Kit Fine develops a Fregean theory of abstraction, and suggests that it may yield a new philosophical foundation for mathematics, one that can account for both our reference to various mathematical objects and our knowledge of various mathematical truths. The Limits of Abstraction breaks new ground both technically and philosophically.
Philosophers have often claimed that general ideas or representations have their origin in abstraction, but it remains unclear exactly what abstraction as a psychological process consists in. We argue that the Lockean aspiration of using abstraction to explain the origins of all general representations cannot work and that at least some general representations have to be innate. We then offer an explicit framework for understanding abstraction, one that treats abstraction as a computational process that operates (...) over an innate quality space of fine-grained general representations. We argue that this framework has important philosophical implications for the nativism-empiricism dispute, for questions about the acquisition of unstructured representations, and for questions about the relation between human and animal minds. (shrink)
This paper presents a formalization of first-order arithmetic characterizing the natural numbers as abstracta of the equinumerosity relation. The formalization turns on the interaction of a nonstandard (but still first-order) cardinality quantifier with an abstraction operator assigning objects to predicates. The project draws its philosophical motivation from a nonreductionist conception of logicism, a deflationary view of abstraction, and an approach to formal arithmetic that emphasizes the cardinal properties of the natural numbers over the structural ones.
I show how omissions lead to robustness and can justify distortions, and I give inferentially relevant explications of abstraction and idealization. Abstraction is explicated as the omission of all and only those claims that use a specific vocabulary; idealization is explicated as the distortion of only those claims that use a specific vocabulary. With these explications, abstraction can justify idealization. As examples of how abstraction justifies idealization and leads to robustness, I discuss Beauchamp and Childress's four (...) principles of biomedical ethics and the qualitative treatment of the Schrödinger equation. (shrink)
Questions concerning the epistemological status of computer science are, in this paper, answered from the point of view of the formal verification framework. State space reduction techniques adopted to simplify computational models in model checking are analysed in terms of Aristotelian abstractions and Galilean idealizations characterizing the inquiry of empirical systems. Methodological considerations drawn here are employed to argue in favour of the scientific understanding of computer science as a discipline. Specifically, reduced models gained by Dataion are acknowledged as Aristotelian (...) abstractions that include only data which are sufficient to examine the interested executions. The present study highlights how the need to maximize incompatible properties is at the basis of both Abstraction Refinement, the process of generating a cascade of computational models to achieve a balance between simplicity and informativeness, and the Multiple Model Idealization approach in biology. Finally, fairness constraints, imposed to computational models to allow fair behaviours only, are defined as ceteris paribus conditions under which temporal formulas, formalizing software requirements, acquire the status of law-like statements about the software systems executions. (shrink)
EDDINGTON frequently insisted on the ' necessity for an outlook beyond physics ' , and was deeply interested in the relations between science and other ways ...
Fitch's basic logic is an untyped illative combinatory logic with unrestricted principles of abstraction effecting a type collapse between properties (or concepts) and individual elements of an abstract syntax. Fitch does not work axiomatically and the abstraction operation is not a primitive feature of the inductive clauses defining the logic. Fitch's proof that basic logic has unlimited abstraction is not clear and his proof contains a number of errors that have so far gone undetected. This paper corrects (...) these errors and presents a reasonably intuitive proof that Fitch's system K supports an implicit abstraction operation. Some general remarks on the philosophical significance of basic logic, especially with respect to neo-logicism, are offered, and the paper concludes that basic logic models a highly intensional form of logicism. (shrink)
What is wrong with abstraction, Michael Potter and Peter Sullivan explain a further objection to the abstractionist programme in the foundations of mathematics which they first presented in their Hale on Caesar and which they believe our discussion in The Reason's Proper Study misunderstood. The aims of the present note are: To get the character of this objection into sharper focus; To explore further certain of the assumptions—primarily, about reference-fixing in mathematics, about certain putative limitations of abstractionist set theory, (...) and about the effects of impredicativity in abstraction principles—which underlie it; and To advance the debate of the issues thereby raised. Thanks for helpful comments to Roy Cook and to an anonymous referee. CiteULike Connotea Del.icio.us What's this? (shrink)
The process of abstraction and concretisation is a label used for an explicative theory of scientific model-construction. In scientific theorising this process enters at various levels. We could identify two principal levels of abstraction that are useful to our understanding of theory-application. The first level is that of selecting a small number of variables and parameters abstracted from the universe of discourse and used to characterise the general laws of a theory. In classical mechanics, for example, we select (...) position and momentum and establish a relation amongst the two variables, which we call Newton’s 2nd law. The specification of the unspecified elements of scientific laws, e.g. the force function in Newton’s 2nd law, is what would establish the link between the assertions of the theory and physical systems. In order to unravel how and with what conceptual resources scientific models are constructed, how they function and how they relate to theory, we need a view of theory-application that can accommodate our constructions of representation models. For this we need to expand our understanding of the process of abstraction to also explicate the process of specifying force functions etc. This is the second principal level at which abstraction enters in our theorising and in which I focus. In this paper, I attempt to elaborate a general analysis of the process of abstraction and concretisation involved in scientific- model construction, and argue why it provides an explication of the construction of models of the nuclear structure. (shrink)
This paper argues for two related theses. The first is that mathematical abstraction can play an important role in shaping the way we think about and hence understand certain phenomena, an enterprise that extends well beyond simply representing those phenomena for the purpose of calculating/predicting their behaviour. The second is that much of our contemporary understanding and interpretation of natural selection has resulted from the way it has been described in the context of statistics and mathematics. I argue for (...) these claims by tracing attempts to understand the basis of natural selection from its early formulation as a statistical theory to its later development by R.A. Fisher, one of the founders of modern population genetics. Not only did these developments put natural selection of a firm theoretical foundation but its mathematization changed the way it was understood as a biological process. Instead of simply clarifying its status, mathematical techniques were responsible for redefining or reconceptualising selection. As a corollary I show how a highly idealised mathematical law that seemingly fails to describe any concrete system can nevertheless contain a great deal of accurate information that can enhance our understanding far beyond simply predictive capabilities. (shrink)
The dangers of character reification for cladistic inference are explored. The identification and analysis of characters always involves theory-laden abstraction—there is no theory-free “view from nowhere.” Given theory-ladenness, and given a real world with actual objects and processes, how can we separate robustly real biological characters from uncritically reified characters? One way to avoid reification is through the employment of objectivity criteria that give us good methods for identifying robust primary homology statements. I identify six such criteria and explore (...) each with examples. Ultimately, it is important to minimize character reification, because poor character analysis leads to dismal cladograms, even when proper phylogenetic analysis is employed. Given the deep and systemic problems associated with character reification, it is ironic that philosophers have focused almost entirely on phylogenetic analysis and neglected character analysis. (shrink)
