In recent years, the e ffort to formalize erotetic inferences (i.e., inferences to and from questions) has become a central concern for those working in erotetic logic. However, few have sought to formulate a proof theory for these inferences. To fill this lacuna, we construct a calculus for (classes of) sequents that are sound and complete for two species of erotetic inferences studied by Inferential Erotetic Logic (IEL): erotetic evocation and regular erotetic implication. While an attempt has been (...) made to axiomatize the former in a sequent system, there is currently no proof theory for the latter. Moreover, the extant axiomatization of erotetic evocation fails to capture its defeasible character and provides no rules for introducing or eliminating question-forming operators. In contrast, our calculus encodes defeasibility conditions on sequents and provides rules governing the introduction and elimination of erotetic formulas. We demonstrate that an elimination theorem holds for a version of the cut rule that applies to both declarative and erotetic formulas and that the rules for the axiomatic account of question evocation in IEL are admissible in our system. (shrink)

It is well known that the circumflex notation used by Russell and Whitehead to form complex function names in Principia Mathematica played a role in inspiring Alonzo Church's “lambda calculus” for functional logic developed in the 1920s and 1930s. Interestingly, earlier unpublished manuscripts written by Russell between 1903–1905—surely unknown to Church—contain a more extensive anticipation of the essential details of the lambda calculus. Russell also anticipated Schönfinkel's combinatory logic approach of treating multiargument functions as functions having other functions (...) as value. Russell’s work in this regard seems to have been largely inspired by Frege’s theory of functions and “value-ranges”. This system was discarded by Russell due to his abandonment of propositional functions as genuine entities as part of a new tack for solving Russell’s paradox. In this article, I explore the genesis and demise of Russell’s early anticipation of the lambda calculus. (shrink)

Gottfried Leibniz embarked on a research program to prove all the Aristotelic categorical syllogisms by diagrammatic and algebraic methods. He succeeded in proving them by means of Euler diagrams, but didn’t produce a manuscript with their algebraic proofs. We demonstrate how key excerpts scattered across various Leibniz’s drafts on logic contained sufficient ingredients to prove them by an algebraic method –which we call the Leibniz-Cayley (LC) system– without having to make use of the more expressive and complex machinery of first-order (...) quantificational logic. In addition, we prove the classic categorical syllogisms again by a relational method –which we call the McColl-Ladd (ML) system– employing categorical relations studied by Hugh McColl and Christine Ladd. Finally, we show the connection of ML and LC with Boolean algebra, proving that ML is a consequence of LC, and that LC is a consequence of the Boolean lattice axioms, thus establishing Leibniz’s historical priority over George Boole in characterizing and applying (a sufficient fragment of) Boolean algebra to effectively tackle categorical syllogistic. (shrink)

This paper aims to address the difficulties of high school students in bridging their computational understanding with their visualization skills in understanding the notion of the limits in their calculus class. This research used a pre-experimental one-group pretest-posttest design research on 62 grade 10 students enrolled in the Science, Technology, and Engineering Program (STEP) in one of the public high schools in Zamboanga del Sur, Philippines. A series of remedial sessions were given to help them understand the function values, (...) one-sided limits, limits of a function, and its discontinuities with the aid of the GeoGebra. Results revealed that there is a significant improvement in their mathematics performance before and after the remediation, t(61) = -19.99, p<.05. The real-time feedback provided by the drawing interface of GeoGebra has provided the students the insights into how objects behave in geometrical space, bridging their understanding in locating the values shown by the computational algebraic processes. Additionally, the significant difference in the achievement reveals a large effect size (Cohen’s =2.54) which could be accounted for by the remediation using interactive activities in the GeoGebra. (shrink)

The purpose of this paper is to outline an alternative approach to introductory logic courses. Traditional logic courses usually focus on the method of natural deduction or introduce predicate calculus as a system. These approaches complicate the process of learning different techniques for dealing with categorical and hypothetical syllogisms such as alternate notations or alternate forms of analyzing syllogisms. The author's approach takes up observations made by Dijkstrata and assimilates them into a reasoning process based on modified notations. The (...) author's model adopts a notation that addresses the essentials of a problem while remaining easily manipulated to serve other analytic frameworks. The author also discusses the pedagogical benefits of incorporating the model into introductory logic classes for topics ranging from syllogisms to predicate calculus. Since this method emphasizes the development of a clear and manipulable notation, students can worry less about issues of translation, can spend more energy solving problems in the terms in which they are expressed, and are better able to think in abstract terms. (shrink)

