Peirce and Dewey were generally more concerned with the process of scientific activity than purely mathematical work. However, their accounts of knowledge production afford some insights into the epistemology of mathematical postulates, especially definition and axioms. Their rejection of rationalist metaphysics and their emphasis on continuity in inquiry provides the pretext for the pragmatic a priori – hypothetical and operational assumptions whose justification relies on their fruitfulness in the long run. This paper focuses on the application of this idea (...) to the epistemology of definitions and an account of progress in mathematics, although it has broader implications for the study of conceptual change and the function and basis of presuppositions in the sciences. (shrink)

Although the concept of uncertainty is as old as Epicurus’s writings, and an excellent quantitative theory, with entropy as the measure of uncertainty having been developed in recent times, there has been little exploration of the qualitative theory. The purpose of the present paper is to give a qualitative axiomatization of uncertainty, in the spirit of the many studies of qualitative comparative probability. The qualitative axioms are fundamentally about the uncertainty of a partition of the probability space of events. (...) Of course, it is common to speak of the uncertainty, or randomness, of a random variable, but only the partition defined by the values of the random variable enter into the definition of uncertainty, not the actual values. It is straightforward to add axioms for decision making following the general line of Savage from the 1950s. Indeed, in the spirit of Epicurus, it is really our intuitive feeling about the uncertainty of the future that motivates much of our thinking about decisions. Here, the distinction between the concepts of probability and uncertainty can be made by citing many familiar examples. Without spelling out the technical details, the axiomatization of qualitative probability with uncertainty as the most important primitive concept, it is possible to raise a different kind of question about bounded rationality. This new question is whether or not one should bound the uncertainty in thinking and investigating any detailed framework of decision making. Discussion of this point is certainly different from the question of bounding rationality by not maximizing expected utility. In practice, we naturally bound uncertainty in our analysis of decision-making problems. As in the case of formulating an alternative for maximizing expected utility, so is the case of rational alternatives to maximizing uncertainty. There are several issues to consider. In the spirit of my other work in qualitative probability, I explore alternatives rather than attempt to give a definitive argument for one single solution. (shrink)

A description of consciousness leads to a contradiction with the postulation from special relativity that there can be no connections between simultaneous event. This contradiction points to consciousness involving quantum level mechanisms. The Quantum level description of the universe is re- evaluated in the light of what is observed in consciousness namely 4 Dimensional objects. A new improved interpretation of Quantum level observations is introduced. From this vantage point the following axioms of consciousness is presented. Consciousness consists of two (...) distinct components, the observed U and the observer I. The observed U consist of all the events I is aware of. A vast majority of these occur simultaneously. Now if I were to be an entity within the space-time continuum, all of these events of U together with I would have to occur at one point in space-time. However, U is distributed over a definite region of space-time (region in brain). Thus, I is aware of a multitude of space-like separated events. It is seen that this awareness necessitates I to be an entity outside the space-time continuum. With I taken as such, a new concept called concept A is introduced. With the help of concept A a very important axiom of consciousness, namely Free Will is explained. Libet s Experiment which was originally seen to contradict Free will, in the light of Concept A is shown to support it. A variation to Libet s Experiment is suggested that will give conclusive proof for Concept A and Free Will. (shrink)

The purpose of this paper is to challenge some widespread assumptions about the role of the modal axiom 4 in a theory of vagueness. In the context of vagueness, axiom 4 usually appears as the principle ‘If it is clear (determinate, definite) that A, then it is clear (determinate, definite) that it is clear (determinate, definite) that A’, or, more formally, CA → CCA. We show how in the debate over axiom 4 two different notions of clarity are in play (...) (Williamson-style "luminosity" or self-revealing clarity and concealeable clarity) and what their respective functions are in accounts of higher-order vagueness. On this basis, we argue first that, contrary to common opinion, higher-order vagueness and S4 are perfectly compatible. This is in response to claims like that by Williamson that, if vagueness is defined with the help of a clarity operator that obeys axiom 4, higher-order vagueness disappears. Second, we argue that, contrary to common opinion, (i) bivalence-preservers (e.g. epistemicists) can without contradiction condone axiom 4 (by adopting what elsewhere we call columnar higher-order vagueness), and (ii) bivalence-discarders (e.g. open-texture theorists, supervaluationists) can without contradiction reject axiom 4. Third, we rebut a number of arguments that have been produced by opponents of axiom 4, in particular those by Williamson. (The paper is pitched towards graduate students with basic knowledge of modal logic.). (shrink)

Spinoza's Ethics self-consciously follows the example of Euclid and other geometers in its use of axioms and definitions as the basis for derivations of hundreds of propositions of philosophical si...

Axiom weakening is a technique that allows for a fine-grained repair of inconsistent ontologies. Its main advantage is that it repairs on- tologies by making axioms less restrictive rather than by deleting them, employing the use of refinement operators. In this paper, we build on pre- viously introduced axiom weakening for ALC, and make it much more irresistible by extending its definitions to deal with SROIQ, the expressive and decidable description logic underlying OWL 2 DL. We extend the (...) definitions of refinement operator to deal with SROIQ constructs, in particular with role hierarchies, cardinality constraints and nominals, and illustrate its application. Finally, we discuss the problem of termi- nation of an iterated weakening procedure. (shrink)

