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

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

The present paper intends to report two results. It is shown that the formula P(x)=∀y∀z[¬G(x, y)→¬M(z)] provides the logic underlying gauge symmetry, where M denotes the predicate of being massive. For the logic of spontaneous symmetry breaking, by Higgs mechanism, we have P(x)=∀y∀z[G(x, y)→M(z)]. Notice that the above two formulas are not logically equivalent. The results are obtained by integrating four components, namely, gauge symmetry and Higgs mechanism in quantum field theory, and Gödel's incompleteness theorem and Tarski's indefinability (...) theorem in mathematical logic. Gödel numbering is the key for arithmetic modeling applied in this paper. (shrink)

An introduction to the special issue on epistemic logic and the foundations of game theory edited by Michael Bacharach and Philippe Mongin. Contributors are Michael Bacharach, Robert Stalnaker, Salvatore Modica and Aldo Rustichini, Luc Lismont and Philippe Mongin, and Hyun-Song Shin and Timothy Williamson.

In computational complexity theory, decision problems are divided into complexity classes based on the amount of computational resources it takes for algorithms to solve them. In theoretical computer science, it is commonly accepted that only functions for solving problems in the complexity class P, solvable by a deterministic Turing machine in polynomial time, are considered to be tractable. In cognitive science and philosophy, this tractability result has been used to argue that only functions in P can feasibly work as computational (...) models of human cognitive capacities. One interesting area of computational complexity theory is descriptive complexity, which connects the expressive strength of systems of logic with the computational complexity classes. In descriptive complexity theory, it is established that only first-order systems are connected to P, or one of its subclasses. Consequently, second-order systems of logic are considered to be computationally intractable, and may therefore seem to be unfit to model human cognitive capacities. This would be problematic when we think of the role of logic as the foundations of mathematics. In order to express many important mathematical concepts and systematically prove theorems involving them, we need to have a system of logic stronger than classical first-order logic. But if such a system is considered to be intractable, it means that the logical foundation of mathematics can be prohibitively complex for human cognition. In this paper I will argue, however, that this problem is the result of an unjustified direct use of computational complexity classes in cognitive modelling. Placing my account in the recent literature on the topic, I argue that the problem can be solved by considering computational complexity for humanly relevant problem solving algorithms and input sizes. (shrink)

Gila Sher interviewed by Chen Bo: -/- I. Academic Background and Earlier Research: 1. Sher’s early years. 2. Intellectual influence: Kant, Quine, and Tarski. 3. Origin and main Ideas of The Bounds of Logic. 4. Branching quantifiers and IF logic. 5. Preparation for the next step. -/- II. Foundational Holism and a Post-Quinean Model of Knowledge: 1. General characterization of foundational holism. 2. Circularity, infinite regress, and philosophical arguments. 3. Comparing foundational holism and foundherentism. 4. A post-Quinean model (...) of knowledge. 5. Intellect and figuring out. 6. Comparing foundational holism with Quine’s holism. 7. Evaluation of Quine’s Philosophy -/- III. Substantive Theory of Truth and Relevant Issues: 1. Outline of Sher’s substantive theory of truth. 2. Criticism of deflationism and treatment of the Liar. 3. Comparing Sher’s substantive theory of truth with Tarski’s theory of truth. -/- IV. A New Philosophy of Logic and Comparison with Other Theories: 1. Foundational account of logic. 2. Standard of logicality, set theory and logic. 3. Psychologism, Hanna’s and Maddy’s conceptions of logic. 4. Quine’s theses about the revisability of logic. -/- V. Epilogue. (shrink)

E. Husserl’s reflections in Logical Investigations on “intentional feelings” and “non-intentional feelings” are significant in both his later ethical explorations and M. Scheler’s thought on ethics. Through the incorporation of the views of Husserl and Scheler, we find that the phenomenology of the intentional feeling-acts is not only the foundation of the non-formal ethics of values in Scheler’s phenomenology, but also at least the constitutive foundation of the ethics of Husserl’s first orientation.

[English] The paper observed that the various ethics that constitute the system of African environmental ethics are not based on or linked to any known African ontology and formal logic. It argued that the contextualisation of African environmental ethics on African ontology and African logic is essential since Western ontology and logic do not serve to adequately explain and provide proper meanings to the various concepts and propositions employed in the African environmental ethics. Therefore, the paper aimed (...) to, and indeed, link and establishes African environmental ethics on a definite and sound African ontology and formal logic based on Ibuanyidanda complementary ontology and Ezumezu integrativist logic. -/- [Annang] Nwed ndun̄ọ ami akondo ada akud ate k’ idem mme ido ukpeme nkan-n̄kuk ke Africa ada nsan nsan ye ọntọlọgy ye lọgik Africa. Nnwed ami abenne awọ́d ate ke ọntọlọgy ye lọgik mfia agwo iwamma ke adinam awan̄a mme nsio nsio akpọ-ikọ, adaha ikọ, mme usem, ye mme edu ake 'adọhọke ke mme ido ukpeme nkan-n̄kuk ke Africa. Ntak ade anam ukpep mkpọ ami anwana ndiben ido ukpeme nkan-n̄kuk Africa n̄ka n̄ke kọọn̄ ke ọntọlọgy ye lọgik afọnnọ nte itiat ikaba, ade anam anye asan̄a 'ke kem ye Ibuanyidanda Ontology ye Ezumezu Logic. (shrink)

The physical foundations of mathematics in the theory of emergent space-time-matter were considered. It is shown that mathematics, including logic, is a consequence of equation which describes the fundamental field. If the most fundamental level were described not by mathematics, but something else, then instead of mathematics there would be consequences of this something else.