For various reasons several authors have enriched classical first order syntax by adding a predicate abstraction operator. “Conservatives” have done so without disturbing the syntax of the formal quantifiers but “revisionists” have argued that predicate abstraction motivates the universal quantifier’s re-classification from an expression that combines with a variable to yield a sentence from a sentence, to an expression that combines with a one-place predicate to yield a sentence. My main aim is to advance the cause of predicate (...)abstraction while cautioning against revisionism. In so doing, however, I shall pursue a secondary aim by conveying mixed blessings to those who hold the view that in the logical sense of “existence” some existing object is such as to exist contingently. Advocates of this view must concede Williamson’s recent contention that the domain of unrestricted objectual quantification could not have been narrower than it is actually, but predicate abstraction affords them some hope of accommodating this concession. (shrink)
Neo-Fregean logicism attempts to base mathematics on abstraction principles. Since not all abstraction principles are acceptable, the neo-Fregeans need an account of which ones are. One of the most promising accounts is in terms of the notion of stability; roughly, that an abstraction principle is acceptable just in case it is satisfiable in all domains of sufficiently large cardinality. We present two counterexamples to stability as a sufficient condition for acceptability and argue that these counterexamples can be (...) avoided only by major departures from the existing neo-Fregean programme. (shrink)
Ethicists of care have objected to traditional moral philosophy's reliance upon abstract universal principles. They claim that the use of abstraction renders traditional theories incapable of capturing morally relevant, particular features of situations. I argue that this objection sometimes conflates two different levels of moral thinking: the level of justification and the level of deliberation. Specifically, I claim that abstraction or attention to context at the level of justification does not entail, as some critics seem to think, a (...) commitment to abstraction or attention to context at the level of deliberation. It follows that critics who reject a theory's use of abstraction at the level of justification have not shown that the theory recommends abstraction at the level of deliberation and that it, therefore, compels the deliberating agent to overlook morally salient details. (shrink)
Experimental philosophers have disagreed about whether "the folk" are intuitively incompatibilists or compatibilists, and they have disagreed about the role of abstraction in generating such intuitions. New experimental evidence using Construal Level Theory is presented. The experiments support the views that the folk are intuitively both incompatibilists and compatibilists, and that abstract mental representations do shift intuitions, but not in a univocal way.
We characterize abstraction in computer science by first comparing the fundamental nature of computer science with that of its cousin mathematics. We consider their primary products, use of formalism, and abstraction objectives, and find that the two disciplines are sharply distinguished. Mathematics, being primarily concerned with developing inference structures, has information neglect as its abstraction objective. Computer science, being primarily concerned with developing interaction patterns, has information hiding as its abstraction objective. We show that abstraction (...) through information hiding is a primary factor in computer science progress and success through an examination of the ubiquitous role of information hiding in programming languages, operating systems, network architecture, and design patterns. (shrink)
The use of “levels of abstraction” in philosophical analysis (levelism) has recently come under attack. In this paper, I argue that a refined version of epistemological levelism should be retained as a fundamental method, called the method of levels of abstraction. After a brief introduction, in section “Some Definitions and Preliminary Examples” the nature and applicability of the epistemological method of levels of abstraction is clarified. In section “A Classic Application of the Method of Abstraction”, the (...) philosophical fruitfulness of the new method is shown by using Kant’s classic discussion of the “antinomies of pure reason” as an example. In section “The Philosophy of the Method of Abstraction”, the method is further specified and supported by distinguishing it from three other forms of “levelism”: (i) levels of organisation; (ii) levels of explanation and (iii) conceptual schemes. In that context, the problems of relativism and antirealism are also briefly addressed. The conclusion discusses some of the work that lies ahead, two potential limitations of the method and some results that have already been obtained by applying the method to some long-standing philosophical problems. (shrink)
Little is known of Edmund Husserl's direct encounter with Georg Cantor's ideas on Platonic idealism and the abstraction of number concepts during the late 19th century, when Husserl's philosophical orientation changed considerably and definitely. Closely analyzing and comparing the two men's writings during that important time in their intellectual careers, I describe the crucial shift in Husserl's views on psychologism and metaphysical idealism as it relates to Cantor's philosophy of arithmetic. I thus establish connections between their ideas which have (...) been until now been virtually unsuspected and contribute to a better understanding of the development of Husserl's thought and of the philosophical and metaphysical ideas within which Cantor chose to frame his theories. (shrink)
We argue against theory-of-mind interpretation of recent false-belief experiments with young infants and explore two other interpretations: enactive and behavioral abstraction approaches. We then discuss the differences between these alternatives.
This paper discusses some aspects of the controversies regarding the operation of the agent intellect on sensory images. I selectively consider views developed between the 13th century and the beginning of the 17th century, focusing on positions which question the need for a (distinct) agent intellect or argue for its essential "inactivity" with respect to phantasms. My aim is to reveal limitations of the Peripatetical framework for analyzing and explaining the mechanisms involved in conceptual abstraction. The first section surveys (...) developments of Aristotelian noetics and abstraction in Ancient and Arabic philosophy. The second section presents a discussion of some "positive" accounts on abstraction and the agent intellect, and some "negative" accounts. (shrink)
This article is an extended critical study of Kit Fine’s The limits of abstraction, which is a sustained attempt to take the measure of the neo-logicist program in the philosophy and foundations of mathematics, founded on abstraction principles like Hume’s principle. The present article covers the philosophical and technical aspects of Fine’s deep and penetrating study.
While Hermann Lotze's philosophy was widely received all over the world, his views on abstraction and Platonic ideas are of particular interest because they were to a large extent adopted by one of the most eminent philosophers of the twentieth century, namely Edmund Husserl. In this paper these views are examined in three distinct aspects. The first of these aspects is to be found in Lotze's thesis that there is a mental process, prior to abstraction, whereby "first universals" (...) are apprehended. The second one lies in his view that there is yet a higher level of apprehension, as found in the process of abstraction itself. According to Lotze, abstraction is not to be identified with the mere removal of particular features, but rather the replacement of these with first universals, resulting in "general images" and ultimately concepts. In addition to Lotze's analysis of the cognition of universals, there is finally a third thesis (an ontological one) which is examined in this paper, namely that the universals are Platonic Ideas in the sense that they have "validity" (Geltung) independently of their corresponding particulars and also of the mind which grasps them. The three claims in question are examined here in detail. Also, an attempt is made to point out some of the connections between Lotze and Husserl on the topic under discussion. (shrink)
Frege suggests that criteria of identity should play a central role in the explanation of reference, especially to abstract objects. This paper develops a precise model of how we can come to refer to a particular kind of abstract object, namely, abstract letter types. It is argued that the resulting abstract referents are ‘metaphysically lightweight’.