In New Essays on Human Understanding, book II, chapter xxi Leibniz presents an interesting picture of the human mind as not only populated by perceptions, volitions and appetitions, but also by endeavours. The endeavours in question can be divided to entelechy and effort; Leibniz calls entelechy as primitive active forces and efforts as derivative forces. The entelechy, understood as primitive active force is to be equated with a substantial form, as Leibniz says: “When an entelechy – i.e. a primary or (...) substantial endeavour – is accompanied by perception, it is a soul” (NE II, xxi, §1; RB, 170). What about efforts, then? One is certainly the will. In NE, II, xxi, §5 Leibniz argues that volition is the effort (conatus) to move towards what one finds good and away from what one finds bad and that this endeavor arises from the perceptions we are aware of. As an endeavour results in action unless it is prevented, from will (which is always directed to the good) and power together follows action. However, this is not so simple. Leibniz argues that there is also a second class of efforts: “There are other efforts, arising from insensible perceptions, which we are not aware of; I prefer to call these ‘appetitions’ rather than volitions” (NE II, xxi, §5; RB, 173). Although there are appetitions of which one can be aware, usually these appetitions arise from the insensible petite perceptions and are consequently affecting us subconsciously. Now, although all minute perceptions are confused perceptions, they are always related to pleasure and displeasure and also to perfection and imperfection. From this follows that there can be different efforts present in the soul at the same time: the will which is directed to apperceived good and several separate appetitions which lead to different goals, both to those which bring about perfection and pleasure of the mind (joy) and those which bring about displeasure and imperfection (sorrow). These efforts are not only in conflict with each other but may also be in conflict with entelechy. A typical case is perceiving a sensual pleasure. Our entelechy which is always directed to final causes (perfection) may be in conflict with several different appetitions which are related in different ways to the sensual pleasure in question. If our understanding is developed enough, our will resists the temptation posed by the pleasure (agreeing with entelechy), but if the temptation is too strong, the appetitions outweigh the will and the resulting action bring about imperfection and sorrow as it is related to imperfection. In this paper I will argue that deliberation in the human soul is a battle of different endeavors described above: the entelechy in the soul strives according to its law-of-the-series towards its telos (perfection) and the will accompanies it by being automatically directed to the good. This thrust towards the apparent good is aided or hindered by the appetitions which can be thought as derivative forces in the Leibnizian dynamics. Depending on whether the predominant appetitions are related to good or bad desires, the deliberation succeeds or fails in achieving the real good which is the goal of human deliberation. The successs can be facilitated beforehand by developing our understanding so that we are less easily swept away by the derivative forces (NE II, xxi, §19). A central role in this task is played by strong willing. As Martha Bolton has noted in her recent paper, an essential feature of the basic, standing endeavors is that they are continuous – although the power balance in the soul changes from moment to moment, something lingers from our previous volitions. That is why Leibniz argues that we pave way for the future deliberations by our previous voluntary actions (NE II, xxi, §23). In contrast, the appetitions are temporary, fliegende Gedanken as Leibniz says in NE II, xxi, §12. Therefore there is a constant, always changing power balance between two kinds of endeavors in the soul: primitive active force versus derivative forces. I will argue that the behavior of the forces in the soul can be understood with a vectorial model which is related to Leibniz’s early ideas of calculus of variations and which was anticipated by Arnauld and Nicole’s Port-Royal Logic. The central idea in the model is that the options are in tension towards each other and the ratio between them at each moment determines the consequent outcome. The proper relationship between the endeavors is not a simple balance, two options which exhaust each other, but a case where different efforts complement each other: “Since the final result is determined by how things weigh against one another, I should think it could happen that he most pressing disquiet did not prevail; for even If it prevailed over each of the contrary endeavours taken singly, it may be outweighed by all of them together.” Leibniz continues : “Everything which then impinges on us weighs in the balance and contributes to determining a resultant direction, almost as in mechanics” (NE II, xxi, §40; RB, 193). The different endeavors can be understood as vectors leading to different directions and the end result is a certain direction that deliberation takes. The dynamical tension between the different endeavors presents a situation where everything affects everything and the following direction, the resulting volition follows more or less automatically. In Theodicy, §325 Leibniz describes the deliberation as follows: “One might, instead of the balance, compare the soul with a force that puts forth an effort on various sides simultaneously, but which acts only at the spot where action is easiest or there is least resistance” (Huggard, 322) This kind of dynamical tension can be understood in terms of the calculus of variations where there are several possible variations available but where the dynamics of the situation results in the decision taking the “easiest” route which is more or less objectively good depending on the level of the deliberator’s understanding. In his comments to Bayle’s note L of “Rorarius” Leibniz says: “The soul, even though it has no parts, has within it, because of the multitude of representations of external things, or rather because of the representation of the universe lodged within it by the creator, a great number, or rather an infinite number, of variations (Woolhouse & Francks (ed.), ‘New System’ and Associated Texts, 101). This kind of deliberation is comparable to God’s choice of the best world with the difference that God’s understanding is infinite which again results in the fact that the choice is the best possible. Whereas in nature the easiest route taken is always optimal as nature is God’s creation, in men the goodness or badness of men’s actions is dependent on their state of wisdom, that is, how developed their understanding is. The more wise men are, the more metaphysical goodness or perfection follows from their actions. (shrink)

In New Essays on Human Understanding, book II, chapter xxi Leibniz presents an interesting picture of the human mind as not only populated by perceptions, volitions and appetitions, but also by endeavours. The endeavours in question can be divided to entelechy and effort; Leibniz calls entelechy as primitive active forces and efforts as derivative forces. The entelechy, understood as primitive active force is to be equated with a substantial form, as Leibniz says: “When an entelechy – i.e. a primary or (...) substantial endeavour – is accompanied by perception, it is a soul” (NE II, xxi, §1; RB, 170). What about efforts, then? One is certainly the will. In NE, II, xxi, §5 Leibniz argues that volition is the effort (conatus) to move towards what one finds good and away from what one finds bad and that this endeavor arises from the perceptions we are aware of. As an endeavour results in action unless it is prevented, from will (which is always directed to the good) and power together follows action. However, this is not so simple. Leibniz argues that there is also a second class of efforts: “There are other efforts, arising from insensible perceptions, which we are not aware of; I prefer to call these ‘appetitions’ rather than volitions” (NE II, xxi, §5; RB, 173). Although there are appetitions of which one can be aware, usually these appetitions arise from the insensible petite perceptions and are consequently affecting us subconsciously. Now, although all minute perceptions are confused perceptions, they are always related to pleasure and displeasure and also to perfection and imperfection. From this follows that there can be different efforts present in the soul at the same time: the will which is directed to apperceived good and several separate appetitions which lead to different goals, both to those which bring about perfection and pleasure of the mind (joy) and those which bring about displeasure and imperfection (sorrow). These efforts are not only in conflict with each other but may also be in conflict with entelechy. A typical case is perceiving a sensual pleasure. Our entelechy which is always directed to final causes (perfection) may be in conflict with several different appetitions which are related in different ways to the sensual pleasure in question. If our understanding is developed enough, our will resists the temptation posed by the pleasure (agreeing with entelechy), but if the temptation is too strong, the appetitions outweigh the will and the resulting action bring about imperfection and sorrow as it is related to imperfection. In this paper I will argue that deliberation in the human soul is a battle of different endeavors described above: the entelechy in the soul strives according to its law-of-the-series towards its telos (perfection) and the will accompanies it by being automatically directed to the good. This thrust towards the apparent good is aided or hindered by the appetitions which can be thought as derivative forces in the Leibnizian dynamics. Depending on whether the predominant appetitions are related to good or bad desires, the deliberation succeeds or fails in achieving the real good which is the goal of human deliberation. The successs can be facilitated beforehand by developing our understanding so that we are less easily swept away by the derivative forces (NE II, xxi, §19). A central role in this task is played by strong willing. As Martha Bolton has noted in her recent paper, an essential feature of the basic, standing endeavors is that they are continuous – although the power balance in the soul changes from moment to moment, something lingers from our previous volitions. That is why Leibniz argues that we pave way for the future deliberations by our previous voluntary actions (NE II, xxi, §23). In contrast, the appetitions are temporary, fliegende Gedanken as Leibniz says in NE II, xxi, §12. Therefore there is a constant, always changing power balance between two kinds of endeavors in the soul: primitive active force versus derivative forces. I will argue that the behavior of the forces in the soul can be understood with a vectorial model which is related to Leibniz’s early ideas of calculus of variations and which was anticipated by Arnauld and Nicole’s Port-Royal Logic. The central idea in the model is that the options are in tension towards each other and the ratio between them at each moment determines the consequent outcome. The proper relationship between the endeavors is not a simple balance, two options which exhaust each other, but a case where different efforts complement each other: “Since the final result is determined by how things weigh against one another, I should think it could happen that he most pressing disquiet did not prevail; for even If it prevailed over each of the contrary endeavours taken singly, it may be outweighed by all of them together.” Leibniz continues : “Everything which then impinges on us weighs in the balance and contributes to determining a resultant direction, almost as in mechanics” (NE II, xxi, §40; RB, 193). The different endeavors can be understood as vectors leading to different directions and the end result is a certain direction that deliberation takes. The dynamical tension between the different endeavors presents a situation where everything affects everything and the following direction, the resulting volition follows more or less automatically. In Theodicy, §325 Leibniz describes the deliberation as follows: “One might, instead of the balance, compare the soul with a force that puts forth an effort on various sides simultaneously, but which acts only at the spot where action is easiest or there is least resistance” (Huggard, 322) This kind of dynamical tension can be understood in terms of the calculus of variations where there are several possible variations available but where the dynamics of the situation results in the decision taking the “easiest” route which is more or less objectively good depending on the level of the deliberator’s understanding. In his comments to Bayle’s note L of “Rorarius” Leibniz says: “The soul, even though it has no parts, has within it, because of the multitude of representations of external things, or rather because of the representation of the universe lodged within it by the creator, a great number, or rather an infinite number, of variations (Woolhouse & Francks (ed.), ‘New System’ and Associated Texts, 101). This kind of deliberation is comparable to God’s choice of the best world with the difference that God’s understanding is infinite which again results in the fact that the choice is the best possible. Whereas in nature the easiest route taken is always optimal as nature is God’s creation, in men the goodness or badness of men’s actions is dependent on their state of wisdom, that is, how developed their understanding is. The more wise men are, the more metaphysical goodness or perfection follows from their actions. (shrink)

