Frege believes (1) that his definition of number is (partly) arbitrary; (2) that it "makes" numbers of certain extensions; (3) that without such a definition we cannot even think or understand arithmetical propositions. this position is part of a view according to which mathematics in general involves the free construction of objects, their properties, and the very contents of mathematical propositions. frege tries to avoid excess subjectivism by the kantian device of treating alternative systems of arithmetic (e.g.) (...) as different appearances of a single realm of unstructured truths. (shrink)

According to the received view, Russell rediscovered about 1900 the logical definition of cardinal number given by Frege in 1884. In the same way, we are told, he stated and developed independently the idea of logicism, using the principle of abstraction as the philosophical ground. Furthermore, the role commonly ascribed in this to Peano was only to invent an appropriate notation to be used as mere instrument. In this paper I hold that the study of Russell's unpublished manuscripts (...) and Peano's (and disciples) writings (as part of a larger investigation only pointed out here) shows, on the contrary, that Russell obtained the method to transform definitions by abstraction into nominal definitions and the general logicist idea from Peano (and his school). The only original insight from Russell, the principle of abstraction, partially derived from Moore's early philosophy, was finally abandoned because it was not of practical use. (shrink)

Joan Weiner has argued that Frege’s definitions of numbers constitute linguistic stipulations that carry no ontological commitment: they don’t present numbers as pre-existing objects. This paper offers a critical discussion of this view, showing that it is vitiated by serious exegetical errors and that it saddles Frege’s project with insuperable substantive difficulties. It is first demonstrated that Weiner misrepresents the Fregean notions of so-called Foundations-content, and of sense, reference, and truth. The discussion then focuses on the role of definitions in (...) Frege’s work, demonstrating that they cannot be understood as mere linguistic stipulations, since they have an ontological aim. The paper concludes with stressing both the epistemological and the ontological aspects of Frege’s project, and their crucial interdependence. (shrink)

The paper focuses on the lectures on the philosophy of mathematics delivered by Wittgenstein in Cambridge in 1939. Only a relatively small number of lectures are discussed, the emphasis falling on understanding Wittgenstein’s views on the most important element of the logicist legacy of Frege and Russell, the definition of number in terms of classes—and, more specifically, by employing the notion of one-to-one correspondence. Since it is clear that Wittgenstein was not satisfied with this definition, the (...) aim of the essay is to propose a reading of the lectures able to clarify why that was the case. This reading shows that his better known views on language and mind expressed in Philosophical Investigations illuminate his conception of mathematics. (shrink)

An objection is offered to the Frege-Russell definition, which identifies the number 1 with the set of all unit sets. It is argued here that the identity conditions for sets require that if any member of a set had not existed, the set itself would not have. Therefore, had any object whatever not existed, the unit set containing it would not have either, and thus the set with which the definition identifies 1 would not have. But then, (...) 1 either would not have existed or would have been a different entity than it is, and it is argued that neither alternative is acceptable. (shrink)

The shortest definition of a number by a first order formula with one free variable, where the notion of a formula defining a number extends a notion used by Boolos in a proof of the Incompleteness Theorem, is shown to be non computable. This is followed by an examination of the complexity of sets associated with this function.

The aim of the paper is to give a new definition of real number. The logical type of any number defined is that of the function B = h(A) which assigns to a class of sets A a class of sets B. I give some conditions which the function h has to fulfill to be considered as number; an intuitive sense of the conditions is as follows: a function, which is number, assigns a class of (...) sets of measure h·m to a class A of sets of equal measure, where m is the measure of A. (shrink)

Several scholars have argued that Wittgenstein held the view that the notion of number is presupposed by the notion of one-one correlation, and that therefore Hume's principle is not a sound basis for a definition of number. I offer a new interpretation of the relevant fragments on philosophy of mathematics from Wittgenstein's Nachlass, showing that if different uses of ‘presupposition’ are understood in terms of de re and de dicto knowledge, Wittgenstein's argument against the Frege-Russell definition (...) of number turns out to be valid on its own terms, even though it depends on two epistemological principles logicist philosophers of mathematics may find too ‘constructivist’. (shrink)

In the history of philosophy, especially its recent history, a number of definitions of necessity have been ventured. Most people, however, find these definitions either circular or subject to counterexamples. I will show that, given a broadly Fregean conception of properties, necessity does indeed have a noncircular counterexample-free definition.