Abstraction is seen as an active process which both enlightens and obscures. Abstractions are not true or false but relatively enlightening or obscuring according to the problem under study; different abstractions may grasp different aspects of a problem. Abstractions may be useless if they can answer questions only about themselves. A theoretical enterprise explores reality through acluster of abstractions that use different perspectives, temporal and horizontal scales, and assumes different givens.
Which abstraction principles are acceptable? A variety of criteria have been proposed, in particular irenicity, stability, conservativeness, and unboundedness. This note charts their logical relations. This answers some open questions and corrects some old answers.
Book Information The Limits of Abstraction. The Limits of Abstraction Kit Fine , Oxford : Clarendon Press , 2002 , x + 203 , £18.99 (cloth). By Kit Fine. Clarendon Press. Oxford. Pp. x + 203. £18.99 (cloth).
In Die Grundlagen der Arithmetik, Frege attempted to introduce cardinalnumbers as logical objects by means of a second-order abstraction principlewhich is now widely known as ``Hume's Principle'' (HP): The number of Fsis identical with the number of Gs if and only if F and G are equinumerous.The attempt miscarried, because in its role as a contextual definition HP fails tofix uniquely the reference of the cardinality operator ``the number of Fs''. Thisproblem of referential indeterminacy is usually called ``the Julius (...) Caesar problem''.In this paper, Frege's treatment of the problem in Grundlagen is critically assessed. In particular, I try to shed new light on it by paying special attention to the framework of his logicism in which it appears embedded. I argue, among other things, that the Caesar problem, which is supposed to stem from Frege's tentative inductive definition of the natural numbers, is only spurious, not genuine; that the genuine Caesar problem deriving from HP is a purely semantic one and that the prospects of removing it by explicitly defining cardinal numbers as objects which are not classes are presumably poor for Frege. I conclude by rejecting two closely connected theses concerning Caesar put forward by Richard Heck: (i) that Frege could not abandon Axiom V because he could not solve the Julius Caesar problem without it; (ii) that (by his own lights) his logicist programme in Grundgesetze der Arithmetik failed because he could not overcome that problem. (shrink)
The logical status of abstraction principles, and especially Hume’s Principle, has been long debated, but the best currently availeble tool for explicating a notion’s logical character—permutation invariance—has not received a lot of attention in this debate. This paper aims to fill this gap. After characterizing abstraction principles as particular mappings from the subsets of a domain into that domain and exploring some of their properties, the paper introduces several distinct notions of permutation invariance for such principles, assessing the (...) philosophical significance of each. (shrink)
Hale proposes a neo-logicist definition of real numbers by abstraction as ratios defined on a complete ordered domain of quantities (magnitudes). I argue that Hale's definition faces insuperable epistemological and ontological difficulties. On the epistemological side, Hale is committed to an explanation of measurement applications of reals which conflicts with several theorems in measurement theory. On the ontological side, Hale commits himself to the necessary and a priori existence of at least one complete ordered domain of quantities, which is (...) extremely implausible because science treats the logical structure of quantities as subject to experimentally and theoretically motivated refinements and revisions. (shrink)
Human rights culture has often been accused of a certain imbalance. For instance, it is often said that the practitioners of human rights (i.e., lawyers, politicians, judges, legislators, intellectual advocates, activists, etc.) are too quick to proclaim the existence of rights and too slow to define or allocate attendant duties. In this article, I address one complaint of this sort: the so-called “claimability objection” to human rights. My central aim is to unearth some of the conceptual complexity underlying that objection. (...) What that analysis reveals, in the broadest of terms, is that claimability is not the obedient philosophical concept that it has been made out to be. On the contrary, its invocation has ramifications that, I suggest, have not been adequately foreseen by its main proponents. To illustrate this point I focus specifically on the work of Onora O’Neill, whose claimability-based critique of welfare rights is at the very center of contemporary debates about this topic. I shall, in particular, challenge two important aspects of O’Neill’s critique. First, in Section III, I question its narrowness. O’Neill understands the claimability of a right to depend on the identification of its duty-bearers. But if we attend to the basic logic of her discussion, it becomes clear that the claimability of a right depends on more than just that; indeed, a whole range of factors – including the determinacy of a right’s weight, content, and holders – become relevant. This undermines O’Neill’s assertion that only certain kinds of purported human rights (i.e., second-generation welfare, as opposed to first-generation liberty rights) are subject to the claimability objection. Moreover, it shifts the natural target of that objection over to the more expansive category of abstract rights. The second challenge that I put to O’Neill’s critique raises fundamental doubts about whether claimability (as O’Neill herself understands it) is a necessary feature of rights at all. This I do in Section V. But before I get to that I discuss (in Section IV) different ways in which the domestic, regional, and international legal practice of human rights is less opaque about deontic matters than might appear at first glance. This claim, I suggest, not only provides us with a possible (if ultimately fragile) way of responding to O’Neill’s critique, it illustrates both the instructive and expressive value that legal practices can have for moral thinking about human rights. Lastly, in Section VI, I reflect more generally on the role of abstraction in the theory and practice of human rights. Thus, by allaying claimability-based concerns about abstraction in Section V, and by illustrating some of the positive functions of abstraction in Section VI, I hope to show that abstract rights are not only coherent but also useful and important. (shrink)
The aim of this essay is to emphasize a number of important points that will provide a better understanding of the history of philosophical thought concerning scientific knowledge. The main points made are: (a) that the principal way of viewing abstraction which has dominated the history of thought and epistemology up to the present is influenced by the original Aristotelian position; (b) that with the birth of modern science a new way of conceiving abstraction came into being which (...) is better characterized by the term idealization, the name that was later, in fact, to be used by scientists to describe their scientific activity; (c) that, however, on account of the influence of empirical and inductive philosophy, scientists have often not had sufficient methodological awareness of this new way of viewing abstraction; (d) that this new concept of abstraction has frequently been expressed in the framework of philosophies that lie outside the mainstream of contemporary epistemology or even exhibit marked anti-scientific tendencies; (e) that the theme of idealization has been taken up again in the last few decades and a great contribution in this direction has been made by the so-called Pozna school of methodology. (shrink)
I claim that Berkeley's main argument against abstraction comes into focus only when we see Descartes as one of its targets. Berkeley does not deploy Winkler's impossibility argument but instead argues that what is impossible is inconceivable. Since Descartes conceives of extension as a determinable, and since determinables cannot exist as such, he falls within the scope of Berkeley's argument.