Chiastic order is an ancient expression for cross-classification. Cross-classification, in turn, is one of many terms used for the operation of conjoining or cross-mapping one domain, class or set of concepts with another. As such, it is the primordial form of non-quantified modelling and combinatory heuristics. This article presents a brief epistemological history of non-quantified modelling: its prehistory in the form of rhetorical chiasmus; its early (pre-symbolic) use by Plato as a cross-order (paradigmatic) modelling method; and its “modern” (symbolic) use (...) by Leibniz as a calculus of concepts. It will also be shown how classification theory itself is built on a cross-classificatory construct involving two fundamental logical/structural relationships: subordination and conjunction. Finally, examples of modern computer-aided, non-quantified modal modelling are presented in the areas of design theory, operational research and decision science. (shrink)

The previous Part I of the paper discusses the option of the Gödel incompleteness statement (1931: whether “Satz VI” or “Satz X”) to be an axiom due to the pair of the axiom of induction in arithmetic and the axiom of infinity in set theory after interpreting them as logical negations to each other. The present Part II considers the previous Gödel’s paper (1930) (and more precisely, the negation of “Satz VII”, or “the completeness theorem”) as a necessary condition for (...) granting the Gödel incompleteness statement to be a theorem just as the statement itself, to be an axiom. Then, the “completeness paper” can be interpreted as relevant to Hilbert mathematics, according to which mathematics and reality as well as arithmetic and set theory are rather entangled or complementary rather than mathematics to obey reality able only to create models of the latter. According to that, both papers (1930; 1931) can be seen as advocating Russell’s logicism or the intensional propositional logic versus both extensional arithmetic and set theory. Reconstructing history of philosophy, Aristotle’s logic and doctrine can be opposed to those of Plato or the pre-Socratic schools as establishing ontology or intensionality versus extensionality. Husserl’s phenomenology can be analogically realized including and particularly as philosophy of mathematics. One can identify propositional logic and set theory by virtue of Gödel’s completeness theorem (1930: “Satz VII”) and even both and arithmetic in the sense of the “compactness theorem” (1930: “Satz X”) therefore opposing the latter to the “incompleteness paper” (1931). An approach identifying homomorphically propositional logic and set theory as the same structure of Boolean algebra, and arithmetic as the “half” of it in a rigorous construction involving information and its unit of a bit. Propositional logic and set theory are correspondingly identified as the shared zero-order logic of the class of all first-order logics and the class at issue correspondingly. Then, quantum mechanics does not need any quantum logics, but only the relation of propositional logic, set theory, arithmetic, and information: rather a change of the attitude into more mathematical, philosophical, and speculative than physical, empirical and experimental. Hilbert’s epsilon calculus can be situated in the same framework of the relation of propositional logic and the class of all mathematical theories. The horizon of Part III investigating Hilbert mathematics (i.e. according to the Pythagorean viewpoint about the world as mathematical) versus Gödel mathematics (i.e. the usual understanding of mathematics as all mathematical models of the world external to it) is outlined. (shrink)

