Elementary geometry can be axiomatized constructively by taking as primitive the concepts of the apartness of a point from a line and the convergence of two lines, instead of incidence and parallelism as in the classical axiomatizations. I first give the axioms of a general plane geometry of apartness and convergence. Constructive projective geometry is obtained by adding the principle that any two distinct lines converge, and affine geometry by adding a parallel line construction, etc. Constructive axiomatization allows solutions to (...) geometric problems to be effected as computer programs. I present a formalization of the axiomatization in type theory. This formalization works directly as a computer implementation of geometry. (shrink)

The conflict between Platonic realism and Constructivism marks a watershed in philosophy of mathematics. Among other things, the controversy over the Axiom of Choice is typical of the conflict. Platonists accept the Axiom of Choice, which allows a set consisting of the members resulting from infinitely many arbitrary choices, while Constructivists reject the Axiom of Choice and confine themselves to sets consisting of effectively specifiable members. Indeed there are seemingly unpleasant consequences of the Axiom of Choice. (...) The non-constructive nature of the Axiom of Choice leads to the existence of non-Lebesgue measurable sets, which in turn yields the Banach-Tarski Paradox. But the Banach-Tarski Paradox is so called in the sense that it is a counter-intuitive theorem. To corroborate my view that mathematical truths are of non-constructive nature, I shall draw upon Gödel’s Incompleteness Theorems. This also shows the limitations inherent in formal methods. Indeed the Löwenheim-Skolem Theorem and the Skolem Paradox seem to pose a threat to Platonists. In this light, Quine/Putnam’s arguments come to take on a clear meaning. According to the model-theoretic arguments, the Axiom of Choice depends for its truth-value upon the model in which it is placed. In my view, however, this is another limitation inherent in formal methods, not a defect for Platonists. To see this, we shall examine how mathematical models have been developed in the actual practice of mathematics. I argue that most mathematicians accept the Axiom of Choice because the existence of non-Lebesgue measurable sets and the Well-Ordering of reals open the possibility of more fruitful mathematics. Finally, after responding to Benacerraf’s challenge to Platonism, I conclude that in mathematics, as distinct from natural sciences, there is a close connection between essence and existence. Actual mathematical theories are the parts of the maximally logically consistent theory that describes mathematical reality. (shrink)

The base units of the SI include six units of continuous quantities and the mole, which is defined as proportional to the number of specified elementary entities in a sample. The existence of the mole as a unit has prompted comment in Metrologia that units of all enumerable entities should be defined though not listed as base units. In a similar vein, the BIPM defines numbers of entities as quantities of dimension one, although without admitting these entities as base units. (...) However, there is a basic ontological distinction between continuous quantities and enumerable aggregates. The distinction is the basis of the difference between real and natural numbers. This paper clarifies the nature of the distinction: (i) in terms of a set of measurement axioms stated by Hölder; and (ii) using the formalism known in metrology as quantity calculus. We argue that a clear and unambiguous scientific distinction should be made between measurement and enumeration. We examine confusion in metrological definitions and nomenclature concerning this distinction, and discuss the implications of this distinction for ontology and epistemology in all scientific disciplines. (shrink)

This paper deals with a foundational aspect of Integrated Information Theory of consciousness: the nature of the relation between the axioms of phenomenology and the postulates of cause-effect power. There has been a lack of clarity in the literature regarding this crucial issue, for which IIT has received much criticism of its axiomatic method and basic tenets. The present contribution elucidates the problem by means of a categorial analysis of the theory’s foundations. Its main results are that: IIT has a (...) set of nine fundamental concepts of reason, called categories, which constitute its categorial lexicon and through which it formulates a system of principles incorporating the axioms, the postulates, and the central identity; and the connection between the axioms and postulates is grounded by their common root in this categorial lexicon, the categories of which find their justification by means of a phenomenological and transcendental deduction. Some further results are the unique origin of axioms and postulates in the categories; the distinction between conceptual and formalized postulates; a clarification of the uniqueness problem of categorial lexica in general; and an IIT account of objectivity by explicating how the physical is defined by means of categories. All of this is put to use against various criticism targeting IIT’s theoretical core. If successful, the proposed interpretation illuminates a central issue in the contemporary study of consciousness and contributes to an environment of mutual understanding between defenders and critics of the theory. (shrink)

Hans Reichenbach introduced two seemingly separate sets of distinctions in his epistemology at different times. One is between the axioms of coordination and the axioms of connections. The other distinction is between the context of discovery and the context of justification. The status and nature of each of these distinctions have been subject-matter of an ongoing debate among philosophers of science. Thus, there is a significant amount of works considering both distinctions separately. However, the relevance of Reichenbach's two distinctions (...) to each other does not seem to have enjoyed the same amount of interest so far. This is what I would like to consider in this paper. In other words, I am concerned with the question: what kind of relationship is there between his two distinctions, if there is any? (shrink)

Benacerraf’s Problem about mathematical truth displays a tension, indeed a seemingly unbridgeable gap, between Platonist foundations for mathematics on the one hand and Hilbert’s ‘finitary standpoint’ on the other. While that standpoint evinces an admirable philosophical unity, it is ultimately an effete rival to Platonism: It leaves mathematical practice untouched, even the highly non-constructive axiom of choice. Brouwer’s intuitionism is a more potent finitist rival, for it engenders significant deviation from standard (classical) mathematics. The essay illustrates three sorts of (...) intuitionistic deviations (weakening, refinement and outright contradiction), and goes on to sketch the technical tools and arguments that engender those clashes with classical mathematics and the philosophical principles underlying those arguments. Those philosophical principles coalesce into a “standpoint” no less unified and no less finitary that Hilbert’s. This intuitionistic standpoint is sufficiently detailed that it itself provides a benchmark for comparing all the rival positions; and it is sufficiently robust that it dissolves the Benacerrafian dichotomy. However, the intuitionistic refutation of the axiom of choice reveals two distinct sub-streams within that intuitionistic standpoint. Distinguishing these streams suggests perhaps that in spite of that robustness, intuitionism has an internal dichotomy parallel to the Benacerrafian split. The Benacerrafian split is a crack in the foundations of mathematics. But I shall argue that this internal intuitionistic dichotomy is not at all a foundational gap; it is rather a special insight about the nature of mathematical thought. (shrink)

In recent years, Reichenbach's 1920 conception of the principles of coordination has attracted increased attention after Michael Friedman's attempt to revive Reichenbach's idea of a "relativized a priori". This paper follows the origin and development of this idea in the framework of Reichenbach's distinction between the axioms of coordination and the axioms of connection. It suggests a further differentiation among the coordinating axioms and accordingly proposes a different account of Reichenbach's "relativized a priori".

One of the most distinctive and intriguing developments of modern set theory has been the realization that, despite widely divergent incentives for strengthening the standard axioms, there is essentially only one way of ascending the higher reaches of infinity. To the mathematical realist the unexpected convergence suggests that all these axiomatic extensions describe different aspects of the same underlying reality.

‘Over a ten year period starting in the mid 90’s I became convinced that all these topics –game theory, economic design, voting theory –belonged to a common area which I called Social Software.’ — Rohit Parikh, [Parikh 05]: p. 252..