Hyperlogic is a hyperintensional system designed to regiment metalogical claims (e.g., “Intuitionistic logic is correct” or “The law of excluded middle holds”) into the object language, including within embedded environments such as attitude reports and counterfactuals. This paper is the first of a two-part series exploring the logic of hyperlogic. This part presents a minimal logic of hyperlogic and proves its completeness. It consists of two interdefined axiomatic systems: one for classical consequence (truth preservation under a classical (...) interpretation of the connectives) and one for “universal” consequence (truth preservation under any interpretation). The sequel to this paper explores stronger logics that are sound and complete over various restricted classes of models as well as languages with hyperintensional operators. (shrink)

The present volume is a sequel to Deontic Logic: Introductory and Systematic Readings : its purpose is to offer a view of some of the main directions of research in contemporary deontic logic. Most of the articles included in Introductory and Systematic Readings represent what may be called the standard modal approach to deontic logic, in which de on tic logic is treated as a branch of modal logic, and the normative concepts of obligation, permission (...) and prohibition are regarded as analogous to the "alethic" modalities necessity, possibility and impossibility. As Simo Knuuttila shows in his contribution to the present volume, this approach goes back to late medieval philosophy. Several 14th century philosophers observed the analogies between deontic and alethic modalities and discussed the deontic interpretations of various laws of modal logic. In contemporary deontic logic the modal approach was revived by G. H. von Wright's classic paper 'Deontic Logic'. Certain analogies between deontic and alethic modalities are obvious and uncontroversial, but the standard approach has often been criticized on the ground that it exaggerates the analogies and tends to ignore those features of normative concepts which distinguish them from other modalities. (shrink)

In 1890 Christian von Ehrenfels published his classic paper "Über 'Gestaltqualitäten'", the first systematic investigation of the philosophy and psychology of Gestalt. Ehrenfels thereby issued an important challenge to the psychological atomism that was still predominant in his day. His paper not only exerted a powerful influence on the philosophy of the Meinong school, it also marked the beginning of the Gestalt tradition in psychology, later associated with the work of Wertheimer, Köhler and Koffka in Berlin. Includes papers by C. (...) Von Ehrenfels, Kurt Grelling and Paul Oppenheim and contributions by K. Mulligan, P.M. Simons and Barry Smith. (shrink)

Fefermans argument is indeed convincing in a certain context, it can be dissolved entirely by modifying the context appropriately.

Logical realism is a view about the metaphysical status of logic. Common to most if not all the views captured by the label ‘logical realism’ is that logical facts are mind- and language-independent. But that does not tell us anything about the nature of logical facts or about our epistemic access to them. The goal of this paper is to outline and systematize the different ways that logical realism could be entertained and to examine some of the challenges that (...) these views face. It will be suggested that logical realism is best understood as a metaphysical view about the logical structure of the world, but this raises an important question: does logical realism collapse into standard metaphysical realism? It will be argued that this result can be accommodated, even if it cannot be altogether avoided. (shrink)

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

A principle, according to which any scientific theory can be mathematized, is investigated. That theory is presupposed to be a consistent text, which can be exhaustedly represented by a certain mathematical structure constructively. In thus used, the term “theory” includes all hypotheses as yet unconfirmed as already rejected. The investigation of the sketch of a possible proof of the principle demonstrates that it should be accepted rather a metamathematical axiom about the relation of mathematics and reality. Its investigation needs philosophical (...) means. Husserl’s phenomenology is what is used, and then the conception of “bracketing reality” is modelled to generalize Peano arithmetic in its relation to set theory in the foundation of mathematics. The obtained model is equivalent to the generalization of Peano arithmetic by means of replacing the axiom of induction with that of transfinite induction. A comparison to Mach’s doctrine is used to be revealed the fundamental and philosophical reductionism of Husserl’s phenomenology leading to a kind of Pythagoreanism in the final analysis. Accepting or rejecting the principle, two kinds of mathematics appear differing from each other by its relation to reality. Accepting the principle, mathematics has to include reality within itself in a kind of Pythagoreanism. These two kinds are called in paper correspondingly Hilbert mathematics and Gödel mathematics. The sketch of the proof of the principle demonstrates that the generalization of Peano arithmetic as above can be interpreted as a model of Hilbert mathematics into Gödel mathematics therefore showing that the former is not less consistent than the latter, and the principle is an independent axiom. An information interpretation of Hilbert mathematics is involved. It is a kind of ontology of information. Thus the problem which of the two mathematics is more relevant to our being is discussed. An information interpretation of the Schrödinger equation is involved to illustrate the above problem. (shrink)

We study a new formal logic LD introduced by Prof. Grzegorczyk. The logic is based on so-called descriptive equivalence, corresponding to the idea of shared meaning rather than shared truth value. We construct a semantics for LD based on a new type of algebras and prove its soundness and completeness. We further show several examples of classical laws that hold for LD as well as laws that fail. Finally, we list a number of open problems. -/- .

