The first beginnings of modern logic in Croatia are recognizable as early as in the middle of the 19th century in Vatroslav Bertić. At the turn of the 20th century, Albin Nagy, who was teaching in Italy, made contributions to algebraic logic and to the philosophy of logic. At that time, a distinctive author Mate Meršić stood out, also working on algebraic logic. In the Croatian academic philosophy, until the publication of Gajo Petrović's textbook (1964) (...) and the contributions by Heda Festini, a critical or indifferent attitude toward modern logic prevailed. The first comprehensive modern logic textbook in Croatian was written by mathematician Vladimir Devidé (1964), known, for example, for his axiomatizations of natural numbers and classical propositional logic. In the Croatian philosophy of the 1980s and 1990s, influential logicians are Goran Švob and mathematician Zvonimir Šikić. At the turn of the 21st century, logical investigations are being intensified and extended to various branches of the contemporary logic. (shrink)

This book treats ancient logic: the logic that originated in Greece by Aristotle and the Stoics, mainly in the hundred year period beginning about 350 BCE. Ancient logic was never completely ignored by modern logic from its Boolean origin in the middle 1800s: it was prominent in Boole’s writings and it was mentioned by Frege and by Hilbert. Nevertheless, the first century of mathematical logic did not take it seriously enough to study the ancient (...) logic texts. A renaissance in ancient logic studies occurred in the early 1950s with the publication of the landmark Aristotle’s Syllogistic by Jan Łukasiewicz, Oxford UP 1951, 2nd ed. 1957. Despite its title, it treats the logic of the Stoics as well as that of Aristotle. Łukasiewicz was a distinguished mathematical logician. He had created many-valued logic and the parenthesis-free prefix notation known as Polish notation. He co-authored with Alfred Tarski’s an important paper on metatheory of propositional logic and he was one of Tarski’s the three main teachers at the University of Warsaw. Łukasiewicz’s stature was just short of that of the giants: Aristotle, Boole, Frege, Tarski and Gödel. No mathematical logician of his caliber had ever before quoted the actual teachings of ancient logicians. -/- Not only did Łukasiewicz inject fresh hypotheses, new concepts, and imaginative modern perspectives into the field, his enormous prestige and that of the Warsaw School of Logic reflected on the whole field of ancient logic studies. Suddenly, this previously somewhat dormant and obscure field became active and gained in respectability and importance in the eyes of logicians, mathematicians, linguists, analytic philosophers, and historians. Next to Aristotle himself and perhaps the Stoic logician Chrysippus, Łukasiewicz is the most prominent figure in ancient logic studies. A huge literature traces its origins to Łukasiewicz. -/- This Ancient Logic and Its Modern Interpretations, is based on the 1973 Buffalo Symposium on Modernist Interpretations of Ancient Logic, the first conference devoted entirely to critical assessment of the state of ancient logic studies. (shrink)

Logical reflection in early modern philosophy (EMP) is marked by the instability of the period, although it is more lasting (the Port-Royal Logic was nevertheless used as a handbook in philosophy courses until the end of the nineteenth century). It started in the sixteenth century and ended in the nineteenth century, a period of 300 years during which there were deep transformations in the conceptions of authority and scientific method. For the history of twentieth-century philosophy, it was the (...) period of “classical logic,” which lasted from the Renaissance to the linguistic turn conducted by Gottlob Frege. The period was used to be thought of as centuries of little or no original contribution to logic, in which conceptions of logic were tainted by rhetoric, epistemology, and psychologism in the worst sense (Kneale and Kneale 1962; Michael 1997). From the last decades of the twentieth century, however, scholars began to regard this period more accurately with respect to reflection and changes in logic and semantics. It has recently become a promising field for historical and conceptual research; today we can say that the legacy of early modern logical reformism has a philosophical, logical, and semantic value in itself. (shrink)

This book is written for those who wish to learn some basic principles of formal logic but more importantly learn some easy methods to unpick arguments and assess their value for truth and validity. -/- The first section explains the ideas behind traditional logic which was formed well over two thousand years ago by the ancient Greeks. Terms such as ‘categorical syllogism’, ‘premise’, ‘deduction’ and ‘validity’ may appear at first sight to be inscrutable but will easily be understood (...) with examples bringing the subjects to life. Traditionally, Venn diagrams have been employed to test arguments. These are very useful but their application is limited and they are not open to quantification. The mid-section of this book introduces a methodology that makes the analysis of arguments accessible with the use of a new form of diagram, modified from those of the mathematician Leonhard Euler. These new diagrammatic methods will be employed to demonstrate an addition to the basic form of syllogism. This includes a refined definition of the terms ‘most’ and ‘some’ within propositions. This may seem a little obscure at the moment but one will readily apprehend these new methods and principles of a more modern logic. (shrink)

This paper intends to explain key differences between Aristotle’s understanding of the relationships between nous, epistêmê, and the art of syllogistic reasoning(both analytic and dialectical) and the corresponding modern conceptions of intuition, knowledge, and reason. By uncovering paradoxa that Aristotle’s understanding of syllogistic reasoning presents in relation to modern philosophical conceptions of logic and science, I highlight problems of a shift in modern philosophy—a shift that occurs most dramatically in the seventeenth century—toward a project of construction, (...) a pervasive desire for rational certainty, and a general insistence on the reducibility of the sciences. The major motivation of this analysis is my intention to show that modern attempts to reduce science/epistêmê to a single science/method of inquiry occlude dialectical and ethico-political dimensions of “reason” and, hence, also impoverish philosophy’s critical capacities. (shrink)

A discussion of the logical role of particular concepts in Robert Pippin's reading Hegel as a theorist of modernity, with special reference to the question whether modernity can be surpassed or left behind.