The distinction has occasioned a long controversy between Carnap and W.V. Quine. The latter distinguishes two sorts of analytic statements: the logical truths, characterized by their remaining true under all reinterpretations of the descriptive terms; and the statements, which reduce to logical truths with the help of definitions or by substitution of synonyms for synonyms. In “Two Dogmas of Empiricism”, he directs his criticism mainly against the latter arguing that the explications so far provided move in a circle, since, (...) in the end, the concept of synonymy is explained in terms of analyticity. He reaches the conclusion that the only form of synonymy not exhibiting this deficiency is that introduced by explicit conventions. But according to him, this procedure is of no avail, because the philosophically interesting content of the general term “analytic in L” cannot be grasped with its help. Carnap’s suggestion of defining a new basic concept “analytic in L0” does not solve the problem since we could just as well designate this with a neutral word, say “K”, thus avoiding the illusion of an explanation. The attempt to define synonymy with the help of semantic rules fails in a similar way: it accomplishes nothing more than to place certain statements under the rubric “Semantic rules” — a notion which is no clearer than the explicandum. Carnap replies that his definition of “analytically true” is given with regard to artificial or formalized languages for which we can determine syntactic and semantic rules in such a way that the concept of truth, in Tarski’s sense, can be applied to them. Analytic statements other than logical truths will thus be distinguished by means of so-called meaning postulates which serve to make the truth-value of such statements independent of empirical matters. I share Carnap’s astonishment that Quine finds the term “semantic rule” unintelligible while formulating no complaint against similar notions such as that of a syntactically well-formed formula, or that of an axiom etc. It appears that he cannot be satisfied with rules, because he expects a theory which would give us an empirical, i.e., a behavioristic, criterion of analyticity for natural languages. His chief dissatisfaction seems to come from the fact that no clear explicandum has been put forward — in contrast, e.g., with the notion of truth, where we have a sufficiently clear pre-theoretical concept. Thus, according to him, there is nothing that could be clarified by an explication. (shrink)

Many believe that the treatment-enhancement distinction marks an important ethical boundary that we should use to shape public policy on biomedical interventions. A common justification for this purported normative force appeals to the idea that, whereas treatments respond to genuine medical needs, enhancements can only satisfy mere preferences or “expensive tastes”. This article offers a critique of that justification, while still accepting the TED as a conceptual tool, as well as some of the key ethical axioms endorsed by its (...) proponents. I begin by laying out the TED, the practical implications that tend to be drawn from it, and the justification just sketched for these implications. Using examples drawn from preventive medicine, biomedical technology, and other categories of biomedical interventions, I then go on to challenge both the presupposition of a fundamental dichotomy between treatments and enhancements, and the assumption that enhancements – barring rare exceptions – cannot serve legitimate medical needs. Finally, I consider some ways in which supporters of the TED might try re-formulating the distinction to blunt the force of my critique. I conclude that such a move cannot fully succeed, and that while the TED does have some degree of normative force, it nevertheless cannot play the role that its advocates expect from it. Seeking to justify a general presumption against enhancements based on the rationale I critique here would mean ignoring their various potential benefits, including medical or therapeutic, and would reflect a prejudice – which I refer to as “treatment fetishism”. (shrink)

Erdős proved that for every infinite \ there is \ with \, such that all pairs of points from Y have distinct distances, and he gave partial results for general a-ary volume. In this paper, we search for the strongest possible canonization results for a-ary volume, making use of general model-theoretic machinery. The main difficulty is for singular cardinals; to handle this case we prove the following. Suppose T is a stable theory, \ is a finite set of formulas of (...) T, \, and X is an infinite subset of M. Then there is \ with \ and an equivalence relation E on Y with infinitely many classes, each class infinite, such that Y is \\)-indiscernible. We also consider the definable version of these problems, for example we assume \ is perfect and we find some perfect \ with all distances distinct. Finally we show that Erdős’s theorem requires some use of the axiom of choice. (shrink)

Church's and Turing's theses dogmatically assert that an informal notion of effective calculability is adequately captured by a particular mathematical concept of computability. I present an analysis of calculability that is embedded in a rich historical and philosophical context, leads to precise concepts, but dispenses with theses. To investigate effective calculability is to analyze symbolic processes that can in principle be carried out by calculators. This is a philosophical lesson we owe to Turing. Drawing on that lesson and recasting work (...) of Gandy, I formulate boundedness and locality conditions for two types of calculators, namely, human computing agents and mechanical computing devices (discrete machines). The distinctive feature of the latter is that they can carry out parallel computations. The analysis leads to axioms for discrete dynamical systems (representing human and machine computations) and allows the reduction of models of these axioms to Turing machines. Cellular automata and a variety of artificial neural nets can be shown to satisfy the axioms for machine computations. (shrink)

The paper argues that, although a distinction between a priori and a posteriori knowledge (or justification) can be drawn, it is a superficial one, of little theoretical significance. The point is not that the distinction has borderline cases, for virtually all useful distinctions have such cases. Rather, it is argued by means of an example, the differences even between a clear case of a priori knowledge and a clear case of a posteriori knowledge may be superficial ones. In (...) both cases, experience plays a role that is more than purely enabling but less than strictly evidential. It is also argued that the cases at issue are not special, but typical of a wide range of others, including knowledge of axioms of set theory and of elementary logical truths. Attempts by Quine and others to make all knowledge a posteriori (‘empirical’) are repudiated. The paper ends with a call for a new framework to be developed for analysing the epistemology of cognitive uses of the imagination. (shrink)

A four-valued matrix is presented which validates all theorems of the implicational fragment, IF, of the classical sentential calculus in which at most two distinct sentence letters occur. The Wajsberg/Diamond-McKinsley Theorem for IF follows as a corollary: every complete set of axioms (with substitution and detachment as rules) must include at least one containing occurrences of three or more distinct sentence letters. Additionally, the matrix validates all IF theses built from nine or fewer occurrences of connectives and letters. So the (...) classic result of Jagkovski for the full sentential calculus -that every complete axiom set must contain either two axioms of length at least nine or else one of length at least eleven-can be improved in the implicational case: every complete axiom set for IF must contain at least one axiom eleven or more characters long. Both results are "best possible", and both apply as well to most subsystems of IF, e.g., the implicational fragments of the standard relevance logics, modal logics, the relatives of implicational intutionism, and logics in the Lukasiewicz family. (shrink)

Towards the end of their reign, the logical positivists found themselves in bitter disagreement as to what extent the methods and axioms of the natural sciences can be justified by our abilities to grunt and point. What began as a project to epistemically ground the natural sciences ended as an argument about cavemen. Although such a story might be a good one, the consensus, among the positivists’ rivals and the positivists themselves, seemed to be that ahead lay a difficult road (...) indeed if grunts clarify physics. Understandably, that road would be one not traveled, and so our weary fact-warriors walked off elsewhere, leaving the detritus in their path and the rest of the world to deal with the consequences. And consequences there were. Take the current, popular idea that the natural sciences need not be governed by moral concerns; or if they do, such moral governance only need be justified by appeals to governmental, technological or, happily in this most learned of eras, divine authority. Although we may not be able to connect such popular ideas directly to the positivist program, it is nevertheless no great leap to say that their program was motivated in strong ways by a blind acceptance of the abilities of technological advancement and the triumphant march of the natural sciences. (shrink)

On the basis of a wide range of historical examples various features of axioms are discussed in relation to their use in mathematical practice. A very general framework for this discussion is provided, and it is argued that axioms can play many roles in mathematics and that viewing them as self-evident truths does not do justice to the ways in which mathematicians employ axioms. Possible origins of axioms and criteria for choosing axioms are also examined. The distinctions introduced aim at (...) clarifying discussions in philosophy of mathematics and contributing towards a more refined view of mathematical practice. (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)

