A method of constructing Hilbert-type axiom systems for standard many-valued propositional logics was offered by Rosser and Turquette. Although this method is considered to be a solution of the problem of axiomatisability of a wide class of many-valued logics, the article demonstrates that it fails to produce adequate axiom systems. The article concerns finitely many-valued propositional logics of Łukasiewicz. It proves that if standard propositional connectives of the Rosser–Turquette axiom systems are definable in terms of the propositional connectives (...) of Łukasiewicz’s logics, and thus, they are normal ones, then every Rosser–Turquette axiom system for a finite-valued Łukasiewicz’s logic is semantically incomplete. (shrink)

The article explores the following question: which among the most often examined in the literature method of constructing Hilbert-type axiom systems for finite-valued propositional logics of Łukasi...

In his 1912 book _The Problems of Philosophy_, Bertrand Russell advocates an indirect realism with regard to physical objects. Only two years later, in his book _Our Knowledge of the External World_ and the paper “The Relation of Sense-Data to Physics”, he changes his method in philosophy. Instead of inferring the existence of physical objects, he now sets out to construct them out of sense-data. As I will argue in this article, the main argument from _The Problems of Philosophy_ (...) can be rationally reconstructed as an inference to the best explanation which infers to unobservable objects. The main motivation for the new approach in _Our Knowledge of the External World_, on the other hand, is to establish a more direct variant of realism, in particular because Russell became skeptical with regard to inferences to unobservable objects. As I will argue, the resulting theory of the physical world loses so much in simplicity that it becomes an unattractive alternative to his earlier position, and Russell’s reason for rejecting simplicity as a criterion of theory choice turns out to be inconsequential. (shrink)

This paper investigates the formal relationship between two prominent approaches to the logic of belief change. The first one uses the idea of "relational partial meet contractions" as developed by Alchourrón, Gärdenfors and Makinson (Journal of Symbolic Logic 1985), the second one uses the concept of "epistemic entrenchment" as elaborated by Gärdenfors and Makinson (in Theoretical Aspects of Reasoning about Knowledge, M. Y. Vardi, Los Altos 1988). The two approaches are shown to be strictly equivalent via direct links between the (...) underlying formal relations. The paper closes with observations about the application of epistemic entrenchment to simple and iterated revisions. (shrink)

We aim at developing a systematic method of separating omniscience principles by constructing Kripke models for intuitionistic predicate logic $\mathbf {IQC}$ and first-order arithmetic $\mathbf {HA}$ from a Kripke model for intuitionistic propositional logic $\mathbf {IPC}$. To this end, we introduce the notion of an extended frame, and show that each IPC-Kripke model generates an extended frame. By using the extended frame generated by an IPC-Kripke model, we give a separation theorem of a schema from a set of schemata (...) in $\mathbf {IQC}$ and a separation theorem of a sentence from a set of schemata in $\mathbf {HA}$. We see several examples which give us separations among omniscience principles. (shrink)

The ancient Greek method of analysis has a rational reconstruction in the form of the tableau method of logical proof. This reconstruction shows that the format of analysis was largely determined by the requirement that proofs could be formulated by reference to geometrical figures. In problematic analysis, it has to be assumed not only that the theorem to be proved is true, but also that it is known. This means using epistemic logic, where instantiations of variables are (...) typically allowed only with respect to known objects. This requirement explains the preoccupation of Greek geometers with questions as to which geometrical objects are ?given?, that is, known or ?data?, as in the title of Euclid's eponymous book. In problematic analysis, constructions had to rely on objects that are known only hypothetically. This seems strange unless one relies on a robust idea of ?unknown? objects in the same sense as the unknowns of algebra. The Greeks did not have such a concept, which made their grasp of the analytic method shaky. (shrink)

En esta discusión abordo la hermenéutica analógica de Mauricio Beuchot comparándola con la hermenéutica filosófica de Hans-Georg Gadamer. Argumento que Beuchot vuelve a la idea clásica de la hermenéutica como método de interpretación y no la juzga, como Gadamer, como una fenomenología de la comprensión. Sin embargo, Beuchot no atiende las razones de Gadamer en contra de concebir la hermenéutica como una metodología. Si se considera como metodología centrada en la analogía, la hermenéutica analógica deja de lado otros recursos interpretativos (...) importantes. Finalmente, Beuchot adopta una teoría correspondentista de la verdad contra la que argumenta Gadamer y que es incompatible con una hermenéutica filosófica. In this paper I analyze Mauricio Beuchot's analogical hermeneutics by contrasting it with Hans-Georg Gadamer's philosophical hermeneutics. I argue that Beuchot goes back to the classical idea of hermeneutics as a method of interpretation, and does not see it, as Gadamer does, as a phenomenology of understanding. However, Beuchot does not address Gadamer's arguments against using hermeneutics as a methodology. Considered as a methodology centered on analogy, analogical hermeneutics overlooks other important interpretation resources. Finally, Beuchot adopts a correspondence theory of truth, which Gadamer objects and is incompatible with philosophical hermeneutics. (shrink)