This paper challenges the first Gettier counterexample to the tripartite account of knowledge. Noting that 'the man who will get the job' is a description and invoking Donnellan's distinction between their 'referential' and 'attributive' uses, I argue that Smith does not actually believe that the man who will get the job has ten coins in his pocket. Smith's ignorance about who will get the job shows that the belief cannot be understood referentially, his ignorance of the coins in his pocket (...) shows that it cannot be understood attributively. An explanation for why Smith appeared to have justified true belief is given by distinguishing between 'belief' and 'belief in truth'. Smith believes the sentence 'the man who will get the job has ten coins in his pocket' to be true (he mistakenly believes that Jones will get the job, of whom he knows that he has ten coins in his pocket) (hence his 'belief'), the sentence is true (hence 'truth'), and he has sufficient reason to assent to it (hence his 'justification'). But he does not believe the proposition expressed. Hence he does not know it either. (shrink)

Science is founded on a method based on critical thinking. A prerequisite for this is not only a sufficient command of language but also the comprehension of the basic concepts underlying our understanding of reality. This constraint implies an awareness of the fact that the truth of the World is not directly accessible to us, but can only be glimpsed through the construction of mod- els designed to anticipate its behaviour. Because the relationship between models and reality rests on the (...) interpretation of founding postulates and in- stantiations of their predictions (and is therefore deeply rooted in language and culture), there can be no demarcation between science and non-science. However, critical thinking is essential to ensure that the link between models and reality is gradually made more adequate to reality, based on what has already been established, thus guaranteeing that science progresses on this basis and excluding any form of relativism. (shrink)

Spinoza scholars have claimed that we are faced with a dilemma: either Spinoza's definitions in his Ethics are real, in spite of indications to the contrary, or the definitions are nominal and the propositions derived from them are false. I argue that Spinoza did not recognize the distinction between real and nominal definitions. Rather, Spinoza classified definitions according to whether they require a priori or a posteriori justification, which is a classification distinct from either the real/nominal (...) or the intensional/extensional classification. I argue that Spinoza uses both a priori and a posteriori definitions in the Ethics and that recognizing both types of definitions allows us to understand Spinoza's geometric method in a new way. We can now understand the geometric method as two methods, one resulting in propositions that Spinoza considers to be absolutely certain and another resulting in propositions that Spinoza does not consider certain. The latter method makes use of a posteriori definitions and postulates, whereas the former method uses only a priori definitions and axioms. (shrink)

Originally titled “Is It Murder in Tennessee to Kill a Chimpanzee,” this article argues in some detail that typical legal definitions of “murder” as involving the intentional killing of “a reasonable being” would require classifying the intentional killing of chimpanzees as murder.

The paper justifies the following theses: The totality can found time if the latter is axiomatically represented by its “arrow” as a well-ordering. Time can found choice and thus information in turn. Quantum information and its units, the quantum bits, can be interpreted as their generalization as to infinity and underlying the physical world as well as the ultimate substance of the world both subjective and objective. Thus a pathway of interpretation between the totality via time, order, choice, and information (...) to the substance of the world is constructed. The article is based only on the well-known facts and definitions and is with no premises in this sense. Nevertheless it is naturally situated among works and ideas of Husserl and Heidegger, linked to the foundation of mathematics by the axiom of choice, to the philosophy of quantum mechanics and information. (shrink)

A description of consciousness leads to a contradiction with the postulation from special relativity that there can be no connections between simultaneous event. This contradiction points to consciousness involving quantum level mechanisms. The Quantum level description of the universe is re- evaluated in the light of what is observed in consciousness namely 4 Dimensional objects. A new improved interpretation of Quantum level observations is introduced. From this vantage point the following axioms of consciousness is presented. Consciousness consists of two (...) distinct components, the observed U and the observer I. The observed U consist of all the events I is aware of. A vast majority of these occur simultaneously. Now if I were to be an entity within the space-time continuum, all of these events of U together with I would have to occur at one point in space-time. However, U is distributed over a definite region of space-time (region in brain). Thus, I is aware of a multitude of space-like separated events. It is seen that this awareness necessitates I to be an entity outside the space-time continuum. With I taken as such, a new concept called concept A is introduced. With the help of concept A a very important axiom of consciousness, namely Free Will is explained. (shrink)

John Corcoran and George Boger. Aristotelian logic and Euclidean geometry. Bulletin of Symbolic Logic. 20 (2014) 131. -/- By an Aristotelian logic we mean any system of direct and indirect deductions, chains of reasoning linking conclusions to premises—complete syllogisms, to use Aristotle’s phrase—1) intended to show that their conclusions follow logically from their respective premises and 2) resembling those in Aristotle’s Prior Analytics. Such systems presuppose existence of cases where it is not obvious that the conclusion follows from the premises: (...) there must be something deductions can show. Corcoran calls a proposition that follows from given premises a hidden consequence of those premises if it is not obvious that the proposition follows from those premises. By a Euclidean geometry we mean an extended discourse beginning with basic premises—axioms, postulates, definitions—1) treating a universe of geometrical figures and 2) resembling Euclid’s Elements. There were Euclidean geometries before Euclid (fl. 300 BCE), even before Aristotle (384–322 BCE). Bochenski, Lukasiewicz, Patzig and others never new this or if they did they found it inconvenient to mention. Euclid shows no awareness of Aristotle. It is obvious today—as it should have been obvious in Euclid’s time, if anyone knew both—that Aristotle’s logic was insufficient for Euclid’s geometry: few if any geometrical theorems can be deduced from Euclid’s premises by means of Aristotle’s deductions. Aristotle’s writings don’t say whether his logic is sufficient for Euclidean geometry. But, there is not even one fully-presented example. However, Aristotle’s writings do make clear that he endorsed the goal of a sufficient system. Nevertheless, incredible as this is today, many logicians after Aristotle claimed that Aristotelian logics are sufficient for Euclidean geometries. This paper reviews and analyses such claims by Mill, Boole, De Morgan, Russell, Poincaré, and others. It also examines early contrary statements by Hintikka, Mueller, Smith, and others. Special attention is given to the argumentations pro or con and especially to their logical, epistemic, and ontological presuppositions. What methodology is necessary or sufficient to show that a given logic is adequate or inadequate to serve as the underlying logi of a given science. (shrink)

