. In this paper, the significance of using general logic-systems and finite consequence operators defined on non-organized languages is discussed. Results are established that show how properties of finite consequence operators are independent from language organization and that, in some cases, they depend only upon one simple language characteristic. For example, it is shown that there are infinitely many finite consequence operators defined on any non-organized infinite language L that cannot be generated from any finite logic-system. On the other (...) hand, it is shown that for any nonempty language L, a set map is a finite consequence operator if and only if it is defined by a general logic-system. Simple logic-system examples that determine specific consequence operator properties are given. (shrink)

This book addresses the hasty development of modern logic, especially its introducing and embracing various kinds of artificial languages and moving from the study of natural languages to that of artificial ones. This shift seemed extremely helpful and managed to elevate logic to a new level of rigor and clarity. However, the change that logic underwent in this way was in no way insignificant, and it is also far from an insignificant matter to determine to what extent the "new logic" (...) only engaged new and more powerful instruments to answer the questions posed by the "old" one, and to what extent it replaced these questions with new ones. Hence, this movement has generated brand new kinds of philosophical problems that have still not been dealt with systematically. Philosophy of Logical Systems addresses these new kinds of philosophical problems that are intertwined with the development of modern logic. Jaroslav Peregrin analyzes the rationale behind the introduction of the artificial languages of logic; classifies the various tools which were adopted to build such languages; gives an overview of the various kinds of languages introduced in the course of modern logic and the motifs of their employment; discusses what can actually be achieved by relocating the problems of logic from natural language into them; and reaches certain conclusions with respect to the possibilities and limitations of this "formal turn" of logic. This book is both an important scholarly contribution to the philosophy of logic and a systematic survey of the standard logical systems that were established during the short history of modern logic. (shrink)

Josiah Royce (1855?1916) was, in addition to being the pre-eminent metaphysician at the turn of the 19th century in the USA, regarded as ?a logician of the first rank?. At the time of his death in 1916, he had begun a substantial and potentially revolutionary project in logic in which he sought to show the connection between logic and ethics, aesthetics, and metaphysics. His system was developed in light of the work of Bertrand Russell and A. B. Kempe and aimed (...) to include Russell's logic as a subsystem. This more comprehensive logic, which Royce called system Σ, brought together an analysis of the character of experience and knowledge with the principles which served to organize interests and direct actions. The result was to be a logical theory framed by ?a profoundly ethical motive as well as a genuinely intellectual one? that would serve as ?a theory of the way in which activities must go on if they go on at all?. This paper examines three stages in the development of Royce's logical system, its relation to the logic of Principia Mathematica, and prospects for its further development. (shrink)

This superb collection of papers focuses on a fundamental question in logic and computation: What is a logical system? With contributions from leading researchers--including Ian Hacking, Robert Kowalski, Jim Lambek, Neil Tennent, Arnon Avron, L. Farinas del Cerro, Kosta Dosen, and Solomon Feferman--the book presents a wide range of views on how to answer such a question, reflecting current, mainstream approaches to logic and its applications. Written to appeal to a diverse audience of readers, What is a Logical (...) System? will excite discussion among students, teachers, and researchers in mathematics, logic, computer science, philosophy, and linguistics. (shrink)

This paper aims at clarifying the nature of Frege's system of logic, as presented in the first volume of the Grundgesetze. We undertake a rational reconstruction of this system, by distinguishing its propositional and predicate fragments. This allows us to emphasise the differences and similarities between this system and a modern system of classical second-order logic.

This paper investigates modular combinations of temporal logic systems. Four combination methods are described and studied with respect to the transfer of logical properties from the component one-dimensional temporal logics to the resulting combined two-dimensional temporal logic. Three basic logical properties are analyzed, namely soundness, completeness, and decidability. Each combination method comprises three submethods that combine the languages, the inference systems, and the semantics of two one-dimensional temporal logic systems, generating families of two-dimensional temporal languages (...) with varying expressivity and varying degrees of transfer of logical properties. The temporalization method and the independent combination method are shown to transfer all three basic logical properties. The method of full join of logic systems generates a considerably more expressive language but fails to transfer completeness and decidability in several cases. So a weaker method of restricted join is proposed and shown to transfer all three basic logical properties. (shrink)

Doubts concerning the validity of logic are as old as the empirical criticism of science. In the last two decades the idea that truth is relative to given sets of basic assumptions has been prominent; and the controversy about the principle of excluded middle has focussed renewed attention upon the nature of logic and its fundamental principles.Recent investigations in formal logic have contributed greatly to the understanding of the principles of logic. It is simply a misunderstanding to conclude from them (...) that the logical principles have been suspended, despite the various formal structures which are called “logics.” The purely formal treatment of problems concerning the foundations of logic, while necessarily undertaken as far as it can go, has essential limitations. (shrink)