This book is an anthology with the following themes. Non-European Tradition: Bussanich interprets main themes of Hindu ethics, including its roots in ritual sacrifice, its relationship to religious duty, society, individual human well-being, and psychic liberation. To best assess the truth of Hindu ethics, he argues for dialogue with premodern Western thought. Pfister takes up the question of human nature as a case study in Chinese ethics. Is our nature inherently good (as Mengzi argued) or bad (Xunzi’s view)? Pfister ob- (...) serves their underlying agreement, that human beings are capable of becoming good, and makes precise the disagreement: whether we achieve goodness by cultivating autonomous feelings or by accepting external precepts. There are political consequences: whether government should aim to respect and em- power individual choices or to be a controlling authority. Early Greek Thinking: Collobert examines the bases of Homeric ethics in fame, prudence, and shame, and how these guide the deliberations of heroes. She observes how, by depending upon the poet’s words, the hero gains a quasi- immortality, although in truth there is no consolation for each person’s inevi- table death. Plato: Santas examines Socratic Method and ethics in Republic 1. There Socrates examines definitions of justice and tests them by comparison to the arts and sciences. Santas shows the similarities of Socrates’ method to John Rawls’ method of considered judgments in reflective equilibrium. McPherran interprets Plato’s religious dimension as like that of his teacher Socrates. McPherran shows how Plato appropriates, reshapes, and extends the religious conventions of his own time in the service of establishing the new enterprise of philosophy. Ac- cording to Taylor, Socrates believes that humans in general have the task of helping the gods by making their own souls as good as possible, and Socrates’ unique ability to cross-examine imposes on him the special task of helping others to become as good as possible. This conception of Socrates’ mission is Plato’s own, consisting in an extension of the traditional conception of piety as helping the gods. Brickhouse and Smith propose a new understanding of Socratic moral psychology—one that retains the standard view of Socrates as an intellectualist, but also recognizes roles in human agency for appetites and passions. They compare and contrast the Socratic view to the picture of moral psychology we get in other dialogues of Plato. Hardy also proposes a new, non-reductive understanding of Socratic eudaimonism—he argues that Socrates invokes a very rich and complex notion of the “Knowledge of the Good and Bad”, which is associated with the motivating forces of the virtues. Rudebusch defends Socrates’ argument that knowledge can never be impotent in the face of psychic passions. He considers the standard objections: that knowledge cannot weigh incom- mensurable human values, and that brute desire, all by itself, is capable of moving the soul to action. Aristotle: Anagnostopoulos interprets Aristotle on the nature and acquisition of virtue. Though virtue of character, aiming at human happiness, requires a complex awareness of multiple dimensions of one’s experience, it is not properly a cognitive capacity. Thus it requires habituation, not education, according to Aristotle, in order to align the unruly elements of the soul with reason’s knowledge of what promotes happiness. Shields explains Aristotle’s doctrine that goodness is meant in many ways as the doctrine that there are different analyses of goodness for different types of circumstance, just as for being. He finds Aristotle to argue for this conclusion, against Plato’s doctrine of the unity of the Good, by applying the tests for homonymy and as an immediate cons- equence of the doctrine of categories. Shields evaluates the issue as unresolved at present. Russell discusses Aristotle’s account of practical deliberation and its virtue, intelligence (phronesis). He relates the account to contemporary philo- sophical controversies surrounding Aristotle’s view that intelligence is neces- sary for moral virtue, including the objections that in some cases it is unnecessary or even impedes human goodness. Frede examines the advantages and disadvantages of Aristotle’s virtue ethics. She explains the general Greek con- ceptions of happiness and virtue, Aristotle’s conception of phronesis and compares the Aristotle’s ethics with modern accounts. Liske discusses the question of whether the Aristotelian account of virtue entails an ethical-psy- chological determinism. He argues that Aristotle’s understanding of hexis allows for free action and ethical responsibility : By making decisions for good actions we are able to stabilize our character (hexis). Hellenistic and Roman: Annas defends an account of stoic ethics, according to which the three parts of Stoicism—logic, physics, and ethics—are integrated as the parts of an egg, not as the parts of a building. Since by this analogy no one part is a foundation for the rest, pedagogical decisions may govern the choice of numerous, equally valid, presentations of Stoic ethics. Piering interprets the Cynic way of life as a distinctive philosophy. In their ethics, Cynics value neither pleasure nor tradition but personal liberty, which they achieve by self-suffi- ciency and display in speech that is frank to the point of insult. Plotinus and Neoplatonism: Gerson outlines the place of ordinary civic virtue as well as philosophically contemplative excellence in Neoplatonism. In doing so he attempts to show how one and the same good can be both action-guiding in human life and be the absolute simple One that grounds the explanation of everything in the universe. Delcomminette follows Plotinus’s path to the Good as the foundation of free will, first in the freedom of Intellect and then in the “more than freedom” of the One. Plotinus postulates these divinities as not outside but within each self, saving him from the contradiction of an external foundation for a truly free will. General Topics: Halbig discusses the thesis on the unity of virtues. He dis- tinguishes the thesis of the identity of virtues and the thesis of a reciprocity of virtues and argues that the various virtues form a unity (in terms of reciprocity) since virtues cannot bring about any bad action. Detel examines Plato’s and Aristotle’s conceptions of normativity : Plato and Aristotle (i) entertained hybrid theories of normativity by distinguishing functional, semantic and ethical normativity, (ii) located the ultimate source of normativity in standards of a good life, and thus (iii) took semantic normativity to be a derived form of normativity. Detel argues that hybrid theories of normativity are—from a mo- dern point of view—still promising. Ho ̈ffe defends the Ancient conception of an art of living against Modern objections. Whereas many Modern philosophers think that we have to replace Ancient eudaimonism by the idea of moral obligation (Pflicht), Ho ̈ffe argues that Eudaimonism and autonomy-based ethics can be reconciled and integrated into a comprehensive and promising theory of a good life, if we enrich the idea of autonomy by the central elements of Ancient eudaimonism. Some common themes: The topics in Chinese and Hindu ethics are perhaps more familiar to modern western sensibilities than Homeric and even Socratic. Anagnostopoulos, Brickhouse and Smith, Frede, Liske, Rudebusch, and Russell all consider in contrasting ways the role of moral character, apart from intellect, in ethics. Brickhouse / Smith, Hardy, and Rudebusch discuss the Socratic con- ception of moral knowledge. Brickhouse / Smith and Hardy retain the standard view of the so called Socratic Intellectualism. Shields and Gerson both consider the question whether there is a single genus of goodness, or if the term is a homonym. Bussanich, McPherran, Taylor, and Delcomminette all consider the relation between religion and ethics. Pfister, Piering, Delcomminette, and Liske all consider what sort of freedom is appropriate to human well-being. Halbig, Detel, and Ho ̈ffe propose interpretations of main themes of Ancient ethics. (shrink)