The axiom of extensionality of set theory states that any two classes that have identical members are identical. Yet the class of persons age i at time t and the class of persons age i + 1 at t + l, both including same persons, possess different demographic attributes, and thus appear to be two different classes. The contradiction could be resolved by making a clear distinction between age groups and cohorts. Cohort is a multitude of individuals, which (...) is constituted within a time interval, and endures throughout part of the time continuum. Age group, on the other hand, is only a reference term to which empirical measurement relates, as in birth or death rates. Accordingly, the two concepts, age group i at t, and age group i + 1 at t + 1, are different. The standard population growth model of Leslie and Lotka, however, does not support such a distinction in age groups. An alternative model, proposed recently, implies precisely such a distinction. (shrink)

The two-fold ontological character of linguistic objects revealed due to the distinction between “type” and “token” introduced by Ch. S. Peirce can be a base of the two-fold, both theoretical and axiomatic, approach to the language. Referring to some ideas included in A. A. Markov’s work [1954] (in Russian) on Theory of Algorithms and in some earlier papers of the author, the problem of formalization of the concrete and abstract words theories raised by J. Słupecki was solved. The construction (...) of the theories presented here has two levels. The axiomatic theory of label-tokens: material, physical linguistic objects, constitutes the first one. Label-types, according to the literature of the subject, are defined on the other level as equivalence classes of equiform label-tokens. Assuming the opposite point of view, one can accept that theory of label-types: abstract labels, formalized on the first level, in which it is possible to define the notion of label-token as well as the derivative notions on the second level, should become the basis of formalization of the theory of linguistic expressions and the theory of language in general. The axioms and definitions of both theories of labels: T k and T p representing the other approach to the ontology of language are included in the sequel of the abstract. The foundations of the theory of labels T k in which the primary assumption as to the label-types existence is superfluous have been referred on the basis of the author's monography "Teorie Języków Syntaktycznie Kategorialnych" ( "The Theories of Syntactically Categorial Languages"), PWN, Warszawa-Wrocław 1985. The basis of the theory of labels T p which takes into account the other position has to be presented here for the first time. Some extended ideas of the paper will also be presented in author's paper "Logiczne podstawy ontologii składni Języka" ("Logical foundations of language syntax ontology), Studia Filozoficzne 6-7 (271-272), (1988), pp. 263-284. (shrink)

The paper discusses various relationships between the concepts mentioned in the title. In Section 1 Todorcevic functions are shown to arise from both morasses and square. In Section 2 the theme is of supplements to morasses which have some of the flavour of square. Distinctions are drawn between differing concepts. In Section 3 forcing axioms related to the ideas in Section 2 are discussed.

The new evidence scholarship addresses three distinct approaches: legal probabilism, Bayesian decision theory and relative plausibility theory. Each has major insights to offer, but none seems satisfactory as it stands. This paper proposes that relative plausibility theory be modified in two substantial ways. The first is by defining its key concept of plausibility, hitherto treated as primitive, by generalising the standard axioms of probability. The second is by complementing the descriptive component of the theory with a normative decision theory adapted (...) to legal process. Because this version of decision theory is based on plausibilities rather than probabilities, it generates plausibilistic expectations as outputs. Because these outputs are comparable, they function as relative plausibilities. Hence the resulting framework is an extension of relative plausibility theory, but it retains deep ties to legal probabilism, through the proposed definition of plausibility, and to Bayesian decision theory, through the normative use of decision theory. (shrink)

We start this paper by arguing that causality should, in analogy with force in Newtonian physics, be understood as a theoretical concept that is not explicated by a single definition, but by the axioms of a theory. Such an understanding of causality implicitly underlies the well-known theory of causal nets and has been explicitly promoted by Glymour. In this paper we investigate the explanatory warrant and empirical content of TCN. We sketch how the assumption of directed cause–effect relations can be (...) philosophically justified by an inference to the best explanation. We then ask whether the explanations provided by TCN are merely post-facto or have independently testable empirical content. To answer this question we develop a fine-grained axiomatization of TCN, including a distinction of different kinds of faithfulness. A number of theorems show that although the core axioms of TCN are empirically empty, extended versions of TCN have successively increasing empirical content. (shrink)

Given Quine's views on philosophical methodology, he should not have taken the axioms of classical mereology to be "self-evident", or "analytic"; but rather, he should have set out to justify them by what might be broadly called an "inference to the best explanation". He does very little to this end. In particular, he does little to examine alternative theories, to see if there might be anything they could explain better than classical mereology can. I argue that there is something important (...) that needs to be explained, namely, the way that properties "travel around in clusters" (eg. we often know that "when and where there is something with such-and-such property, there is also something with so-and-so other property", and so on). I argue that these clusterings of properties can be given various subtle (broadly "commonsense") explanations using a version of mereology that denies the classical axiom of "extensionality" (that is, denying that two distinct things must have distinct parts). I offer a challenge to the Quinean metaphysics: to show that these "non-extensional" explanations can be replaced by better explanations that use only classical, extensional mereology and set theory. (shrink)

Since the time of Aristotle's students, interpreters have considered Prior Analytics to be a treatise about deductive reasoning, more generally, about methods of determining the validity and invalidity of premise-conclusion arguments. People studied Prior Analytics in order to learn more about deductive reasoning and to improve their own reasoning skills. These interpreters understood Aristotle to be focusing on two epistemic processes: first, the process of establishing knowledge that a conclusion follows necessarily from a set of premises (that is, on the (...) epistemic process of extracting information implicit in explicitly given information) and, second, the process of establishing knowledge that a conclusion does not follow. Despite the overwhelming tendency to interpret the syllogistic as formal epistemology, it was not until the early 1970s that it occurred to anyone to think that Aristotle may have developed a theory of deductive reasoning with a well worked-out system of deductions comparable in rigor and precision with systems such as propositional logic or equational logic familiar from mathematical logic. When modern logicians in the 1920s and 1930s first turned their attention to the problem of understanding Aristotle's contribution to logic in modern terms, they were guided both by the Frege-Russell conception of logic as formal ontology and at the same time by a desire to protect Aristotle from possible charges of psychologism. They thought they saw Aristotle applying the informal axiomatic method to formal ontology, not as making the first steps into formal epistemology. They did not notice Aristotle's description of deductive reasoning. Ironically, the formal axiomatic method (in which one explicitly presents not merely the substantive axioms but also the deductive processes used to derive theorems from the axioms) is incipient in Aristotle's presentation. Partly in opposition to the axiomatic, ontically-oriented approach to Aristotle's logic and partly as a result of attempting to increase the degree of fit between interpretation and text, logicians in the 1970s working independently came to remarkably similar conclusions to the effect that Aristotle indeed had produced the first system of formal deductions. They concluded that Aristotle had analyzed the process of deduction and that his achievement included a semantically complete system of natural deductions including both direct and indirect deductions. Where the interpretations of the 1920s and 1930s attribute to Aristotle a system of propositions organized deductively, the interpretations of the 1970s attribute to Aristotle a system of deductions, or extended deductive discourses, organized epistemically. The logicians of the 1920s and 1930s take Aristotle to be deducing laws of logic from axiomatic origins; the logicians of the 1970s take Aristotle to be describing the process of deduction and in particular to be describing deductions themselves, both those deductions that are proofs based on axiomatic premises and those deductions that, though deductively cogent, do not establish the truth of the conclusion but only that the conclusion is implied by the premise-set. Thus, two very different and opposed interpretations had emerged, interestingly both products of modern logicians equipped with the theoretical apparatus of mathematical logic. The issue at stake between these two interpretations is the historical question of Aristotle's place in the history of logic and of his orientation in philosophy of logic. This paper affirms Aristotle's place as the founder of logic taken as formal epistemology, including the study of deductive reasoning. A by-product of this study of Aristotle's accomplishments in logic is a clarification of a distinction implicit in discourses among logicians--that between logic as formal ontology and logic as formal epistemology. (shrink)