This essay endeavors to define the concept of indefinite extensibility in the setting of category theory. I argue that the generative property of indefinite extensibility for set-theoretic truths in category theory is identifiable with the Grothendieck Universe Axiom and the elementary embeddings in Vopenka's principle. The interaction between the interpretational and objective modalities of indefinite extensibility is defined via the epistemic interpretation of two-dimensional semantics. The semantics can be defined intensionally or hyperintensionally. By characterizing the modal profile of $\Omega$-logical validity, (...) and thus the generic invariance of mathematical truth, modal coalgebras are further capable of capturing the notion of definiteness for set-theoretic truths, in order to yield a non-circular definition of indefinite extensibility. (shrink)

How do axioms, or first principles, in ethics compare to those in mathematics? In this companion piece to G.C. Field's 1931 "On the Role of Definition in Ethics", I argue that there are similarities between the cases. However, these are premised on an assumption which can be questioned, and which highlights the peculiarity of normative inquiry.

Critical examination of Alchourrón and Bulygin’s set-theoretic definition of normative system shows that deductive closure is not an inevitable property. Following von Wright’s conjecture that axioms of standard deontic logic describe perfection-properties of a norm-set, a translation algorithm from the modal to the set-theoretic language is introduced. The translations reveal that the plausibility of metanormative principles rests on different grounds. Using a methodological approach that distinguishes the actor roles in a norm governed interaction, it has been shown that metanormative (...) principles are directed second-order obligations and, in particular, that the requirement related to deductive closure is directed to the norm-applier role rather than to the norm-giver role. The approach has been applied to the case of pure derogation yielding a new result, namely, that an independence property is a perfection-property of a norm-set in view of possible derogation. This paper in a polemical way touches upon several points raised by Kristan in his recent paper. (shrink)

This essay describes and evaluates the conception of mereological structure that underpins Deleuze’s account of ontogenesis in Difference and Repetition. A theory of mereology is a theory of composition: it asks what it is to be a part making a whole, what it is to be a whole collecting its parts; in short, in what the relation of making or composing consists. The locus classicus for modern mereology is the third of Husserl’s Logical Investigations (‘On the Theory of Wholes and (...) Parts’), which deduces the definitions, axioms and principles governing the relation of parthood and associated notions, such as wholeness, separation, simplicity and complexity. My question is this: what happens to these notions when they are no longer the terms of a ‘calculus of individuals’ and, instead, become the terms of a differential calculus of individuation? (shrink)

How the arguments of Spinoza's Ethics work might seem obvious. Even if Spinoza's exposition is not perfect, and some suppressed premises might have to be recovered, it seems clear enough that the demonstrations are supposed to show, in Euclidian fashion, how truths about the basic structure of nature—as well as truths about how to live—follow from axioms and uncontroversial definitions. If readers keep their imagination and emotions from sullying their reasoning, they will see the force of the demonstrations (...) and be convinced.In his engaging, highly original book, Garver argues that the Ethics is not a linear march through timeless truths, but rather a complicated drama that works precisely because its "characters,"... (shrink)

The iterative conception of set is typically considered to provide the intuitive underpinnings for ZFCU (ZFC+Urelements). It is an easy theorem of ZFCU that all sets have a definite cardinality. But the iterative conception seems to be entirely consistent with the existence of “wide” sets, sets (of, in particular, urelements) that are larger than any cardinal. This paper diagnoses the source of the apparent disconnect here and proposes modifications of the Replacement and Powerset axioms so as to allow for (...) the existence of wide sets. Drawing upon Cantor’s notion of the absolute infinite, the paper argues that the modifications are warranted and preserve a robust iterative conception of set. The resulting theory is proved consistent relative to ZFC + “there exists an inaccessible cardinal number.”. (shrink)

Discusses Frege's formal definitions and characterizations of infinite and finite sets. Speculates that Frege might have discovered the "oddity" in Dedekind's famous proof that all infinite sets are Dedekind infinite and, in doing so, stumbled across an axiom of countable choice.

Discusses Frege's formal definitions and characterizations of infinite and finite sets. Speculates that Frege might have discovered the "oddity" in Dedekind's famous proof that all infinite sets are Dedekind infinite and, in doing so, stumbled across an axiom of countable choice.