Gideon Rosen and Robert Schwartzkopff have independently suggested (variants of) the following claim, which is a varian of Hume's Principle: -/- When the number of Fs is identical to the number of Gs, this fact is grounded by the fact that there is a one-to-one correspondence between the Fs and Gs. -/- My paper is a detailed critique of the proposal. I don't find any decisive refutation of the proposal. At the same time, it has some consequences which many will (...) find objectionable. (shrink)

The purpose of this paper is to examine the status of logic from a metaphysical point of view – what is logic grounded in and what is its relationship with metaphysics. There are three general lines that we can take. 1) Logic and metaphysics are not continuous, neither discipline has no bearing on the other one. This seems to be a rather popular approach, at least implicitly, as philosophers often skip the question altogether and go about their (...) business, be it logic or metaphysics. However, it is not a particularly plausible view and it is very hard to maintain consistently, as we will see. 2) Logic is prior to metaphysics and has metaphysical implications. The extreme example of this kind of approach is the Dummettian one, according to which metaphysical questions are reducible to the question of which logic to adopt. 3) Metaphysics is prior to logic, and your logic should be compatible with your metaphysics. This approach suggests an answer to the question of what logic is grounded in, namely, metaphysics. Here I will defend the third option. (shrink)

For centuries, science was considered as something radically different from religion. Yet, the foundations of true science are deeply religious in nature. This paper seeks to show how religion is the only foundation needed for the formulation of scientific theories, since it provides the core principles on which the building of exact sciences is based upon. Our need to understand the cosmos and our faith in us being able to do so, are the main prerequisites for conducting science; prerequisites (...) that are derived from our belief in us being the sons of God and, thus, being able to read His mind. From its birth on 7 March 1277 up to today, science seems to be the only logical attitude of religious people towards the unknown cosmos. (shrink)

Epstein and Carnielli's fine textbook on logic and computability is now in its second edition. The readers of this journal might be particularly interested in the timeline `Computability and Undecidability' added in this edition, and the included wall-poster of the same title. The text itself, however, has some aspects which are worth commenting on.

The present article arises from the reflection on the failure of external educational models applied in emerging contexts without considering contextual factors of each town. According to Carnoy (1974) there is a strong tradition between underdeveloped countries of coping the cultural forms and successful models from first world societies, without having prepared beforehand the objective and subjective conditions that would determine its success or failure. The aim of this paper is to analyze the philosophical basis of success behind the Finnish (...) educational system, with the purpose of identify in it, references to overcome problems of educational reality in emerging societies. In this way contribute to the discussion about the failure on adapting external educational models to emerging contexts. It is an exploratory, descriptive and comparative review, and the character of it is bibliographic and qualitative; in addition, it will make use of logical-scientific methods (deductive-inductive) with their respective procedures, supported by phenomenological and hermeneutical philosophical methods. The results of the copy and implementation of changes in education based on successful models from other latitudes, without considering the philosophical and epistemological substratum that sustains it, or preparing the ground for its adaptation, has resulted in poor educational systems that do not respond to the concrete needs of the educational reality of emerging societies, and limit and frustrate the autonomous and original creation of contextualized and relevant educational models too. (shrink)

I provide an analysis of sentences of the form ‘To be F is to be G’ in terms of exact truth-maker semantics—an approach that identifies the meanings of sentences with the states of the world directly responsible for their truth-values. Roughly, I argue that these sentences hold just in case that which makes something F is that which makes it G. This approach is hyperintensional, and possesses desirable logical and modal features. These sentences are reflexive, transitive and symmetric, and, if (...) they are true, then they are necessarily true, and it is necessary that all and only Fs are Gs. I close by defining an asymmetric and irreflexive notion of analysis in terms of the reflexive and symmetric one. (shrink)

In Mathematics is megethology. Philosophia Mathematica, 1, 3–23) David K. Lewis proposes a structuralist reconstruction of classical set theory based on mereology. In order to formulate suitable hypotheses about the size of the universe of individuals without the help of set-theoretical notions, he uses the device of Boolos’ plural quantification for treating second order logic without commitment to set-theoretical entities. In this paper we show how, assuming the existence of a pairing function on atoms, as the unique assumption non (...) expressed in a mereological language, a mereological foundation of set theory is achievable within first order logic. Furthermore, we show how a mereological codification of ordered pairs is achievable with a very restricted use of the notion of plurality without plural quantification. (shrink)