Abstract: Laws of computer science are prescriptive in nature but can have descriptive analogs in the physical sciences. Here, we describe a law of conservation of information in network programming, and various laws of computational motion (invariants) for programming in general, along with their pedagogical utility. Invariants specify constraints on objects in abstract computational worlds, so we describe language and data abstraction employed by software developers and compare them to Floridi's concept of levels of abstraction. We also consider (...) Floridi's structural account of reality and its fit for describing abstract computational worlds. Being abstract, such worlds are products of programmers' creative imaginations, so any "laws" in these worlds are easily broken. The worlds of computational objects need laws in the form of self-prescribed invariants, but the suspension of these laws might be creative acts. Bending the rules of abstract reality facilitates algorithm design, as we demonstrate through the example of search trees. (shrink)
This article presents an historical and conceptual overview on different approaches to logical abstraction. Two main trends concerning abstraction in the history of logic are highlighted, starting from the logical notions of concept and function. This analysis strictly relates to the philosophical discussion on the nature of abstract objects. I develop this issue further with respect to the procedure of abstraction involved by (typed) λ-systems, focusing on the crucial change about meaning and predicability. In particular, the analysis (...) of the nature of logical types in the context of Constructive Type Theory allows elucidation of the role of the previously introduced notions. Finally, the connection to the analysis of abstraction in computer science is drawn, and the methodological contribution provided by the notion of information is considered, showing its conceptual and technical relevance. Future research shall focus on the notion of information in distributed systems, analysing the paradigm of information hiding in dependent type theories. (shrink)
Berkeley and Hume object to Locke's account of abstraction. Abstraction is separating in the mind what cannot be separated in reality. Their objection is that if a is inseparable in reality from b, then the idea of a is inseparable from the idea of b. The former inseparability is the reason for the latter. In most interpretations, however, commentators leave the former unexplained in explaining the latter. This article assumes that Berkeley and Hume present a unified front against (...) Locke. Hume supplements Berkeley's argument just where there are gaps. In particular, Hume makes explicit something Berkeley leaves implicit: The argument against Locke depends on the principle that things are inseparable if and only if they are identical. Abstraction is thinking of one of an inseparable pair while not thinking of the other. But doing so entails thinking of something while not thinking of it. This is the fundamental objection. (shrink)
Peano's axiomatizations of geometry are abstract and non-intuitive in character, whereas Peano stresses his appeal to concrete spatial intuition in the choice of the axioms. This poses the problem of understanding the interrelationship between abstraction and intuition in his geometrical works. In this article I argue that axiomatization is, for Peano, a methodology to restructure geometry and isolate its organizing principles. The restructuring produces a more abstract presentation of geometry, which does not contradict its intuitive content but only puts (...) it into a particular form. (shrink)
It is proved in this paper that the positive abstraction scheme is consistent with extensionality only if one drops equality out of the language. The theory obtained is then compared with GPK, a wellknown set theory based on an extended positive comprehension scheme.
One of the main problems plaguing neo-logicism is the Bad Company challenge: the need for a well-motivated account of which abstraction principles provide legitimate definitions of mathematical concepts. In this article a solution to the Bad Company challenge is provided, based on the idea that definitions ought to be conservative. Although the standard formulation of conservativeness is not sufficient for acceptability, since there are conservative but pairwise incompatible abstraction principles, a stronger conservativeness condition is sufficient: that the class (...) of acceptable abstraction principles be strictly logically symmetrically class conservative . The article concludes with an examination of which classes of abstraction principles meet this criteria. (shrink)
Several modern accounts of explanation acknowledge the importance of abstraction and idealization for our explanatory practice. However, once we allow a role for abstraction, questions remain. I ask whether the relation between explanations at different theoretical levels should be thought of wholly in terms of abstraction, and argue that changes of variable between theories can lead to novel explanations that are not merely abstractions of some more detailed picture. I use the example of phase transitions as described (...) by statistical mechanics and thermodynamics to illustrate this, and to demonstrate some details of the relationship between abstraction, idealization, and novel explanation. (shrink)
In empirical science, hypostatic abstraction posits an entity defined by its assumed physical relation to a known phenomenon. If the assumed relation is real, the posited entity is physically real and is not an ens rationis. The posited entity, being identified indirectly, by its relation to something else, may be the agreed-upon subject of mutually incommensurable theories, and this is a key to understanding the history of science. Natural kinds may be introduced by hypostatic abstraction, and this explains (...) why, contrary to received doctrine, concepts of natural kinds can never be vague in the sense of being fuzzy, though they can be vague in the sense of lacking specificity. Terms defined by hypostatic abstraction are rigid designators in Kripke's sense, but show how rigid designation is consistent with the Fregean theory of reference. (shrink)
Objects which philosophers have traditionally categorized as abstract are standardly referred to by complex noun phrases of certain canonical forms, such as ‘the set of Fs’, ‘the number of Fs’, ‘the proposition that P’, and ‘the property of being F’. It is no accident that such noun phrases are well-suited to appear in ‘Fregean’ identity-criteria, or ‘abstraction’ principles, for which Frege’s criterion of identity for cardinal numbers provides the paradigm. Notoriously, such principlesare apt to create paradoxes, and the most (...) intuitively plausible ‘Fregean’ identity-criterion for properties is afflicted by this problem. In this case, it may be possible to overcome the difficulty by modifying the criterion in a way which requires an independent account of the existence-conditions of properties, but it appears that such a strategy demands acceptance of the doctrine of immanent realism—the view that a property exists only if it is exemplified by some object. (shrink)