Logical works of this period, beginning with Generales Inquisitiones and ending wi th the two dated pieces of 1 Aug. 1690 and 2 Aug. 1690 , are read as a sustained effort, finally successful, to develop a set of axioms and an appropriate schema for the expression of categorical propositions faithful to traditional syllogistic. This same set of axioms is shown to be comprehensive of the propositional calculus of Principia Mathematica, providing that ‘Some A is A’ is not a (...) thesis in an unrestricted sense. There is no indication in the works of this period that Leibniz understood just how significant is this logicalsystem he developed. But it is undeniable that he held tenaciously to this particular set of axioms throughout the period, a set of axioms of great power. (shrink)

The main objective o f this descriptive paper is to present the general notion of translation between logical systems as studied by the GTAL research group, as well as its main results, questions, problems and indagations. Logical systems here are defined in the most general sense, as sets endowed with consequence relations; translations between logical systems are characterized as maps which preserve consequence relations (that is, as continuous functions between those sets). In this sense, (...) logics together with translations form a bicomplete category of which topological spaces with topological continuous functions constitute a full subcategory. We also describe other uses of translations in providing new semantics for non-classical logics and in investigating duality between them. An important subclass of translations, the conservative translations, which strongly preserve consequence relations, is introduced and studied. Some specific new examples of translations involving modal logics, many-valued logics, para- consistent logics, intuitionistic and classical logics are also described. (shrink)

This book is focused on the first three parts of Bolzano's Theory of Science and introduces a more systematic reconsideration of Bolzano's logical thought. In undertaking this task, the book is intended as an exploration, not so much of the more specifically discursive aspects of Bolzano's logical thought - already amply studied - as much as on identifying the singularly coherent and systematic nature of the logic presented in Bolzano's work.

We introduce a methodology whereby an arbitrary logic system L can be enriched with temporal features to create a new system T(L). The new system is constructed by combining L with a pure propositional temporal logic T (such as linear temporal logic with Since and Until) in a special way. We refer to this method as adding a temporal dimension to L or just temporalising L. We show that the logic system T(L) preserves several properties of the original temporal logic (...) like soundness, completeness, decidability, conservativeness and separation over linear flows of time. We then focus on the temporalisation of first-order logic, and a comparison is make with other first-order approaches to the handling of time. (shrink)

According to Suszko’s Thesis, there are but two logical values, true and false. In this paper, R. Suszko’s, G. Malinowski’s, and M. Tsuji’s analyses of logical twovaluedness are critically discussed. Another analysis is presented, which favors a notion of a logical system as encompassing possibly more than one consequence relation. [A] fundamental problem concerning many-valuedness is to know what it really is. [13, p. 281].

이 글은 묵자의 철학 가운데 소통문제에 대한 연구이다. 먼저 묵자의 소통을 위한 논증적 이론체계의 구축과정을 살펴보고, 그 논증을 통하여 도출된 이론들을 실천에 옮기는 소통에 집중하고자 하였다. 묵자는 천의지에 의한 의를 소통적 본질로 하여 의존 관계인 사와 실을 소통 방법의 기능으로 삼아 백성들의 삶을 안정시키는 것에 중점을 둔다. 그리고 소통을 형성하는 관계적 언어를 다스림인 치에 두면서, 그 다스림을 내외적 다스림의 논증적 사유체계로 확장시켜 나간다. 이것은 인간이 사물과 접촉함으로써 발현되는 앎(知)으로 나타나 실용적 가치인 이(利)를 창출하는 것이다. 묵자의 소통을 위한 논리는 실리주의를 기초로 (...) 하여 경험적이며, 결과적 사유방식을 체계적으로 이론화 했다. 소통의 핵심인 의는 백성들의 기본적 욕구를 저해하는 삼환을 해결하여 교상리로 귀착 시킨다. 특히 묵자는 인식론을 실의 물드림으로 비유하고, 인간의 이기적 사유를 소통의 이론체계(義-事-實-治-利-染)로 물들게 하여 안생생의 사회, 즉 겸애와 교상리의 이론으로 전쟁을 방비하여 백성들의 안정된 삶과 평화를 펼치는 것을 목적으로 한다. (shrink)

This thorough investigation of the implications, both logical and philosophical, of Lesniewski's systems should do much to arouse interest in this neglected Polish philosopher. The author takes great pains to relate Lesniewski's ideas to those prevailing today in "orthodox" circles of logicians. There is an excellent bibliography.—A. E. F.

Bernard Bolzano was born in Prague in 1781 and died in 1848 in the same city1. His major work Wissenschaftslehre was published in 1837 in Sulzbach, Germany.Since the publication of Jan Berg’s Bolza...