This paper is about Poincaré’s view of the foundations of geometry. According to the established view, which has been inherited from the logical positivists, Poincaré, like Hilbert, held that axioms in geometry are schemata that provide implicit definitions of geometric terms, a view he expresses by stating that the axioms of geometry are “definitions in disguise.” I argue that this view does not accord well with Poincaré’s core commitment in the philosophy of geometry: the view that geometry is the (...) study of groups of operations. In place of the established view I offer a revised view, according to which Poincaré held that axioms in geometry are in fact assertions about invariants of groups. Groups, as forms of the understanding, are prior in conception to the objects of geometry and afford the proper definition of those objects, according to Poincaré. Poincaré’s view therefore contrasts sharply with Kant’s foundation of geometry in a unique form of sensibility. According to my interpretation, axioms are not definitions in disguise because they themselves implicitly define their terms, but rather because they disguise the definitions which imply them. (shrink)

I investigate the role of geometric intuition in Frege’s early mathematical works and the significance of his view of the role of intuition in geometry to properly understanding the aims of his logicist project. I critically evaluate the interpretations of Mark Wilson, Jamie Tappenden, and Michael Dummett. The final analysis that I provide clarifies the relationship of Frege’s restricted logicist project to dominant trends in German mathematical research, in particular to Weierstrassian arithmetization and to the Riemannian conceptual/geometrical tradition at Göttingen. (...) Concurring with Tappenden, I hold that Frege’s logicism should not be understood as a continuing a project of reductionist arithmetization. However, Frege does not quite take up the Riemannian banner either. His logicism supports a hierarchical understanding of the structure of mathematical knowledge, according to which arithmetic is applicable to geometry but not vice versa because the former is more general, as revealed by the strictly logical nature of its objects in comparison to the intuitional nature of geometric objects. I suggest, in particular, that Frege intended that foundational work would show the use of geometric intuition in complex analysis, a source of error for Riemann that Weierstrass was proud to have uncovered, to be inessential. (shrink)

This thesis develops an account of ethics called the Linguistic Perspective, which is realist in a practical, non-theoretical sense, and is rooted Wittgenstein’s 'On Certainty'. On this account, normativity is intrinsic to human action and language; the norms of ethics are the logical limits of the most basic, unassailable concepts that practical reasoning requires for intelligibility. Part I lays the groundwork for this account by developing a Tractarian Reading of 'On Certainty'. Here, I contend that 'On Certainty' is primarily concerned (...) with the logical requirements and limits of language, and like the 'Tractatus' it develops these concerns in a realist direction. 'On Certainty', I argue, does so by advancing three key claims about logic: Everything descriptive of a language-game is part of logic; the boundary between logical and empirical propositions is not sharp; logic is founded on action. I investigate these claims, with particular emphasis on how natural human reactions provide logical conditions for the possibility of language. Hinge propositions, I argue, are rooted in these reactions; they point us to the limits of intelligibility of our concepts, though these limits are vague and inarticulable. In Part II, I discuss the practical syllogism and its validity. I argue that practical reasoning is distinct from theoretical reasoning and embodies a different standard of logical validity for inference. This standard is not strict and narrow, but creative and wide open; furthermore, it is an intrinsically ethical standard. Then, I advance my Linguistic Perspective on ethics, according to which human reactions provide the conditions for ethical normativity, just as they do for language. I argue that we can access ethical norms by reflecting on those linguistic concepts that are rooted in the most basic patterns of action, and by identifying the norms of behaviour that ensure the development of those concepts without contradiction. (shrink)

Contemporary psychologism has been amended for most of the objections by its opponents over a century ago. However, some authors still raise doubts about its ability to account for some peculiar properties of logic. In particular, it is argued that the psychological universality of patterns of inferential behavior is not sufficient to account for the normativity of logic. In this paper, I deal with the issue and offer three alternative solutions that do not rely on mere empirical universality. (...) I will use the works of Laurence Jonathan Cohen, Diego Marconi and Marcello D'Agostino, adapting them for the purpose of defending logical psychologism. I will therefore argue that, although more refined work on the subject is needed, contemporary psychologism has the key resources to retain its place in the philosophical debate on the foundations of logic. (shrink)

We argue that Wittgenstein’s philosophical perspective on Gödel’s most famous theorem is even more radical than has commonly been assumed. Wittgenstein shows in detail that there is no way that the Gödelian construct of a string of signs could be assigned a useful function within (ordinary) mathematics. — The focus is on Appendix III to Part I of Remarks on the Foundations of Mathematics. The present reading highlights the exceptional importance of this particular set of remarks and, more specifically, (...) emphasises its refined composition and rigorous internal structure. (shrink)

Full applicability of physics to human biology does not necessarily imply that one can uncover a comprehensive, algorithmic correlation between physical brain states and corresponding mental states. The argument takes into account that information processing is finite in principle in a finite world. Presumbly the brain-mind-relation cannot be resolved in all essential aspects, particularly when high degrees of abstraction or self-analytical processes are involved. Our conjecture plausibly unifies the universal validity of physics and a logical limitation of human thought, and (...) it does not regard consciousness -the most basic human experience - as a marginal phenomenon. (shrink)