This article first examines a number of different definitions of lying, from Aldert Vrij, Warren Shibles, Sissela Bok, the Oxford English Dictionary, Linda Coleman and Paul Kay, and Joseph Kupfer. It considers objections to all of them, and then defends Kupfer’s definition, as well as a modified version of his definition, as the best of those so far considered. Next, it examines five other definitions of lying, from Harry G. Frankfurt, Roderick M. Chisholm and Thomas D. Feehan, (...) David Simpson, Thomas Carson, and Don Fallis. It finds reason to reject these definitions, in favor of the two definitions of lying previously defended, namely:(i) To lie (to another person) = df. to make a believed-false statement (to another person) with the intention that that statement be believed to be true (by the other person).(ii) To lie (to another person) = df. to make a believed-false statement (to another person), either with the intention that that statement be believed to be true (by the other person), or with the intention that it be believed (by the other person) that that statement is believed to be true (by the person making the statement), or with both intentions. (shrink)

This paper concerns one of the undecided disputes of modern moral philosophy: the possibility of moral dilemmas. Whereas proponents of the possibility of moral dilemmas often appeal to moral experience, many opponents refer to ethical theory and deontic logic. My aim in this paper is to clarify some of the tension between moral experience and ethical theory with respect to moral dilemmas. In Part One I try to show that a number of logical arguments against the possibility of moral (...) dilemmas, though apparently very different, turn out to be basically the same, as they are all based on the following concept of ought: if A ought to be done, doing B is impermissible and doing A itself is permissible. In Part Two I present an overview of several definitions of moral dilemmas that have been given by proponents of moral dilemmas: definitions that define moral dilemmas in terms of oughts and definitions that define them in terms of reasons. I conclude that, while reason is to weak, ought is too strong a concept to define moral dilemmas with. In this way, the arguments from Part One create a logical problem for proponents of the possibility of moral dilemmas to define moral dilemmas. (shrink)

Following Asher Peres’s observation that, as in classical physics, in quantum theory, too, a given physical object considered “has a precise position and a precise momentum,” this article examines the question of the definition of quantum variables, and then the new type (as against classical physics) of relationships between mathematics and physics in quantum theory. The article argues that the possibility of the precise definition and determination of quantum variables depends on the particular nature of these relationships.

In positive logic the negation of a propositionA is defined byA X whereX is some fixed proposition. A number of standard properties of negation, includingreductio ad absurdum, can then be proved, but not the law of noncontradiction so that this forms a paraconsistent logic. Various stronger paraconsistent logics are then generated by putting in particular propositions forX. These propositions range from true through contingent to false.

In this paper, I show that, in a number of publications in the early 1950s, Margaret Macdonald argues that art does not admit of definition, that art is—in the sense associated with Wittgenstein—a family resemblance concept, and that definitions of art are best understood as confused or poorly expressed contributions to art criticism. This package of views is most typically associated with a famous paper by Morris Weitz from 1956. I demonstrate that Macdonald advanced that package prior to (...) Weitz, indeed, prior to any other philosopher of art of the period. Despite this, Macdonald’s contribution is nowhere to be found in the subsequent literature on the definition of art. In closing, I raise the prospect that Macdonald was in fact the primary influence on the development of Weitz’s critique of the definitional project. (shrink)