This article had its start with another article, concerned with measuring the speed of gravitational waves - "The Measurement of the Light Deflection from Jupiter: Experimental Results" by Ed Fomalont and Sergei Kopeikin (2003) - The Astrophysical Journal 598 (1): 704–711. This starting-point led to many other topics that required explanation or naturally seemed to follow on – Unification of gravity with electromagnetism and the 2 nuclear forces, Speed of electromagnetic waves, Energy of cosmic rays and UHECRs, Digital string theory, (...) Dark energy+gravity+binary digits, Cosmic strings and wormholes from Figure-8 Klein bottles, Massless and massive photons and gravitons, Inverse square+quantum entanglement = God+evolution, Binary digits projected to make Prof. Greene’s cosmic holographic movie, Renormalization of infinity, Colliding subuniverses, Unifying cosmic inflation, TOE (emphasizing “EVERYthing”) = Bose-Einstein renormalized. The text also addresses (in a nonmathematical way) the wavelength of electromagnetic waves, the frequency of gravitational waves, gravitational and electromagnetic waves having identical speed, the gamma-ray burst designated GRB 090510, the smoothness of space, and includes these words – “Gravity produces electromagnetism. Retrocausally (by means of humans travelling into the past of this subuniverse with their electronics); this “Cosmic EM Background” produces base-2 mathematics, which produces gravity. EM interacts with gravity to produce particles, mass – gravity/EM could be termed “the Higgs field” - and the nuclear forces associated with those particles. It makes gravity using BITS that copy the principle of magnetism attracting and repelling, before pasting it into what we call the strong force and dark energy.” . (shrink)

Logic has pride of place in mathematics and its 20th century offshoot, computer science. Modern symbolic logic was developed, in part, as a way to provide a formal framework for mathematics: Frege, Peano, Whitehead and Russell, as well as Hilbert developed systems of logic to formalize mathematics. These systems were meant to serve either as themselves foundational, or at least as formal analogs of mathematical reasoning amenable to mathematical study, e.g., in Hilbert’s consistency program. Similar efforts (...) continue, but have been expanded by the development of sophisticated methods to study the properties of such systems using proof and model theory. In parallel with this evolution of logical formalisms as tools for articulating mathematical theories (broadly speaking), much progress has been made in the quest for a mechanization of logical inference and the investigation of its theoretical limits, culminating recently in the development of new foundational frameworks for mathematics with sophisticated computer-assisted proof systems. In addition, logical formalisms developed by logicians in mathematical and philosophical contexts have proved immensely useful in describing theories and systems of interest to computer scientists, and to some degree, vice versa. Three examples of the influence of logic in computer science are automated reasoning, computer verification, and type systems for programming languages. (shrink)

This paper sets out to evaluate the claim that Aristotle’s Assertoric Syllogistic is a relevance logic or shows significant similarities with it. I prepare the grounds for a meaningful comparison by extracting the notion of relevance employed in the most influential work on modern relevance logic, Anderson and Belnap’s Entailment. This notion is characterized by two conditions imposed on the concept of validity: first, that some meaning content is shared between the premises and the conclusion, and second, (...) that the premises of a proof are actually used to derive the conclusion. Turning to Aristotle’s Prior Analytics, I argue that there is evidence that Aristotle’s Assertoric Syllogistic satisfies both conditions. Moreover, Aristotle at one point explicitly addresses the potential harmfulness of syllogisms with unused premises. Here, I argue that Aristotle’s analysis allows for a rejection of such syllogisms on formal grounds established in the foregoing parts of the Prior Analytics. In a final section I consider the view that Aristotle distinguished between validity on the one hand and syllogistic validity on the other. Following this line of reasoning, Aristotle’s logic might not be a relevance logic, since relevance is part of syllogistic validity and not, as modern relevance logic demands, of general validity. I argue that the reasons to reject this view are more compelling than the reasons to accept it and that we can, cautiously, uphold the result that Aristotle’s logic is a relevance logic. (shrink)

In this paper we present an overview, with historical and critical remarks, of two articles by S. Jaśkowski ([20, 21] 1948 and [22, 23] 1949), which contain the oldest known formulation of a paraconsistent logic. Jaśkowski has built the logic – he termed discussive (D2) – by defining two new connectives and by introducing a modal translation map from D2 systems into Lewis’ modal logic S5. Discussive systems, for their formal details and their original philosophical justification, have (...) attracted discrete attention among experts. Indeed, in what follows, after having introduced Jaśkowski methodology of building D2 and his main philosophical motivations for providing such a system, we will explore some of the main contributions to the development of D2. (shrink)

The present study defines the function of the foundation of science in early modern Dutch philosophy, from the first introduction of Cartesian philosophy in Utrecht University by Henricus Regius to the acceptance of Newtonian physics by Willem Jacob ‘s Gravesande. My main claim is that a foundation of science was required because the conceptual premises of new ways in thinking had to be justified not only as alternatives to the established philosophical paradigms or as an answer to the “sceptical (...) crisis” signalled by Richard Popkin, but also in order to be introduced in the academia. This being my general thesis, different kinds of foundation of scientific knowledge are analysed in the light of the specific problems philosophers had to face. To this end, foundational theories are analysed in the light of 1) the overall aim of their philosophy, 2) the relations of disciplines they thought out, 3) what they regarded as scientific knowledge. Through this analysis, I assess that the foundation of science was carried out by metaphysical, logical and theological arguments, which fitted the needs of answering the didactical and introductory requirements of new paradigms, assessed the reliability of new methodology in leading us to grasp the truth, and provided philosophy with its conceptual apparatus. Finally, I demonstrate that a philosophy of science as a reflection on the principles, methods, concepts and application of modern science emerged in the Dutch Republic. (shrink)

The criticism of Immanuel Kant’s logic commenced with the advent of the so-called ‘new logic’ in the 20th century. One particular passage from the second preface to the Critique of Pure Reason has been a source of contention, where Kant asserted that logic has not taken a step forward or backward since Aristotle (B VIII). In Kant scholarship, one current strategy to avoid this criticism is to relocate Kant within the domain of philosophy of logic or (...) by segregating his general logic from modern formal systems. In this paper, it will be contended that this strategy is too weak, given that the B-preface has currently been analyzed in a markedly divergent manner by the so-called ‘methodological interpretation’. In his methodology and history of science of the B-preface, Kant means something different by progress and regression than what his 20th-century critics assume he meant. By examining what Kant and his critics considered to be progress and regress in science and logic, good reasons can be put forward for the argument that Kant was correct in his assertion that logic has not gone a step forward or backward. (shrink)