On the one hand, the absence of contraction is a safeguard against the logical (property theoretic) paradoxes; but on the other hand, it also disables inductive and recursive definitions, in its most basic form the definition of the series of natural numbers, for instance. The reason for this is simply that the effectiveness of a recursion clause depends on its being available after application, something that is usually assured by contraction. This paper presents a way of overcoming this problem within (...) the framework of a logic based on inclusion and unrestricted abstraction, without any form of extensionality. (shrink)
Dorion Cairns correctly interprets the preconstituted stratum of Edmund Husserl’s Fifth Cartesian Meditation to be the primordial ego and not the social world, as was thought by Alfred Schutz, who considered Husserl to be insufficiently attentive to the social world’s hold upon us. Following Cairns’s interpretation, which involves recovering and reconstructing strata that may never exist independently, one better understands how the transfer of sense animate organism involves automatic association, or somatic apprehension. This sense-transfer extends to any animate organism, not (...) just humans, and draws on extensive unreflected-upon similarities despite the distinctive fact that the other’s body is never given to oneself as is one’s own. Following Cairns’s interpretation, one can also understand the second epoché as an imaginative, reconstructive abstraction rather than as an example of failed ascesis. Consequently, Husserl appears as less intellectualized in his approach to empathy than often thought to be and more confident in the phenomenologist’s capacity to imagine and attend selectively to experience. (shrink)
In this paper I argue that the idea ‘becoming-woman’ is an attempt to transform embodied experience but, because it is unable to concern itself with mechanisms, structures and processes of sexual differentiation, fails in this task. In the first section I elaborate the relationship between becoming-woman and Deleuze's ‘superior’ or ‘transcendental’ empiricism and suggest that problems can be traced back to an underlying Humean empiricism. Along with Hume, Deleuze, it seems, presumes a bundle model of the object which dissolves things (...) into episodic objects of perception and leaves the subject unable to distinguish between fanciful objects, erroneous perception and any other thing. The empiricist ontology thus has consequences for epistemology and leaves us unable to question the conservative tendencies of common sense. As an alternative to transcendental empiricism, the second section considers how transcendental realism, with its ontological commitment to the mind-independent character of things, may provide a more fruitful and productive line of enquiry. Given that there is such a choice, in the third section I speculate as to the specific desires that drive such philosophical abstraction; abstraction which culminates in the non sex-specific figure becoming-woman whilst disguising the mind-independent character of the mechanisms, structures and objects that affect the subject. So I conclude that, despite all appearances of radicalism, the philosophical model ‘becoming-woman’ – aligned as it is with schizo-processes and the philosophical loss of mind-independent things – is more of the same and sexual difference remains a hidden term. Due to this, I believe that feminists should view it with suspicion. (shrink)
The paper is a study of the logic of existence, negation, and order in the Neoplatonic tradition. The central idea is that Neoplatonists assume a logic in which the existence predicate is a comparative adjective and in which monadic predicates function as scalar adjectives that nest the background order. Various scalar predicate negations are then identifiable with various Neoplatonic negations, including a privative negation appropriate for the lower orders of reality and a hyper-negation appropriate for the higher. Reversion to the (...) One can then be explained as the logical inference of hyper-negations from mundane knowledge. Part I develops the relevant linguistic and logical theory, and Part II defends Wolfson and the scalar interpretation against the more traditional Aristotelian understanding of Whittaker and others of reversion as intensional abstraction. (shrink)
In 1870 Jordan proved that the composition factors of two composition series of a group are the same. Almost 20 years later Hölder (1889) was able to extend this result by showing that the factor groups, which are quotient groups corresponding to the composition factors, are isomorphic. This result, nowadays called the Jordan-Hölder Theorem, is one of the fundamental theorems in the theory of groups. The fact that Jordan, who was working in the framework of substitution groups, was able to (...) prove only a part of this theorem is often used to emphasize the importance and even the necessity of the abstract conception of groups, which was employed by Hölder. However, as a little-known paper from 1873 reveals, Jordan had all the necessary ingredients to prove the Jordan-Hölder Theorem at his disposal (namely, composition series, quotient groups, and isomorphisms), and he also noted a connection between composition factors and corresponding quotient groups. Thus, I argue that the answer to the question posed in the title is “Yes.” It was not the lack of the abstract notion of groups which prevented Jordan from proving the Jordan-Hölder Theorem, but the fact that he did not ask the right research question that would have led him to this result. In addition, I suggest some reasons why this has been overlooked in the historiography of algebra, and I argue that, by hiding computational and cognitive complexities, abstraction has important pragmatic advantages. (shrink)
I present a general theory of abstraction operators which treats them as variable-binding term- forming operators, and provides a reasonably uniform treatment for definite descriptions, set abstracts, natural number abstraction, and real number abstraction. This minimizing, extensional and relational theory reveals a striking similarity between definite descriptions and set abstracts, and provides a clear rationale for the claim that there is a logic of sets (which is ontologically non- committal). The theory also treats both natural and real (...) numbers as answering to a two-fold process of abstraction. The first step, of conceptual abstraction, yields the object occupying a particular position within an ordering of a certain kind. The second step, of objectual abstraction, yields the number sui generis, as the position itself within any ordering of the kind in question. (shrink)
We develop a functional abstraction principle for the type-free algorithmic logic introduced in our earlier work. Our approach is based on the standard combinators but is supplemented by the novel use of evaluation trees. Then we show that the abstraction principle leads to a Curry fixed point, a statement C that asserts C ⇒ A where A is any given statement. When A is false, such a C yields a paradoxical situation. As discussed in our earlier work, this (...) situation leaves one no choice but to restrict the use of a certain class of implicational rules including modus ponens. (shrink)
With the aid of a non-standard (but still first-order) cardinality quantifier and an extra-logical operator representing numerical abstraction, this paper presents a formalization of first-order arithmetic, in which numbers are abstracta of the equinumerosity relation, their properties derived from those of the cardinality quantifier and the abstraction operator.