It is commonly thought that such topics as Impossibility, Incompleteness, Paraconsistency, Undecidability, Randomness, Computability, Paradox, Uncertainty and the Limits of Reason are disparate scientific physical or mathematical issues having little or nothing in common. I suggest that they are largely standard philosophical problems (i.e., language games) which were resolved by Wittgenstein over 80 years ago. -/- Wittgenstein also demonstrated the fatal error in regarding mathematics or language or our behavior in general as a unitary coherent logical ‘system,’ rather than as (...) a motley of pieces assembled by the random processes of natural selection. “Gödel shows us an unclarity in the concept of ‘mathematics’, which is indicated by the fact that mathematics is taken to be a system” and we can say (contra nearly everyone) that is all that Gödel and Chaitin show. Wittgenstein commented many times that ‘truth’ in math means axioms or the theorems derived from axioms, and ‘false’ means that one made a mistake in using the definitions, and this is utterly different from empirical matters where one applies a test. Wittgenstein often noted that to be acceptable as mathematics in the usual sense, it must be useable in other proofs and it must have real world applications, but neither is the case with Godel’s Incompleteness. Since it cannot be proved in a consistent system (here Peano Arithmetic but a much wider arena for Chaitin), it cannot be used in proofs and, unlike all the ‘rest’ of PA it cannot be used in the real world either. As Rodych notes “…Wittgenstein holds that a formal calculus is only a mathematical calculus (i.e., a mathematical language-game) if it has an extra- systemic application in a system of contingent propositions (e.g., in ordinary counting and measuring or in physics) …” Another way to say this is that one needs a warrant to apply our normal use of words like ‘proof’, ‘proposition’, ‘true’, ‘incomplete’, ‘number’, and ‘mathematics’ to a result in the tangle of games created with ‘numbers’ and ‘plus’ and ‘minus’ signs etc., and with -/- ‘Incompleteness’ this warrant is lacking. Rodych sums it up admirably. “On Wittgenstein’s account, there is no such thing as an incomplete mathematical calculus because ‘in mathematics, everything is algorithm [and syntax] and nothing is meaning [semantics]…” -/- I make some brief remarks which note the similarities of these ‘mathematical’ issues to economics, physics, game theory, and decision theory. -/- Those wishing further comments on philosophy and science from a Wittgensteinian two systems of thought viewpoint may consult my other writings -- Talking Monkeys--Philosophy, Psychology, Science, Religion and Politics on a Doomed Planet--Articles and Reviews 2006-2019 3rd ed (2019), The Logical Structure of Philosophy, Psychology, Mind and Language in Ludwig Wittgenstein and John Searle 2nd ed (2019), Suicide by Democracy 4th ed (2019), The Logical Structure of Human Behavior (2019), The Logical Structure of Consciousness (2019, Understanding the Connections between Science, Philosophy, Psychology, Religion, Politics, and Economics and Suicidal Utopian Delusions in the 21st Century 5th ed (2019), Remarks on Impossibility, Incompleteness, Paraconsistency, Undecidability, Randomness, Computability, Paradox, Uncertainty and the Limits of Reason in Chaitin, Wittgenstein, Hofstadter, Wolpert, Doria, da Costa, Godel, Searle, Rodych, Berto, Floyd, Moyal-Sharrock and Yanofsky (2019), and The Logical Structure of Philosophy, Psychology, Sociology, Anthropology, Religion, Politics, Economics, Literature and History (2019). (shrink)

I discuss Aristotle's anomalous terminology in Physics A.1 (involving "universals" and "particulars") and its coherence with Aristotle's notion of scientific demonstration.

The link between the high-order metaphysics and abstractions, on the one hand, and choice in the foundation of set theory, on the other hand, can distinguish unambiguously the “good” principles of abstraction from the “bad” ones and thus resolve the “bad company problem” as to set theory. Thus it implies correspondingly a more precise definition of the relation between the axiom of choice and “all company” of axioms in set theory concerning directly or indirectly abstraction: the principle of abstraction, (...) axiom of comprehension, axiom scheme of specification, axiom scheme of separation, subset axiom scheme, axiom scheme of replacement, axiom of unrestricted comprehension, axiom of extensionality, etc. (shrink)

In the article ”Inconsistency of N from a not-finitist point of view” we have shown the inconsistency of N, going through a denial. Here we delete this indirect step and essentially repeat the same proof. Contextually we find a contradiction about natural number definition. Then we discuss around the rejection of infinity.

This paper provides an overview of the contemporary debate on the concepts and definitions of Corporate Social Responsibility (CSR) and Corporate Sustainability (CS). The conclusions, based on historical perspectives, philosophical analyses, impact of changing contexts and situations and practical considerations, show that "one solution fits all"-definition for CS(R) should be abandoned, accepting various and more specific definitions matching the development, awareness and ambition levels of organizations.

This paper and its sequel “look under the hood” of the usual sorts of proof-theoretic systems for certain well-known intuitionistic and classical propositional modal logics. Section 1 is preliminary. Of most importance: a marked formula will be the result of prefixing a formula in a propositional modal language with a step-marker, for this paper either 0 or 1. Think of 1 as indicating the taking of “one step away from 0.” Deductions will be constructed using marked formulas. Section 2 presents (...) the model-theoretic concepts, based on those in [7], that guide the rest of this paper. Section 3 presents Natural Deduction systems IK and CK, formalizations of intuitionistic and classical one-step versions of K. In these systems, occurrences of step-markers allow deductions to display deductive structure that is covered over in familiar “no step” proof-theoretic systems for such logics. Box and Diamond are governed by Introduction and Elimination rules; the familiar K rule and Necessitation are derived (i.e. admissible) rules. CK will be the result of adding the 0-version of the Rule of Excluded Middle to the rules which generate IK. Note: IK is the result of merely dropping that rule from those generating CK, without addition of further rules or axioms (as was needed in [7]). These proof-theoretic systems yield intuitionistic and classical consequence relations by the obvious definition. Section 4 provides some examples of what can be deduced in IK. Section 5 defines some proof-theoretic concepts that are used in Section 6 to prove the soundness of the consequence relation for IK (relative to the class of models defined in Section 2.) Section 7 proves its completeness (relative to that class). Section 8 extends these results to the consequence relation for CK. (Looking ahead: Part 2 will investigate one-step proof-theoretic systems formalizing intuitionistic and classical one-step versions of some familiar logics stronger than K.). (shrink)

Stochastic independence has a complex status in probability theory. It is not part of the definition of a probability measure, but it is nonetheless an essential property for the mathematical development of this theory. Bayesian decision theorists such as Savage can be criticized for being silent about stochastic independence. From their current preference axioms, they can derive no more than the definitional properties of a probability measure. In a new framework of twofold uncertainty, we introduce preference axioms that (...) entail not only these definitional properties, but also the stochastic independence of the two sources of uncertainty. This goes some way towards filling a curious lacuna in Bayesian decision theory. (shrink)