We argue that the extant evidence for Stoic logic provides all the elements required for a variable-free theory of multiple generality, including a number of remarkably modern features that straddle logic and semantics, such as the understanding of one- and two-place predicates as functions, the canonical formulation of universals as quantified conditionals, a straightforward relation between elements of propositional and first-order logic, and the roles of anaphora and rigid order in the regimented sentences that express multiply (...) general propositions. We consider and reinterpret some ancient texts that have been neglected in the context of Stoic universal and existential propositions and offer new explanations of some puzzling features in Stoic logic. Our results confirm that Stoic logic surpasses Aristotle’s with regard to multiple generality, and are a reminder that focusing on multiple generality through the lens of Frege-inspired variable-binding quantifier theory may hamper our understanding and appreciation of pre-Fregean theories of multiple generality. (shrink)

Broadly speaking, two views of modernity are prevalent in contemporary debates. According to the first view, i.e. “modernization theory,” there is one single form of modernity, which is tantamount to liberal, capitalist modernity. The West has already and fully achieved modernity; non-Western societies have lagged behind and must simply catch up with the West. In contrast, according to the second view, “post-colonial theory,” there is no such thing as modernity. What the West erroneously calls “modernity” is nothing but a highly (...) parochial way of thought, and of economic and political organization, that developed by accident first in the West and then by the exercise of military violence or economic power was subsequently “universalized.” In this chapter, I argue that Hegel, read properly, provides a third conception of modernity, one which, contrary to the modernization theory, respects differences between societies, and yet, contrary to post-colonial theory, does not engage in dividing the human species into fundamentally different or incommensurate groups. I argue that Hegel’s theory of modernity must be sought at the conjunction of the Science of Logic (specifically the logic of the Concept) and the Philosophy of Right. I then elaborate on Hegel’s theory of modernity, thus construed. (shrink)

This paper analyses a hitherto unknown technique of using logic diagrams to create argument maps in eristic dialectics. The method was invented in the 1810s and -20s by Arthur Schopenhauer, who is considered the originator of modern eristic. This technique of Schopenhauer could be interesting for several branches of research in the field of argumentation: Firstly, for the field of argument mapping, since here a hitherto unknown diagrammatic technique is shown in order to visualise possible situations of arguments (...) in a dialogical controversy. Secondly, the art of controversy or eristic, since the diagrams do not analyse the truth of judgements and the validity of inferences, but the persuasiveness of arguments in a dialogue. (shrink)

Modern categorical logic as well as the Kripke and topological models of intuitionistic logic suggest that the interpretation of ordinary “propositional” logic should in general be the logic of subsets of a given universe set. Partitions on a set are dual to subsets of a set in the sense of the category-theoretic duality of epimorphisms and monomorphisms—which is reflected in the duality between quotient objects and subobjects throughout algebra. If “propositional” logic is thus seen (...) as the logic of subsets of a universe set, then the question naturally arises of a dual logic of partitions on a universe set. This paper is an introduction to that logic of partitions dual to classical subset logic. The paper goes from basic concepts up through the correctness and completeness theorems for a tableau system of partition logic. (shrink)

Classical logic is usually interpreted as the logic of propositions. But from Boole's original development up to modern categorical logic, there has always been the alternative interpretation of classical logic as the logic of subsets of any given (nonempty) universe set. Partitions on a universe set are dual to subsets of a universe set in the sense of the reverse-the-arrows category-theoretic duality--which is reflected in the duality between quotient objects and subobjects throughout algebra. Hence (...) the idea arises of a dual logic of partitions. That dual logic is described here. Partition logic is at the same mathematical level as subset logic since models for both are constructed from (partitions on or subsets of) arbitrary unstructured sets with no ordering relations, compatibility or accessibility relations, or topologies on the sets. Just as Boole developed logical finite probability theory as a quantitative treatment of subset logic, applying the analogous mathematical steps to partition logic yields a logical notion of entropy so that information theory can be refounded on partition logic. But the biggest application is that when partition logic and the accompanying logical information theory are "lifted" to complex vector spaces, then the mathematical framework of quantum mechanics is obtained. Partition logic models indefiniteness (i.e., numerical attributes on a set become more definite as the inverse-image partition becomes more refined) while subset logic models the definiteness of classical physics (an entity either definitely has a property or definitely does not). Hence partition logic provides the backstory so the old idea of "objective indefiniteness" in QM can be fleshed out to a full interpretation of quantum mechanics. (shrink)

2nd edition. The theory of logical consequence is central in modern logic and its applications. However, it is mostly dispersed in an abundance of often difficultly accessible papers, and rarely treated with applications in mind. This book collects the most fundamental aspects of this theory and offers the reader the basics of its applications in computer science, artificial intelligence, and cognitive science, to name but the most important fields where this notion finds its many applications.

In this article, I outline a logic of design of a system as a specific kind of conceptual logic of the design of the model of a system, that is, the blueprint that provides information about the system to be created. In section two, I introduce the method of levels of abstraction as a modelling tool borrowed from computer science. In section three, I use this method to clarify two main conceptual logics of information inherited from modernity: Kant’s (...) transcendental logic of conditions of possibility of a system, and Hegel’s dialectical logic of conditions of in/stability of a system. Both conceptual logics of information analyse structural properties of given systems. Strictly speaking, neither is a conceptual logic of information about the conditions of feasibility of a system, that is, neither is a logic of information as a logic of design. So, in section four, I outline this third conceptual logic of information and then interpret the conceptual logic of design as a logic of requirements, by introducing the relation of “sufficientisation”. In the conclusion, I argue that the logic of requirements is exactly what we need in order to make sense of, and buttress, a constructionist approach to knowledge. (shrink)

It is my intention in this article to present some consequences of Quine’s thesis on the dependence of ontology on ideology (Quine, 1980), seeking an argument for my own thesis on the dependence (theoretical) existence of entities on identity type or ontology dependence on logic and language.If Quine's thesis is correct, then we can expand the resolution of this conclusion and say that ontology depends on the identity or on identification of the "identity criteria for conceptual schemes" (Davidson, 2001) (...) which is constructed in the theory. Consequently I will speak about types of identity which adapts choice of ontology and of which depends ontology of a theory. Here I want to connect the different types of use of the term identity in Aristotle's writings and the different types of predications that are based on them with the concept of identity as the equivalence of symbols in modern logic. I want to reinterpret Quine's statement: "There is no entity without identity " in the form of imlication "What (kind of) identity such (kind of ) entity." . (shrink)