A fundamental problem in artificial intelligence is that nobody really knows what intelligence is. The problem is especially acute when we need to consider artificial systems which are significantly different to humans. In this paper we approach this problem in the following way: we take a number of well known informal definitions of human intelligence that have been given by experts, and extract their essential features. These are then mathematically formalised to produce a general measure of intelligence for arbitrary (...) machines. We believe that this equation formally captures the concept of machine intelligence in the broadest reasonable sense. We then show how this formal definition is related to the theory of universal optimal learning agents. Finally, we survey the many other tests and definitions of intelligence that have been proposed for machines. (shrink)

The cybernetic definition of a living individual proposed previously (Korzeniewski, 2001) is very abstract and therefore describes the essence of life in a very formal and general way. In the present article this definition is reformulated in order to determine clearly the relation between life in general and a living individual in particular, and it is further explained and defended. Next, the cybernetic definition of a living individual is confronted with the real world. It is demonstrated that (...) numerous restrictions imposed on the cybernetic definition of life by physical reality imply a number of particular properties of life that characterize present life on Earth, namely: (1) a living individual must be a dissipative structure (and therefore a low-entropy thermodynamic system out of the state of equilibrium); (2) spontaneously-originated life must be based on organic compounds; (3) evolutionarily stable self-dependent, free-living individuals must have some minimal level of complexity of structure and function; (4) a living individual must have a record of identity separated from an executive machinery; (5) the identity of living individuals must mutate and may evolve; (6) living individuals may collect and accumulate information in subsequent generations over very long periods of time; (7) the degree of complexity of a living individual reflects the degree of complexity of its environment (ecological niche) and (8) living individuals are capable of supple adaptation to varying environmental conditions. Thus, the cybernetic definition of a living individual, when confronted with the real physical world, generates most of the general properties of the present life on Earth. (shrink)

The concept of randomness as applied to number sequences is important to the study of the relationship between the foundations of mathematics and physics. A reason is that while randomness is often defined in mathematical-logical terms, the only way one has to generate random number sequences is by means of repetitive physical processes. This paper will examine the question: What definition of randomness is correct in the sense of being the weakest allowable? Why this question is so (...) important will become clear during the course of the discussion.The main body of this paper is divided into three sections. Section 1. discusses the use of probability theory to describe various statistical processes and some of the alternative definitions of randomness that have been proposed.In Section 2. a criterion which a definition of randomness should satisfy is proposed. (shrink)

In the insightful and searching paper of Mills and Beatty the following definition of ‘fitness’, as the term figures in the theory of natural selection, is offered:The [individual] fitness of an organism x in environment E equals n =dfn is the expected number of descendants which x will leave in E.

The general definition of mental disorder stated in the fifth edition of the Diagnostic and Statistical Manual of Mental Disorders seems to identify a mental disorder with a harmful dysfunction. However, the presence of distress or disability, which may be bracketed as the presence of harm, is taken to be merely usual, and thus not a necessary requirement: a mental disorder can be diagnosed as such even if there is no harm at all. In this paper, we focus on (...) the harm requirement. First, we clarify what it means to say that the harm requirement is not necessary for defining the general concept of mental disorder. In this respect, we briefly examine the two components of harm, distress and disability, and then trace a distinction between mental disorder tokens and mental disorder types. Second, we argue that the decision not to regard the harm requirement as a necessary criterion for mental disorder is tenable for a number of practical and theoretical reasons, some pertaining to conceptual issues surrounding the two components of harm and others pertaining to the problem of false negatives and the status of psychiatry vis-à-vis somatic medicine. However, we believe that the harm requirement can be maintained among the specific diagnostic criteria of certain individual mental disorders. More precisely, we argue that insofar as the harm requirement is needed among the specific diagnostic criteria of certain individual mental disorders, it should be unpacked and clarified. (shrink)