The principle of maximal entropy (further abbreviated as “MaxEnt”) can be founded on the formal mechanism, in which future transforms into past by the mediation of present. This allows of MaxEnt to be investigated by the theory of quantum information. MaxEnt can be considered as an inductive analog or generalization of “Occam’s razor”. It depends crucially on choice and thus on information just as all inductive methods of reasoning. The essence shared by Occam’s razor and MaxEnt is for the relevant (...) data known till now to be postulated as an enough fundament of conclusion. That axiom is the kind of choice grounding both principles. Popper’s falsifiability (1935) can be discussed as a complement to them: That axiom (or axiom scheme) is always sufficient but never necessary condition of conclusion therefore postulating the choice in the base of MaxEnt. Furthermore, the abstraction axiom (or axiom scheme) relevant to set theory (e.g. the axiom scheme of specification in ZFC) involves choice analogically. (shrink)

Although the invariance criterion of logicality first emerged as a criterion of a purely mathematical interest, it has developed into a criterion of considerable linguistic and philosophical interest. In this paper I compare two different perspectives on this criterion. The first is the perspective of natural language. Here, the invariance criterion is measured by its success in capturing our linguistic intuitions about logicality and explaining our logical behavior in natural-linguistic settings. The second perspective is more theoretical. Here, the invariance criterion (...) is used as a tool for developing a theoretical foundation of logic, focused on a critical examination, explanation, and justification of its veridicality and modal force. (shrink)

The current definition of Constructive mathematics as “mathematics within intuitionist logic” ignores two fundamental issues. First, the kind of organization of the theory at issue. I show that intuitionist logic governs a problem-based organization, whose model is alternative to that of the deductive-axiomatic organization, governed by classical logic. Moreover, this dichotomy is independent of that of the kind of infinity, either potential or actual, to which respectively correspond constructive mathematical and classical mathematical tools. According to this view (...) a mathematical theory is based on the choices regarding these two dichotomies. As an example of this kind of foundation, arithmetic is rationally re-founded on constructive mathematical tools and the model of the problem-based organization. In conclusion, constructive mathematics is not only mathematics making use of constructive tools in intuitionist logic but also organized according to around a basic problem, solved by a method discovered using intuitionist logic. (shrink)

This review essay analyzes the book by Giuseppe Primiero, On the foundations of computing. Oxford: Oxford University Press (ISBN 978-0-19-883564-6/hbk; 978-0-19-883565-3/pbk). xix, 296 p. (2020). It gives a critical view from the perspective of physical computing as a foundation of computing and argues that the neglected pillar of material computation (Stepney) should be brought centerstage and computing recognized as the fourth great domain of science (Denning).

Categorical logic has shown that modern logic is essentially the logic of subsets (or "subobjects"). Partitions are dual to subsets so there is a dual logic of partitions where a "distinction" [an ordered pair of distinct elements (u,u′) from the universe U ] is dual to an "element". An element being in a subset is analogous to a partition π on U making a distinction, i.e., if u and u′ were in different blocks of π. Subset (...) logic leads to finite probability theory by taking the (Laplacian) probability as the normalized size of each subset-event of a finite universe. The analogous step in the logic of partitions is to assign to a partition the number of distinctions made by a partition normalized by the total number of ordered pairs |U|² from the finite universe. That yields a notion of "logical entropy" for partitions and a "logical information theory." The logical theory directly counts the (normalized) number of distinctions in a partition while Shannon's theory gives the average number of binary partitions needed to make those same distinctions. Thus the logical theory is seen as providing a conceptual underpinning for Shannon's theory based on the logical notion of "distinctions.". (shrink)

The view that contradictions cannot be true has been part of accepted philosophical theory since at least the time of Aristotle. In this regard, it is almost unique in the history of philosophy. Only in the last forty years has the view been systematically challenged with the advent of dialetheism. Since Graham Priest introduced dialetheism as a solution to certain self-referential paradoxes, the possibility of true contradictions has been a live issue in the philosophy of logic. Yet, despite the (...) arguments advanced by dialetheists, many logicians and philosophers still hold the opinion that contradictions cannot be true. -/- Rather than advocating the truth of certain contradictions, this thesis offers a different challenge to the classical logician. By showing that it can be philosophically coherent to propose that true contradictions are metaphysically possible, the thesis suggests that the classical logician must do more than she currently has to justify her confidence in the impossibility of true contradictions. Simply fighting off the dialetheist’s putative examples of true contradictions at the actual world isn’t enough to justify the classical logician’s conclusion that true contradictions are impossible. -/- To aid the thesis dialectically, we introduce a new position, absolutism, which hypothesises that it’s metaphysically possible for at least one contradiction to be true, contrasting with the dialetheic hypothesis that some contradictions are true in the actual world. We demonstrate that absolutism can be given a philosophically coherent interpretation, an appropriate logic, and that certain criticisms are completely toothless against absolutism. The challenge put to the classical logician is then: On what logical or philosophical grounds can we rule out the metaphysical possibility of true contradictions? (shrink)