As a grand narrative of progress, the utopian project of modernity is primarily concerned with notions of rationalism, universalism, and the development of a metalanguage. The triumph of the Moderate Enlightenment has seen logics of domination, accumulation and individualism incorporated into the project of modernity, with these logics giving rise to globalised capitalism as the metalanguage of modernity and neoliberal economics as the grand narrative of rational progress. The project of modernity is all but complete, requiring only the formality of (...) an end. However, rather than utopia, the foreseeable endpoint of modernity is environmental collapse, with neoliberal economics also serving as a grand narrative of environmental destruction. As an anti-modern response, postmodernism has been a triumphant failure. While there is much to be gained from the postmodern critique of modernity, its incredulity towards metanarratives has left it incapable of forming an adequate response to modernity, especially in regards to action on climate change. Postmodernism is better characterized as a crisis located within modernity itself and it will be argued that rather than the pursuit of the modern or the post-modern, we need to re-imagine ourselves as proto- historical to overcome the impasse of late-capitalism. (shrink)

ABSTRACT: An introduction to Stoic logic. Stoic logic can in many respects be regarded as a fore-runner of modern propositional logic. I discuss: 1. the Stoic notion of sayables or meanings (lekta); the Stoic assertibles (axiomata) and their similarities and differences to modern propositions; the time-dependency of their truth; 2.-3. assertibles with demonstratives and quantified assertibles and their truth-conditions; truth-functionality of negations and conjunctions; non-truth-functionality of disjunctions and conditionals; language regimentation and ‘bracketing’ devices; Stoic basic (...) principles of propositional logic; 4. Stoic modal logic; 5. Stoic theory of arguments: two premisses requirement; validity and soundness; 6. Stoic syllogistic or theory of formally valid arguments: a reconstruction of the Stoic deductive system, which consisted of accounts of five types of indemonstrable syllogisms, which function as nullary argumental rules that identify indemonstrables or axioms of the system, and four deductive rules (themata) by which certain complex arguments can be reduced to indemonstrables and thus shown to be formally valid themselves; 7. arguments that were considered as non-syllogistically valid (subsyllogistic and unmethodically concluding arguments). Their validity was explained by recourse to formally valid arguments. (shrink)

The Bounds of Logic presents a new philosophical theory of the scope and nature of logic based on critical analysis of the principles underlying modern Tarskian logic and inspired by mathematical and linguistic development. Extracting central philosophical ideas from Tarski’s early work in semantics, Sher questions whether these are fully realized by the standard first-order system. The answer lays the foundation for a new, broader conception of logic. By generally characterizing logical terms, Sher establishes a (...) fundamental result in semantics. Her development of the notion of logicality for quantifiers and her work on branching are of great importance for linguistics. Sher outlines the boundaries of the new logic and points out some of the philosophical ramifications of the new view of logic for such issues as the logicist thesis, ontological commitment, the role of mathematics in logic, and the metaphysical underpinning of logic. She proposes a constructive definition of logical terms, reexamines and extends the notion of branching quantification, and discusses various linguistic issues and applications. (shrink)

Everyone will find something interesting in this book, and many will find something or other that they completely disagree with. William Demopoulos was no fan o.

In ancient philosophy, there is no discipline called “logic” in the contemporary sense of “the study of formally valid arguments.” Rather, once a subfield of philosophy comes to be called “logic,” namely in Hellenistic philosophy, the field includes (among other things) epistemology, normative epistemology, philosophy of language, the theory of truth, and what we call logic today. This entry aims to examine ancient theorizing that makes contact with the contemporary conception. Thus, we will here emphasize the theories (...) of the “syllogism” in the Aristotelian and Stoic traditions. However, because the context in which these theories were developed and discussed were deeply epistemological in nature, we will also include references to the areas of epistemological theorizing that bear directly on theories of the syllogism, particularly concerning “demonstration.” Similarly, we will include literature that discusses the principles governing logic and the components that make up arguments, which are topics that might now fall under the headings of philosophy of logic or non-classical logic. This includes discussions of problems and paradoxes that connect to contemporary logic and which historically spurred developments of logical method. For example, there is great interest among ancient philosophers in the question of whether all statements have truth-values. Relevant themes here include future contingents, paradoxes of vagueness, and semantic paradoxes like the liar. We also include discussion of the paradoxes of the infinite for similar reasons, since solutions have introduced sophisticated tools of logical analysis and there are a range of related, modern philosophical concerns about the application of some logical principles in infinite domains. Our criterion excludes, however, many of the themes that Hellenistic philosophers consider part of logic, in particular, it excludes epistemology and metaphysical questions about truth. Ancient philosophers do not write treatises “On Logic,” where the topic would be what today counts as logic. Instead, arguments and theories that count as “logic” by our criterion are found in a wide range of texts. For the most part, our entry follows chronology, tracing ancient logic from its beginnings to Late Antiquity. However, some themes are discussed in several eras of ancient logic; ancient logicians engage closely with each other’s views. Accordingly, relevant publications address several authors and periods in conjunction. These contributions are listed in three thematic sections at the end of our entry. (shrink)

Buddhist philosophers have investigated the techniques and methodologies of debate and argumentation which are important aspects of Buddhist intellectual life. This was particularly the case in India, where Buddhism and Buddhist philosophy originated. But these investigations have also engaged philosophers in China, Japan, Korea and Tibet, and many other parts of the world that have been influenced by Buddhism and Buddhist philosophy. Several elements of the Buddhist tradition of philosophy are thought to be part of this investigation. -/- There are (...) interesting reasoning patterns discernible in the writings of Buddhist philosophers. For instance, the Mādhyamika philosopher Nāgārjuna presents arguments for the emptiness of all things in terms of catuṣkoṭi (four ‘corners’: roughly speaking, truth, false, both, neither). There are also the Indian vāda (debate) literature and Tibetan bsdus grwa (collected topics) that list techniques of debates. -/- The main interest here is the tradition of Buddhist philosophy, sometimes referred to as Pramāṇavāda, whose central figures are Dignāga (approx. 480–540 CE) and Dharmakīrti (6th– 7th CE). This philosophy is understood to be the Buddhist school of logic-epistemology. ‘Pramāṇavāda’ is not a doxographical term traditionally used to refer to a recognised school of Buddhist philosophy. It is a conventional term that is sometimes used in modern literature. Nevertheless, in what follows, we think of it as a tradition within Buddhist philosophy and ‘Buddhist logic’ refers to what is developed in this tradition. -/- Buddhist logicians have systematically analysed the kind of reasoning involved in acquiring knowledge. They hold that there are valid ways to reason that are productive of knowledge. In what follows, some of the main elements of their analyses will be described. Note, however, that, while exegetical studies of Buddhist texts are important, we must step back from them and consider what is involved in taking a Buddhist approach to logic in light of modern formal logic. This analysis will be done against the backdrop of the contemporary literature on logic and related subjects in contemporary philosophy to make sense of the logical studies by Buddhist logicians. (shrink)