PLEASE NOTE: This is the corrected 2nd eBook edition, 2021. ●●●●● _Critique of Impure Reason_ has now also been published in a printed edition. To reduce the otherwise high price of this scholarly, technical book of nearly 900 pages and make it more widely available beyond university libraries to individual readers, the non-profit publisher and the author have agreed to issue the printed edition at cost. ●●●●● The printed edition was released on September 1, 2021 and is now available through (...) all booksellers, including Barnes & Noble, Amazon, and brick-and-mortar bookstores under ISBN 978-0-578-88646-6. ●●●●● In light of the length of this book, readers who would like to have a more detailed description of the book's objectives and method may find it helpful to read the detailed and clearly written Wikipedia entry about this work: From the Wikipedia search page, use the search phrase "Critique of Impure Reason". At least at the time of this writing (11/29/2021), the Wikipedia entry is well-researched and accurate. ●●●●● In addition, a "Primer on Bartlett's CRITIQUE OF IMPURE REASON" has been made available by the author. It is available under its title through PhilPapers and other philosophy online archives. ●●●●● COMMENDATIONS OF THIS WORK, from the back cover of the published edition: ●●●●● “I admire its range of philosophical vision.” – Nicholas Rescher, Distinguished University Professor of Philosophy, University of Pittsburgh, author of more than 100 books. ●●●●● “Bartlett’s _Critique of Impure Reason_ is an impressive, bold, and ambitious work. Careful scholarship is balanced by original analyses that lead the reader to recognize the limits of meaning, knowledge, and conceptual possibility. The work addresses a host of traditional philosophical problems, among them the nature of space, time, causality, consciousness, the self, other minds, ontology, free will and determinism, and others. The book culminates in a fascinating and profound new understanding of relativity physics and quantum theory.” – Gerhard Preyer, Professor of Philosophy, Goethe-University, Frankfurt am Main, Germany, author of many books including _Concepts of Meaning_, _Beyond Semantics and Pragmatics_, _Intention and Practical Thought_, and _Contextualism in Philosophy_. ●●●●● “[This work’s] goal is of a unique and difficult species: Dr. Bartlett seeks to develop a formal logical calculus on the basis of transcendental philosophical arguments; in fact, he hopes that this calculus will be the formal expression of the transcendental foundation of knowledge.... I consider Dr. Bartlett’s work soundly conceived and executed with great skill.” – C. F. von Weizsäcker, philosopher and physicist, former Director, Max-Planck-Institute, Starnberg, Germany. ●●●●● “Bartlett has written an American “Prolegomena to All Future Metaphysics.” He aims rigorously to eliminate meaningless assertions, reach bedrock, and place philosophy on a firm foundation that will enable it, like science and mathematics, to produce lasting results that generations to come can build on. This is a great book, the fruit of a lifetime of research and reflection, and it deserves serious attention.” — Martin X. Moleski, former Professor, Canisius College, Buffalo, NY, studies of scientific method, the presuppositions of thought, and the self-referential nature of epistemology. ●●●●● “Bartlett has written a book on what might be called the underpinnings of philosophy. It has fascinating depth and breadth, and is all the more striking due to its unifying perspective based on the concepts of reference and self-reference.” – Don Perlis, Professor of Computer Science, University of Maryland, author of numerous publications on self-adjusting autonomous systems and philosophical issues concerning self-reference, mind, and consciousness. ●●●●● ●●●●● The _Critique of Impure Reason: Horizons of Possibility and Meaning_ comprises a major and important contribution to philosophy. Thanks to the generosity of its publisher, this massive 885-page volume has been published as a free open access eBook (3.75MB) as well as an open access printed edition. It inaugurates a revolutionary paradigm shift in philosophical thought by providing compelling and long-sought-for solutions to a wide range of philosophical problems. In the process, the work fundamentally transforms the way in which the concepts of reference, meaning, and possibility are understood. The book includes a Foreword by the celebrated German philosopher and physicist Carl Friedrich von Weizsäcker. ●●●●● In Kant’s _Critique of Pure Reason_ we find an analysis of the preconditions of experience and of knowledge. In contrast, but yet in parallel, the new _Critique_ focuses upon the ways—unfortunately very widespread and often unselfconsciously habitual—in which many of the concepts that we employ _conflict_ with the very preconditions of meaning and of knowledge. ●●●●● This is a book about the boundaries of frameworks and about the unrecognized conceptual confusions in which we become entangled when we attempt to transgress beyond the limits of the possible and meaningful. We tend either not to recognize or not to accept that we all-too-often attempt to trespass beyond the boundaries of the frameworks that make knowledge possible and the world meaningful. ●●●●● The _Critique of Impure Reason_ proposes a bold, ground-breaking, and startling thesis: that a great many of the major philosophical problems of the past can be solved through the recognition of a viciously deceptive form of thinking to which philosophers as well as non-philosophers commonly fall victim. For the first time, the book advances and justifies the criticism that a substantial number of the questions that have occupied philosophers fall into the category of “impure reason,” violating the very conditions of their possible meaningfulness. ●●●●● The purpose of the study is twofold: first, to enable us to recognize the boundaries of what is referentially forbidden—the limits beyond which reference becomes meaningless—and second, to avoid falling victims to a certain broad class of conceptual confusions that lie at the heart of many major philosophical problems. As a consequence, the boundaries of _possible meaning_ are determined. ●●●●● Bartlett, the author or editor of more than 20 books, is responsible for identifying this widespread and delusion-inducing variety of error, _metalogical projection_. It is a previously unrecognized and insidious form of erroneous thinking that undermines its own possibility of meaning. It comes about as a result of the pervasive human compulsion to seek to transcend the limits of possible reference and meaning. ●●●●● Based on original research and rigorous analysis combined with extensive scholarship, the _Critique of Impure Reason_ develops a self-validating method that makes it possible to recognize, correct, and eliminate this major and pervasive form of fallacious thinking. In so doing, the book provides at last provable and constructive solutions to a wide range of major philosophical problems. ●●●●● CONTENTS AT A GLANCE ▪▪▪▪▪ Preface ▪▪▪▪▪ Foreword by Carl Friedrich von Weizsäcker ▪▪▪▪▪ Acknowledgments ▪▪▪▪▪ Avant-propos: A philosopher’s rallying call ▪▪▪▪▪ Introduction ▪▪▪▪▪ A note to the reader ▪▪▪▪▪ A note on conventions ▪▪▪▪▪ ▪▪▪▪▪ ▪▪▪▪▪ PART I ▪▪▪▪▪ ▪▪▪▪▪ WHY PHILOSOPHY HAS MADE NO PROGRESS AND HOW IT CAN ▪▪▪▪▪ 1 Philosophical-psychological prelude ▪▪▪▪▪ 2 Putting belief in its place: Its psychology and a needed polemic ▪▪▪▪▪ 3 Turning away from the linguistic turn: From theory of reference to metalogic of reference ▪▪▪▪▪ 4 The stepladder to maximum theoretical generality ▪▪▪▪▪ ▪▪▪▪▪ ▪▪▪▪▪ PART II ▪▪▪▪▪ THE METALOGIC OF REFERENCE ▪▪▪▪▪ A New Approach to Deductive, Transcendental Philosophy ▪▪▪▪▪ 5 Reference, identity, and identification ▪▪▪▪▪ 6 Self-referential argument and the metalogic of reference ▪▪▪▪▪ 7 Possibility theory ▪▪▪▪▪ 8 Presupposition logic, reference, and identification ▪▪▪▪▪ 9 Transcendental argumentation and the metalogic of reference ▪▪▪▪▪ 10 Framework relativity ▪▪▪▪▪ 11 The metalogic of meaning ▪▪▪▪▪ 12 The problem of putative meaning and the logic of meaninglessness ▪▪▪▪▪ 13 Projection ▪▪▪▪▪ 14 Horizons ▪▪▪▪▪ 15 De-projection ▪▪▪▪▪ 16 Self-validation ▪▪▪▪▪ 17 Rationality: Rules of admissibility ▪▪▪▪▪ ▪▪▪▪▪ ▪▪▪▪▪ PART III ▪▪▪▪▪ PHILOSOPHICAL APPLICATIONS OF THE METALOGIC OF REFERENCE ▪▪▪▪▪ Major Problems and Questions of Philosophy and the Philosophy of Science ▪▪▪▪▪ 18 Ontology and the metalogic of reference ▪▪▪▪▪ 19 Discovery or invention in general problem-solving, mathematics, and physics ▪▪▪▪▪ 20 The conceptually unreachable: “The far side” ▪▪▪▪▪ 21 The projections of the external world, things-in-themselves, other minds, realism, and idealism ▪▪▪▪▪ 22 The projections of time, space, and space-time ▪▪▪▪▪ 23 The projections of causality, determinism, and free will ▪▪▪▪▪ 24 Projections of the self and of solipsism ▪▪▪▪▪ 25 Non-relational, agentless reference and referential fields ▪▪▪▪▪ 26 Relativity physics as seen through the lens of the metalogic of reference ▪▪▪▪▪ 27 Quantum theory as seen through the lens of the metalogic of reference ▪▪▪▪▪ 28 Epistemological lessons learned from and applicable to relativity physics and quantum theory ▪▪▪▪▪ ▪▪▪▪▪ PART IV ▪▪▪▪▪ HORIZONS ▪▪▪▪▪ 29 Beyond belief ▪▪▪▪▪ 30 _Critique of Impure Reason_: Its results in retrospect ▪▪▪▪▪ ▪▪▪▪▪ SUPPLEMENT ▪▪▪▪▪ The Formal Structure of the Metalogic of Reference ▪▪▪▪▪ APPENDIX I ▪▪▪▪▪ The Concept of Horizon in the Work of Other Philosophers ▪▪▪▪▪ APPENDIX II ▪▪▪▪▪ Epistemological Intelligence ▪▪▪▪▪ References ▪▪▪▪▪ Index ▪▪▪▪▪ About the author. 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Quasi-Nelson logic (QNL) was recently introduced as a common generalisation of intuitionistic logic and Nelson's constructive logic with strong negation. Viewed as a substructural logic, QNL is the axiomatic extension of the Full Lambek Calculus with Exchange and Weakening by the Nelson axiom, and its algebraic counterpart is a variety of residuated lattices called quasi-Nelson algebras. Nelson's logic, in turn, may be obtained as the axiomatic extension of QNL by the double negation (or involutivity) axiom, and intuitionistic logic as (...) the extension of QNL by the contraction axiom. A recent series of papers by the author and collaborators initiated the study of fragments of QNL, which correspond to subreducts of quasi-Nelson algebras. In the present paper we focus on fragments that contain the connectives forming a residuated pair (the monoid conjunction and the so-called strong Nelson implication), these being the most interesting ones from a substructural logic perspective. We provide quasi-equational (whenever possible, equational) axiomatisations for the corresponding classes of algebras, obtain twist representations for them, study their congruence properties and take a look at a few notable subvarieties. Our results specialise to the involutive case, yielding characterisations of the corresponding fragments of Nelson's logic and their algebraic counterparts. (shrink)