Physical dimensions like “mass”, “length”, “charge”, represented by the symbols [M], [L], [Q], are not numbers, but used as numbers to perform dimensional analysis in particular, and to write the equations of physics in general, by the physicist. The law of excluded middle falls short of explaining the contradictory meanings of the same symbols. The statements like “m tends to 0”, “r tends to 0”, “q tends to 0”, used by the physicist, are inconsistent on dimensional grounds because “m”, “r”, (...) “q” represent quantities with physical dimensions of [M], [L], [Q] respectively and “0” represents just a number—devoid of physical dimension. Consequently, due to the involvement of the statement “q tends to 0'', where q is the test charge” in the definition of electric field leads to either circular reasoning or a contradiction regarding the experimental verification of the smallest charge in the Millikan–Fletcher oil drop experiment. Considering such issues as problematic, by choice, I make an inquiry regarding the basic language in terms of which physics is written, with an aim of exploring how truthfully the verbal statements can be converted to the corresponding physico-mathematical expressions, where “physico-mathematical” signifies the involvement of physical dimensions. Such investigation necessitates an explanation by demonstration of “self inquiry”, “middle way”, “dependent origination”, “emptiness/relational existence”, which are certain terms that signify the basic tenets of Buddhism. In light of such demonstration I explain my view of “definition”; the relations among quantity, physical dimension and number; meaninglessness of “zero quantity” and the associated logico-linguistic fallacy; difference between unit and unity. Considering the importance of the notion of electric field in physics, I present a critical analysis of the definitions of electric field due to Maxwell and Jackson, along with the physico-mathematical conversions of the verbal statements. The analysis of Jackson’s definition points towards an expression of the electric field as an infinite series due to the associated “limiting process” of the test charge. However, it brings out the necessity of a postulate regarding the existence of charges, which nevertheless follows from the definition of quantity. Consequently, I explain the notion of undecidable charges that act as the middle way to resolve the contradiction regarding the Millikan–Fletcher oil drop experiment. In passing, I provide a logico-linguistic analysis, in physico-mathematical terms, of two verbal statements of Maxwell in relation to his definition of electric field, which suggests Maxwell’s conception of dependent origination of distance and charge ) and that of emptiness in the context of relative vacuum. This work is an appeal for the dissociation of the categorical disciplines of logic and physics and on the large, a fruitful merger of Eastern philosophy and Western science. Nevertheless, it remains open to how the reader relates to this work, which is the essence of emptiness. (shrink)

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

In this paper, I attempt to clarify the heart of Dewey’s philosophy: his method (denotative method (DM) / pattern of inquiry (PI)). Despite the traditional understanding of Dewey as anti-foundationalist, I want to show that Dewey did have metaphysical foundations for his method: the principle of continuity or theory of emergentism. I also argue that Dewey’s metaphysical position is better named as ‘cultural emergentism’, rather than his own term ‘cultural naturalism’. What Dewey called ‘common sense’ in his Logic, (...) Husserl termed as the ‘life-world’ in his Crisis. I compare two perspectives of dealing with the phenomenon and conclude that for Dewey, the difference between natural sciences and the common sense inquiry is that of subject-matter but not of method. Thus, the goal is to find the unified method to be applied in both domains. Whereas Husserl was more pessimistic: for him, the difference was not only in subject-matter, but in the very methods. Following that discussion, I also attempt to reformulate the hard problem of consciousness in Deweyan terms. In the end, I compare Dewey’s DM / PI with Popper’s understandings of scientific method and conclude that there is no significant difference between the two and that Dewey’s method could also be looked at as hypothetic-deductive method, with the only difference in emphases. (shrink)

I propose that an irreducible property of physical space — consistency — is the origin of logic. I propose that an inconsistent space is inconceivable and that this inconceivability can be recognized as the force behind logical propositions. The implications of this argument are briefly explored and then applied to address two paradoxes: Zeno of Elea’s paradox regarding the race between Achilles and the Tortoise, and Lewis Carroll’s paradox regarding the Tortoise’s conversation with Achilles after the race. I conclude (...) that Achilles would have won on both accounts, in the race and the argument, by invoking the consistency of space as the foundation of both movement across space and logical argument. (shrink)

This paper presents a new system of logic, LF, that is intended to be used as the foundation of the formalization of science. That is, deductive validity according to LF is to be used as the criterion for assessing what follows from the verdicts, hypotheses, or conjectures of any science. In work currently in progress, we argue for the unique suitability of LF for the formalization of logic, mathematics, syntax, and semantics. The present document specifies the language and (...) rules of LF, lays out some notational conventions, and states some basic technical facts about the system. (shrink)

The Aristotelian origins of formal systems are outlined, together with Aristotle's use of causal terms in describing syllogisms. The precision and exactness of a formalism, based on the projection of logical forms into perceptive signs, is contrasted with foundational, abstract concepts, independent of any formalism, which are presupposed for the understanding of a formal language. The definition of a formal system by means of a Turing machine is put in the context of Wittgenstein's general considerations of a machine understood as (...) a sign. A modification of Łukasiewicz's logic Ł3 with the inclusion of justification terms is proposed in order to formally analyse some features of formalistic reasoning as a mechanical, causal affair within a wider context of "indeterminacy". (shrink)

A key question in the philosophy of logic is how we have epistemic justification for claims about logical entailment (assuming we have such justification at all). Justification holism asserts that claims of logical entailment can only be justified in the context of an entire logical theory, e.g., classical, intuitionistic, paraconsistent, paracomplete etc. According to holism, claims of logical entailment cannot be atomistically justified as isolated statements, independently of theory choice. At present there is a developing interest in—and endorsement of—justification (...) holism due to the revival of an abductivist approach to the epistemology of logic. This paper presents an argument against holism by establishing a foundational entailment-sentence of deduction which is justified independently of theory choice and outside the context of a whole logical theory. (shrink)

The foundational ideas of David Hilbert have been generally misunderstood. In this dissertation prospectus, different aims of Hilbert are summarized and a new interpretation of Hilbert's work in the foundations of mathematics is roughly sketched out. Hilbert's view of the axiomatic method, his response to criticisms of set theory and intuitionist criticisms of the classical foundations of mathematics, and his view of the role of logical inference in mathematical reasoning are briefly outlined.