In this paper, I address a topic that has been mostly neglected in Leibniz scholarship: Leibniz’s conception of number. I argue that Leibniz thinks of numbers as a certain kind of relation, and that as such, numbers have a privileged place in his metaphysical system as entities that express a certain kind of possibility. Establishing the relational view requires reconciling two seemingly inconsistent definitions of number in Leibniz’s corpus; establishing where numbers fit in Leibniz’s ontology requires confronting a (...) challenge from the well-known nominalist reading of Leibniz most forcefully articulated in Mates. While my main focus is limited to the positive integers, I also argue that Leibniz intends to subsume them under a more general conception of number. (shrink)

This paper is concerned with an intuitive contrast that arises when we consider sentences containing empty definite descriptions. A sentence like ‘The king of France is bald’ appears neither true nor false, while a sentence like ‘My friend was visited by the king of France’ appears false. Recently, Stephen Yablo has suggested an account of this intuitive contrast. Yablo’s account is particularly interesting, since it has important consequences for the ontological commitments of number sentences like ‘The number of (...) planets is even’. However, the paper argues that Yablo’s account is not convincing and that it can thus not establish these consequences. Further, it develops a Strawsonian account of the intuitive contrast. The developed account allows us to draw important conclusions regarding the correct analysis of definite descriptions and of existence sentences containing definite descriptions. (shrink)

Many of the contemporary disagreements regarding privacy are conceptual in nature. They concern the meaning or definition of privacy and the analytic basis of distinguishing privacy rights from other kinds of rights recognized within moral, political, or legal theories. The two main alternatives within this debate include reductionist views, which seek a narrow account of the kinds of invasions or intrusions distinctly involving privacy losses, and anti-reductionist theories, which treat a much broader array of interferences with a person as (...) separate and irreducible kinds of privacy invasions. Other theorists have expressed doubts about the prospects for achieving greater analytical precision even within a fairly expansive anti-reductionist approach. However, a reductionist privacy definition is defended in this article, and its primary theoretical virtues are its ability to unify and explain the insights of several competing definitions and its role in developing an account of privacy rights that is both internally coherent and consistent with a plausible understanding of the theoretical basis for a number of related rights. (shrink)

We review briefly the attempts to define random sequences. These attempts suggest two theorems: one concerning the number of subsequence selection procedures that transform a random sequence into a random sequence; the other concerning the relationship between definitions of randomness based on subsequence selection and those based on statistical tests.

In [S, pp. 77–88], Smullyan introduced the class of rudimentary relations, and showed that they form a basis for the recursively enumerable sets. He also asked [S, p. 81] if the addition and multiplication relations were rudimentary. In this note we answer one of these questions by showing that the addition relation is rudimentary. This result was communicated to Smullyan orally in 1960 and is announced in [S, p. 81, footnote 1]. However, the proof has not yet appeared in print. (...) Let us begin by reviewing Smullyan's definition [S, p. 10] of dyadic notation for the positive integers. Each positive integerais identified with the unique stringanan−1…a1a0of 1's and 2's such thata= Σin=0ai2iBecause of this identification, we are able to speak of typographical properties of numbers. (shrink)

In this special issue, our objective is to clarify what biosemioticians may mean insofar as they claim that living systems are capable of making choices or that biosemiotic interpretations are partially indeterminate. A number of different senses of the term “chance” are discussed as we move toward a consensus. We find that biosemiosic chance may arise out of conditions involving quantum indeterminacy, randomness, deterministic chaos, or unpredictability, but biosemiosic chance is mainly due to the fact that living entities invest (...) their environments with different meanings and values, which are not inherent in the material aspects. Accordingly, interpretive responses are in part self-determined. A self-determined interpretation may be thought of as a process in which localized biases constrain the probabilities of the outcomes of the interactions between an agent and its environment. The agent’s internal formal constraints that are defined by similarity and proximity are relative and, as such, vague, and thus interpretation may be somewhat underdetermined and involve an element of chance. (shrink)