We study a fragment of Intuitionistic Linear Logic combined with non-normal modal operators. Focusing on the minimal modal logic, we provide a Gentzen-style sequent calculus as well as a semantics in terms of Kripke resource models. We show that the proof theory is sound and complete with respect to the class of minimal Kripke resource models. We also show that the sequent calculus allows cut elimination. We put the logical framework to use by instantiating it as a logic (...) of agency. In particular, we apply it to reason about the resource-sensitive use of artefacts. (shrink)

This book aims to develop certain aspects of Gottlob Frege’s theory of meaning, especially those relevant to intensional logic. It offers a new interpretation of the nature of senses, and attempts to devise a logical calculus for the theory of sense and reference that captures as closely as possible the views of the historical Frege. (The approach is contrasted with the less historically-minded Logic of Sense and Denotation of Alonzo Church.) Comparisons of Frege’s theory with those of Russell and (...) others are given. It is in the end shown that developing Frege’s theory in these ways reveals serious problems hitherto largely unnoticed, including those possibly rendering a Fregean intensional logic inconsistent even if his naïve class theory is excluded. (shrink)

Bilattices, introduced by Ginsberg as a uniform framework for inference in artificial intelligence, are algebraic structures that proved useful in many fields. In recent years, Arieli and Avron developed a logical system based on a class of bilattice-based matrices, called logical bilattices, and provided a Gentzen-style calculus for it. This logic is essentially an expansion of the well-known Belnap–Dunn four-valued logic to the standard language of bilattices. Our aim is to study Arieli and Avron’s logic from the perspective of (...) abstract algebraic logic . We introduce a Hilbert-style axiomatization in order to investigate the properties of the algebraic models of this logic, proving that every formula can be reduced to an equivalent normal form and that our axiomatization is complete w.r.t. Arieli and Avron’s semantics. In this way, we are able to classify this logic according to the criteria of AAL. We show, for instance, that it is non-protoalgebraic and non-self-extensional. We also characterize its Tarski congruence and the class of algebraic reducts of its reduced generalized models, which in the general theory of AAL is usually taken to be the algebraic counterpart of a sentential logic. This class turns out to be the variety generated by the smallest non-trivial bilattice, which is strictly contained in the class of algebraic reducts of logical bilattices. On the other hand, we prove that the class of algebraic reducts of reduced models of our logic is strictly included in the class of algebraic reducts of its reduced generalized models. Another interesting result obtained is that, as happens with some implicationless fragments of well-known logics, we can associate with our logic a Gentzen calculus which is algebraizable in the sense of Rebagliato and Verdú . We also prove some purely algebraic results concerning bilattices, for instance that the variety of distributive bilattices is generated by the smallest non-trivial bilattice. This result is based on an improvement of a theorem by Avron stating that every bounded interlaced bilattice is isomorphic to a certain product of two bounded lattices. We generalize it to the case of unbounded interlaced bilattices. (shrink)

General Introduction: In [1] a Calculus of Qualia (CQ) was proposed. The key idea is that, for example, blackness is radically different than █. The former term, “blackness” refers to or is about a quale, whereas the latter term, “█” instantiates a quale in the reader’s mind and is non-referential; it does not even refer to itself. The meaning and behavior of these terms is radically different. All of philosophy, from Plato through Descartes through Chalmers, including hieroglyphics and emojis, (...) used referential terms up until CQ. This paper in this series of papers addresses the question of why is there something rather than nothing, at length, and gives a radical new answer. It has to do with the fact in CQ that, while blackness is contingently possible, █ is necessarily actual, as it was in this sentence. (shrink)

In this paper we define intensional models for the classical theory of types, thus arriving at an intensional type logic ITL. Intensional models generalize Henkin's general models and have a natural definition. As a class they do not validate the axiom of Extensionality. We give a cut-free sequent calculus for type theory and show completeness of this calculus with respect to the class of intensional models via a model existence theorem. After this we turn our attention to applications. (...) Firstly, it is argued that, since ITL is truly intensional, it can be used to model ascriptions of propositional attitude without predicting logical omniscience. In order to illustrate this a small fragment of English is defined and provided with an ITL semantics. Secondly, it is shown that ITL models contain certain objects that can be identified with possible worlds. Essential elements of modal logic become available within classical type theory once the axiom of Extensionality is given up. (shrink)

Although the cognition is significant in strategic reasoning, its role has been weakly analyzed, because only the average intelligence is usually considered. For example, prisoner's dilemma in game theory, would have different outcomes for persons with different intelligence. I show how various levels of intelligence influence the quality of reasoning, decision, or the probability of psychosis. I explain my original methodology developed for my MA thesis in clinical psychology in 1998, and grant research in 1999, demonstrating the bias of the (...) classic IQ method, and how the intelligence limits thinking. Based on that I defined Personality Model, providing insight into understanding of psychosis (schizophrenia, bi-polar), which has not been explained yet by psychology or psychiatry. In addition, it enables to analyze and assess non-linear problems, utilizable in computer programming, visualization (animation) or other fields including Baduk game. I've already applied some principles in complex information system www.each.co.uk, and video-animations exhibited in London, Germany, Tokyo. I need to mention my experience in chess composition between 1994 and 2000, winning a few international prizes and inventing a special class of fairy rules redefining the mate. The chess composition principles or patterns show the way to organize logical series to higher advanced mechanisms (like calculus), applicable to other fields. One of such principles is a logical aesthetic innovation: new strategy, defined by Italian composers. Finally I show how the simple redefinition of the classic utility concept links economics and psychology to explain irrational / destructive behavior. All presented results (from the research) can be repeated. (shrink)