N. Wiener's negative definition of information is well known: it states what information is not. According to this definition, it is neither matter nor energy. But what is it? It is shown how one can follow the lead of dialectical logic as expounded by G.W.F. Hegel in his main work -- "The Science of Logic" -- to answer this and some related questions.

Does it make sense to employ modern logical tools for ancient philosophy? This well-known debate2 has been re-launched by the indologist Piotr Balcerowicz, questioning those who want to look at the Eastern school of Jainism with Western glasses. While plainly acknowledging the legitimacy of Balcerowicz's mistrust, the present paper wants to propose a formal reconstruction of one of the well-known parts of the Jaina philosophy, namely: the saptabhangi, i.e. the theory of sevenfold predication. Before arguing for this formalist approach (...) to philosophy, let us return to the reasons to be reluctant at it. (shrink)

Analytic philosophy has been perhaps the most successful philosophical movement of the twentieth century. While there is no one doctrine that defines it, one of the most salient features of analytic philosophy is its reliance on contemporary logic, the logic that had its origin in the works of George Boole and Gottlob Frege and others in the mid‐to‐late nineteenth century. Boolean algebra, the heart of Boole's contributions to logic, has also come to represent a cornerstone of (...) class='Hi'>modern computing. Frege had broad philosophical interests, and his writings on the nature of logical form, meaning and truth remain the subject of intense theoretical discussion, especially in the analytic tradition. Frege's works, and the powerful new logical calculi developed at the end of the nineteenth century, influenced many of its most seminal figures, such as Bertrand Russell, Ludwig Wittgenstein and Rudolf Carnap. (shrink)

Logical and Spiritual Reflections is a collection of six shorter philosophical works, including: Hume’s Problems with Induction; A Short Critique of Kant’s Unreason; In Defense of Aristotle’s Laws of Thought; More Meditations; Zen Judaism; No to Sodom. Of these works, the first set of three constitutes the Logical Reflections, and the second set constitutes the Spiritual Reflections. Hume’s Problems with Induction, which is intended to describe and refute some of the main doubts and objections David Hume raised with regard to (...) inductive reasoning. It replaces the so-called problem of induction with a principle of induction. David Hume’s notorious skepticism was based on errors of observation and reasoning, with regard to induction, causation, necessity, the self and freewill. These are here pointed out and critically analyzed in detail – and more accurate and logical theories are proposed. The present work also includes refutations of Hempel’s and Goodman’s alleged paradoxes of induction. A Short Critique of Kant’s Unreason, which is a brief critical analysis of some of the salient epistemological and ontological ideas and theses in Immanuel Kant’s famous Critique of Pure Reason. It shows that Kant was in no position to criticize reason, because he neither sufficiently understood its workings nor had the logical tools needed for the task. Kant’s transcendental reality, his analytic-synthetic dichotomy, his views on experience and concept formation, and on the forms of sensibility (space and time) and understanding (his twelve categories), are here all subjected to rigorous logical evaluation and found deeply flawed – and more coherent theories are proposed in their stead. In Defense of Aristotle’s Laws of Thought, which addresses, from a phenomenological standpoint, numerous modern and Buddhist objections and misconceptions regarding the basic principles of Aristotelian logic. Many people seem to be attacking Aristotle’s Laws of Thought nowadays, some coming from the West and some from the East. It is important to review and refute such ideas as they arise. More Meditations, which is a sequel to the author’s earlier work, Meditations. It proposes additional practical methods and theoretical insights relating to meditation and Buddhism. It also discusses certain often glossed over issues relating to Buddhism – notably, historicity, idolatry, messianism, importation to the West. Zen Judaism, which is a frank reflection on the tensions between reason and faith in today’s context of knowledge, and on the need to inject Zen-like meditation into Judaism. This work also treats some issues in ethics and theodicy. No to Sodom, which is an essay against homosexuality, using biological, psychological, spiritual, ethical and political arguments. (shrink)

In this article I attempt to overcome extant obstacles in deriving fundamental, objective and logically deduced definitions of personhood and their rights, by introducing an a priori paradigm of beings and morality. I do so by drawing a distinction between entities that are sought as ends and entities that are sought as means to said ends. The former entities, I offer, are the essence of personhood and are considered precious by observers possessing a logical system of valuation. The latter entities (...) – those sought only as a means to an end – I term ‘materials.’ Materials are sought for their conditional value: Important for achieving sought ends, they are not considered precious in and of themselves. A normative system for how this dichotomy of entities should interact is consequently derived and introduced. This paradigm has applicability for modern humanism and beyond. Assuming societal technological progression whereby human bodies and their surrounding infrastructures continue to evolve and integrate, the distinction between beings and their supporting materials, and a moral code for their interactions, will become ever more relevant. (shrink)

Description Logics (DLs) are a family of well-known terminological knowledge representation formalisms in modern semantics-based systems. This research focuses on analysing how our developed Occurrence Logic (OccL) can conceptually and logically support the development of a description logic. OccL is integrated into the alternative theory of natural language syntax in `Deviational Syntactic Structures' under the label `EFA(X)3' (or the third version of Epi-Formal Analysis in Syntax, EFA(X), which is a radical linguistic theory). From the logical point of (...) view, OccL is a formal logic that mainly deals with the occurrences of symbols as well as with their priorities within linguistic descriptions, i.e. natural language syntax, semantics and phonology. In this article---based on our OccL-based definitions of the concepts of `strong implication' and `occurrence value' as well as of the logical concept `identical occurrence constructor (IDOC)' that is the most fundamental logical concept in our formalism---we will model Occurrence Description Logic ($\mathcal{ODL}$). Accordingly, we will formally-logically analyse `occurrence(s) of symbol(s)' within descriptions of the world in $\mathcal{ODL}$. In addition, we will analyse and assess the logical concepts of `occurrence' and `occurrence priority' in $\mathcal{ODL}$. This research can make a strong logical background for our future research in the development of a Modal Occurrence Description Logic. (shrink)