In a forthcoming paper in this journal, entitled “Bad company objection to Joongol Kim’s adverbial theory of numbers”, Namjoong Kim presents an ingenious Russell-style paradox based on an analogue of Kim’s definition of the number 1, and argues that Kim’s theory needs to provide a criterion of demarcation between acceptable and unacceptable definitions of adverbial entities. This paper addresses this ‘bad company’ objection and some other related issues concerning Kim’s adverbial theory by clarifying the purposes and uses of (...) the formal framework of the theory. (shrink)

In his classic 1936 paper Tarski sought to motivate his definition of logical consequence by appeal to the inference form: P(0), P(1), . . ., P(n), . . . therefore ∀nP(n). This is prima facie puzzling because these inferences are seemingly first-order and Tarski knew that Gödel had shown first-order proof methods to be complete, and because ∀nP(n) is not a logical consequence of P(0), P(1), . . ., P(n), . . . by Taski's proposed definition. An attempt (...) to resolve the puzzle due to Etchemendy is considered and rejected. A second attempt due to Gómez-Torrente is accepted as far as it goes, but it is argued that it raises a further puzzle of its own: it takes the plausibility of Tarski's claim that his definition captures our common concept of logical consequence to depend upon our common concept being a reductive conception. A further interpretation of what Tarski had in mind when he offered the example is proposed, using materials well known to Tarski at the time. It is argued that this interpretation makes the motivating example independent of reductive definitions which take natural numbers to be higher-order set theoretic entities, and it also explains why he did not regard the distinction between defined and primitive terms as pressing, as was the distinction between logical and nonlogical terms. (shrink)

What is a number? Answering this will answer questions about its philosophical foundations - rational numbers, the complex numbers, imaginary numbers. If we are to write or talk about something, it is helpful to know whether it exists, how it exists, and why it exists, just from a common-sense point of view [Quine, 1948, p. 6]. Generally, there does not seem to be any disagreement among mathematicians, scientists, and logicians about numbers existing in some way, but currently, in the (...) mainstream arena only definitions, descriptions of properties, and effects are presented as evidence. Enough historical description of numbers in history provides an empirical basis of number, although a case can be made that numbers do not exist by themselves empirically. Correspondingly, numbers exist as abstractions. All the while, though, these "descriptions" beg the question of what numbers are ontologically. Advocates for numbers being the ultimate reality have the problem of wrestling with the nature of reality. I start on the road to discovering the ontology of number by looking at where people have talked about numbers as already existing: history. Of course, we need to know not only what ontology is but the problems of identifying one, leading to the selection between metaphysics and provisional approaches. While we seem to be dimensionally limited, at least we can identify a more suitable bootstrapping ontology than mere definitions, leading us to the unity of opposites. The rest of the paper details how this is done and modifies Peano's Postulates. (shrink)

A number of non-equivalent definitions of slavery have been offered by historians, sociologists, bodies of international governance, and legal scholars. None is clearly adequate. Here I review extant definitions of slavery found in or suggested by Lovejoy, Patterson, Honoré, Bales, Ingram, and the League of Nations 1926 Slavery Convention, and argue that each is subject to counterexample. I then attempt to formulate and defend a more adequate definition, one focusing on consent, control, and the intentions of the slaveholder, (...) and relevant for the present as well as the past. (shrink)

The paper puts forward an analytic definition of “love” that is intended to characterize the use of the word in expressions of the form “He/she loves...” It is proposed that when “love” is used in such contexts, it denotes “a disposition towards either (i) desiring a perceived good or desiring union with it, either as an end itself or with it being a source of delight in itself or (ii) desiring good for a particular object for its own sake.” (...) The first of these might be referred to as unitive love and the second as contributory love. A defense of the definition is offered by considering its application to a number of seemingly problematic cases. Many of the puzzling features of “love” arise from the disjunctive nature of its use – that we use “love” when either unitive or contributory love is present. Confusions arise when we take what can be said about unitive love, and apply this to contributory love, or vice versa, or when we assume all uses of “love” entail both unitive and contributory aspects. Consideration is given to the implications of the proposed definition and distinctions for cultivating love within society, marriages, political life, and media. (shrink)