Al-Fârâbî’s thought on intellect was known to the Latin West through the translation of his Letter on the Intellect, through the Long Commentary on the De Anima by Averroes and through some other works. Al-Fârâbî identified the active power of intellect in Aristotle’s De Anima 3.5 as the unique and separately existing Agent Intellect, but the role of the Agent Intellect in forming intelligibles in act in the human soul is by no means unequivocally clear. Further, the apprehension of intelligibles (...) by human beings and the intellectual development of the soul, oftentimes described as an activity of abstracting (intaza`a), seems to be a genuineabstraction from experience, yet it somehow involves the emanative power of the Agent Intellect. This paper works to provide a coherent explanation of the natureof abstraction and the role of Agent Intellect in that activity. (shrink)
In theories that idealize the object of study, falsity is inserted somehow. However, the actual propositions by which the idealization takes place need not be false at all. An example from physics illustrates that the Ideal Gas Law and Boyle's Law are respective idealizations of the van der Waals Law. The idealizational procedures involved in reasoning from the latter to the former can be repeated at a higher level of abstraction than that of the laws as we know these (...) from physics textbooks. Thus, idealization and abstraction can be seen as relatively independent methods of reasoning, the one to be carried out with or without the other at the same time. This underlines Uskali Mäki's taxonomy, which shows that horizontal isolations in economic reasoning form a procedure completely different from vertical isolations. In contrast, however, to his taxonomy, I propose to define idealization as horizontal isolation. The lessons for the policy relevance of science ? and of economics in particular ? are that the use of ideal models does not necessarily imply a total lack of their external validity. Further, I show that abstraction in theorizing, under certain conditions, may increase the policy relevance of theories, rather than that it is decreased. Abstract theories tend to count the actual social world ? where policymakers try to intervene ? among their models more easily than concrete theories. Finally, this paper also deals with one very problematic aspect of the common use of clauses in order to hedge economics hypotheses. Many idealizational clauses have a propensity for imprecise reference, due to which it is impossible to judge the external validity of economic models. I shall indicate how this problem relates to issues of interdisciplinarity in social science. (shrink)
λ-calculi are of interest to logicians and computer scientists but have largely escaped philosophical commentary, perhaps because they appear narrowly technical or uncontroversial or both. I argue that even within logic λ-expressions need to be understood correctly, as functors signifying functions in intension within a categorical or typed language. λ-expressions are not names but pure viable binders generating functors, and as such they are of use in giving explicit definitions. But λ is applicable outside logic and computer science, anywhere where (...) the notions of complex whole, substitution, abstraction and structure make sense. To illustrate this, two domains are considered. One is somewhat frivolous: the study of flags; the other is very serious: manufacturing engineering. In each case we can employ λ-abstraction to describe substitutions within a structure, and in the latter case there is even a practical need for such a notation. (shrink)
An informal theory is set forth of relations between abstract entities, includingcolors, physical quantities, times, andplaces in space, and the concrete things thathave them, or areat orin them, based on the assumption that there are close analogies between these relations and relations between abstractsets and the concrete things that aremembers of them. It is suggested that even standard scientific usage of these abstractions presupposes principles that are analogous to postulates of abstraction, identity, and other fundamental principles of set theory. (...) Also discussed is the significance of important disanalogies between sets and physical abstractions, including especiallymodal andtemporal aspects of physical abstractions, which is related to the problem of the characterizingconstancy, of colors, physical attributes, and locations in space. (shrink)
This paper analyzes both philosophical and practical assumptions underlying claims for the dual nature of software, including software as a machine made of text, and software as a concrete abstraction. A related view of computer science as a branch of pure mathematics is analyzed through a comparative examination of the nature of abstraction in mathematics and computer science. The relationship between the concrete and the abstract in computer programs is then described by exploring a taxonomy of approaches borrowed (...) from philosophy of mind. (shrink)
This paper presents a new algorithm to find an appropriate similarityunder which we apply legal rules analogically. Since there may exist a lotof similarities between the premises of rule and a case in inquiry, we haveto select an appropriate similarity that is relevant to both thelegal rule and a top goal of our legal reasoning. For this purpose, a newcriterion to distinguish the appropriate similarities from the others isproposed and tested. The criterion is based on Goal-DependentAbstraction (GDA) to select a (...) similarity such that an abstraction basedon the similarity never loses the necessary information to prove the ground (purpose of legislation) of the legal rule. In order to cope withour huge space of similarities, our GDA algorithm uses some constraintsto prune useless similarities. (shrink)
Given the complexity and generalizability of motor skills, it is difficult to account for learning in this area without incorporating the concept of unconscious abstraction. A model based solely on association does not seem to account for data in this domain; specifically, instances that require learners to execute a practiced motor skill in a novel situation.
Cases where analogy has played a significant role in the formation of a new scientific concept are well-documented. Yet, how is it that genuinely new representations can be constructed from existing representations? It is argued that the process of âgeneric modelingâ enables abstraction of features common to both the domain of the source of the analogy and of the target phenomena. The analysis focuses on James Clerk Maxwell's construction of the electromagnetic field concept. The mathematical representation Maxwell constructed turned (...) out to be a system of abstract laws that when applied to electromagnetic systems yield laws of a dynamical system that will not map back onto the mechanicals domains used in their construction. (shrink)
Economics has been persistently criticized for its heavy reliance on unrealistic assumptions. Some people reply to this criticism by saying that the unrealistic assumptions of economics result from abstraction from unimportant details, and abstraction is necessary for knowledge of a complex real world. So, far from unrealistic assumptions detracting from the epistemic worth of economics, such assumptions are essential for economic knowledge. I call this line of argument ?the Abstractionist Defense?. After clarifying abstraction, unrealistic assumptions and kindred (...) notions, I show that the Abstractionist Defense does not successfully rebut the position of those who criticize economics for its unrealistic assumptions. (shrink)
In the philosophy of science, abstraction has usually been analyzed in terms of the interface between our experience and the design of our concepts. The often implicit assumption here is that such interface has a definite identifiable and universalizable structure, determining the epistemic correctness of any abstraction. Our claim is that, on the contrary, the epistemic grounding of abstraction should not be reduced to the structural norms of such interface but is also related to the constraints on (...) the cognitive processes of specific abstractions. This suggests that we should understand abstraction as embodied in different kinds of abstraction practices. (shrink)
This paper addresses Klima’s charge of inconsistancy against John Buridan in a book recently published on the subject. Klima argues that Buridan’s theoryof abstraction commits him to the aspectuality of substantial concepts. However, his semantics of absolute terms and concepts prevents him from accepting anyaspectuality of substantial concepts. In light of this problem, the paper gives a detailed reconstruction of Buridan’s account of abstraction, beginning with sensoryperception and singular cognition and ending with the formation of substantial concepts that (...) have a universal signification. Then, from this reconstruction, someBuridanian responses are given to Klima’s critique, which explain at least why Buridan did not see the problem himself. Finally, the conclusion comes down in favor of Klima and, in light of the discussion, highlights some fundamental problems with the nominalist project. (shrink)
We investigate the class of strongly distributive pregroups, a common abstraction of MV-algebras and Abelian l-groups which was introduced by E.Casari. The main result of the paper is a representation theorem which yields both Chang's representation of MV-algebras and Clifford's representation of Abelian l-groups as immediate corollaries.