This interesting and imaginative monograph is based on the author’s PhD dissertation supervised by Saul Kripke. It is dedicated to Timothy Smiley, whose interpretation of PRIOR ANALYTICS informs its approach. As suggested by its title, this short work demonstrates conclusively that Aristotle’s syllogistic is a suitable vehicle for fruitful discussion of contemporary issues in logical theory. Aristotle’s syllogistic is represented by Corcoran’s 1972 reconstruction. The review studies Lear’s treatment of Aristotle’s logic, his appreciation of the Corcoran-Smiley paradigm, and his (...) understanding of modern logical theory. In the process Corcoran and Scanlan present new, previously unpublished results. Corcoran regards this review as an important contribution to contemporary study of PRIOR ANALYTICS: both the book and the review deserve to be better known. (shrink)

Early Modern German Philosophy (1690-1750) makes some of the key texts of early German thought available in English, in most cases for the first time. The translations range from texts by the most important figures of the period, including Christian Thomasius, Christian Wolff, Christian August Crusius, and Georg Friedrich Meier, as well as texts by consequential but less familiar thinkers such as Dorothea Christiane Erxleben, Theodor Ludwig Lau, Friedrich Wilhelm Stosch, and Joachim Lange. The topics covered range across a (...) number of areas of theoretical philosophy, including metaphysics (the immortality of the soul, materialism and its refutation, the pre-established harmony), epistemology (the principle of sufficient reason, the limits of reason with respect to matters of faith), and logic (the role of prejudices in cognition and the doctrine of truth). (shrink)

Modern science generally assumes that the same laws of logic apply to mechanical, chemical and biological entities alike because they are all ultimately material objects. This may seem to be so obvious that there would be no need to validate it -- experimentally or logically. In this article we would like to critically examine this assumption and show that from an experiential/observational level, as well as from a rational/logical level, it is not valid. This becomes apparent, for instance, (...) when we consider the simple observation in which we distinguish animate from inanimate objects: those objects that seem to spontaneously move themselves and those that move only when impelled by some applied force outside or beyond the object. This distinction may be valid at the macroscopic level more than at the level of theoretical atomic particles. Thus the detailed nature of spontaneous movement must also be understood. (shrink)

In this paper, we make distinctions between Classical Logic (where the propositions are 100% true, or 100 false) and the Neutrosophic Logic (where one deals with partially true, partially indeterminate and partially false propositions) in order to respond to K. Georgiev’s criticism [1]. We recall that if an axiom is true in a classical logic system, it is not necessarily that the axiom be valid in a modern (fuzzy, intuitionistic fuzzy, neutrosophic etc.) logic system.

There is an exegetical quandary when it comes to interpreting Locke's relation to logic.On the one hand, over the last few decades a substantive amount of literature has been dedicated to explaining Locke's crucial role in the development of a new logic in the seventeenth and eighteenth centuries. John Yolton names this new logic the "logic of ideas," while James Buickerood calls it "facultative logic."1 Either way, Locke's Essay is supposedly its "most outspoken specimen" or (...) "culmination."2 Call this reading the 'New Logic interpretation.'On the other hand, from the typical standpoint of a philosopher accustomed to the modern conception of logic, whatever Locke—indeed, whatever most of the... (shrink)

Proposition are the material of our reasoning. Proposition are the basic building blocks of the world/thought. Proposition have intense relation with the world. World is a series of atomic facts and these facts are valued by the proposition although sentences explain the world of reality but can’t have any truth values, only proposition have truth values to describe the world in terms of assertions. Propositions are truth value bearers, the only quality of proposition is truth & falsity, that they are (...) either true or false. Proposition mirrors the world and explains how world is arranged in an orderly manner. It scans the world(object) and are composed of atomic facts experienced and can be analyzed into propositions. Propositions are the basic units of logic. The truth (affirm) and falsity (nego) are the qualities of the propositions and universality (generality) and Particularity are the quantities of the propositions. There are propositions which are neither true nor false, they are called Pseudo-Propositions and their quality are ipso-facto i.e meaningless. Propositions are used in computers with the modifications brought by the modern logicians in the form of statements or logical sentences. The truth table of the logical gates and binary operations (1,s or 0,s are due to the revolution of the modern logic or mathematical logic. It is a fact that proposition cannot change the word but it shows the relation between the object and of the word. Objectives: The objectives of this research is to explore the importance and need of propositions in logic. It also shows the analysis of propositions and how a philosopher thoughts in terms of propositions or concepts. In this research problem it is shown that propositions had been described in many ways by most of the philosophers and logicians from Aristotle to contemporary philosophers. It also analysis the contribution of the philosophers towards proposition and its relation to the world of reality. This research also describes the definition and nature of proposition. (shrink)

This paper contends that Stoic logic (i.e. Stoic analysis) deserves more attention from contemporary logicians. It sets out how, compared with contemporary propositional calculi, Stoic analysis is closest to methods of backward proof search for Gentzen-inspired substructural sequent logics, as they have been developed in logic programming and structural proof theory, and produces its proof search calculus in tree form. It shows how multiple similarities to Gentzen sequent systems combine with intriguing dissimilarities that may enrich contemporary discussion. Much (...) of Stoic logic appears surprisingly modern: a recursively formulated syntax with some truth-functional propositional operators; analogues to cut rules, axiom schemata and Gentzen’s negation-introduction rules; an implicit variable-sharing principle and deliberate rejection of Thinning and avoidance of paradoxes of implication. These latter features mark the system out as a relevance logic, where the absence of duals for its left and right introduction rules puts it in the vicinity of McCall’s connexive logic. Methodologically, the choice of meticulously formulated meta-logical rules in lieu of axiom and inference schemata absorbs some structural rules and results in an economical, precise and elegant system that values decidability over completeness. (shrink)

