition of logic consists in a gradual strengthening of logical principles, starting from a weak principle of contradiction that tolerates (...) a sort of contradictions in predication, and then proceeding to the (constructive) principle of sufficient reason, and to a classical-like logic, which includes the principle of excluded middle. A first-order formalisation is applied to this reconstruction, which reveals implicit modalities in Kant's account of logic, and confirms the implementability of Kant's logic into a sound and complete formal system. (shrink)

We illustrate the glorious history of logical foundations and discuss the uncertain future.

The aim of the dissertation is to show that the logical foundations of the Marxian Theory of Value are reasonably sound. The first chapter presents a concise history of MTV and formulates in a detailed way the problem of foundations in MTV. This is the problem of proving the existence of labor-values in a general economy in which joint products, alternative techniques, and heterogeneous labor are allowed. ;In the second chapter the problem is tackled through the demonstration (...) that there is a measurement of abstract labor in such general systems, where abstract labor is understood as a binary relation over the set of labor expenditures induced by a system of prices, in a commodity economy, that satisfies certain properties. It is also shown that whenever the rate of profit is uniform the price system satisfies such properties. ;Using the results of the second chapter, in the last chapter an axiomatization of MTV is provided and two of the most typical models of MTV are reconstructed in terms of abstract labor in a very detailed way. The first of these is the classical Leontief model and the second one is a more general model due to John K. Roemer. The conclusion of the dissertation is that the foundations of MTV are as sound as those of Neo-classical Economics. (shrink)

Mathematical logic is a branch of mathematics that takes axiom systems and mathematical proofs as its objects of study. This book shows how it can also provide a foundation for the development of information science and technology. The first five chapters systematically present the core topics of classical mathematical logic, including the syntax and models of first-order languages, formal inference systems, computability and representability, and Gödel’s theorems. The last five chapters present extensions and developments of classical mathematical logic, particularly the (...) concepts of version sequences of formal theories and their limits, the system of revision calculus, proschemes (formal descriptions of proof methods and strategies) and their properties, and the theory of inductive inference. All of these themes contribute to a formal theory of axiomatization and its application to the process of developing information technology and scientific theories. The book also describes the paradigm of three kinds of language environments for theories and it presents the basic properties required of a meta-language environment. Finally, the book brings these themes together by describing a workflow for scientific research in the information era in which formal methods, interactive software and human invention are all used to their advantage. The second edition of the book includes major revisions on the proof of the completeness theorem of the Gentzen system and new contents on the logic of scientific discovery, R-calculus without cut, and the operational semantics of program debugging. This book represents a valuable reference for graduate and undergraduate students and researchers in mathematics, information science and technology, and other relevant areas of natural sciences. Its first five chapters serve as an undergraduate text in mathematical logic and the last five chapters are addressed to graduate students in relevant disciplines. (shrink)

This paper explores proof-theoretic aspects of hybrid type-logical grammars, a logic combining Lambek grammars with lambda grammars. We prove some basic properties of the calculus, such as normalisation and the subformula property and also present both a sequent and a proof net calculus for hybrid type-logical grammars. In addition to clarifying the logical foundations of hybrid type-logical grammars, the current study opens the way to variants and extensions of the original system, including but not limited (...) to a non-associative version and a multimodal version incorporating structural rules and unary modes. (shrink)

This essay presents a philosophical and computational theory of the representation of de re, de dicto, nested, and quasi-indexical belief reports expressed in natural language. The propositional Semantic Network Processing System (SNePS) is used for representing and reasoning about these reports. In particular, quasi-indicators (indexical expressions occurring in intentional contexts and representing uses of indicators by another speaker) pose problems for natural-language representation and reasoning systems, because--unlike pure indicators--they cannot be replaced by coreferential NPs without changing the meaning of the (...) embedding sentence. Therefore, the referent of the quasi-indicator must be represented in such a way that no invalid coreferential claims are entailed. The importance of quasi-indicators is discussed, and it is shown that all four of the above categories of belief reports can be handled by a single representational technique using belief spaces containing intensional entities. Inference rules and belief-revision techniques for the system are also examined. (shrink)