The actual/virtual distinction is used to give an alternative account of quantum interference by way of a new theory of probability. The new theory is obtained by changing one of the axioms of the canonical theory of probability while keeping the other axioms fixed. It is used to give an alternative account of constructive quantum interference in the two-slit experiment. The account crucially involves a distinction between actual and virtual probabilities. Although actual probabilities are operational and virtual probabilities (...) are not, there is a substantial connection between them. Virtual probabilities may be obtained indirectly from actual probabilities. In showing this, interpretive considerations are brought to bear including some having to do with quantum non-locality. Directions for future research are discussed in closing.La distinción actual/virtual se utiliza para dar una explicación alternativa de la interferencia cuántica a través de una nueva teoría de la probabilidad. La nueva teoría se ha obtenido cambiando uno de los axiomas de la teoría canónica de la probabilidad y manteniendo los otros axiomas fijos. Se usa para dar una explicación alternativa de la interferencia cuántica constructiva en el experimento de la doble rendija. El informe fundamentalmente implica una distinción entre las probabilidades actuales y virtuales. Aunque las probabilidades actuales son operativas y las probabilidades virtuales no, hay una relación sustancial entre ellas. Probabilidades virtuales pueden obtenerse indirectamente a partir de las probabilidades reales. Para demostrar esta interpretación hay que tener en cuenta las consideraciones de interpretación que se ejercen entre ellos y que tienen relación con la no-localidad cuántica. Directrices para futuras investigaciones se abordan al final del artículo. (shrink)

The symmetries of a physical theory are often associated with two things: conservation laws and representational redundancies. But how can a physical theory's symmetries give rise to interesting conservation laws, if symmetries are transformations that correspond to no genuine physical difference? In this article, I argue for a disambiguation in the notion of symmetry. The central distinction is between what I call "analytic" and "synthetic" symmetries, so called because of an analogy with analytic and synthetic propositions. "Analytic" symmetries are (...) the turning of idle wheels in a theory's formalism, and correspond to no physical change; "synthetic" symmetries cover all the rest. I argue that analytic symmetries are distinguished because they act as fixed points or constraints in any interpretation of a theory, and as such are akin to Poincaré's conventions or Reichenbach's 'axioms of co-ordination', or 'relativized constitutive a priori principles'. (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)

In classical game theory the idea that players randomize between their actions according to a particular optimal probability distribution has always been viewed as puzzling. In this paper, we establish a fundamental connection between n-person normal form games and quantum mechanics, which eliminates the conceptual problems of these random strategies. While the two theories have been regarded as distinct, our main theorem proves that if we do not give any other piece of information to a player in a game, than (...) the payoff matrix—the axiom of “no-supplementary data” holds—then the state of mind of a rational player is algebraically isomorphic to a pure quantum state. The “no supplementary data” axiom is captured in a Lukasiewicz’s three-valued Kripke semantics wherein statements about whether a strategy or a belief of a player is rational are initially indeterminate i.e. neither true, nor false. As a corollary, we show that in a mixed Nash equilibrium, the knowledge structure of a player implies that probabilities must verify the standard “Born rule” postulate of QM. The puzzling “indifference condition” wherein each player must be rationally indifferent between all the pure actions of the support of his equilibrium strategy is resolved by his state of mind being described by a “quantum superposition” prior a player is asked to make a definite choice in a “measurement”. Finally, these results demonstrate that there is an intrinsic limitation to the predictions of game theory, on a par with the “irreducible randomness” of quantum physics. (shrink)

Part of the ambiguity lies in the various points of view from which this question might be considered. The crudest di erence lies between the point of view of the working mathematician and that of the logician concerned with the foundations of mathematics. Now some of my fellow mathematical logicians might protest this distinction, since they consider themselves to be just more of those \working mathematicians". Certainly, modern logic has established itself as a very respectable branch of mathematics, and (...) there are quite a few highly technical journals in logic, such as The Journal of Sym-. (shrink)

Contrary to what has been alleged, the moral equivalence of combatants (MEC) is not a doctrine that was expressly developed by the traditional theorists of just war. Working from the axiom that just cause is unilateral, they did not embrace a conception of public war that included MEC. Indeed, MEC was introduced in the early fifteenth century as a challenge to the then reigning just war paradigm. It does not follow, however, that the distinction between private and public (...) war had no place in the traditional teaching. Thomas Aquinas and his successors did not analyse just war by extrapolation from the related idea of self-defense. Rather, they likened just war to a legal proceeding that could solely be undertaken by persons possessed of legitimate authority. For this reason, just war was first and foremost public war. Private war was deemed ?war? only in a secondary and reduced sense of the term. It was accordingly understood that public war should be waged and its morality judged by reference to a set of norms that are not directly reducible to those governing private self (and other)-defense. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Historical Violence, Censorship, and the Serial Killer: The Case of American PsychoCarla Freccero (bio)R.L.: Do you believe in God?B.E.E.: Are you asking me if I was raised in a religious family or if I go to church? I was raised an agnostic. I don’t know—I hate to fly, I have a fear of flying. That means either that I have no faith in air traffic controllers or that I’ve (...) done something really bad, and this is God’s way of getting at me. Maybe I’m caught in the middle.... But no, I don’t believe in God. That’s such a strange thing to admit in an interview.—Robert Love, “Psycho Analysis,” Rolling Stone interview with Bret Easton EllisBefore Capitol Hill acts on these dimwitted proposals, we should remember what the writer Bret Easton Ellis might have said, but didn’t: “Doing bad stuff is wrong.”—Ted Rall, “Meet the Cusp Kids”US mass media has become a much-publicized target of censorious commentary within American public culture in recent years. Censorship, as E. S. Burt notes, may take at least two forms: philosophical censorship, such as that discussed by Leo Strauss’s Persecution and the Art of Writing, and the more commonly applied form of US censorship enacted against “pornography” or “obscenity.” Philosophical censorship, in this reading, entails the state’s persecution of message-bearing texts or utterances that counter the doxa and threaten to disrupt the state, while the second form of censorship involves “a language use that falls outside the interpretive scheme defining the work as the intentional act of a moral being” [Burt 25]. In the United States, state censorship of the first kind may perhaps be practiced through the “incitement to riot” escape clause provision in the First Amendment, while obscenity censorship—in certain forms—is explicitly written into legislation as excluded from speech protection.Extending Strauss’s argument to the liberal state, Burt suggests in effect that these two forms of censorship become related insofar as the state’s censorious intent is directed not so much at the content of the expression as at the degree to which it confirms the axiom, what Burt calls the “very fundamental, convening assumption of the... liberal state” [25] that “virtue is knowledge” [see Strauss 25]. Thus obscenity is a form of expression that cannot be interpreted as productive knowledge but instead as a certain repetitive excess, “dirt for dirt’s sake” [Burt, quoting Judge Woolsey 28]. The problem here, of course, is that such distinctions do not apply comfortably to that category of utterance termed the literary, and especially to the use of figurative language, where “art for art’s sake” can be [End Page 44] seen precisely to resemble the nonproductivity of obscenity. The text I am going to discuss in depth here, Bret Easton Ellis’s American Psycho, has been attacked on both grounds—as an active (and violent) disruption of the doxa that is also obscenely nonproductive of knowledge—and as such, it serves as an illustrative, and I will argue, noncoincidental, case of “censorship” in the liberal state.It may be useful here to begin by surveying and making distinctions among the various popular and popularized censorships in recent US media: Ice-T’s “Cop Killer” was probably the most authentic case of “philosophical” censorship, even though its silencing did not derive directly from the state (the liberal state often does not function in some of the ways supposed when the definition of censorship is restricted to state censorship). “Cop Killer” is a song about killing policemen, and thus came closest—in the list I am going to propose of silenced works—to tempting the state’s retaliation. It was written and sung by a black man (a black man who often plays a gangster and a cop in popular films), in a genre already regarded as threateningly disruptive to the state: rap. The result is that the CD containing this song was pulled from the shelves of music stores and can no longer be purchased. There was no trial. The author withdrew his work from public circulation in the face of... (shrink)