Translations from Lambda calculi into combinatory logics can be used to avoid some implementational problems of the former systems. However, this scheme can only be efficient if the translation produces short output with a small number of combinators, in order to reduce the time and transient storage space spent during reduction of combinatory terms. In this paper we present a combinatory system and an abstraction algorithm, based on the original bracket abstraction operator of Schonfinkel [9]. The algorithm introduces (...) at most one combinator for each abstraction in the initial Lambda term. This avoids explosive term growth during successive abstractions and makes the system suitable for practical applications. We prove the correctness of the algorithm and establish some relations between the combinatory system and the Lambda calculus. (shrink)
A bracket abstraction algorithm is a means of translating λ-terms into combinators. Broda and Damas, in [1], introduce a new, rather natural set of combinators and a new form of bracket abstraction which introduces at most one combinator for each λ-abstraction. This leads to particularly compact combinatory terms. A disadvantage of their abstraction process is that it includes the whole Schonfinkel [4] algorithm plus two mappings which convert the Schonfinkel abstract into the new abstract. This paper (...) shows how the new abstraction can be done more directly, in fact, using only 2n - 1 algorithm steps if there are n occurrences of the variable to be abstracted in the term. Some properties of the Broda-Damas combinators are also considered. (shrink)
From the advent of general purpose, Turing-complete machines, the relation between operators, programmers and users with computers can be observed as interconnected informational organisms (inforgs), henceforth analysed with the method of levels of abstraction (LoAs), risen within the philosophy of information (PI). In this paper, the epistemological levellism proposed by L. Floridi in the PI to deal with LoAs will be formalised in constructive terms using category theory, so that information itself is treated as structure-preserving functions instead of Cartesian (...) products. The milestones in the history of modern computing are then analysed through constructive levellism to show how the growth of system complexity lead to more and more information hiding. (shrink)
A debated topic in Avicennan psychology is whether for Avicenna abstraction is a metaphor for emanation or to be taken literally. This issue stems from the deeper philosophical question of whether humans acquire intelligibles externally from an emanation by the Active Intellect, which is a separate substance, or internally from an inherently human cognitive process, which prepares us for an emanation from the Active Intellect. I argue that the tension between thesedoctrines is only apparent. In his logical works Avicenna (...) limns an account where through the internal human process of abstraction accidents accruing to an essence existing in matter are extracted, thus preparing the essence for new accidents emanating externally from the Active Intellect, which make the essence something conceptualized in the intellect. This study, then, outlines the epistemological and metaphysical framework presented in the logical works that underpins Avicenna’s theory of abstraction presented in his psychological works. (shrink)
A debated topic in Avicennan psychology is whether for Avicenna abstraction is a metaphor for emanation or to be taken literally. This issue stems from the deeper philosophical question of whether humans acquire intelligibles externally from an emanation by the Active Intellect, which is a separate substance, or internally from an inherently human cognitive process, which prepares us for an emanation from the Active Intellect. I argue that the tension between thesedoctrines is only apparent. In his logical works Avicenna (...) limns an account where through the internal human process of abstraction accidents accruing to an essence existing in matter are extracted, thus preparing the essence for new accidents emanating externally from the Active Intellect, which make the essence something conceptualized in the intellect. This study, then, outlines the epistemological and metaphysical framework presented in the logical works that underpins Avicenna’s theory of abstraction presented in his psychological works. (shrink)
Machine generated contents note: -- List of figures -- Acknowledgements -- Introduction -- Convention -- Seeing and the Experience of Pictures -- A Theory of Depiction -- Resemblance -- Transparency and Resemblance -- Realism -- Varieties of Realism -- Abstraction -- Notes -- Index.
In this chapter we will investigate the nature of abstraction in detail, its entwinement with logical thinking, and the general role it plays for the mind. We find that non-logical capabilities are not only important for input processing, but also for output processing. Human beings jointly use analytic and embodied capacities for thinking and acting, where analytic thinking mirrors reflection and logic, and where abstraction is the form in which embodied thinking is revealed to us. We will follow (...) the philosophical analyses of Heidegger and Polanyi to elaborate the fundamental difference between abstraction and logics and how they come together in the mind. If computational approaches to mind are to be successful, they must be able to recognize meaningful and salient elements of a context and engage in abstraction. Computational minds must be able to imagine and volitionally blend abstractions as a way of recognizing gestalt contexts. And it must be able to discern the validity of these blendings in ways that, in humans, arise from a sensus communis. (shrink)
It is illegitimate to read any ontology about "race" off of biological theory or data. Indeed, the technical meaning of "genetic variation" is fluid, and there is no single theoretical agreed-upon criterion for defining and distinguishing populations (or groups or clusters) given a particular set of genetic variation data. Thus, by analyzing three formal senses of "genetic variation"—diversity, differentiation, and heterozygosity—we argue that the use of biological theory for making epistemic claims about "race" can only seem plausible when it relies (...) on the user’s own assumptions about race; the move from biological measures to claims about “race” inevitably amounts to a pernicious reification. We also excavate assumptions in the history of the technical discourse over the concept of "race" (e.g., Livingstone's and Dobzhansky's 1962 exchange, Edwards' 2003 response to Lewontin 1972, as well as contemporary discussions of cladistic "race", and "races" as clusters). We show that claims about the existence (or non-existence) of "race" are underdetermined by biological facts, methods, and theories. Biological theory does not force the concept of "race" upon us; our social discourse, social ontology, and social expectations do. We become prisoners of our abstractions at our own hands, and at our own expense. (shrink)
This paper looks at the history of the problem of individuation from Plato to Whitehead. Part I takes as its point of departure Reiner Wiehl’s interpretation of the different meanings of “abstract” in the metaphysics of Alfred North Whitehead and arrives at a corresponding taxonomy of different ways things can be called concrete. Part II compares the way philosophers in different periods understand the relation between thought and intuition. The view mostly associated with ancient philosophy is that thought and sense-perception (...) target different kinds of objects. The view mostly associated with modern philosophy (although it was introduced by the Stoics) is that thought and sense-perception are different ways of targeting the same objects. These differences have specific consequences for theories of individuation, which are assessed historically in Part III and then applied to Whitehead’s difficult texts in part IV. (shrink)
In this paper I examine a new variant of the well-known idea that the self is an abstract object. I propose a simple model of the self as a property of temporal slices of a body's history. I argue that this model, when combined with even a modest realism with regard to properties, implies that the self has many of the chief features traditionally attributed to selves. I conclude that this model allows one to reconcile the full reality of the (...) self with even the most deflationary materialistic theories of consciousness. (shrink)
With a focus on the question of visuality in Heidegger's sustained involvement with Daoist and Zen thought, this paper discusses the interchange between Heidegger and Hisamatsu at a 1958 colloquium. In light of the key concerns – visuality, art, and the empty origin of manifestation – it interrogates three texts,The Origin of the Work of Art,Parmenides, andArt and Space,concerning visuality, the play of the glance, writing, space and place, and the Graeco-Asian though of phainesthai. In conclusion, it addresses the opening (...) for a philosophical consideration of abstract painting that these analyses provide. (shrink)
This book offers an historically-informed critical assessment of Dummett's account of abstract objects, examining in detail some of the Fregean presuppositions whilst also engaging with recent work on the problem of abstract entities.