The paper discusses properties of 4QL, a DATALOG⌉⌉-like query language, originally outlined by Maluszyński and Szalas (Maluszyński & Szalas, 2011). 4QL allows one to use rules with negation in heads and bodies of rules. It is based on a simple and intuitive semantics and provides uniform tools for “lightweight” versions of known forms of nonmonotonic reasoning. Negated literals in heads of rules may naturally lead to inconsistencies. On the other hand, rules do not have to attach meaning to some literals. (...) Therefore 4QL is founded on a four-valued semantics, employing the logic introduced in (Maluszyński et al., 2008; Vitória et al., 2009) with truth values: ‘true’, ‘false’, ‘inconsistent’ and ‘unknown’. In addition, 4QL is tractable w.r.t. data complexity and captures PTIME queries. Even though DATALOG⌉⌉ is known as a concept for the last 30 years, to our best knowledge no existing approach enjoys these properties.In the current paper we:investigate properties of well-supported models of 4QLprove the correctness of the algorithm for computing well-supported modelsshow that 4QL has PTIME data complexity and captures PTIME. (shrink)

This essay presents a philosophical and computationol theory of the representation of de re, de dlcto, nested, and quasi-indexical belief reports expressed in natural language. The propositional Semantic Network Processing System (SNePS) is used for representing and reasoning about these reports. In particular, quasi-indicators (indexical expressions occurring in intentional contexts and representing uses of indicators by another speaker) pose problems for natural language representation and reasoning systems, because--unlike pure indicators --they cannot be replaced by coreferential NPs without changing the meaning (...) of the embedding sentence. Therefore, the referent of the quasi-indicator must be represented in such a way that no invalid coreferential claims are entailed. The importance of quasi-indicators is discussed, and it is shown that all four of the above categories of belief reports can be handled by a single representational technique using belief spaces containing intensional entities. Inference rules and belief-revision techniques for the system are also examined. (shrink)

This article attempts to show points of contact between current analytical philosophy and the philosophy of Bernard Lonergan. It does so by focusing on the philosophy of logic. Debates concerning the adequacy and/or completeness of the various logical systems which have been or are being elaborated are an important aspect of analytical philosophy today. Lonergan was also preoccupied with the foundations of logic. This is manifest not only in his 1957 Boston College lectures on the subject, but throughout (...) his philosophical writing. In the article we shall see how Lonergan's cognitional theory can throw light on issues in the philosophy of logic which are of importance to analytical philosophers. The discussion, which will be non-technical in nature, will examine contemporary debates on the principles of excluded middle and non-contradiction, and will also focus on Lonergan's unique contribution to a philosophical analysis of both the importance and the limitations of symbolism in philosophical argument. /// O presente artigo pretende mostrar a existência de pontos de contacto entre a Filosofia Analítica corrente e o pensamento de Bernard Lonergan. Em ordem a isso, o autor do artigo concentra a sua atenção na filosofia da lógica. Com efeito, os debates acerca da adequação e/ou completude dos vários sistemas lógicos já elaborados ou em processo de elaboração constituem um importante aspecto da Filosofia Analítica mais recente. Por seu lado, Bernard Lonergan também se preocupou com os fundamentos da Lógica. Isso é o que se pode ver não apenas mediante as suas conferências pronunciadas no Boston College em 1957 acerca desta matéria, mas ao longo de todos os seus escritos filosóficos. Neste sentido, o artigo pretende sobretudo demonstrar de que modo a teoria cognitiva de Lonergan pode projectar luz sobre diversos problemas candentes em filosofia da lógica, problemas esses que são de grande importância para os filósofos analíticos. Esta discussão, que aqui se apresenta sem recorrer a procedimentos técnicos, examina os debates contemporâneos acerca dos princípios do terceiro excluído e de não-contradição, chamando especialmente a atenção para o contributo de Lonergan para a análise filosófica quer da importância quer dos limites do simbolismo na argumentação filosófica. (shrink)

In this book the author gives a broad overview of different areas of research in nonmonotonic reasoning, and presents some new results and ideas based on his research. The guiding principles are: clarification of the different research activities in the area, which have sometimes been undertaken independently of each other; and appreciation of the fact that these research activities often represent different means to the same ends, namely sound theoretical foundations and efficient computation. The book begins with a discussion (...) of the various types of nonmonotonic reasoning, their applications and their logics. Theorem proving techniques for these logics are also described. The following chapters deal with formulations of nonmonotonic inheritance, and nonmonotonic reasoning based on nonmonotonic rules. The final chapter discusses the achievements in the field in the light of the Yale shooting example. The book will be welcomed by researchers in theoretical computer science and artificial intelligence. (shrink)