In order to modelize the reasoning of intelligent agents represented by a poset T, H. Rasiowa introduced logic systems called “Approximation Logics”. In these systems a set of constants constitutes a fundamental tool. In this paper, we consider logic systems called L'T without this kind of constants but limited to the case where T is a finite poset. We study the tableau method for this system and we prove its completeness for a class of formulas with respect (...) to an algebraic semantics. (shrink)

Stanislaw Lesniewski’s interests were, for the most part, more philosophical than mathematical. Prior to taking his doctorate at Jan Kazimierz University in Lvov, Lesniewski had spent time at several continental universities, apparently becoming relatively attached to the philosophy of one of his teachers, Hans Comelius, to the chapters of John Stuart Mill’s System of Logic that dealt specifically with semantics, and, in general, to studies of general grammar and philosophy of language. In these several early interests are already to be (...) found the roots of the work that was to occupy Lesniewski’s life: a search for a definitive doctrine of what sorts of things there are in the world, or better, of what language must be like if it is adequately and efficiently to represent the world. (shrink)

In this paper we define and study a generalized notion of a logical system that covers on an equal formal basis sentential, equational and sequential systems. We develop a general theory of equivalence between generalized logics that provides, first, a conception of algebraizable logic , second, a formal concept of equivalence between sequential systems and, third, a notion of equivalence between sentential and sequential systems. We also use our theory of equivalence for developing a general algebraic (...) approach to conjunctive non-pseudo-axiomatic self-extensional sentential logics. Finally, we consider within the framework of the mentioned approach various sequential formulations for some well-known sentential logics. (shrink)

As part of our small contribution in dialogue toward better peace development and reconciliation studies, and following Toffler & Toffler’s War and Antiwar (1993), the present article delves into a realm of logic beyond the traditional confines of negation and the excluded middle principle, exploring the nuances of "Otherness" that transcend classical and Nagatomo logics. Departing from the foundational premises of classical Aristotelian logic systems, this exploration ventures into alternative realms of reasoning, specifically examining Neutrosophic Logic and Klein bottle (...) logic (cf. Smarandache, 2005). The study challenges conventional boundaries and explores the implications of embracing paradoxes and self-reference in logic systems, aiming to redefine approaches to understanding truth and reasoning. The paper investigates how these alternative logics open avenues for philosophical inquiry, redefining entropy, and potentially influencing innovative perspectives in free energy systems. Through this exploration, it seeks to expand the discourse on logic, welcoming a broader spectrum of thought beyond established frameworks; and we also discuss shortly a number of possible implementations including in risk management and also Klein bottle entropy redefinition (Tang et al, 2018). (shrink)

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)

For three hundred years two conflicting views of the world (1) have provided the overall frameworks for thought in western culture. The present paper shows neither view is sufficient for human understanding of many important systems and behaviors. A third view which appears sufficient is presented. Illustrations of the third view show increased understanding is obtained in many problems. The sufficiency of the historic views and the route to the third view are provided through discussion of the issue of (...) multi-disciplinarity, the question of whether it is possible to base everything we know on one discipline, or, on the contrary, if there is a logical necessity for using principles and concepts from many disciplines to achieve human understanding of the world. The present article provides three distinct proofs of the logical necessity of multi-disciplinarity. The proofs proceed via study of: (1) the hierarchical structure of proto-typical systems in various areas of human concern; (2) the use of “integrated-control-information” by life forms and human artifacts; and (3) an extension of the theory of dimensions. The three proofs interlock, confirm, and extend each other. (shrink)

Say that a property is topological if and only if it is invariant under homeomorphism. Homeomorphism would be a successful criterion for the equivalence of logical systems only if every logically significant property of every logical system were topological. Alas, homeomorphisms are sometimes insensitive to distinctions that logicians value: properties such as functional completeness are not topological. So logics are not just devices for exploring closure topologies. One still wonders, though, how much of logic is topological. This (...) essay examines some logically significant properties that are topological (or are topological in some important class). In the process, we learn something about the conditions under which the meaning of a connective can be "given by the connective's role in inference.". (shrink)

This paper offers a two dimensional variation of Standard Deontic Logic SDL, which we call 2SDL. Using 2SDL we can show that we can overcome many of the difficulties that SDL has in representing linguistic sets of Contrary-to-Duties (known as paradoxes) including the Chisholm, Ross, Good Samaritan and Forrester paradoxes. We note that many dimensional logics have been around since 1947, and so 2SDL could have been presented already in the 1970s. Better late than never! As a detailed case study (...) illustrating the power of 2SDL, we examine the system DL of Deontic Logic of Andrew Jones and Ingmar Pörn offered in 1985 to solve the Chisholm paradox of Contrary to Duties. The critical examination is done using logics and methods available in 1985 and solutions are proposed using what was available in 1985. (shrink)