In this paper I present a new way of understanding Dutch Book Arguments: the idea is that an agent is shown to be incoherent iff he would accept as fair a set of bets that would result in a loss under any interpretation of the claims involved. This draws on a standard definition of logical inconsistency. On this new understanding, the Dutch Book Arguments for the probability axioms go through, but the Dutch Book Argument for Reflection fails. The question (...) of whether we have a Dutch Book Argument for Conditionalization is left open. (shrink)

Edward Nieznanski developed in 2007 and 2008 two different systems in formal logic which deal with the problem of evil. Particularly, his aim is to refute a version of the logical problem of evil associated with a form of religious determinism. In this paper, we revisit his first system to give a more suitable form to it, reformulating it in first-order modal logic. The new resulting system, called N1, has much of the original basic structure, and many axioms, (...) class='Hi'>definitions, and theorems still remain; however, some new results are obtained. If the conclusions attained are correct and true, then N1 solves the problem of evil through the refutation of a version of religious determinism, showing that the attributes of God in Classical Theism, namely, those of omniscience, omnipotence, infallibility, and omnibenevolence, when adequately formalized, are consistent with the existence of evil in the world. We consider that N1 is a good example of how formal systems can be applied in solving interesting philosophical issues, particularly in Philosophy of Religion and Analytic Theology, establishing bridges between such disciplines. (shrink)

The paper surveys the currently available axiomatizations of common belief (CB) and common knowledge (CK) by means of modal propositional logics. (Throughout, knowledge- whether individual or common- is defined as true belief.) Section 1 introduces the formal method of axiomatization followed by epistemic logicians, especially the syntax-semantics distinction, and the notion of a soundness and completeness theorem. Section 2 explains the syntactical concepts, while briefly discussing their motivations. Two standard semantic constructions, Kripke structures and neighbourhood structures, are introduced in Sections (...) 3 and 4, respectively. It is recalled that Aumann's partitional model of CK is a particular case of a definition in terms of Kripke structures. The paper also restates the well-known fact that Kripke structures can be regarded as particular cases of neighbourhood structures. Section 3 reviews the soundness and completeness theorems proved w.r.t. the former structures by Fagin, Halpern, Moses and Vardi, as well as related results by Lismont. Section 4 reviews the corresponding theorems derived w.r.t. the latter structures by Lismont and Mongin. A general conclusion of the paper is that the axiomatization of CB does not require as strong systems of individual belief as was originally thought- only monotonicity has thusfar proved indispensable. Section 5 explains another consequence of general relevance: despite the "infinitary" nature of CB, the axiom systems of this paper admit of effective decision procedures, i.e., they are decidable in the logician's sense. (shrink)

Debates over the reality of race often rely on arguments about the connection between race and science—those who deny that race is real argue that there is no significant support from science for our ordinary race concepts; those who affirm that race is real argue that our ordinary race concepts are supported by scientific findings. However, there is arguably a more fundamental concern here: How should we define race concepts in the first place? The reason I claim that this definitional (...) question is more fundamental is that our handling of the underlying definitional problem often determines the scientific support our ordinary race concepts need, and importantly the likelihood of finding such support. In short, the defini- tional question, “How do we define race?” often undercuts the question of whether race is scientifically meaningful. (shrink)

We show how removing faith-based beliefs in current philosophies of classical and constructive mathematics admits formal, evidence-based, definitions of constructive mathematics; of a constructively well-defined logic of a formal mathematical language; and of a constructively well-defined model of such a language. -/- We argue that, from an evidence-based perspective, classical approaches which follow Hilbert's formal definitions of quantification can be labelled `theistic'; whilst constructive approaches based on Brouwer's philosophy of Intuitionism can be labelled `atheistic'. -/- We then adopt (...) what may be labelled a finitary, evidence-based, `agnostic' perspective and argue that Brouwerian atheism is merely a restricted perspective within the finitary agnostic perspective, whilst Hilbertian theism contradicts the finitary agnostic perspective. -/- We then consider the argument that Tarski's classic definitions permit an intelligence---whether human or mechanistic---to admit finitary, evidence-based, definitions of the satisfaction and truth of the atomic formulas of the first-order Peano Arithmetic PA over the domain N of the natural numbers in two, hitherto unsuspected and essentially different, ways. -/- We show that the two definitions correspond to two distinctly different---not necessarily evidence-based but complementary---assignments of satisfaction and truth to the compound formulas of PA over N. -/- We further show that the PA axioms are true over N, and that the PA rules of inference preserve truth over N, under both the complementary interpretations; and conclude some unsuspected constructive consequences of such complementarity for the foundations of mathematics, logic, philosophy, and the physical sciences. -/- . (shrink)

A proof of Fermat’s last theorem is demonstrated. It is very brief, simple, elementary, and absolutely arithmetical. The necessary premises for the proof are only: the three definitive properties of the relation of equality (identity, symmetry, and transitivity), modus tollens, axiom of induction, the proof of Fermat’s last theorem in the case of n = 3 as well as the premises necessary for the formulation of the theorem itself. It involves a modification of Fermat’s approach of infinite descent. The infinite (...) descent is linked to induction starting from n = 3 by modus tollens. An inductive series of modus tollens is constructed. The proof of the series by induction is equivalent to Fermat’s last theorem. As far as Fermat had been proved the theorem for n = 4, one can suggest that the proof for n ≥ 4 was accessible to him. An idea for an elementary arithmetical proof of Fermat’s last theorem (FLT) by induction is suggested. It would be accessible to Fermat unlike Wiles’s proof (1995). (shrink)