2nd edition. Many-valued logics are those logics that have more than the two classical truth values, to wit, true and false; in fact, they can have from three to infinitely many truth values. This property, together with truth-functionality, provides a powerful formalism to reason in settings where classical logic—as well as other non-classical logics—is of no avail. Indeed, originally motivated by philosophical concerns, these logics soon proved relevant for a plethora of applications ranging from switching theory to cognitive modeling, (...) and they are today in more demand than ever, due to the realization that inconsistency and vagueness in knowledge bases and information processes are not only inevitable and acceptable, but also perhaps welcome. The main modern applications of (any) logic are to be found in the digital computer, and we thus require the practical knowledge how to computerize—which also means automate—decisions (i.e. reasoning) in many-valued logics. This, in turn, necessitates a mathematical foundation for these logics. This book provides both these mathematical foundation and practical knowledge in a rigorous, yet accessible, text, while at the same time situating these logics in the context of the satisfiability problem (SAT) and automated deduction. The main text is complemented with a large selection of exercises, a plus for the reader wishing to not only learn about, but also do something with, many-valued logics. (shrink)

Abstract. As a general theory of reasoning—and as a general theory of what holds true under every possible circumstance—logic is supposed to be ontologically neutral. It ought to have nothing to do with questions concerning what there is, or whether there is anything at all. It is for this reason that traditional Aristotelian logic, with its tacit existential presuppositions, was eventually deemed inadequate as a canon of pure logic. And it is for this reason that modern (...) quantification theory, too, with its residue of existentially loaded theorems and patterns of inference, has been claimed to suffer from a defect of logical purity. The law of non-contradiction rules out certain circumstances as impossible—circumstances in which a statement is both true and false, or perhaps circumstances where something both is and is not the case. Is this to be regarded as a further ontological bias? (shrink)

This paper presents an analysis of Seneca's 58th letter to Lucilius and Porphyry's Isagoge, which were the origin of the tree diagrams that became popular in philosophy and logic from the early Middle Ages onwards. These diagrams visualise the extent to which a concept can be understood as a category, genus, species or individual and what the method of dihairesis (division) means. The paper explores the dissimilarities between Seneca's and Porphyry's tree structures, scrutinising them through the perspective of (...) class='Hi'>modern graph theory. Although traditional trees can be interpreted in a modern way, their terms and definitions defy modern standardisation. These distinctions are not only useful for differentiating between Seneca's and Porphyry's trees, but also for comprehending the evolution of this philosophical tradition. Thus the paper highlights the continued significance of these tree diagrams in analysing concepts and categories in philosophy and structuring objects and information in modern fields such as artificial intelligence, computer science, bioinformatics, graph theory, and many more. (shrink)

“Second-order Logic” in Anderson, C.A. and Zeleny, M., Eds. Logic, Meaning, and Computation: Essays in Memory of Alonzo Church. Dordrecht: Kluwer, 2001. Pp. 61–76. -/- Abstract. This expository article focuses on the fundamental differences between second- order logic and first-order logic. It is written entirely in ordinary English without logical symbols. It employs second-order propositions and second-order reasoning in a natural way to illustrate the fact that second-order logic is actually a familiar part of our (...) traditional intuitive logical framework and that it is not an artificial formalism created by specialists for technical purposes. To illustrate some of the main relationships between second-order logic and first-order logic, this paper introduces basic logic, a kind of zero-order logic, which is more rudimentary than first-order and which is transcended by first-order in the same way that first-order is transcended by second-order. The heuristic effectiveness and the historical importance of second-order logic are reviewed in the context of the contemporary debate over the legitimacy of second-order logic. Rejection of second-order logic is viewed as radical: an incipient paradigm shift involving radical repudiation of a part of our scientific tradition, a tradition that is defended by classical logicians. But it is also viewed as reactionary: as being analogous to the reactionary repudiation of symbolic logic by supporters of “Aristotelian” traditional logic. But even if “genuine” logic comes to be regarded as excluding second-order reasoning, which seems less likely today than fifty years ago, its effectiveness as a heuristic instrument will remain and its importance for understanding the history of logic and mathematics will not be diminished. Second-order logic may someday be gone, but it will never be forgotten. Technical formalisms have been avoided entirely in an effort to reach a wide audience, but every effort has been made to limit the inevitable sacrifice of rigor. People who do not know second-order logic cannot understand the modern debate over its legitimacy and they are cut-off from the heuristic advantages of second-order logic. And, what may be worse, they are cut-off from an understanding of the history of logic and thus are constrained to have distorted views of the nature of the subject. As Aristotle first said, we do not understand a discipline until we have seen its development. It is a truism that a person's conceptions of what a discipline is and of what it can become are predicated on their conception of what it has been. (shrink)

The article identifies real losses of foreign economic activity of industrial enterprises of Ukraine under the influence of modern challenges. The main purpose of the study is to implement a scientific and methodological approach to assess the impact of modern challenges on the development of foreign economic activity of enterprises. A critical analysis of the content of scientific publications on the studied issues revealed the lack of an integrated approach to establishing the real losses of foreign economic activity (...) of enterprises under the influence of modern challenges. The urgency of solving a certain problem is due to the scale of the impact of modern challenges on the dynamics of foreign economic activity of industrial enterprises, which requires an assessment of such an impact for the further development of measures that will be aimed at minimizing the negative impact of modern challenges. The methodological basis is the system of methods used to obtain the results of the study: theoretical generalization and analytical methods ‒ to clarify scientific approaches in determining the impact of modern challenges on the development of foreign economic activity of enterprises; generalization, grouping and integral evaluation ‒ to determine the level of influence of modern challenges on the development of foreign economic activity of industrial enterprises in 2013‒2021; relative and absolute deviations, coefficient analysis of changes in foreign trade turnover ‒ to establish the volume of losses of foreign economic activity of industrial enterprises of Ukraine under the influence of modern challenges; logical generalization ‒ to obtain conclusions and develop recommendations based on research results. The object of research in the work is the foreign economic activity of enterprises. The article presents the results of assessing the impact of modern challenges on the development of foreign economic activity of industrial enterprises, which became possible due to the developed and implemented author's scientific and methodological approach. Analysis and logical generalization of the results of assessing the impact of modern challenges on the development of foreign economic activity of industrial enterprises became the basis for determining the real volumes of losses of their foreign trade turnover. The results of the study have practical value for the Ministry of Economy of Ukraine and the Entrepreneurship and export promotion office, because they can serve as the basis for developing a program of recommendations for activating the export initiative of industrial enterprises under modern conditions. (shrink)