This book is both a history of philosophy of logic told from the Kantian viewpoint and a reconstruction of Kant’s theory of logic from a historical perspective. Kant’s theory represents a turning point in a history of philosophical debates over the following questions. (1) Is logic a science, instrument, standard of assessment, or mixture of these? (2) If logic is a science, what is the subject matter that differentiates it from other sciences, particularly metaphysics? (3) If logic is a necessary (...) instrument to all philosophical inquiries, how is it so entitled? (4) If logic is both a science and an instrument, how are these two roles related? Kant’s official answer to these questions centers on three distinctions: general versus particular logic; pure versus applied logic; pure general logic versus transcendental logic. The true meaning and significance of each distinction becomes clear, I argue, only if we consider two factors. First, Kant was mindful of various historical views on how logic relates to other branches of philosophy (viz. metaphysics and physics) and to the workings of common human understanding. Second, he first coined ‘transcendental logic’ while struggling to secure metaphysics as a proper “science,” and this conceptual innovation would in turn have profound implications for his mature theory of logic. Against this backdrop, I reassess the place of Kant’s theory in the history of philosophy of logic and highlight certain issues that are still debated today, such as normativity of logic and the challenges posed by logical pluralism. In Chapter 1, “Kant and a Philosophical History of Logic – Methodological Reflections,” I discuss certain exegetical challenges posed by Kant’s logic corpus and argue for a “history of philosophical problems” method by which to reconstruct a Kantian theory of logic. In Chapter 2, “The Nature and Place of Logic – A History of Controversies,” I construct a partial history of philosophy of logic that revolves around the (supposedly) scientific status of logic on the one hand and its value or utility on the other. In Chapter 3, “The Making of a Scientific Logic from Bacon to Wolff,” I examine how four representative early modern philosophers – Francis Bacon, John Locke, G.W. Leibniz, and Christian Wolff – approached the four questions mentioned above. In Chapter 4, “Kant on the Way to His Own Philosophy of Logic,” I consider how Kant, in the decade between around mid-1760s and mid-1770s, navigated between existing accounts of logic until he finally found his own voice. I highlight the breakthroughs that would mark his critical departures from previous views and pave the way for the final articulation of his own view. In Chapter 5, “Logic and the Demands of Kantian ‘Science’,” I zero in on Kant’s official theory of logic in his monumental publication, Critique of Pure Reason. I foreground both what makes the theory original and what leaves it vulnerable to criticisms from post-Kantian thinkers. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Kant on the Method of MathematicsEmily Carson1. INTRODUCTIONThis paper will touch on three very general but closely related questions about Kant’s philosophy. First, on the role of mathematics as a paradigm of knowledge in the development of Kant’s Critical philosophy; second, on the nature of Kant’s opposition to his Leibnizean predecessors and its role in the development of the Critical philosophy; and finally, on the specific role of (...) intuition in Kant’s philosophy of mathematics. One of the points that I want to make is that recognizing the importance of these first two issues is essential to giving an adequate account of the third. That is, only by appreciating how Kant uses the example of mathematical knowledge, and in particular the objections he had to previous accounts of such knowledge, can we understand the philosophical role that he ascribes to intuition in his own account of mathematics.I don’t propose to explain the details of Kant’s philosophy of mathematics, but rather to compare his views on mathematics in the pre-Critical Prize Essay of 1764 to those of the Critique of Pure Reason, with the aim of showing how Kant’s doctrine of construction in pure intuition arises out of a combination of two factors: first, his attempt to ground the distinction between the respective methods of mathematics and philosophy, and second, his concern to defend the reality of mathematical knowledge against the views of those he called ‘the metaphysicians.’ The first factor demands an account of the certainty of mathematics, the second requires an account of its content. I shall argue that there is a possible conflict between Kant’s accounts of these features in the Prize Essay, a possibility which is finally ruled out only by means of the Critical doctrine of pure intuition. So I hope to show that there is a significant philosophical role for intuition in Kant’s account of mathematical knowledge (over and above the more logical role attributed to it on readings [End Page 629] such as those of Friedman, Beth, and Hintikka1) in reconciling the content of mathematics and its certainty.2. THE MATHEMATICIANS VS. THE METAPHYSICIANSThroughout his early writings, Kant appealed to the evidence and certainty of mathematics; his attitude towards metaphysics, however, underwent a dramatic shift. In the Physical Monadology of 1756, Kant asked how metaphysics and geometry can be united,[f] or the former peremptorily denies that space is infinitely divisible, while the latter, with its usual certainty, asserts that it is infinitely divisible. Geometry contends that empty space is necessary for free motion, while metaphysics hisses the idea off the stage. Geometry holds universal attraction or gravitation to be hardly explicable by mechanical causes but shows that it derives from the forces which are inherent in bodies at rest and which act at a distance, whereas metaphysics dismisses the notion as an empty delusion of the imagination [1:475–6].2He attempted in this work to reconcile the respective positions of metaphysics and geometry regarding infinite divisibility, thereby demonstrating that “it is neither the case that the geometer is mistaken nor that the opinion to be found among metaphysicians deviates from the truth” [1:480]. But in a series of works from 1762–3, Kant reveals a markedly different attitude towards metaphysics. For example, in “The only possible argument in support of a demonstration of the existence of God,” Kant says:the mania for method and the imitation of the mathematician, who advances with a sure step along a well-surfaced road, have occasioned a large number of such mishaps on the slippery ground of metaphysics. These mishaps are constantly before one’s eyes, but there is little hope that people will be warned by them, or that they will learn to be more circumspect as a result [2:71].In contrast, Kant describes the geometer as uncovering “with the greatest certainty” the most secret properties of that which is extended [2:70]. But the less conciliatory attitude to metaphysics comes out most clearly in Kant’s actual procedure in “The only possible argument.” In the Seventh Reflection, he presents an attempt to explain the origin of... (shrink)