The concept of quantum information is introduced as both normed superposition of two orthogonal sub-spaces of the separable complex Hilbert space and in-variance of Hamilton and Lagrange representation of any mechanical system. The base is the isomorphism of the standard introduction and the representation of a qubit to a 3D unit ball, in which two points are chosen. The separable complex Hilbert space is considered as the free variable of quantum information and any point in it (a wave function describing (...) a state of a quantum system) as its value as the bound variable. A qubit is equivalent to the generalization of ‘bit’ from the set of two equally probable alternatives to an infinite set of alternatives. Then, that Hilbert space is considered as a generalization of Peano arithmetic where any unit is substituted by a qubit and thus the set of natural number is mappable within any qubit as the complex internal structure of the unit or a different state of it. Thus, any mathematical structure being reducible to set theory is re-presentable as a set of wave functions and a subspace of the separable complex Hilbert space, and it can be identified as the category of all categories for any functor represents an operator transforming a set (or subspace) of the separable complex Hilbert space into another. Thus, category theory is isomorphic to the Hilbert-space representation of set theory & Peano arithmetic as above. Given any value of quantum information, i.e. a point in the separable complex Hilbert space, it always admits two equally acceptable interpretations: the one is physical, the other is mathematical. The former is a wave function as the exhausted description of a certain state of a certain quantum system. The latter chooses a certain mathematical structure among a certain category. Thus there is no way to be distinguished a mathematical structure from a physical state for both are described exhaustedly as a value of quantum information. This statement in turn can be utilized to be defined quantum information by the identity of any mathematical structure to a physical state, and also vice versa. Further, that definition is equivalent to both standard definition as the normed superposition and in-variance of Hamilton and Lagrange interpretation of mechanical motion introduced in the beginning of the paper. Then, the concept of information symmetry can be involved as the symmetry between three elements or two pairs of elements: Lagrange representation and each counterpart of the pair of Hamilton representation. The sense and meaning of information symmetry may be visualized by a single (quantum) bit and its interpretation as both (privileged) reference frame and the symmetries of the Standard model. (shrink)

When reusing existing ontologies for publishing a dataset in RDF (or developing a new ontology), preference may be given to those providing extensive subcategorization for important classes (denoted as focus classes). The subcategories may consist not only of named classes but also of compound class expressions. We define the notion of focused categorization power of a given ontology, with respect to a focus class and a concept expression language, as the (estimated) weighted count of the categories that can be built (...) from the ontology’s signature, conform to the language, and are subsumed by the focus class. For the sake of tractable initial experiments we then formulate a restricted concept expression language based on existential restrictions, and heuristically map it to syntactic patterns over ontology axioms (so-called FCE patterns). The characteristics of the chosen concept expression language and associated FCE patterns are investigated using three different empirical sources derived from ontology collections: first, the concept expression pattern frequency in class definitions; second, the occurrence of FCE patterns in the Tbox of ontologies; and last, for class expressions generated from the Tbox of ontologies (through the FCE patterns); their ‘meaningfulness’ was assessed by different groups of users, yielding a ‘quality ordering’ of the concept expression patterns. The complementary analyses are then compared and summarized. To allow for further experimentation, a web-based prototype was also implemented, which covers the whole process of ontology reuse from keyword-based ontology search through the FCP computation to the selection of ontologies and their enrichment with new concepts built from compound expressions. (shrink)

Basic Formal Ontology (BFO) is a top-level ontology used in hundreds of active projects in scientific and other domains. BFO has been selected to serve as top-level ontology in the Industrial Ontologies Foundry (IOF), an initiative to create a suite of ontologies to support digital manufacturing on the part of representatives from a number of branches of the advanced manufacturing industries. We here present a first draft set of axioms and definitions of an IOF upper ontology descending from (...) BFO. The axiomatization is designed to capture the meanings of terms commonly used in manufacturing and is designed to serve as starting point for the construction of the IOF ontology suite. (shrink)

The philosophy of science of Patrick Suppes is centered on two important notions that are part of the title of his recent book (Suppes 2002): Representation and Invariance. Representation is important because when we embrace a theory we implicitly choose a way to represent the phenomenon we are studying. Invariance is important because, since invariants are the only things that are constant in a theory, in a way they give the “objective” meaning of that theory. Every scientific theory gives a (...) representation of a class of structures and studies the invariant properties holding in that class of structures. In Suppes’ view, the best way to define this class of structures is via axiomatization. This is because a class of structures is given by a definition, and this same definition establishes which are the properties that a single structure must possess in order to belong to the class. These properties correspond to the axioms of a logical theory. In Suppes’ view, the best way to characterize a scientific structure is by giving a representation theorem for its models and singling out the invariants in the structure. Thus, we can say that the philosophy of science of Patrick Suppes consists in the application of the axiomatic method to scientific disciplines. What I want to argue in this paper is that this application of the axiomatic method is also at the basis of a new approach that is being increasingly applied to the study of computer science and information systems, namely the approach of formal ontologies. The main task of an ontology is that of making explicit the conceptual structure underlying a certain domain. By “making explicit the conceptual structure” we mean singling out the most basic entities populating the domain and writing axioms expressing the main properties of these primitives and the relations holding among them. So, in both cases, the axiomatization is the main tool used to characterize the object of inquiry, being this object scientific theories (in Suppes’ approach), or information systems (for formal ontologies). In the following section I will present the view of Patrick Suppes on the philosophy of science and the axiomatic method, in section 3 I will survey the theoretical issues underlying the work that is being done in formal ontologies and in section 4 I will draw a comparison of these two approaches and explore similarities and differences between them. (shrink)