Individuals differ in their ability to feel their own and others’ internal states, with those that have more autistic and less empathic traits clustering at the clinical end of the spectrum. However, when we consider semantic competence, this group could compensate with a higher capacity to imagine the meaning of words referring to emotions. This is indeed what we found when we asked people with different levels of autistic and empathic traits to rate the degree of imageability of various kinds (...) of words. But this was not the whole story. Individuals with marked autistic traits demonstrated outstanding ability to imagine theoretical concepts, i.e., concepts that are commonly grasped linguistically through their definitions. This distinctive characteristic was so pronounced that, using tree-based predictive models, it was possible to accurately predict participants’ inclination to manifest autistic traits, as well as their adherence to autistic profiles – including whether they fell above or below the diagnostic threshold – from their imageability ratings. We speculate that this quasi-perceptual ability to imagine theoretical concepts represents a specific cognitive pattern that, while hindering social interaction, may favor problem solving in abstract, non-socially related tasks. This would allow people with marked autistic traits to make use of perceptual, possibly visuo-spatial, information for “higher” cognitive processing. (shrink)

The word cosmopolitanism is derived from “cosmos” and “polites” . The cosmopolite is a citizen of the world. The Stoics elaborate on the theme, using the ideas of oikeiosis and sympathy as its basis, thus drawing from their physics. Particularly, Epictetus defends cosmopolitanism on the assumption that man is akin to God, whereas Marcus Aurelius highlights the common possession of mind and that man is by nature able for communal life. For the Stoics man is a social being who can (...) be perfected only within the society of other human beings. The brotherhood of men is grounded on the indubitable axiom that the human soul is the source of the unique good, which is virtue. The distinctive parameter for creating a community is virtue, which is an objective for everyone but also an inherent and ecumenical capacity. (shrink)

Axiomatic characterization results in social choice theory are usually compared either regarding the normative plausibility or regarding the logical strength of the axioms involved. Here, instead, we propose to compare axiomatizations according to the language used for expressing the axioms. In order to carry out such a comparison, we suggest a formalist approach to axiomatization results which uses a restricted formal logical language to express axioms. Axiomatic characterization results in social choice theory then turn into definability results of formal logic. (...) The advantages of this approach include the possibility of non-axiomatizability results, a distinction between absolute and relative axiomatizations, and the possibility to ask how rich a language needs to be to express certain axioms. We argue for formal minimalism, i.e., for favoring axiomatizations in the weakest language possible. (shrink)

One of the more distinctive features of Bob Hale and Crispin Wright’s neologicism about arithmetic is their invocation of Frege’s Constraint – roughly, the requirement that the core empirical applications for a class of numbers be “built directly into” their formal characterization. In particular, they maintain that, if adopted, Frege’s Constraint adjudicates in favor of their preferred foundation – Hume’s Principle – and against alternatives, such as the Dedekind-Peano axioms. In what follows we establish two main claims. First, we show (...) that, if sound, Hale and Wright’s arguments for Frege’s Constraint at most establish a version on which the relevant application of the naturals is transitive counting – roughly, the counting procedure by which numerals are used to answer “how many”-questions. Second, we show that this version of Frege’s Constraint fails to adjudicate in favor of Hume’s Principle. If this is the version of Frege’s Constraint that a foundation for arithmetic must respect, then Hume’s Principle no more – and no less – meets the requirement than the Dedekind-Peano axioms do. (shrink)

In the present study of Descartes’ epistemological investigations, I have tried to show that his renowned principle of clarity and distinctness is not, in fact, one but two axioms. Most interpreters and critics have taken the two formulations of such a principle here considered as successive moments of it. At best, this position is insufficient, for each “version” of the principle of clarity and distinctness guarantees different kinds of cognitive content. Moreover, while the validity of one “version” is not dependent (...) on the thesis of God’s veracity, no such thing can be asserted of the validity of the other. These two formulations of the principle of clarity and distinctness are: 1. Whatever is clearly and distinctly perceived is true; 2. whatever we perceive clearly and distinctly as belonging to the nature of something can indeed be predicated of the thing in question. The fust formula corresponds to what I have characterized as “presentative” knowledge; the second one expresses the guarantee of “representative” knowledge. This distinction is all-important for solving the question of whether Descartes’ proofs of God’s existence and veracity---both the a priori and the a posteriori proofs that we find in the Cartesian corpus-are circular. On the basis of such a distinction, it is possible to argue that at least the ontological argument---and possibly as well the proof “par les effets”---is not at all dependent on the principle of clarity and distinctness, which in turn draws its ultimate validity from God’s faithfulness. In other words, as suggested above, only the second “version” needs to be guaranteed by God’s veracity. On the other hand, the first “version” has no normative value, for it merely describes what is the case whenever a clear and distinct cognition occurs. An example of this is our knowledge of God as the most perfect being. (shrink)