This article presents modal versions of resource-conscious logics. We concentrate on extensions of variants of linear logic with one minimal non-normal modality. In earlier work, where we investigated agency in multi-agent systems, we have shown that the results scale up to logics with multiple non-minimal modalities. Here, we start with the language of propositional intuitionistic linear logic without the additive disjunction, to which we add a modality. We provide an interpretation of this language on a class of Kripke resource models (...) extended with a neighbourhood function: modal Kripke resource models. We propose a Hilbert-style axiomatisation and a Gentzen-style sequent calculus. We show that the proof theories are sound and complete with respect to the class of modal Kripke resource models. We show that the sequent calculus admits cut elimination and that proof-search is in PSPACE. We then show how to extend the results when non-commutative connectives are added to the language. Finally, we put the l.. (shrink)

General Introduction: In [1] a Calculus of Qualia (CQ) was proposed. The key idea is that, for example, blackness is radically different than █. The former term, “blackness” refers to or is about a quale, whereas the latter term, “█” instantiates a quale in the reader’s mind and is non-referential; it does not even refer to itself. The meaning and behavior of these terms is radically different. All of philosophy, from Plato through Descartes through Chalmers, including hieroglyphics and emojis, (...) used referential terms up until CQ. This paper in this series of papers includes a relatively short but comprehensive discussion with the AI Claude 3.5 Sonnet on how the CQ impacts various theories of consciousness, including Materialism, Dualism, Idealism, and 14 others. (shrink)

Analytical philosophy is ruled by the alliance of logic, linguistics and mathematics since its beginnings in the syllogistic calculus of terms and premises in Aristotle's Analytica protera, in the theories of medieval logic that dealt with what are Proprietatis Terminorum (significatio, suppositio, appellatio), in the theological apologetics of argumentation with the combinatorics of symbols by Raymundus Llullus in the work Ars Magna, Generalis et Ultima (1305-08), in what is presented as Theologia Combinata (cf. Tomus II.p.251) in Ars Magna Sciendi (...) sive Combinatoria by Athanasius Kircher (1669), in analytical combinatorics based in Leibniz's dissertation Ars Combinatoria (1666), where language is understood as a lingua characteristica and as a calculus ratiocinator, in the combinatorics of classes that have the meanings 1 or 0 given in the idea of Boolean functions (1854), in Frege's work Begriffschrift, eine der arithmeticeschen nachgebildete Fomelsprache des reinen Denkens (Concept letter, the language of formulas of pure thought made according to the model in arithmetical language), in Cantor's study of set theory / Grundlagen einer allgemeinen Mannigfaltigkeitstheorie (1883) and up to algorithms of fuzzy logic as a "means of approximate thinking that is in the human prior" (Zadeh, 1990) and fuzzy linguistics in computer semantics (Burghard B. Rieger, 1993). These are all the philosophical ideas of mathematicians and logicians that founded what we call logical pragmatism here as a demand that logic be absolutely used in every way in which it will dominate the grammar of language until logic itself becomes the philosophical grammar of the mind!!! (shrink)

The idea of this paper is to put actual qualia into equations (broadly understood) to get what might be called qualations. Qualations arguably have different meanings and truth behaviors than the analogous equations. For example, the term ‘ black ’ arguably has a different meaning and behavior than the term ‘ █ ’. This is a step in the direction of a ‘calculus of qualia’ and of expanding science to include 1st-person phenomena.

We explain and explore class-theoretic potentialism—the view that one can always individuate more classes over a set-theoretic universe. We examine some motivations for class-theoretic potentialism, before proving some results concerning the relevant potentialist systems (in particular exhibiting failures of the $\mathsf {.2}$ and $\mathsf {.3}$ axioms). We then discuss the significance of these results for the different kinds of class-theoretic potentialists.

The basic idea is to put qualia into equations (broadly understood) to get what might as well be called qualations. Qualations arguably have different truth behaviors than the analogous equations. Thus ‘black’ has a different behavior than ‘ █ ’. This is a step in the direction of a ‘calculus of qualia’. It might help clarify some issues.

A general class of labeled sequent calculi is investigated, and necessary and sufficient conditions are given for when such a calculus is sound and complete for a finite -valued logic if the labels are interpreted as sets of truth values. Furthermore, it is shown that any finite -valued logic can be given an axiomatization by such a labeled calculus using arbitrary "systems of signs," i.e., of sets of truth values, as labels. The number of labels needed is logarithmic (...) in the number of truth values, and it is shown that this bound is tight. (shrink)

We introduce translations between display calculus proofs and labeled calculus proofs in the context of tense logics. First, we show that every derivation in the display calculus for the minimal tense logic Kt extended with general path axioms can be effectively transformed into a derivation in the corresponding labeled calculus. Concerning the converse translation, we show that for Kt extended with path axioms, every derivation in the corresponding labeled calculus can be put into a special (...) form that is translatable to a derivation in the associated display calculus. A key insight in this converse translation is a canonical representation of display sequents as labeled polytrees. Labeled polytrees, which represent equivalence classes of display sequents modulo display postulates, also shed light on related correspondence results for tense logics. (shrink)

One of the premises of the argument from design is the need for the order of a designer. This premise has been explained in various ways, one of which is the calculus of probabilities. It posits that the likelihood of an ordered thing coming into existence by chance is so low that human beings consider it nearly impossible; therefore, this order requires an intelligent designer. However, some contemporary Muslim philosophers, such as Ayatollah Jawādī Āmulī, along with certain Western philosophers, (...) have contested the validity of the calculus of probabilities. They argue that the calculus of probabilities does not reveal reality but is merely practically useful. They assert that undesigned states have the same probability as designed states. In this article, I address these objections, refute them, and ultimately demonstrate the soundness of the argument from design based on the calculus of probabilities. (shrink)

A number of authors have objected to the application of non-classical logic to problems in philosophy on the basis that these non-classical logics are usually characterised by a classical metatheory. In many cases the problem amounts to more than just a discrepancy; the very phenomena responsible for non-classicality occur in the field of semantics as much as they do elsewhere. The phenomena of higher order vagueness and the revenge liar are just two such examples. The aim of this paper is (...) to show that a large class of non-classical logics are strong enough to formulate their own model theory in a corresponding non-classical set theory. Specifically I show that adequate definitions of validity can be given for the propositional calculus in such a way that the metatheory proves, in the specified logic, that every theorem of the propositional fragment of that logic is validated. It is shown that in some cases it may fail to be a classical matter whether a given sentence is valid or not. One surprising conclusion for non-classical accounts of vagueness is drawn: there can be no axiomatic, and therefore precise, system which is determinately sound and complete. (shrink)

Inhalt: Vorbemerkung Kommentar Titel Vorrede Erster Teil Zweiter Teil Dritter Teil Benutzte Literatur a)Zeitgenossen und Vorläufer Fichtes b)Moderne Interpreten Sachregister zum Fichte-Text Verzeichnis der zitierten Arbeiten Fichtes.