In this chapter, I discuss some problems of Kant’s logic corpus while recognizing its richness and potential value. I propose and explain a methodic way to approach it. I then test the proposal by showing how we may use various mate- rials from the corpus to construct a Kantian demonstration of the formal rules of thinking (or judging) that lie at the base of Kant’s Metaphysical Deduction. The same proposal can be iterated with respect to other topics. The said demonstration (...) will have cleared the path for such iterations. (shrink)

In this paper, we show that the predicate logics of consistent extensions of Heyting's Arithmetic plus Church's Thesis with uniqueness condition are complete $\Pi _{2}^{0}$. Similarly, we show that the predicate logic of HA*, i.e. Heyting's Arithmetic plus the Completeness Principle (for HA*) is complete $\Pi _{2}^{0}$. These results extend the known results due to Valery Plisko. To prove the results we adapt Plisko's method to use Tennenbaum's Theorem to prove 'categoricity of interpretations' under certain assumptions.

The era in which Abū Naṣr al-Fārābī emerged as a canonical scientist significantly contributed to his education and shaped his scientific worldview. The formation of al-Farabi’s spiritual worldview and his ideas is directly associated with embracing the ancient philosophical tradition, more precisely, Aristotle’s philosophy and logic. The focus of the article is alFarabi’s analysis of Aristotle’s principles in the study of logic and their further development. Al-Farabi’s worldwide reputation as the Second Teacher after Aristotle, the First Teacher, in the East (...) is directly rooted in his mastery and advancement of the science of logic. In simple terms, the philosopher states that logic leads a person to a straight path to being capable of radically discerning between good and bad, truth and lies, and honesty and deceit. This, in turn, steers humanity away from succumbing to mistakes or getting into trouble. Al-Farabi constructs a comprehensive scientific analysis of the socio-cultural foundations of Aristotle’s doctrine of logic and the ability to adapt it in his unique way to society. The study of the theoretical basis of al-Farabi’s teachings on logic and its significance involves employing hermeneutic, comparative, and stylistic methods in the research of the texts. (shrink)

In a first part, I defend that formal semantics can be used as a guide to ontological commitment. Thus, if one endorses an ontological view \(O\) and wants to interpret a formal language \(L\) , a thorough understanding of the relation between semantics and ontology will help us to construct a semantics for \(L\) in such a way that its ontological commitment will be in perfect accordance with \(O\) . Basically, that is what I call constructing formal semantics from an (...) ontological perspective. In the rest of the paper, I develop rigorously and put into practice such a method, especially concerning the interpretation of second-order quantification. I will define the notion of ontological framework: it is a set-theoretical structure from which one can construct semantics whose ontological commitments correspond exactly to a given ontological view. I will define five ontological frameworks corresponding respectively to: (i) predicate nominalism, (ii) resemblance nominalism, (iii) armstrongian realism, (iv) platonic realism, and (v) tropism. From those different frameworks, I will construct different semantics for first-order and second-order languages. Notably I will present different kinds of nominalist semantics for second-order languages, showing thus that we can perfectly quantify over properties and relations while being ontologically committed only to individuals. I will show in what extent those semantics differ from each other; it will make clear how the disagreements between the ontological views extend from ontology to logic, and thus why endorsing an ontological view should have an impact on the kind of logic one should use. (shrink)

We present a new method of constructing a model of \SCH+\AP.

This paper concerns Aristotle and Galen’s scientific method and the place of philosophy in their natural scientific endeavors as manifested in their discussions on the role of the heart in the formation of the embryo. I will begin first by discussing Aristotle’s conception of natural sciences and his discussion on the role of the heart in the body and embryo. Following this is Galen’s critique of the role of the heart in the formation of the embryo. Galen had considerably (...) more knowledge about anatomy and the construction of the embryo, yet his scientific method is not radically different from Aristotle’s as he also utilizes both logical arguments and observational data. Nevertheless, he attempts to banish philosophers from discussing anatomical issues, thus opening the path to specialization. (shrink)