Background: Ontologies and taxonomies are among the most important computational resources for molecular biology and bioinformatics. A series of recent papers has shown that the Gene Ontology (GO), the most prominent taxonomic resource in these fields, is marked by flaws of certain characteristic types, which flow from a failure to address basic ontological principles. As yet, no methods have been proposed which would allow ontology curators to pinpoint flawed terms or definitions in ontologies in a systematic way. Results: We (...) present computational methods that automatically identify terms and definitions which are defined in a circular or unintelligible way. We further demonstrate the potential of these methods by applying them to isolate a subset of 6001 problematic GO terms. By automatically aligning GO with other ontologies and taxonomies we were able to propose alternative synonyms and definitions for some of these problematic terms. This allows us to demonstrate that these other resources do not contain definitions superior to those supplied by GO. Conclusion: Our methods provide reliable indications of the quality of terms and definitions in ontologies and taxonomies. Further, they are well suited to assist ontology curators in drawing their attention to those terms that are ill-defined. We have further shown the limitations of ontology mapping and alignment in assisting ontology curators in rectifying problems, thus pointing to the need for manual curation. (shrink)

François Recanati; IV*—Contextual Dependence and Definite Descriptions, Proceedings of the Aristotelian Society, Volume 87, Issue 1, 1 June 1987, Pages 57–74, h.

In this paper we analyze the uses and misuses of argumentation schemes from verbal classification, and show how argument from definition supports argumentation based on argument from verbal classification. The inquiry has inevitably included the broader study of the concept of definition. The paper presents the schemes for argument from classification and for argument from definition, and shows how the latter type of argument so typically supports the former. The problem of analyzing arguments based on classification is framed in a (...) structure that reveals the crucial role it plays in the persuasion process. The survey of the literature includes the work of Hastings, Perelman, Kienpointner and Schiappa, but still finds much of value in Aristotle. Lessons drawn from Aristotle’s Topics are shown to be useful for developing new tools for assessing definitions and arguments from definition. (shrink)

This essay provides a condensed introductory ‘snapshot’ of just a few of the many and profound correlations existing between early (pre-Abhidhamma) Pāḷi Buddhism and Transcendental Phenomenology, by focusing on what is arguably the most central and essential ‘philosophical problem’ in both traditions: the true nature and significance of the ‘I’ of subjective intentional consciousness. It argues that the Buddhist axiom of ‘not-self’ (anattā) is by no means incompatible with the fundamental phenomenological irreducibility, and necessity, of transcendental subjectivity – or, as (...) Husserl also puts it, of the ‘pure’ or ‘transcendental ‘I’’ – a structure evidently essential to intentional consciousness as ‘consciousness-of’. On the one hand, Husserl recognises (and struggles with) the peculiar ‘emptiness’ of the ‘pure ‘I’’. On the other hand, a fundamental distinction must clearly be drawn between genuine intentional subjectivity – which even Buddhas and Arahants must of necessity possess – and the erroneous bases upon which the concept of ‘self’ (attā) that Buddhism rejects is constituted: the feeling of ‘I am’ (‘asmī’ti), the sense of ‘I am this’ (‘ayam-aham-asmī’ti), and the concept/conceit of ‘I am’ (asmi-māna) – all of which Buddhas and Arahants by definition do not possess. Hence, it is argued that, while the ‘pure I’ does not refer to some permanent ‘entity’ called ‘self’, nor is it merely an empty, non-referring, conventional linguistic marker: it has not merely a ‘use’, but a genuine meaning, which derives from the intrinsic, irreducible, and ‘pre-linguistic’ experiential structure of ‘consciousness-of’ itself. What is more, this meaning is not only recognised and admitted, but actively utilised, within the doctrine and methodology of early Buddhism, without any sense of contradicting the axiom of anattā. (shrink)

The classification of a state of affairs under a legal category can be considered as a kind of con- densed decision that can be made explicit, analyzed, and assessed us- ing argumentation schemes. In this paper, the controversial conflict of opinions concerning the nature of “marriage” in Obergefell v. Hodges is analyzed pointing out the dialecti- cal strategies used for addressing the interpretive doubts. The dispute about the same-sex couples’ right to marry hides a much deeper disa- greement not only (...) about what mar- riage is, but more importantly about the dialectical rules for defining it. (shrink)

With the advent of the semantic web, the problem of ambiguity is becoming more and more urging. Semantic analysis is necessary for explaining and resolving some sorts of ambiguity by inquiring into the relation between possibilities of predication and definition of a concept in order to solve problems such as interpretation and ambiguity. If computing is now approaching such problems of linguistic analysis, what is worth inquiring into is how the development of linguistic studies can be useful for developing the (...) theoretical background of ontologies. Our proposal is to develop a theory of definition alternative to the traditional metaphysical approach and the modern relativistic account. We interpret the ancient notion of essential definition in a dialectical perspective, and show how the dialectical definition by genus and difference corresponds to the semantic analysis of the definiendum. The dialectical definition is shown to be grounded on the deepest endoxa (shared knowledge) of a community, and to be the argumentatively strongest definition. After presenting the most common types of definition used in argumentation, the linguistic and logical characteristics of the notion of definition by genus and difference are set out in a pragmatic framework. (shrink)

The concept of formal transcendentalism is utilized. The fundamental and definitive property of the totality suggests for “the totality to be all”, thus, its externality (unlike any other entity) is contained within it. This generates a fundamental (or philosophical) “doubling” of anything being referred to the totality, i.e. considered philosophically. Thus, that doubling as well as transcendentalism underlying it can be interpreted formally as an elementary choice such as a bit of information and a quantity corresponding to the number of (...) elementary choices to be defined. This is the quantity of information defined both transcendentally and formally and thus, philosophically and mathematically. If one defines information specifically, as an elementary choice between finiteness (or mathematically, as any natural number of Peano arithmetic) and infinity (i.e. an actually infinite set in the meaning of set theory), the quantity of quantum information is defined. One can demonstrate that the so-defined quantum information and quantum information standardly defined by quantum mechanics are equivalent to each other. The equivalence of the axiom of choice and the well-ordering “theorem” is involved. It can be justified transcendentally as well, in virtue of transcendental equivalence implied by the totality. Thus, all can be considered as temporal as far anything possesses such a temporal counterpart necessarily. Formally defined, the frontier of time is the current choice now, a bit of information, furthermore interpretable as a qubit of quantum information. (shrink)