The relationship between the One and the Multiple in mystic philosophy is a profound and central theme that explores the nature of existence, the cosmos, and the divine. This theme is present in various mystical traditions, including those of the East and West, and it addresses the paradoxical coexistence of the unity and multiplicity of all things. -/- In mystic philosophy, the **One** often represents the ultimate reality, the source from which all things emanate and to which all things return. (...) It is the absolute, the infinite, and the unchanging. The One is beyond all attributes and is often associated with the divine or the absolute truth. -/- The **Multiple**, on the other hand, represents the manifest world, the diversity of forms, and the realm of change and plurality. It is the world we experience through our senses, the domain of time and space, where differentiation and individuality are apparent. -/- The relationship between the One and the Multiple is not one of opposition but of emanation and unity. The Multiple is seen as a reflection, expression, or manifestation of the One. In this sense, the diversity of the world doesn’t contradict the unity of the One but rather demonstrates it in a myriad of forms. -/- Mystics seek to understand and experience this relationship through various practices and insights. They aim to transcend the illusion of separation and duality to experience the non-dual reality where the One and the Multiple are recognized as inseparable. -/- This concept can be illustrated by the metaphor of the ocean and its waves. The ocean represents the One—vast, deep, and all-encompassing—while the waves represent the Multiple—distinct, diverse, and ever-changing. Each wave is unique, yet it is not separate from the ocean. The wave’s existence is dependent on and made of the same substance as the ocean. In the same way, each individual entity in the Multiple is an expression of the One. -/- In summary, the relationship between the One and the Multiple in mystic philosophy is a dynamic interplay that challenges the conventional understanding of separation. It invites a deeper exploration of reality, where the apparent multiplicity of the world is a direct expression of the singular, underlying essence of all that is. -/- The relationship between the One and the Multiple in the context of mathematical philosophy is a profound topic that touches upon the very foundations of existence and knowledge. It’s a theme that has been explored by philosophers and mathematicians alike, often leading to the contemplation of unity and diversity within the structures of reality. -/- In mathematics, the concept of the One can be seen as the basis of unity from which all numbers derive. It’s the identity element in multiplication, the starting point in counting, and the foundation of dimension in geometry. The One is often associated with the concept of monism in philosophy, which posits that there is a single, underlying substance or principle that constitutes reality. -/- On the other hand, the Multiple represents the infinite variety and diversity of forms and numbers that arise from the One. It’s the embodiment of plurality and the complex interplay of different entities that mathematics seeks to understand and describe. This reflects the philosophical stance of pluralism, which acknowledges the existence of multiple realities or truths. -/- The interplay between the One and the Multiple can be seen in the mathematical concept of sets. A set can be thought of as a unity, a whole composed of distinct elements. Yet, each element within the set also retains its individuality, contributing to the diversity of the set’s composition. This duality is mirrored in the philosophical exploration of the universal and the particular, where the universal represents the One, and the particulars represent the Multiple. -/- In the realm of mathematical philosophy, this relationship often leads to questions about the nature of mathematical objects: Are they discovered as part of an objective reality (the One), or are they constructed by the human mind from a multitude of experiences (the Multiple)? This debate resonates with the philosophical inquiry into the nature of truth and reality. -/- Reflecting on the One and the Multiple can also lead to a deeper understanding of the self and the cosmos. Just as the number one is integral to the existence of all other numbers, the individual self can be seen as a unique expression of the universal whole. Similarly, the cosmos can be viewed as a grand unity composed of a multiplicity of forms and phenomena. -/- Gödel’s incompleteness theorems have a fascinating connection to the philosophical concepts of the One and the Multiple. These theorems, which are pivotal in mathematical logic and philosophy of mathematics, articulate the inherent limitations of formal axiomatic systems, particularly those sufficient to express the arithmetic of natural numbers. -/- Gödel’s incompleteness theorems have a fascinating connection to the philosophical concepts of the One and the Multiple. These theorems, which are pivotal in mathematical logic and philosophy of mathematics, articulate the inherent limitations of formal axiomatic systems, particularly those sufficient to express the arithmetic of natural numbers⁴. -/- The **first incompleteness theorem** reveals that within any such consistent system, there are propositions that are true but cannot be proven within the system itself. This reflects the idea of the Multiple in that there is an abundance of mathematical truths, a multiplicity that exceeds the unifying framework of any one system. It suggests that the realm of mathematical truth is more extensive than any single formal system can fully capture. -/- The **second incompleteness theorem** extends this by showing that a system cannot prove its own consistency. This relates to the concept of the One, as it implies that a system’s complete self-understanding, its unity and coherence, is unattainable from within. It must look beyond itself, to an external vantage point, to ascertain its consistency. -/- In the context of the One and the Multiple, Gödel’s theorems imply that the One (a consistent formal system) is inherently incomplete and cannot encompass the Multiple (the totality of mathematical truths). This resonates with philosophical discussions about the relationship between unity and plurality. Just as a single philosophical system cannot capture the entirety of truth, a single formal mathematical system cannot encapsulate all mathematical truths. -/- Moreover, Gödel’s work suggests that the pursuit of a single, unified theory of everything in mathematics—a One that encompasses all Multiples—is an inherently Sisyphean task. There will always be more truths (Multiples) than can be derived from any given set of axioms (the One). -/- In essence, Gödel’s incompleteness theorems provide a formal underpinning to the philosophical notion that the One cannot exist without the Multiple, and vice versa. They are interdependent, with the One giving rise to the Multiple, and the Multiple necessitating the One for its expression and comprehension. This interplay is a dance of limitations and possibilities, where the boundaries of logic, mathematics, and philosophy blur into one another. -/- The relationship between the One and the Multiple in the context of the mathematics of zero is a deeply philosophical inquiry that bridges the abstract world of numbers with the existential questions of being and non-being. -/- Zero, in mathematics, is a symbol of absence, a representation of nothingness, yet it holds a pivotal position as a number. It is the void from which all things emerge and to which they return. In the philosophy of mathematics, zero is the paradoxical junction where the One and the Multiple converge and diverge. -/- From the perspective of the One, zero can be seen as the origin—the singular point that precedes the existence of numbers. It is the empty set, the foundation upon which the edifice of mathematics is constructed. As the identity element in addition, zero maintains the integrity of numbers, for adding zero to any number leaves it unchanged, reflecting the immutable nature of the One. -/- Conversely, when we consider the Multiple, zero represents the infinite potentiality of creation. It is the canvas upon which the integers, both positive and negative, express their multitude. Zero is the balance point, the fulcrum around which the symphony of numbers dances. It embodies the plurality of possibilities, the beginning of the number line that stretches infinitely in both directions. -/- Philosophically, zero challenges our understanding of existence. It is both something and nothing—a number that quantifies the absence of quantity. This duality echoes the philosophical struggle to comprehend how the One gives rise to the Multiple. How does the unity of being manifest the diversity of the cosmos? Zero offers a mathematical metaphor for this mystery, as it encapsulates the transition from non-existence to existence, from the undifferentiated One to the differentiated Multiple. -/- In the realm of set theory, zero corresponds to the empty set—a set with no elements. This set is unique in that it is the only set that contains nothing, yet it is the foundation upon which all other sets are built. The empty set is the mathematical embodiment of the One, and all other sets, containing multiple elements, arise from it. -/- Reflecting on zero in the context of Gödel’s incompleteness theorems, we find a resonance with the idea that the One (a consistent formal system) cannot capture the entirety of the Multiple (the totality of mathematical truths). Zero, as the foundation of numbers, similarly suggests that from the nothingness of the One, the infinite complexity of the Multiple emerges—yet it can never be fully encompassed or expressed by any single system. -/- In conclusion, the mathematics of zero offers a profound reflection on the relationship between the One and the Multiple. It serves as a bridge between the abstract and the concrete, the known and the unknowable, challenging us to ponder the origins of existence and the nature of reality itself. -/- . (shrink)

The distinction between analytic and synthetic propositions, and with that the distinction between a priori and a posteriori truth, is being abandoned in much of analytic philosophy and the philosophy of most of the sciences. These distinctions should also be abandoned in the philosophy of mathematics. In particular, we must recognize the strong empirical component in our mathematical knowledge. The traditional distinction between logic and mathematics, on the one hand, and the natural sciences, on the other, should (...) be dropped. Abstract mathematical objects, like transcendental numbers or Hilbert spaces, are theoretical entities on a par with electromagnetic fields or quarks. Mathematical theories are not primarily logical deductions from axioms obtained by reflection on concepts but, rather, are constructions chosen to solve some collection of problems while fitting smoothly into the other theoretical commitments of the mathematician who formulates them. In other words, a mathematical theory is a scientific theory like any other, no more certain but also no more devoid of content. (shrink)

ABSTRACT The author proposes to convert Ayn Rand’s theory of man into a sixth branch of her Objectivist philosophy called the metaphysics of man. This branch would be distinct from both the metaphysics of reality and epistemology. Along with consolidating all the axioms about the fundamental nature of man, this new framework will simplify and clarify the structure of Objectivism.