Wajsberg and Jankov provided us with methods of constructing a continuum of logics. However, their methods are not suitable for super-intuitionistic and modal predicate logics. The aim of this paper is to present simple ways of modification of their methods appropriate for such logics. We give some concrete applications as generic examples. Among others, we show that there is a continuum of logics (1) between the intuitionistic predicate logic and the logic of constant domains, (2) between a predicate extension ofS4 (...) andS4 with the Barcan formula. Furthermore, we prove that (3) there is a continuum of predicate logics with equality whose equality-free fragment is just the intuitionistic predicate logic. (shrink)

Between inventing the concept of a universal computer in 1936 and breaking the German Enigma code during World War II, Alan Turing, the British founder of computer science and artificial intelligence, came to Princeton University to study mathematical logic. Some of the greatest logicians in the world--including Alonzo Church, Kurt Gödel, John von Neumann, and Stephen Kleene--were at Princeton in the 1930s, and they were working on ideas that would lay the groundwork for what would become known as computer science. (...) Though less well known than his other work, Turing's 1938 Princeton PhD thesis, "Systems of Logic Based on Ordinals," which includes his notion of an oracle machine, has had a lasting influence on computer science and mathematics. This book presents a facsimile of the original typescript of the thesis along with essays by Andrew Appel and Solomon Feferman that explain its still-unfolding significance. A work of philosophy as well as mathematics, Turing's thesis envisions a practical goal--a logical system to formalize mathematical proofs so they can be checked mechanically. If every step of a theorem could be verified mechanically, the burden on intuition would be limited to the axioms. Turing's point, as Appel writes, is that "mathematical reasoning can be done, and should be done, in mechanizable formal logic." Turing's vision of "constructive systems of logic for practical use" has become reality: in the twenty-first century, automated "formal methods" are now routine. Presented here in its original form, this fascinating thesis is one of the key documents in the history of mathematics and computer science. (shrink)

Kleene [5] mentions two ways of extending the constructive ordinals. The first is by relativizing the setOof notations for the constructive ordinals, using fundamental sequences which are partial recursive inO. In this way we obtain the setOOwhich provides notations for the ordinals less than ω1O. Continuing the process, the sequenceO,OO,, … and the corresponding ordinalsare obtained. A second possibility is to define higher number classes in which partial recursive functions are used at limit ordinals to provide an “accessibility” mapping from (...) a previously defined number class. The relationship between the ordinals obtained by the two methods of extension has been an open problem. Methods developed in this article are used to show that the two ways of extending the constructive ordinals are equivalent, provided the sets of notations for the higher number classes satisfy certain natural conditions. Equivalence is obtained, not only with respect to ordinals, but also with respect to the forms of the sets of notations for the higher number classes. Specifically, the fundamental fact that the sets of notations for the constructive ordinals are complete Π11sets generalizes to suitably defined higher number classes. As an application we prove that the ordinals of the Addison and Kleene [1] constructive third number class are exactly the ordinals less than ω1Oand the setof notations for their third number class is recursively isomorphic toOO. (shrink)

This article reflects one scholar's attempt to locate herself within emerging ethical methodologies given a specific concern with cross-cultural women's moral praxis. The field of comparative ethics's debt to past debates over methodology is considered through a typology of three waves of methodological invention. The article goes on to describe a specific research focus on U.S. Catholic and Iranian Shii women that initiated a search for a distinct method. This method of comparative ethics, which focuses on the production (...) of ethical knowledge through the interaction of discursive logics of various moral agents, is described. The conclusion turns to how methodological invention can itself become a constructive project through the way it (1) locates the scholar in relation to her subject of study and (2) allows for isolation of tactics within specific moral discourses. (shrink)

In this paper, we interpret a 19th century diagram, which is meant to visualise G.W.F. Hegel’s entire method of the `Science of Logic' on the basis of bitwise operations. For the interpretation of the diagram we use a binary numeral system, and discuss whether the anti-Hegelian argument associated with it is valid or not. The reinterpretation is intended to make more precise rules of construction, a stricter binary code and a review of strengths and weaknesses of the critique.

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)

Preface -- Part I: Philosophical logic and philosophy of language -- Rules versus theorems : a new approach for mediation -- Between intuitionistic and two-valued logic -- On the relation between the partition of a whole into parts and the attribution of properties to an object -- Basic objectives of dialogic logic in historical perspective -- Pragmatic and semiotic prerequisites for predication : a dialogue model -- Pragmatics and semiotics : the peircean version of ontology and epistemology -- Intentionality and (...) its language-dependency -- Meaning postulates and rules of argumentation : remarks concerning the pragmatic tie between meaning (of terms) and truth (of propositions) -- What do language games measure? -- Features of Indian logic -- Part II: Methods in philosophy, in art, and in science -- The concept of science : some remarks on the methodological issue construction versus description in the philosophy of science -- Is and ought revisited -- Competition and cooperation : are they antagonistic or complementary? -- Another version of methodological dualism -- The pre-established harmony between the two Adams -- On the way to conceptual and perceptual knowledge -- Self and other : remarks on human nature and human culture -- On the concept of symmetry -- Procedural principles of the Eerlangen School : on the interrelation between the principles of method, of dialogue, and of reason. (shrink)