I have spent over 50 years of dedicated research on this theme. These are my overall conclusions. I think that science-as-it-is has come to too-limited conclusions, largely because, by in large, they are using an outmoded, "Newtonian" model of reality, (echoing Penrose, D'Espagnat, Maturana). Twentieth century physics has changed that, and it is time to apply its results to the rest of our world picture. I contend that the "materialists", as exemplified by Dennett, are archeological artifacts! There still remains room (...) for the human mind and the human soul -and my book shows how! This book proposes an entirely new perspective and an actual solution to the problem based on modern biology,(Maturana), modern physics, (Bohr, D'Espagnat, Einstein), modern logic, (Quine) and modern philosophy, (Cassirer), but it is still firmly grounded in Realism. It's conclusion is that the existence of "mind" is a necessary consequence of realism. (shrink)

In this work, author evaluated past theories and perspectives behind the definitions of science and/or branches of science. Also some of the philosophers of science and their specific philosophical interests were expressed. Author considered some type of interactions between some disciplines to determine, to solve the philosophical/scientific problems and to define the possible solutions. The purposes of this article are: (i) to define new synthesis method, (ii) to define new perspective for the philosophy of science, (iii) to define relation (...) between new philosophy perspective and philosophy of science, (iv) to define and organize name, number, relations, and correct structure between special science branches and philosophy of science, (v) to define necessary and sufficient number of branches for philosophy of science, (vi) to define and express the importance and place of new philosophy of science perspective in the new system, (vii) to extend the definition/limits of philosophy of science, (viii) to re-define meanings of some philosophical/scientific theories, (ix) to define systematic solution for the conflicts, problems, confusions about philosophy of science, sciences and branches of science, (x) to define new branches of science, (xi) to re-construct branches and hierarchy of science, (xii) to define new theories about science and branches of science. Author considered R-Synthesis as a method for the evaluation of the philosophy, philosophy of science, sciences and branches of science. This R-Synthesis includes evaluation of eight categories of general/specific perspective, 21-dimensions, and 12 general subjects (with related scope and contents) for the past 12,000 years. It is a kind of synthesis of science and non-science, physical science and non-physical science, religious science and non-religious science, and others. In this article, author defined 27 possible definitive/certain result cases for this new synthesis. Author defined the possible formation stages shortly to express new disciplines, new constructional and/or complementary theories. These theories are considered to define 21 major effective disciplines. New philosophy perspective is defined (R-Philosophy) shortly. New perspective and sub branches are defined for the philosophy of science. Major sciences are defined due to new basic philosophies. 42-basic components are defined for each science branch. New and/or re-constructed sciences, branches of science, basic sciences, and new hierarchy of science are defined with figure. Electromagnetic sciences, information sciences, and system sciences are defined specifically. Hybrid Sciences, New Era Science, and Ideal Scientific System are defined with general/specific figure. Relation between the some old branches and new branches of science was expressed generally due to new perspective of philosophy of science. (shrink)

The idea of science being the best – or the only – way to reach the truth about our cosmos has been a major belief of modern civilization. Yet, science has grown tall on fragile legs of clay. Every scientific theory uses axioms and assumptions that by definition cannot be proved. This poses a serious limitation to the use of science as a tool to find the truth. The only way to search for the latter is to redefine the (...) former to its original glory. In the days well before Galileo and Newton, science and religion were not separated. They worked together to discover the truth and while the latter had God as its final destination, the former had God as its starting point. Science is based on the irrational (unproven) belief that the world is intelligible along many other assumptions. This poses a serious limitation to science that can only be overcome if we accept the irrationality of the cosmos. The motto “Credo quia absurdum” holds more truth than one can ever realize at first glance. There is nothing logical in logic, whereas there is deep wisdom in the irrational. For while the former tries to build castles on moving sand, the latter digs deep inside the depths of existence itself in order to build on the most concrete foundations that there can be: the cosmos itself. The only way forward is backwards. Backwards to a time when religion led the quest for knowledge by accepting what we cannot know, rather than trying to comprehend what we do not. Science was anyway based on that in the first place. (shrink)

The two manuscripts of the Korte Verhanedling that were discovered in the mid-nineteenth century contain two appendices. These appendices are even more enigmatic than the KV itself, and it is the first appendix that is the subject of this study. Unfortunately, there are very few studies of this text, and its precise nature seems to be still in question after more than a century and a half of scholarship. It is commonly assumed that the appendices were written after the body (...) of the KV and I am not going to challenge this assumption. The first appendix is written in a geometric manner, and it contains seven axioms and four propositions. Strikingly, it does not include any definitions. This is in sharp contrast with Spinoza's Ethics and his 1663 book on Descartes' Principles of Philosophy, which are similarly written in a geometric manner but include both definitions and axioms. One could perhaps suspect that the text we currently have is merely a fragment from a more extensive work which included definitions. In my paper, I will show that this is not the case, and that the first appendix belongs to a genuine work of Spinoza that never contained definitions. I would further argue that the first appendix is most probably the earliest draft we currently have of Spinoza's magnum opus, the Ethics, but we have a long way to go before we reach that conclusion. (shrink)