This book was completed by the early 1960s and published in 1984 but it has not lost its topicality, for it contains an important re-assessment of the relations of two main streams of contemporary philosophy - the Analytical and the Dialectic. Adherents and critics of these traditions tend to assurnethat they are diametrically opposed, that their roots, concerns and approaches contradict each other, and that no reconciliation is possible. In contradistinction Russell derives both traditions from the common root of the (...) dissatisfaction with the arguments against speculative philosophy. These according to the author leave a lacuna - certain elementsof our Weltanschaaung have been removed, but they cannot be removed without replacement lest we have an incomplete world view, so incomplete in fact that it cannot be viable. According to Russell part of this vacuum is taken up by the analytical tradition but this tradition is not capable of taking up the remainder of it. That portion of the vacant space is however taken up by the dialectical tradi tion, which in turn cannot itself handle the whole of the problem. Thus the two reactions to the demise of speculative philosophy appear to be complementary in at least this sense. But the author goes further, for according to hirn the analytical arguments themselves clearly point to the emergence of dialectical problems, and the dialectical problems themselves need some such background to arise. (shrink)

We review our approach to quantum mechanics adding also some new interesting results. We start by giving proof of two important theorems on the existence of the A(Si) and i,±1 N Clifford algebras. This last algebra gives proof of the von Neumann basic postulates on the quantum measurement explaining thus in an algebraic manner the wave function collapse postulated in standard quantum theory. In this manner we reach the objective to expose a self-consistent version of quantum mechanics. In detail we (...) realize a bare bone skeleton of quantum mechanics recovering all the basic foundations of this theory on an algebraic framework. We give proof of the quantum like Heisenberg uncertainty relations using only the basic support of the Clifford algebra. In addition we demonstrate the well known phenomenon of quantum Mach Zender interference using the same algebraic framework, as well as we give algebraic proof of quantum collapse in some cases of physical interest by direct application of the theorem that we derive to elaborate the i,±1 N algebra. We also discuss the problem of time evolution of quantum systems as well as the changes in space location, in momentum and the linked invariance principles. We are also able to re-derive the basic wave function of standard quantum mechanics by using only the Clifford algebraic approach. In this manner we obtain a full exposition of standard quantum mechanics using only the basic axioms of Clifford algebra. We also discuss more advanced features of quantum mechanics. In detail, we give demonstration of the Kocken-Specher theorem, and also we give an algebraic formulation and explanation of the EPR paradox only using the Clifford algebra. By using the same approach we also derive Bell inequalities. Our formulation is strongly based on the use of idempotents that are contained in Clifford algebra. Their counterpart in quantum mechanics is represented by the projection operators that, as it is well known, are interpreted as logical statements, following the basic von Neumann results. Von Neumann realized a matrix logic on the basis of quantum mechanics. Using the Clifford algebra we are able to invert such result. According to the results previously obtained by Orlov in 1994, we are able to give proof that quantum mechanics derives from logic. We show that indeterminism and quantum interference have their origin in the logic. Therefore, it seems that we may conclude that quantum mechanics, as it appears when investigated by the Clifford algebra, is a two-faced theory in the sense that it looks from one side to “matter per se”, thus to objects but simultaneously also to conceptual entities. We advance the basic conclusion of the paper: There are stages of our reality in which we no more can separate the logic ( and thus cognition and thus conceptual entity) from the features of “matter per se”. In quantum mechanics the logic, and thus the cognition and thus the conceptual entity-cognitive performance, assume the same importance as the features of what is being described. We are at levels of reality in which the truths of logical statements about dynamic variables become dynamic variables themselves so that a profound link is established from its starting in this theory between physics and conceptual entities. Finally, in this approach there is not an absolute definition of logical truths. Transformations , and thus … “redefinitions”…. of truth values are permitted in such scheme as well as the well established invariance principles, clearly indicate . (shrink)

George Boole emerged from the British tradition of the “New Analytic”, known for the view that the laws of logic are laws of thought. Logicians in the New Analytic tradition were influenced by the work of Immanuel Kant, and by the German logicians Wilhelm Traugott Krug and Wilhelm Esser, among others. In his 1854 work An Investigation of the Laws of Thought on Which are Founded the Mathematical Theories of Logic and Probabilities, Boole argues that the laws of (...) thought acquire normative force when constrained to mathematical reasoning. Boole’s motivation is, first, to address issues in the foundations of mathematics, including the relationship between arithmetic and algebra, and the study and application of differential equations (Durand-Richard, van Evra, Panteki). Second, Boole intended to derive the laws of logic from the laws of the operation of the human mind, and to show that these laws were valid of algebra and of logic both, when applied to a restricted domain. Boole’s thorough and flexible work in these areas influenced the development of model theory (see Hodges, forthcoming), and has much in common with contemporary inferentialist approaches to logic (found in, e.g., Peregrin and Resnik). (shrink)