Charles Sanders Peirce’s first rule of logic (EP 2.48, 1898) identifies the inception point of human inquiry. Taking a closer look at this principle, we find at its core a necessary relationship between emptiness and desire that underlies all genuine instances of human learning and adaptation. This composite relationship plays a critical role in the function or failure of learning but has received scant attention in the literature. As a result, the complexities of the first rule of logic are not (...) well understood, often being mistakenly conflated with the rule’s famous corollary, ‘do not block the way of inquiry’, or passed over with cursory definitions, including ‘wonder’, ‘doubt’ and ‘the will to learn’. Following a background discussion highlighting the nature of reflexive inquiry and fallibilism that situate human consciousness both within and beyond animal being, I draw on multiple layers of evidence from a range of disciplines to better reveal the complex dynamics intrinsic to the first rule of logic. These layers include a closer reading and exegesis of the original passage and surrounding text; a semiotic reanalysis of this reading in light of recent advances in the semiotic theory of learning; a resituation of these distinctions within broader contemporary discussions of emptiness ontology to which I contribute in part via an original semantic/rhetorical analysis of a linguistic construction in Laozi; the introduction of a closely related pedagogical tool under development in the context of my own university-level teaching in ethnography and research methods; and the dialogic situation of this diagram within discourses of psychotherapy, philosophy and literature. Building on these principles and distinctions, the paper closes with a perspective shift on obstacles and desire in human learning and an expanded reformulation of the first rule of logic. (shrink)

This book, written by one of the most distinguished of contemporary philosophers of mathematics, is a fully rewritten and updated successor to the author's earlier The Unprovability of Consistency. Its subject is the relation between provability and modal logic, a branch of logic invented by Aristotle but much disparaged by philosophers and virtually ignored by mathematicians. Here it receives its first scientific application since its invention. Modal logic is concerned with the notions of necessity and possibility. What George Boolos does (...) is to show how the concepts, techniques, and methods of modal logic shed brilliant light on the most important logical discovery of the twentieth century: the incompleteness theorems of Kurt Godel and the 'self-referential' sentences constructed in their proof. The book explores the effects of reinterpreting the notions of necessity and possibility to mean provability and consistency. (shrink)

The primary goal of this chapter is to present a new method—called Logical Argument Mapping —for the analysis of framing processes as they occur in any communication, but especially in conflicts. I start with a distinction between boundary setting, meaning construction, and sensemaking as three forms or aspects of framing, and argue that crucial for the resolution of frame-based controversies is our ability to deal with those “webs” of mutually supporting beliefs that determine sensemaking processes. Since any analysis (...) of framing in conflicts and communication is itself influenced by sensemaking—there is no “frame-neutrality”—the main problem for an analyst is to cope with his or her own cognitive limitations. LAM offers a solution to this problem. The method will be exemplified with an analysis of two conflicting interpretations of how the international community should deal with Hamas after its election victory in 2006. (shrink)

This article examines the ‘integrated conception of competence’ as conceived by Paul Hager and David Beckett and suggests that its characterization in terms intended to distance it from behaviouristic and reductionist notions of competence is not sufficient to differentiate it from other models. Taking up Hager and Beckett’s idea that competence must be inferred from behaviour, it is suggested that this indicates how the integrated conception is more properly distinguished by virtue of the method used rather than what it (...) is that is assessed. Drawing on the work of Wittgenstein and Donald Davidson, it is argued that it is possible to discern two logically distinct methodological approaches to competence assessment, allowing a clear distinction to be made between the integrated conception and the kind of approach which predominates in the UK’s framework of vocational qualifications. While the latter is shown to be rightly criticized for its deficiencies, in contrast the integrated conception is seen to suggest a methodological approach that is capable of acknowledging the full richness of occupational practice. (shrink)

The paper is devoted to an approach to analytic tableaux for propositional logic, but can be successfully extended to other logics. The distinguishing features of the presented approach are:(i) a precise set-theoretical description of tableau method; (ii) a notion of tableau consequence relation is defined without help of a notion of tableau, in our universe of discourse the basic notion is a branch;(iii) we define a tableau as a finite set of some chosen branches which is enough to check; (...) hence, in our approach a tableau is only a way of choosing a minimal set of closed branches;(iv) a choice of tableau can be arbitrary, it means that if one tableau starting with some set of premisses is closed in the defined sense, then every branch in the power set of the set of formulas, that starts with the same set, is closed. (shrink)