Application 1. The case against Materialism and Illusionism Application 2. Ineffability Application 3. Hard Problems Application 4. Knowledge Argument questions Application 5. Argument for A-theories of time Application 6. Possible qualia are necessary.

The global water crisis is being exacerbated by climate change, even in the United States. Recycled water is a feasible alternative to alleviate the water shortage, but it is constrained by humans’ perceptions. The current study examines how residents’ water scarcity awareness and climate change belief influence their willingness to use recycled water directly and indirectly. Bayesian Mindsponge Framework (BMF) analytics was employed on a dataset of 1831 residents in Albuquerque, New Mexico, an arid inland region in the US. We (...) discovered that residents’ willingness to use direct recycled potable water is positively affected by their awareness of water scarcity, but only if they believe in the impacts of climate change on the water cycle. Meanwhile, the willingness to use indirect recycled potable water is influenced by water scarcity awareness, and the belief in climate change further enhances this effect. These findings implicate that fighting climate change denialism and informing the public of the water scarcity situation in the region can contribute to the effectiveness and sustainability of long-term water conservation and climate change alleviation efforts. (shrink)

The question of the relationship between ontology and theology, the main problem of the interpretation in volume 3, is also the guiding question of our last volume. The history of metaphysics is a history of the efforts towards an outlook on the world and life, which are about the meaning and connection of fundamental concepts: being, life, intellect, unity, truth, goodness. From these, the concept of divinity is derived. As in the previous volumes, a rich material of original texts and (...) interpretations is spread out. (shrink)

With the third volume, it is invited to enter the intellectual environment of Aristotle. The most relevant sources are given in full (with English translation), so that the commentary is also a reader documenting the disputationes metaphysicae of the 4th century BC. For the undeniable contradictions in the Metaphysics, a new genetic explanation is offered.

In this short piece I provide an interpretation of David Lewis' thesis of Composition as Identity.

This article identifies and formalizes the logical features of analogous terms that justify their use in deduction. After a survey of doctrines in Aristotle, Aquinas, and Cajetan, the criteria of “analogy of proper proportionality” are symbolized in first-order predicate logic. A common genus justifies use of a common term, but does not provide the inferential link required for deduction. Rather, the respective differentiae foster this link through their identical proportion. A natural-language argument by analogy is formalized so as to exhibit (...) these criteria, thereby showing the validity of analogical deduction. (shrink)

1. First section notes: subtraction -/- Evidently -/- (1) black – black = empty-set -/- because anything minus itself is the empty set, but -/- (2) █ – █ = █ + █ -/- because there are two instances of black on the left and two instances of black on the right...

The ontological pictures underpinning David Lewis's Parts of Classes and On the Plurality of Worlds are in some tension. One tension concerns whether the sets and classes of Parts of Classes can be found in Lewis's modal space, since they cannot in general be parts of any possible world. The second is that the atoms that are the mathematical ontology of Parts of Classes seem to meet the criteria for being possible worlds themselves, and so fail (...) to be the material Parts of Classes needs. Two different responses to these tensions are discussed: one is unappealing in several ways but keeps more of Lewis's doctrines, while the other requires some modifications to the system of Parts of Classes. (shrink)

The naive theory of properties states that for every condition there is a property instantiated by exactly the things which satisfy that condition. The naive theory of properties is inconsistent in classical logic, but there are many ways to obtain consistent naive theories of properties in nonclassical logics. The naive theory of classes adds to the naive theory of properties an extensionality rule or axiom, which states roughly that if two classes have exactly the same members, they are (...) identical. In this paper we examine the prospects for obtaining a satisfactory naive theory of classes. We start from a result by Ross Brady, which demonstrates the consistency of something resembling a naive theory of classes. We generalize Brady’s result somewhat and extend it to a recent system developed by Andrew Bacon. All of the theories we prove consistent contain an extensionality rule or axiom. But we argue that given the background logics, the relevant extensionality principles are too weak. For example, in some of these theories, there are universal classes which are not declared coextensive. We elucidate some very modest demands on extensionality, designed to rule out this kind of pathology. But we close by proving that even these modest demands cannot be jointly satisfied. In light of this new impossibility result, the prospects for a naive theory of classes are bleak. (shrink)

We give an argument for temporal A-theories based on the calculus of qualia.

This paper introduces non-referential terms into language as used in the Calculus of Qualia (CQ). It is possible to write “blackness” to refer to the experience of the subjective, internal quale of blackness. It is possible to write █ as another word for “blackness,” another referential term. But that is not its meaning in CQ. In CQ the word “blackness” is used as usual, as a symbol that refers to something other than itself. But █ is the immediate subjective (...) experience (quale) that arises in the reader’s mind. “Blackness” refers to the color or quale. But █ in no way refers, _not even to itself_, it is strictly non-referential. It’s presentation instantiates it. This is no some trivial definitional trick but is a paradigm shift. Examples, traps, and concequences for theories of truth and meaning are briefly explored. (shrink)

We generalize the Kolmogorov axioms for probability calculus to obtain conditions defining, for any given logic, a class of probability functions relative to that logic, coinciding with the standard probability functions in the special case of classical logic but allowing consideration of other classes of "essentially Kolmogorovian" probability functions relative to other logics. We take a broad view of the Bayesian approach as dictating inter alia that from the perspective of a given logic, rational degrees of belief are (...) those representable by probability functions from the class appropriate to that logic. Classical Bayesianism, which fixes the logic as classical logic, is only one version of this general approach. Another, which we call Intuitionistic Bayesianism, selects intuitionistic logic as the preferred logic and the associated class of probability functions as the right class of candidate representions of epistemic states (rational allocations of degrees of belief). Various objections to classical Bayesianism are, we argue, best met by passing to intuitionistic Bayesianism—in which the probability functions are taken relative to intuitionistic logic—rather than by adopting a radically non-Kolmogorovian, for example, nonadditive, conception of (or substitute for) probability functions, in spite of the popularity of the latter response among those who have raised these objections. The interest of intuitionistic Bayesianism is further enhanced by the availability of a Dutch Book argument justifying the selection of intuitionistic probability functions as guides to rational betting behavior when due consideration is paid to the fact that bets are settled only when/if the outcome bet on becomes known. (shrink)