The so-called Paradox of Serious Possibility is usually regarded as showing that the standard axioms of belief revision do not apply to belief sets that are introspectively closed. In this article we argue to the contrary: we suggest a way of dissolving the Paradox of Serious Possibility so that introspective statements are taken to express propositions in the standard sense, which may thus be proper members of belief sets, and accordingly the normal axioms of belief revision apply to them. Instead (...) the paradox is avoided by making explicit, for any occurrence of an introspective modality in the object language, the belief state to which this occurrence refers; this will make it impossible for any doxastic modality to refer to two distinct belief sets within one and the same context of doxastic appraisal. By this move the standard derivation of a contradiction from the theory of belief revision in the presence of introspectively closed belief sets does not go through any more, and indeed the premisses of the Paradox of Serious Possibility become jointly consistent once they are reformulated with our amended introspective modalities only. Additionally, we present a probabilistic version of the Paradox of Serious Possibility which can be avoided in a perfectly analogous manner. (shrink)

The best and most popular argument for probabilism is the accuracy-dominance argument, which purports to show that alethic considerations alone support the view that an agent’s degrees of belief should always obey the axioms of probability. I argue that extant versions of the accuracy-dominance argument face a problem. In order for the mathematics of the argument to function as advertised, we must assume that every omniscient credence function is classically consistent; there can be no worlds in the set of dominance-relevant (...) worlds that obey some other logic. This restriction cannot be motivated on alethic grounds unless we’re also willing to accept that rationality requires belief in every metaphysical necessity, as the distinction between a priori logical necessities and a posteriori metaphysical ones is not an alethic distinction. To justify the restriction to classically consistent worlds, non-alethic motivation is required. And thus, if there is a version of the accuracy-dominance argument in support of probabilism, it isn’t one that is grounded in alethic considerations alone. (shrink)

We propose an axiomatization of aversion to incomplete preferences. Some prevailing models of incomplete preferences rely on the hypothesis that incompleteness is temporary and that by keeping their opportunity set open individuals reveal a preference for flexibility. We consider that the maintenance of incomplete preference is also aversive. Our model allows us to show how incompleteness induces an aversive attitude in two different ways: intrinsic and instrumental. Intrinsic aversion holds when one instance of incomplete preference in the set suffices to (...) decrease its utility. Instrumental aversion holds only insofar dominating options are affected by incompleteness. Given two partially overlapping sets of axioms on the binary relation over sets we formalize their consistency with the two types of aversion to incompleteness. Finally, we relate our model to the classical Sen’s distinction between tentative and assertive incompleteness. The spelling out of this distinction in the terms of our approach uncovers to what extent aversion to incompleteness may be compatible with a preference for flexibility. (shrink)

The Pigou-Dalton (PD) principle recommends a non-leaky, non-rank-switching transfer of goods from someone with more goods to someone with less. This Article defends the PD principle as an aspect of distributive justice—enabling the comparison of two distributions, neither completely equal, as more or less just. It shows how the PD principle flows from a particular view, adumbrated by Thomas Nagel, about the grounding of distributive justice in individuals’ “claims.” And it criticizes two competing frameworks for thinking about justice that less (...) clearly support the principle: the veil-of-ignorance framework, and Larry Temkin’s proposal that fairer distributions are those concerning which individuals have fewer “complaints.” -/- The Article also clarifies the relation between the PD principle and prioritarianism. Prioritarians will surely endorse the PD principle (with the “good” individual well-being), but they are also committed to a distinct axiom of separability: the moral value of someone’s well-being change does not depend upon her position relative to others. The PD principle neither implies separability, nor is implied by it. Although prioritarianism is very plausible, the case for the PD principle is yet more compelling than for the combination of that principle with separability. In discussing prioritarianism, we should differentiate between these two, logically independent aspects of the view. -/- . (shrink)

One cannot really speak of a school of personalistic economists. Moreover, there is a wide gulf between the economic philosophy of the personalists and the mathematical context of economic science. Since the thirties, philosophers such as Alexandre Marc, Jacques Maritain, Emmanuel Mounier and many others have been searching, on the basis of a personalistic view of man, for a `third way' between individualistic capitalism and statist socialism , but there was seldom interest from the side of the scientific economists.Fortunately, there (...) are some notable exceptions. François Perroux, Kenneth Boulding, Ernst Schumacher, Serge Christophe Kolm, and Amartya Sen are economists who, in various ways, have attempted to bridge the gulf between the personalistic view of man and economic rationality. By carrying out an immanent critique of some of the standard economic assumptions, they opened a path for a normative economics that was clearly distinct from a purely positive economics as well as from Pareto's welfare economics. One can affix various labels to these attempts: humanistic economics , socio-economics , moral economics . In what follows, I shall interpret these attempts as initiatives in the direction of a personalistic economics. I would also situate the current research in `economics and ethics', which is becoming part of the mainstream of economic thought, and which has received official recognition with the recent Nobel prize award to Sen, within this tendency towards a more personalistic economics.Attention to moral feelings, human rights, altruistic motives, justice and solidarity, non-profit organizations, etc., contribute to situating the economic problem in a field where economics, politics and ethics no longer function as three distinct domains but where their interaction is central.Yet does this interactive economics fit into a personalistic way of thinking? Is a generalized system theory not more attractive? System theory has a different focus than personalistic economics. System theory considers the axioms of system equilibrium and sustainable growth as the ultimate horizon, whereas a personalist sees them as merely important conditions for a good society and economy. However refined it may be, system theory views the person from an external point of view, while the personalist refers to the internal experience of human freedom and meaning as something that transcends and regulates the system's functionality.One could object that neo-classical economists do not use system theory as their point of departure. In their eyes, system equilibrium is the result of the utility-maximizing behaviour of individuals. These individuals subjectively determine what their preferences are and then act accordingly. Is this openness for a free, utility-maximizing actor — where there is room for both material and immaterial, egoistic and altruistic preferences — not sufficient to speak of a personalistic economics ? If this were true, then the discourse of personalism would be more or less superfluous.I take the term `personalistic economics' to refer to a normative economics that does not simply reconstruct the problem of efficient allocation as an individual or social problem of utility, but in the first instance as a problem of human dignity and social justice. To reduce the problem of human choice to a problem of subjective utility is in itself a far-reaching normative standpoint. As soon as one realizes this, there is no reason to exclude other normative assumptions, such as personalistic ones. (shrink)

The distinction between the syntactic and the semantic approach to scientific theories emerged in formal philosophy of science. The semantic approach is commonly considered more advanced and more successful than the syntactic one, but the transition from the one approach to the other was not brought about without any loss. In essence, it is the formal analysis of atomic propositions and the analysis of deductive reasoning that dropped out of consideration in at least some of the elaborated versions of (...) the semantic approach. In structuralist theory of science, as founded by Sneed and Stegmüller, the focus is on global propositions concerning the question of whether or not certain empirical systems satisfy a set-theoretic predicate that encodes the axioms of a scientific theory. Hence, an analysis of deductive reasoning from atomic premisses with the help of a given theory is missing. The objective of the present paper is to develop a deductive system on the basis of the structuralist framework. This system comes with a novel formulation of empirical propositions in structuralism. (shrink)