The semantic valuations of classical logic, strong Kleene logic, the logic of paradox and the logic of first-degree entailment, all respect the Dunn conditions: we call them Dunn logics. In this paper, we study the interpolation properties of the Dunn logics and extensions of these logics to more expressive languages. We do so by relying on the \ calculus, a signed tableau calculus whose rules mirror the Dunn conditions syntactically and which characterizes the Dunn logics in a uniform way. In (...) terms of the \ calculus, we first introduce two different interpolation methods, each of which uniformly shows that the Dunn logics have the interpolation property. One of the methods is closely related to Maehara’s method but the other method, which we call the pruned tableau method, is novel to this paper. We provide various reasons to prefer the pruned tableau method to the Maehara-style method. We then turn our attention to extensions of Dunn logics with so-called appropriate implication connectives. Although these logics have been considered at various places in the literature, a study of the interpolation properties of these logics is lacking. We use the pruned tableau method to uniformly show that these extended Dunn logics have the interpolation property and explain that the same result cannot be obtained via the Maehara-style method. Finally, we show how the pruned tableau method constructs interpolants for functionally complete extensions of the Dunn logics. (shrink)

In 1950, Alan Turing proposed his iconic imitation game, calling it a ‘test’, an ‘experiment’, and the ‘the only really satisfactory support’ for his view that machines can think. Following Turing’s rhetoric, the ‘Turing test’ has been widely received as a kind of crucial experiment to determine machine intelligence. In later sources, however, Turing showed a milder attitude towards what he called his ‘imitation tests’. In 1948, Turing referred to the persuasive power of ‘the actual production of machines’ rather than (...) that of a controlled experiment. Observing this, I propose to distinguish the logical structure from the rhetoric of Turing’s argument. I argue that Turing’s proposal of a crucial experiment may have been a concession to meet the standards of his interlocutors more than his own, while his construction of machine intelligence rather reveals a method of successive idealizations and exploratory experiments. I will draw a parallel with Galileo’s construction of idealized fall in a void and the historiographical controversies over the role of experiment in Galilean science. I suggest that Turing, like Galileo, relied on certain kinds of experiment, but also on rhetoric and propaganda to inspire further research that could lead to convincing scientific and technological progress. (shrink)

John Corcoran. 1979 Review of Hintikka and Remes. The Method of Analysis (Reidel, 1974). Mathematical Reviews 58 3202 #21388. -/- The “method of analysis” is a technique used by ancient Greek mathematicians (and perhaps by Descartes, Newton, and others) in connection with discovery of proofs of difficult theorems and in connection with discovery of constructions of elusive geometric figures. Although this method was originally applied in geometry, its later application to number played an important role in (...) the early development of algebra [Jacob Klein, English translation, Greek mathematical thought and the origin of algebra, especially pp. 154–157, M.I.T. Press, Cambridge, Mass., 1968]. -/- It is universally agreed that the method of analysis begins by “assuming the thing sought after” (e.g., in geometry, the truth of the proposition to be proved or the existence of the geometric figure to be constructed). Aside from this, little else can be taken for granted. There is disagreement concerning the “direction of analysis”, i.e. whether one is to seek implications of the assumption or whether one is to seek implicants of it. There is also disagreement concerning what is to be “anatomized” (analyzed), i.e., whether one analyzes mathematical objects (figures), mathematical propositions (the axioms, known theorems, and analytic assumption) or an imagined proof (of the analytic assumption from axioms and known theorems). (shrink)

The two volumes in this advanced textbook present results, proof methods, and translations of motivational and philosophical considerations to formal constructions. In this Vol. I the author explains preferential structures and abstract size. In the associated Vol. II he presents chapters on theory revision and sums, defeasible inheritance theory, interpolation, neighbourhood semantics and deontic logic, abstract independence, and various aspects of nonmonotonic and other logics. In both volumes the text contains many exercises and some solutions, and the author limits (...) the discussion of motivation and general context throughout, offering this only when it aids understanding of the formal material, in particular to illustrate the path from intuition to formalisation. Together these books are a suitable compendium for graduate students and researchers in the area of computer science and mathematical logic. (shrink)

In this thesis I investigate the implications, for one's account of mathematics, of holding an anti-realist view. The primary aim is to appraise the scope of revision imposed by anti-realism on classical inferential practice in mathematics. That appraisal has consequences both for our understanding of the nature of mathematics and for our attitude towards anti-realism itself. If an anti-realist position seems inevitably to be absurdly revisionary then we have grounds for suspecting the coherence of arguments canvassed in favour of anti-realism. (...) I attempt to defend the anti-realist position by arguing, i) that it is not internally incoherent for anti-realism to be a potentially revisionary position, and ii) that an anti-realist position can, plausibly, be seen to result in a stable intuitionistic position with regard to the logic it condones. The use of impredicative methods in classical mathematics is a site of traditional intuitionistic attacks. I undertake an examination of what the anti-realist attitude towards such methods should be. This question is of interest both because such methods are deeply implicated in classical mathematical theory of analysis and because intuitionistic semantic theories make use of impredicative methods. I attempt to construct the outlines of a set theory which is anti-realistically acceptable but which, although having no antecedent repugnance for impredicative methods as such, appears to be too weak to offer an anti-realistic vindication of impredicative methods in general. I attempt to exonerate intuitionistic semantic theories in their use of impredicative methods by showing that a partial order relying on the nature of our grasp of the intuitionistic meaning stipulations for the logical constants precludes a possible circularity. (shrink)