This thesis introduces the "method of structural refinement", which serves as a means of transforming the relational semantics of a modal and/or constructive logic into an 'economical' proof system by connecting two proof-theoretic paradigms: labelled and nested sequent calculi. The formalism of labelled sequents has been successful in that cut-free calculi in possession of desirable proof-theoretic properties can be automatically generated for large classes of logics. Despite these qualities, labelled systems make use of a complicated syntax that explicitly incorporates (...) the semantics of the associated logic, and such systems typically violate the subformula property to a high degree. By contrast, nested sequent calculi employ a simpler syntax and adhere to a strict reading of the subformula property, making such systems useful in the design of automated reasoning algorithms. However, the downside of the nested sequent paradigm is that a general theory concerning the automated construction of such calculi (as in the labelled setting) is essentially absent, meaning that the construction of nested systems and the confirmation of their properties is usually done on a case-by-case basis. The refinement method connects both paradigms in a fruitful way, by transforming labelled systems into nested (or, refined labelled) systems with the properties of the former preserved throughout the transformation process. To demonstrate the method of refinement and some of its applications, we consider grammar logics, first-order intuitionistic logics, and deontic STIT logics. The introduced refined labelled calculi will be used to provide the first proof-search algorithms for deontic STIT logics. Furthermore, we employ our refined labelled calculi for grammar logics to show that every logic in the class possesses the effective Lyndon interpolation property. (shrink)

Many anthropologists interpret neoliberalism as a radical break from and dangerous rupture in post-war societies that featured Keynesian economic policies and welfare provision. The allure of a mythic welfare state has boosted John Maynard Keynes’s popularity to many who embrace certain facets of socialism. Many critical social scientists have embraced Keynesianism in ways that overlook how the US used Keynesian policies to reengineer and redeploy state power. Keynes’s liberal synthesis inspired managers in the US Treasury Department to understand depression-era problems (...) of unemployment and poverty in ways that were consonant with the expansion of corporate power. For understanding Keynesianism, as it actually existed during the Cold War, we must analyse how the US Treasury and State Departments used Keynesian principles to rebuild the social reproductive capacities necessary for capitalist accumulation both domestically and in Western Europe. I focus on how the architects of postwar capitalism used full employment policies, labour laws and welfare provision to renovate the nexus of political practices and institutional structures in ways that formed a benevolent and caring image of ‘the state’ and the myth of a class compromise. Through these reforms, governmental planners and administrators used the ‘state idea’ to reorganize capital accumulation as if the post-war economy would represent ordinary people’s best interests. In the process, these sophisticated practices of power became reified as the ‘welfare state’ and the ‘Keynesian compromise’ in ways that endow these institutions and policies with a character divorced from practices of power. The post-war state embodied a dialectic of repression and reform that combined criminalizing dissent with full employment policies and welfare provision. Taking these aspects of power into account, we can see post-war Keynesianism in ways that inspire a robust and far-reaching criticism of the contemporary predicament of economic uncertainty, political instability and environmental degradation. (shrink)

We employ a recently developed methodology -- called "structural refinement" -- to extract nested sequent systems for a sizable class of intuitionistic modal logics from their respective labelled sequent systems. This method can be seen as a means by which labelled sequent systems can be transformed into nested sequent systems through the introduction of propagation rules and the elimination of structural rules, followed by a notational translation. The nested systems we obtain incorporate propagation rules that are parameterized with formal grammars, (...) and which encode certain frame conditions expressible as first-order Horn formulae that correspond to a subclass of the Scott-Lemmon axioms. We show that our nested systems are sound, cut-free complete, and admit hp-admissibility of typical structural rules. (shrink)

This paper presents a critical analysis of dual-class share structures (DCSS) within an ethical and logical framework for ownership and control. Moreover, it contends that the existence of DCSS constitutes a de facto infringement on the ownership rights of shareholders. By adopting a conceptual approach rooted in symbolic logic, the paper demonstrates that DCSS falls short of the ethical precept of “one share, one vote” and is therefore an unjust and inefficient equity arrangement. The paper contributes to the existing literature (...) on DCSS by focusing on the ethical and logical basis of such share structures. (shrink)

The present book, “Rights of Depressed Classes: A Constitutional Approach “is the fourth e-book of the Centre which includes the essence of the occasional papers presented in several seminars. Human Rights is one of the majors subjects for discussion in academics as well as in social sector and has an international approach to social issues and problems. The struggle to promote, protect and preserve human rights changes and holds continuity in every generation in our society. The concept and practice (...) of human rights is the hallmark of any modern society. Marginalized Groups of India includes SCs/STs, Women and Children etc. The Constitution of India provides equal rights to all its citizens’ right to live with equality, honour and dignity. But the caste system and untouchability somehow and others are still playing negative roles from different parts of the society. Talhan, Meham, Dulina, Gohana, Saalwan, Chakwada, Khairlanji, Khandamal and Mirchpur are some of the recent instances of atrocities against Dalits in India. Atrocities against Dalits thus continue to exist even today, despite constitutional safeguards, and various legislative measures. Empowerment is the process of increasing the capacity of individuals or groups to make choices and to transform those choices into desires actions and outcomes. (shrink)

In this paper we introduce a Gentzen calculus for (a functionally complete variant of) Belnap's logic in which establishing the provability of a sequent in general requires \emph{two} proof trees, one establishing that whenever all premises are true some conclusion is true and one that guarantees the falsity of at least one premise if all conclusions are false. The calculus can also be put to use in proving that one statement \emph{necessarily approximates} another, where necessary approximation is a (...) natural dual of entailment. The calculus, and its tableau variant, not only capture the classical connectives, but also the `information' connectives of four-valued Belnap logics. This answers a question by Avron. (shrink)

The study developed and determined instant melon (Citrulluscolocynthis) soup powder’s shelf life for working-class mothers in Lagos, Nigeria. The study adopted research and development design. The population for the study was 2,077. The sample size for the study was 341. This consisted of three groups made up of 25 lecturers, 192 mothers working in banks and 124 mothers working. Seven sets of instruments were used for data collection. Data was analyzed using mean and standard deviation for the research questions. Findings (...) of the study indicated that the shelf life determinants included free fatty acid (FFA), moisture content and total viable count measured over a period of 22 weeks. From the analysis, FFA content slightly appreciated over the weeks; from 0.60 at the start to 0.7 5 at 4weeks; 0.86 at 8weeks; 0.95 at 12 weeks; 1.70 at 16 weeks and 4.06 at 22 weeks. The analysis showed that the moisture content of the developed melon soup powder slightly decreased over the weeks; from 9.41 at the start to 9.30 at 4 weeks; 9.00 at 8 weeks; 8.82 at 12 weeks; 8.60 at 16 weeks and 8.30 at 22 weeks. The total viable count (TVC) of the developed melon soup powder increased over the weeks. At the start of the experiment, the TVC of the developed melon soup powder was 8; TVC increased to 10 at 4 weeks; 14 at 8 weeks; 15 at 12 weeks; 18 at 16 weeks and 29 at 22 weeks. (shrink)