The partial structures program of da Costa, French and others offers a unified framework within which to handle a wide range of issues central to contemporary philosophy of science. I argue that the program is inadequately equipped to account for simple cases where idealizations are used to construct abstract, mathematical models of physical systems. These problems show that da Costa and French have not overcome the objections raised by Cartwright and Suárez to using model-theoretic techniques in the philosophy of science. (...) However, my concerns arise independently of the more controversial assumptions that Cartwright and Suárez have employed. (shrink)
There is a growing consensus that the mental lexicon contains both abstract and word-specific acoustic information. To investigate their relative importance for word recognition, we tested to what extent perceptual learning is word specific or generalizable to other words. In an exposure phase, participants were divided into two groups; each group was semantically biased to interpret an ambiguous Mandarin tone contour as either tone1 or tone2. In a subsequent test phase, the perception of ambiguous contours was dependent on the exposure (...) phase: Participants who heard ambiguous contours as tone1 during exposure were more likely to perceive ambiguous contours as tone1 than participants who heard ambiguous contours as tone2 during exposure. This learning effect was only slightly larger for previously encountered than for not previously encountered words. The results speak for an architecture with prelexical analysis of phonological categories to achieve both lexical access and episodic storage of exemplars. (shrink)
A quick look into the index of Brentano’s Psychology from an Empirical Standpoint reveals that all references to “abstract terms” occur only in the appendix (taken from Brentano’s “Nachlass” essays). What should we make of this? Was it the case that the inquiry into abstract, as well as non-existent, objects came as an afterthought to Brentano? Or was he all too aware of the consequences of such investigations? Furthermore, was it largely the absence of such inquirythat prompted Husserl and his (...) early students in Göttingen, such as Daubert and Reinach, to develop a deep ontological commitment to entities he refers to as “abstract” or “ideal”? (shrink)
Animals detect and acquire resources through a sequence of shape changes. This process is tightly coupled to the sensory and mechanical ecology of the animal. Building physical models allow us to prescind from modeling these aspects of the environment, which may not yet be described or suitably abstracted. The significance of this hybrid of physical modeling and experimentation to the acquisition of scientific knowledge is discussed.
For conventional economics things have value only to the degree that they give pleasure to individual human beings. In response to continuing environmental deterioration several alternatives have been offered for valuing resources and allocating them between generations. Most of these approaches are highly abstract. The deterioration of the Earth and the mistreatment of its inhabitants will not be stemmed by abstractions. Neither will abstract ideas direct us to the best use of our resources. We need to foster personal relationships between (...) human beings and particular portions of the Earth. (shrink)
The phenomena of human consciousness and subjectivity are explored from the perspective of affect-logic, a comprehensive meta-theory of the interactions between emotion and cognition based mainly on cognitive and social psychology, psychopathology, neurobiology Piaget?s genetic epistemology, psychoanalysis, and evolutionary science. According to this theory, overt or covert affective-cognitive interactions are obligatorily present in all mental activity, seemingly ?neutral? thinking included. Emotions continually exert numerous so-called operator-effects, both linear and nonlinear, on attention, on memory and on comprehensive thought, or logic in (...) a broad sense. They deeply ?affect? also consciousness and subjectivity, as showed by the analysis of four crucially involved phenomena, namely (1) attention, (2) abstraction, (3) language, and (4) the prevailing affective state. The conclusion is that neither consciousness nor subjectiovity can be adequately understood without fully considering their emotional aspects. (shrink)
This English translation of the author's original work has been thoroughly revised, expanded and updated. The book covers logical systems known as type-free or self-referential . These traditionally arise from any discussion on logical and semantical paradoxes. This particular volume, however, is not concerned with paradoxes but with the investigation of type-free sytems to show that: (i) there are rich theories of self-application, involving both operations and truth which can serve as foundations for property theory and formal semantics; (ii) these (...) theories provide a new outlook on classical topics, such as inductive definitions and predicative mathematics; (iii) they are particularly promising with regard to applications. Research arising from paradoxes has moved progressively closer to the mainstream of mathematical logic and has become much more prominent in the last twenty years. A number of significant developments, techniques and results have been discovered. Academics, students and researchers will find that the book contains a thorough overview of all relevant research in this field. (shrink)
The goal is to sketch a nominalist approach to mathematics which just like neologicism employs abstraction principles, but unlike neologicism is not committed to the idea that mathematical objects exist and does not insist that abstraction principles establish the reference of abstract terms. It is well-known that neologicism runs into certain philosophical problems and faces the technical difficulty of finding appropriate acceptability criteria for abstraction principles. I will argue that a modal and iterative nominalist approach to (...) class='Hi'>abstraction principles circumvents those difficulties while still being able to put abstraction principles to a foundational use. (shrink)
The first amounts, roughly, to "It is necessarily the case that any President of the U.S. is a citizen of the U.S." But the second says, "the person who in fact is the President of the U.S, has the property of necessarily being a citizen of the U.S," Thus, while (2) is clearly true, it would be reasonable to consider (3) false.
On the ground of Kant’s reformulation of the principle of con- tradiction, a non-classical logic KC and its extension KC+ are constructed. In KC and KC+, \neg( \phi \wedge \neg\phi ), \phi \rightarrow (\neg\phi \rightarrow \phi), and \phi \vee \neg\phi are not valid due to specific changes in the meaning of connectives and quantifiers, although there is the explosion of derivable consequences from { \phi, ¬\phi } (the deduc- tion theorem lacking). KC and KC+ are interpreted as fragments of an S5-based first-order (...) modal logic M. The quantification in M is combined with a “subject abstraction” device, which excepts predicate letters from the scope of modal operators. Derivability is defined by an appropriate labelled tableau system rules. Informally, KC is mainly ontologically motivated (in contrast, for example, to Jaśkowski’s discussive logic), relativizing state of affairs with respect to conditions such as time. (shrink)
The purpose of this paper is to assess the prospects for a neo-logicist development of set theory based on a restriction of Frege's Basic Law V, which we call (RV): PQ[Ext(P) = Ext(Q) [(BAD(P) & BAD(Q)) x(Px Qx)]] BAD is taken as a primitive property of properties. We explore the features it must have for (RV) to sanction the various strong axioms of Zermelo–Fraenkel set theory. The primary interpretation is where ‘BAD’ is Dummett's ‘indefinitely extensible’. 1 Background: what and why? (...) 2 Framework 3 GOOD candidates, indefinite extensibility 4 The framework of (RV) alone, or almost alone 5 The axioms 6 Brief closing. (shrink)
Stewart Shapiro and Alan Weir have argued that a crucial part of the demonstration of Frege's Theorem (specifically, that Hume's Principle implies that there are infinitely many objects) fails if the Neo-logicist cannot assume the existence of the empty property, i.e., is restricted to so-called Aristotelian Logic. Nevertheless, even in the context of Aristotelian Logic, Hume's Principle implies much of the content of Peano Arithmetic. In addition, their results do not constitute an objection to Neo-logicism so much as a clarification (...) regarding the view of logic that the Neo-logicist must take. (shrink)
We correct a misunderstanding by Hale and Wright of an objection we raised in 'Hale on Caesar' to their abstractionist programme for rehabilitating logicism in the foundations of mathematics.
A co-authored article with Roy T. Cook forthcoming in a special edition on the Caesar Problem of the journal Dialectica. We argue against the appeal to equivalence classes in resolving the Caesar Problem.