This new and revised edition of Peter Kreeft's Socratic Logic is updated, adding new exercises and more complete examples, all with Kreeft's characteristic clarity and wit. Since its introduction in the spring of 2004, Socratic Logic has proven to be a different type of logic text: This is the only complete system of classical Aristotelian logic in print. The "old logic" is still the natural logic of the four language arts. Symbolic, or "mathematical," logic is not for the humanities. This (...) book is simple and user-friendly. It is highly interactive, with a plethora of exercises and a light, engaging style. It is practical. It is designed for do-it-yourselfers as well as classrooms. It emphasizes topics in proportion to probable student use: e.g., interpreting ordinary language, not only analyzing but also constructing effective arguments, smoking out hidden assumptions, making "argument maps," and using Socratic method in various circumstances. It is philosophical. Its exercises expose students to many classical quotations, and additional chapters introduce philosophical issues in a Socratic manner and from a commonsense, realistic point of view. It prepares students for reading Great Books rather than Dick and Jane, and models Socrates as the beginner's ideal teacher and philosopher. (shrink)

This is the second part of a paper devoted to the study of the maximal intermediate propositional logics with the disjunction property , whose first part has appeared in this journal with the title “A method to single out maximal propositional logics with the disjunction property I”. In the first part we have explained the general results upon which a method to single out maximal constructive logics is based and have illustrated such a method by exhibiting the (...) Kripke semantics of maximal constructive logics extending the logic ST of Scott, for which, in turn, a semantical characterization in terms of Kripke frames has been given. In the present part we complete the illustration of the method of the first part, having in mind some aspects which might be interesting for a classification of the maximal constructive logics, and an application of the heuristic content of the method to detect the nonmaximality of apparently maximal constructive logics. Thus, on the one hand we introduce the logic AST , which is compared with ST and is seen as a logic “alternative” to it, in a sense which will be precisely explained. We provide a Kripke semantics for AST and show that there are maximal constructive logics which neither are extensions of ST nor are extensions of AST. Finally, we give a further application of the results of the first part by exhibiting the Kripke semantics of a maximal constructive logic extending AST. On the other hand, we compare the maximal constructive logics presented in both parts of the paper with a constructive logic introduced by Maksimova , which has been conjectured to be maximal by Chagrov and Zacharyashchev ; from this comparison a disproof of the conjecture arises. (shrink)

A method for constructing continua of logics squeezed between some intermediate predicate logics, developed by Suzuki [8], is modified and applied to intervals of the form [L, L+ ¬¬S], where Lis a predicate logic, Sis a closed predicate formula. This solves one of the problems from Suzuki's paper.

The issue of the emergence of formal axiomatic logical systems due to the emergence of logical antinomies in formal axiomatic systems, specifically the issue of developing formal logical axiomatics in the calculus of considerations was investigated in the considered research. At the same time, in order to determine the characteristics of the implementation of the logical-methodological principles and provisions of the deductive reasoning obviously, conceptual-logical foundations of the calculus of considerations was studied and the main (...) propositions of the calculus of considerations and the analysis of the initial logical operations on them were given in the study. The basic laws of logic and the expression of one logical act with another one were also explained in the research work by giving the concepst of proportional form and tautology in the calculus of considerations. At the same time, main properties and the procedure of setting of the formal deductive theory in the calculus of considerations were studied in the article. In order to provide an adequate characterization of the conceptual and methodological bases of the calculus of considerations in the research work, the basic logical laws of the calculus of considerations developed by the German mathematician and thinker P. Hilbert were also given. In addition, the main principles and methodological provisions of the deductive method were investigated, and the rules of deriving new conclusions from the axioms were interpreted. At the end of the reviewed article, the main principles and methods of the formal axiomatic mathematical systems built on a deductive basis, specifically the calculus of considerations, were studied and a logical-methodological analysis of the calculus of considerations was carried out on this basis. In this framework, the syntactic and semantic analysis of formalized language in formal axiomatic logical systems was carried out separately, the constituent parts of those systems, including the construction scheme, were given. (shrink)