The axiom of infinity states that infinite sets exist. I will argue that this axiom lacks justification. I start by showing that the axiom is not self-evident, so it needs separate justification. Following Maddy’s :481–511, 1988) distinction, I argue that the axiom of infinity lacks both intrinsic and extrinsic justification. Crucial to my project is Skolem’s From Frege to Gödel: a source book in mathematical logic, 1879–1931, Cambridge, Harvard University Press, pp. 290–301, 1922) distinction (...) between a theory of real sets, and a theory of objects that theory calls “sets”. While Dedekind’s argument fails, his approach was correct: the axiom of infinity needs a justification it currently lacks. This epistemic situation is at variance with everyday mathematical practice. A dilemma ensues: should we relax epistemic standards or insist, in a skeptical vein, that a foundational problem has been ignored? (shrink)

Definitions Axioms Prop. I. Substance is by nature prior to its modifications Prop. II. Two substances, whose attributes are different, have nothing in common Prop III. Things, which have nothing in common, cannot be one the cause of the other Prop. IV. Two or more distinct things are distinguished one from the other either by the difference of the attributes of the substance, or by the differences of their modifications Prop. V. There cannot exist in the universe two or more (...) substances having the same nature or attribute Prop. VI. One substance cannot be produced by another substance Prop. VII. Existence belongs to the nature of substance Prop. VIII. Every substance is necessarily infinite Prop. IX. The more reality or being a thing has, the greater the number of its attributes Prop. X. Each particular attribute of the one substance must be conceived through itself Prop. XI. God, or substance consisting of infinite attributes, of which each expresses eternal and infinite essentiality, necessarily exists Prop. XII. No attribute of substance can be conceived, from which it would follow that substance can be divided Prop. XIII. Substance absolutely infinite is indivisible Prop. XIV. Besides God no substance can be granted or conceived Prop. XV. Whatsoever is, is in God, and without God nothing can be, or be conceived Prop. XVI. From the necessity of the divine nature must follow an infinite number of things in infinite ways--that is, all things which fall within the sphere of infinite intellect Prop. XVII. God acts solely by the laws of his own nature and is not constrained by anyone Prop. XVIII. God is the indwelling and not the transient cause of all things Prop. XIX. God and all the attributes of God are eternal Prop. XX. The existence of God and his essence are one and the same Prop. XXI. All things, which follow from the absolute nature of any attribute of God, must always exist and be infinite, or, in other words, are eternal and infinite through the said attribute Prop.. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:BOOK REVIEWS The Immutability of God in the Theology of Hans Urs von Balthasar. By GERARD F. O'HANLON, S.J. Cambridge: Cambridge University Press, 1990. Pp. 246. $59.95 (cloth). O'Hanlon unfolds Balthasar's theology in four main chapters, which treat the question of immutability in terms, respectively, of Christ· ology; creation; time and eternity; and inner trinitarian life in God. In Chapter 5, O'Hanlon compares Balthasar's approach with some English-speaking authors (...) (B. Davies, A. Kenny, E. Stump, Kretzmann, P. Geach, J. Kvanvig, D. H. Mellor, A. E. Williams). Finally, in Chapter 6, O'Hanlon ofiers a summary and assessment of Balthasar's proposals. (1) Regarding his treatment of Christology in Chapter 1, O'Hanlon discusses how Balthasar's attempt to find a "middle way" between simple mutability and a traditional sense of immutability in God lies in uncovering the trinitarian presuppositions of Christology. Whether God is open to being afiected by dialogue with creatures hinges decisively on how we understand Jesus Christ or, more precisely, on how we understand the humanity of Jesus to reveal what God is like (9-10). This is an approach difierent from.that of those who would criticize the traditional axiom of divine immutability on the basis of certain philosophical assumptions regarding the priority of becoming over being (10-11). Balthasar's approach is first theological: when we con· sider the soteriological and eschatological dimensions of Christ's work for us, we are forced to ask the question about the nature of God: "does he too not sufier, is he not afiected by sin, can he he simply apathetic to the prospect of any final loss of his creatures to hell? " (9). At the heart of Ba1thasar's position, then, lies an affirmation of a kenosis within the inner life of God, which already provides a foundation for the possibility of the (second) kenosis revealed in the incarnation and the cross (11-14). That is, the "kind of emptying and self. giving we see in the incarnation is both the image and the effect of the eternal ' externalization ' of God that is involved in the intra-trinitarian life" (14). It is imperative that one see the nuances of this affirmation. On the one hand, Balthasar is careful to avoid a simple identification of the second with the first kenosis: this would make the incarna· 335 336 BOOK REVIEWS tion and cross natural and necessary to God, thus entailing theopaschism (14-15). On the other hand, the suffering love of Christ does truly reveal God's sovereign power to be absolute love and self-giving, because there " is a certain analogy between the divine and human natures of Christ which is due to the identity of the person, the pre-existent Logos" (14). Thus Balthasar is led to qualify the traditional understanding of divine immutability (15) : there is " some real kenosis within God which has ontological status and is not merely functional, soteriological, or a simple addition which does not affect God" {16). O'Hanlon makes clear how Balthasar is at pains always to respect the mysterious nature of the issue, as well as the limits set by the great Christian tradition. At the heart of the matter lies a return to the person -nature distinction as set by Chalcedon. Traditionally, the affirmation has been of a divine nature in Christ which remains immutable and a human nature which changes and suffers. Balthasar goes beyond this affirmation. On the basis of Philippians 2:5-11, he arrives at the necessity of positing a real kenosis in God ; and from his recognition of the ontological, personal identity of the Logos as the subject who unites the two distinct natures in Christ, he refuses to limit the change and suffering which Christ experiences to his human nature alone. In fact, the tendency to understand the human nature of Christ as an instrumentum conjunctum which does not affect the divine person he considers as Nestorian in character. Yet, at the same time, Balthasar affirms the incommensurable difference between God and the world, between the divine and human " unmixed " natures of Christ. The relation between the two natures is precisely analogous (after the manner... (shrink)

We present $\in_I$-Logic (Epsilon-I-Logic), a non-Fregean intuitionistic logic with a truth predicate and a falsity predicate as intuitionistic negation. $\in_I$ is an extension and intuitionistic generalization of the classical logic $\in_T$ (without quantifiers) designed by Sträter as a theory of truth with propositional self-reference. The intensional semantics of $\in_T$ offers a new solution to semantic paradoxes. In the present paper we introduce an intuitionistic semantics and study some semantic notions in this broader context. Also we enrich the quantifier-free language by (...) the new connective < that expresses reference between statements and yields a finer characterization of intensional models. Our results in the intuitionistic setting lead to a clear distinction between the notion of denotation of a sentence and the here-proposed notion of extension of a sentence (both concepts are equivalent in the classical context). We generalize the Fregean Axiom to an intuitionistic version not valid in $\in_I$. A main result of the paper is the development of several model constructions. We construct intensional models and present a method for the construction of standard models which contain specific (self-)referential propositions. (shrink)