This paper offers a critical analysis of two central issues in Luigi Ferrajoli’s Principia iuris , and more generally of his theory of rights. One is the way in which ‘expectations’ play a crucial role in his deontic theory by establishing the logical basis for his guarantee-based conception of law and rights. The axiomatic way in which Ferrajoli arrives at his conception of fundamental rights is questioned, for it fails to give a full account of the nature of subjective (...) rights. The other issues discussed here is Ferrajoli’s own defence of the universality of fundamental rights, and how this is made to depend on a ‘normative technique’ that one can associate with legal discourse and modern law in Western societies. This, however, poses the problem of how this formal universalism can be reconciled with the intercultural dialogue that Ferrajoli also advocates. (shrink)

We study the foundation of space-time theory in the framework of first-order logic (FOL). Since the foundation of mathematics has been successfully carried through (via set theory) in FOL, it is not entirely impossible to do the same for space-time theory (or relativity). First we recall a simple and streamlined FOL-axiomatization Specrel of special relativity from the literature. Specrel is complete with respect to questions about inertial motion. Then we ask ourselves whether we can prove the usual relativistic properties of (...) accelerated motion (e.g., clocks in acceleration) in Specrel. As it turns out, this is practically equivalent to asking whether Specrel is strong enough to “handle” (or treat) accelerated observers. We show that there is a mathematical principle called induction (IND) coming from real analysis which needs to be added to Specrel in order to handle situations involving relativistic acceleration. We present an extended version AccRel of Specrel which is strong enough to handle accelerated motion, in particular, accelerated observers. Among others, we show that~the Twin Paradox becomes provable in AccRel, but it is not provable without IND. (shrink)

This paper proposes a logic, motivated by modal interpretations, in which every quantum mechanics propositions has a truth-value. This logic is completely classical, hence violates the conditions of the Kochen-Specker theorem. It is shown how the violation occurs, and it is argued that this violation is a natural and acceptable consequence of modal interpretations. It is shown that despite its classicality, the proposed logic is empirically indistinguishable from quantum logic.

As the subtitle suggests, the book is organized around the themes of judgment, inference and truth. Material for the first two topics is largely taken from the second edition of Bradley's Principles of Logic. The discussion of his conception of truth relies on essays written in reply to various authors. In general, the book is to be welcomed by students of Bradley for its remarkably clear and unpretentious exposition of central themes in these difficult topics.Much of the book is taken (...) up with discussion of the views of Bradley's precursors and contemporaries in British and German idealism, material that is important for understanding Bradley's own views. Hegel scholars may cringe, however, at the oft-repeated claim, stated without qualification, that Hegel regarded thought and reality as identical. Allard argues that Hegel must maintain this because his view of deductive inference required that objects be self-determining, and self-determining objects require the identity of thought and being. The argument is particularly puzzling in the light of the fact that Bradley's view of inference also requires the self-development of its objects, although he denied that thought and reality are identical. (shrink)

This paper proposes a logic, motivated by modal interpretations, in which every quantum mechanics propositions has a truth-value. This logic is completely classical, hence violates the conditions of the Kochen-Specker theorem. It is shown how the violation occurs, and it is argued that this violation is a natural and acceptable consequence of modal interpretations. It is shown that despite its classicality, the proposed logic is empirically indistinguishable from quantum logic.

This book is a study in the logic of questions (sometimes called erotetic logic). The central topics in erotetic logic have been the structure of questions and the question-answer relationship. This book doesn't neglect these problems, but much of it is focussed on other issues. The main subject is the logical analysis of certain relations between questions and the contexts of their appearance. And our aim is to elaborate the conceptual apparatus of the inferential approach to the logic of (...) questions. Questions are asked for many reasons and for different purposes. Yet, before a question is asked or posed, a questioner must arrive at it. In many cases arriving at a question resembles coming to a conclusion: there are some premises involved and some inferential thought processes take place. If we agree that a conclusion need not be "conclusive", we may say that sometimes questions can play the role of conclusions. But questions can also perform the role of premises: we often pass from some "initial" question to another question. In other words, there are inferential thought processes - we shall call them erotetic inferences - in which questions play the roles of conclusions or conclusions and premises. The inferential approach to the logic of questions focusses its attention on the analysis of erotetic inferences. This book consists of eight chapters. (shrink)