This new and revised edition of Peter Kreeft's Socratic Logic is updated, adding new exercises and more complete examples, all with Kreeft's characteristic clarity and wit. Since its introduction in the spring of 2004, Socratic Logic has proven to be a different type of logic text: This is the only complete system of classical Aristotelian logic in print. The "old logic" is still the natural logic of the four language arts. Symbolic, or "mathematical," logic is not for the humanities. This (...) book is simple and user-friendly. It is highly interactive, with a plethora of exercises and a light, engaging style. It is practical. It is designed for do-it-yourselfers as well as classrooms. It emphasizes topics in proportion to probable student use: e.g., interpreting ordinary language, not only analyzing but also constructing effective arguments, smoking out hidden assumptions, making "argument maps," and using Socratic method in various circumstances. It is philosophical. Its exercises expose students to many classical quotations, and additional chapters introduce philosophical issues in a Socratic manner and from a commonsense, realistic point of view. It prepares students for reading Great Books rather than Dick and Jane, and models Socrates as the beginner's ideal teacher and philosopher. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Reviewed by:Hegel's Concept of Life: Self-Consciousness, Freedom, Logic by Karen NgMarina F. BykovaKaren Ng. Hegel's Concept of Life: Self-Consciousness, Freedom, Logic. Oxford: Oxford University Press, 2020. Pp. iii + 319. Hardback, $85.00.In her insightful book, Karen Ng defends the fundamental significance of Hegel's concept of life, which she considers "constitutive" not merely of his dynamic account of reason but also of his "idealist program" itself (3–4), the very core (...) structure that grounds his idealism (151, 291). Integrating this concept into his philosophy allows Hegel to resolve three systematic problems indicated in the subtitle of the book: self-consciousness, freedom or self-determination, and logical form.The book offers a systematic account of the logical concept of life by defending three interconnected claims, helpfully summarized in chapter 1. The first, rather general claim clarifies Ng's interpretive point of departure. She maintains that the "core tenets" of Hegel's philosophy can be understood through the "purposiveness theme" inherited from Kant's a priori principle of purposiveness in his Critique of Judgment (5, 43). Ng advances this claim in chapter 2, where she argues that it is working through Kant's concept of inner purposiveness of natural ends from the teleological power of judgment that leads Hegel to develop his own account of Concept, which is foundational for his entire philosophical system (61–64).Chapter 3 continues the examination of Hegel's immediate predecessors in relation to the interpretation of his "speculative identity thesis" (8), which constitutes the second claim. According to Ng, Hegel first introduces this thesis—which remains central to all his works—in the Differenzschift. In this work, Hegel, siding with Schelling, criticizes Fichte's subjective version of idealism, more specifically for his explanation of the relation between subject and object as constructed and facilitated by subjective consciousness itself, thus rendering "nature lifeless and dead" (88). Opposing this approach, in the Phenomenology of Spirit, Hegel interprets the identity of object and subject as facilitated by the relation and opposition between self-consciousness and life. Thus, the speculative identity thesis assumes a fundamental significance of life for self-consciousness. Unfolding this thesis is at the core of Hegel's absolute method in the Science of Logic as well. On Ng's reading, Hegel's speculative thesis along with his account of self-consciousness also have their roots in the purposiveness theme inherited from Kant. What is at work here is Kant's claim that the principle of purposiveness is a necessary condition for judgment, which Hegel develops into a claim about the concept of life as being constitutive for self-consciousness.In the rest of the book (chapters 4–8), devoted to a close analysis of Hegel's Science of Logic, Ng advances her third and perhaps most original claim: understanding Hegel's [End Page 527] Subjective Logic "as his own version of a 'critique of judgment'" (18). She demonstrates that this part of the Logic offers a positive account of the concept of life that Hegel develops in his critical interaction with Kant. The most illustrative in this respect is the "Subjectivity" section of the Logic, in which the form of life is presented as the ground and presupposition of Hegel's theory of concepts and judgments. According to Ng, such a reading makes evident that "life opens up the space of reasons" (234, 281).Chapter 4 takes up Hegel's immanent deduction of the Concept argument, revealing that the key to understanding it is the concept of reciprocity (Wechselwirkung) (127), most fully recognizable as an internal purposiveness of the Concept itself.Ng traces the purposiveness theme through the entirety of the Subjective Logic, establishing its significance for understanding the deduction of the Concept and for the transition to the Idea. In her analysis, she closely follows Hegel's division of the Subjective Logic and addresses Subjectivity, Objectivity, and the Idea in chapters 5–8, respectively. With chapter 5, Ng moves to a discussion of how Hegel finds judgment to arise from the "original judgement of life," the initial unity and division of subject and object within life (172). Chapter 6 extends this discussion by elucidating the tension Hegel... (shrink)

Regions-based theories of space aim—among others—to define points in a geometrically appealing way. The most famous definition of this kind is probably due to Whitehead. However, to conclude that the objects defined are points indeed, one should show that they are points of a geometrical or a topological space constructed in a specific way. This paper intends to show how the development of mathematical tools allows showing that Whitehead’s method of extensive abstraction provides a construction of objects that are (...) fundamental building blocks of specific topological spaces. (shrink)

This chapter examines the role and nature of Thomas Hobbes's method for science in its historical context to clarify how his theoretical and practical philosophies are unified. Hobbes's politics is commonly studied independently of his method and science. Civil philosophy, for Hobbes, primarily concerns the commonwealth and the duties plus rights of its subjects. Methodical philosophizing produces scientia, i.e., valid causal syllogisms, in the shortest way possible. Hobbes's use of mechanical analogies give the impression that civil science employs (...) a method of mechanical construction at odds with method in physics. The bifurcation of philosophical method into a universal ordering of subject matter and a particular method of demonstration was commonplace in the logic textbooks of Hobbes's intellectual context. Method aimed at unqualified knowledge uses resolution/analysis and composition/synthesis in the Scholastic sense to holistically know as much as possible about body. (shrink)