Many philosophers have attempted to offer a logical theory of objects, employing different techniques. Thus R. Carnap tried to “reconstruct” logically the world by using the modern symbolic logic, while N. Goodman “constructed” the world with the help of the calculus of individuals or the logic of part-whole relations. W.V. Quine helped to steer the attention toward the question of ontological commitment and toward a theory of objects produced by a logical analysis of natural languages. Recently, there have (...) been controversial discussions concerning the metaphysical and logical aspects of objects in actual and possible worlds as well as new contributions to the theory of nonexistent objects, rekindling the old Russell-Meinong controversy. It seems that even the proposed theories of events, such as that of R. M. Martin, maintain the great significance of the notion of an object. Then there is a plethora of the so-called intensional objects, such as the alleged objects of propositional attitudes, the story of which is tied with the names of Frege, Russell, Carnap and many others. Is it possible to give an all-embracing theory of objects which would satisfy all the parties involved? Although such possibility may be very questionable, the instrumental role of logic for the development of the metaphysical theories of objects is indisputable. Two factors are, however, involved in the constitution of a logical theory of objects: the basis or the given data from which the theory starts; the logical apparatus employed, i.e., the logical axioms and rules of inference, just as the relevant semantical rules which are formulated in the metalanguage. While the logical apparatus is usually very clear, at least in the case of the formally well-construed theories, the given basis of a metaphysical theory T—as we may call it—might contain many tacit assumptions which would need a lot of clarification. It seems that philosophers fight more about the presuppositions and assertions of their systems than about the logic involved. I would like to discuss the interplay between the above two factors, starting with the realm of the given objects. As to the second factor—the logical apparatus involved—the contributions of first-order predicate logic and of the calculus of individuals will be considered. (shrink)

The aim of this book is to present the fundamental theoretical results concerning inference rules in deductive formal systems. Primary attention is focused on: admissible or permissible inference rules the derivability of the admissible inference rules the structural completeness of logics the bases for admissible and valid inference rules. There is particular emphasis on propositional non-standard logics (primary, superintuitionistic and modal logics) but general logical consequence relations and classical first-order theories are also considered. The book is basically self-contained (...) and special attention has been made to present the material in a convenient manner for the reader. Proofs of results, many of which are not readily available elsewhere, are also included. The book is written at a level appropriate for first-year graduate students in mathematics or computer science. Although some knowledge of elementary logic and universal algebra are necessary, the first chapter includes all the results from universal algebra and logic that the reader needs. For graduate students in mathematics and computer science the book is an excellent textbook. (shrink)

Logical pluralism is a general idea that there is more than one correct logic. Carnielli and Rodrigues [2019a] defend an epistemic interpretation of the paraconsistent logic N4, according to which an argument is valid in this logic just in case it necessarily preserves evidence. The authors appeal to this epistemic interpretation to briefly motivate a kind of logical pluralism: “different accounts of logical consequence may preserve different properties of propositions”. The aim of this paper is to study (...) the prospect of a logical pluralism based on different interpretations of logical systems. First, we give our analysis of what it means to interpret a logic – and make some hopefully useful distinctions along the way. Second, we present what we call an interpretational logical pluralism: there is more than one correct logic and a logic is correct only if it has some adequate interpretation. We consider four variants of this idea, bring up some possible objections, and try to find plausible solutions on behalf of the pluralist. We will argue that interpretations of logical systems provide a promising – albeit not unproblematic – route to logical pluralism. (shrink)

Section 1 contains a survey of options in constructing a formal system of dialogue rules. The distinction between material and formal systems is discussed (section 1.1). It is stressed that the material systems are, in several senses, formal as well. In section 1.2 variants as to language form (choices of logical constants and logical rules) are pointed out. Section 1.3 is concerned with options as to initial positions and the permissibility of attacks on elementary statements. The (...) problem of ending a dialogue, and of infinite dialogues, is treated in section 1.4. Other options, e.g., as to the number of attacks allowed with respect to each statement, are listed in section 1.5. Section 1.6 explains the concept of a chain of arguments.From section 2 onward four types of dialectic systems are picked out for closer study: D, E, Di and Ei. After a preliminary section on dialogue sequents and winning strategies, the equivalence of derivability in intuitionistic logic and the existence of a winning strategy (for the Proponent) on the strength of Ei is shown by simple inductive proofs. (shrink)

Lots of things are usefully modelled in science as dynamical systems: growing populations, flocking birds, engineering apparatus, cognitive agents, distant galaxies, Turing machines, neural networks. We argue that relevant logic is ideal for reasoning about dynamical systems, including interactions with the system through perturbations. Thus, dynamical systems provide a new applied interpretation of the abstract Routley-Meyer semantics for relevant logic: the worlds in the model are the states of the system, while the (in)famous ternary relation is a (...) combination of perturbation and evolution in the system. Conversely, the logic of the relevant conditional provides sound and complete laws of dynamical systems. (shrink)