From the beginning of the 16th century to the end of the 18th century, there were not less than ten philosophers who focused extensively on Venn’s ostensible analytical diagrams, as noted by modern historians of logic (Venn, Gardner, Baron, Coumet et al.). But what was the reason for early modern philosophers to use logic or analytical diagrams? Among modern historians of logic one can find two theses which are closely connected to each other: M. (...) Gardner states that since the Middle Ages certain logic diagrams were used just in order to teach “dull-witted students”. Therefore, logic diagrams were just a means to an end. According to P. Bernhard, the appreciation of logic diagrams had not started prior to the 1960s, therefore the fact that logic diagrams become an end the point of research arose very late. The paper will focus on the question whether logic resp. analytical diagrams were just means in the history of (early) modern logic or not. In contrast to Gardner, I will argue that logic diagrams were not only used as a tool for “dull-witted students”, but rather as a tool used by didactic reformers in early modern logic. In predating Bernhard’s thesis, I will argue that in the 1820s logic diagrams had already become a value in themselves in Arthur Schopenhauer’s lectures on logic, especially in proof theory. (shrink)

A critical study of marronage is urgent since certain anthropological and historical analyses have painted an image of slavery—and therefore of marronage as well—as something of the past. This text will demonstrate that marronage, rather than being simply flight from the plantation in a literal sense, also refers to flight from oppressive institutions through permanent institutional reconfiguration as well as to an existential state of Being. In this text I sketch out ways in which marronage is made manifest within and (...) outside Euromodernity, which I call maroon logics. Maroon logics consists of two interdependent modes of embodying freedom and struggling for liberation. On the one hand there is sociogenic marronage, elaborated by political theorist Neil Roberts in his book Freedom as Marronage (2015), which refers to permanent institutional change as a manifestation of marronage. On the other hand, there is what I call analectical marronage, which denotes resistance to the coloniality of being. (shrink)

It is a received view that Kant’s formal logic (or what he calls “pure general logic”) is thoroughly intensional. On this view, even the notion of logical extension must be understood solely in terms of the concepts that are subordinate to a given concept. I grant that the subordination relation among concepts is an important theme in Kant’s logical doctrine of concepts. But I argue that it is both possible and important to ascribe to Kant an objectual notion (...) of logical extension according to which the extension of a concept is the multitude of objects falling under it. I begin by defending this ascription in response to three reasons that are commonly invoked against it. First, I explain that this ascription is compatible with Kant’s philosophical reflections on the nature and boundary of a formal logic. Second, I show that the objectual notion of extension I ascribe to Kant can be traced back to many of the early modern works of logic with which he was more or less familiar. Third, I argue that such a notion of extension makes perfect sense of a pivotal principle in Kant’s logic, namely the principle that the quantity of a concept’s extension is inversely proportional to that of its intension. In the process, I tease out two important features of the Kantian objectual notion of logical extension in terms of which it markedly differs from the modern one. First, on the modern notion the extension of a concept is the sum of the objects actually falling under it; on the Kantian notion, by contrast, the extension of a concept consists of the multitude of possible objects—not in the metaphysical sense of possibility, though—to which a concept applies in virtue of being a general representation. While the quantity of the former extension is finite, that of the latter is infinite—as is reflected in Kant’s use of a plane-geometrical figure (e.g., circle, square), which is continuum as opposed to discretum, to represent the extension in question. Second, on the modern notion of extension, a concept that signifies exactly one object has a one-member extension; on the Kantian notion, however, such a concept has no extension at all—for a concept is taken to have extension only if it signifies a multitude of things. This feature of logical extension is manifested in Kant’s claim that a singular concept (or a concept in its singular use) can, for lack of extension, be figuratively represented only by a point—as opposed to an extended figure like circle, which is reserved for a general concept (or a concept in its general use). Precisely on account of these two features, the Kantian objectual extension proves vital to Kant’s theory of logical quantification (in universal, particular and singular judgments, respectively) and to his view regarding the formal truth of analytic judgments. (shrink)

I take advantage of two recent results: 1) the recognition of an alternative theoretical organization to the deductive-axiomatic one; it is characterized by a sequence of four logical steps belonging to intuitionist logic; 2) the recognition of the logical content of Cusanus’ philosophical works; also this content pertains to intuitionist logic, which Cusanus anticipated by even identifying some its logical laws. Many Cusanus’ books present the alternative theoretical organization; whose yet he did not apply in a clear way (...) its last two steps, the ad absurdum arguing and the application of the principle of sufficient reason. For this reason in the last books his thinking transcended to a metaphysical level. Hegel’s philosophy of logic shares Cusanus’ philosophical effort for exiting out classical logic. He seems to have assumed Cusanus’ anticipation of the new theoretical organization; all goes as if Hegel wanted to improve Cusanus’ poor method, by introducing a process of additions of two negations which however does not agree with the intuitionist logic. Moreover, Hegel presupposes, like Cusanus in his last years, the free application of the principle of sufficient reason to the entire subject of his philosophy of logic without awareness of the logical constraints to which is subject a correct application of this principle; hence, he introduced a metaphysical arguing which according to modern mathematical logic has no consistent content. Also Hegel’s supposed prolongation of Cusanus’ logical discoveries was an essentially metaphysical view. (shrink)

ABSTRACT The logic of humor has been acknowledged as an essential dimension of every joke. However, what is the logic of jokes, exactly? The modern theories of humor maintain that jokes are characterized by their own logic, dubbed “pseudo,” “playful,” or “local,” which has been the object of frequent criticisms. This article intends to address the limitations of the current perspectives on the logic of jokes by proposing a rhetorical approach to humorous texts. Building on (...) the traditional development of Aristotle’s almost neglected view of jokes as surprising enthymemes, the former are analyzed as rhetorical arguments. Like enthymemes, jokes are characterized by natural inferences that can be represented as topics, and quasi-formalized in argumentation theory as argumentation schemes. Like rhetorical arguments, jokes express a reason in support of different types of conclusions and proceed from distinct kinds of reasoning and semantic relations. (shrink)