This paper analyses the abstractionist account of quantity championed by Leibniz, especially in the 1680s. Leibniz introduced the notion of quantity in an indirect way, via an abstraction principle. In the first part of the paper, I identify the context in which this approach arose in light of Leibniz's criticism of his earlier dream of an ‘alphabet of human thought’. Recognising the impossibility of such a project led him to realise that, when dealing with terms referring to abstract objects, (...) we should always consider them within the true sentences in which they occur. In the second part, I describe this approach in detail. This allows us to look at some key concepts of Leibniz's theory of quantity. In particular, I raise the problem of the relationship between the two sides of the abstraction principle: how should we think of the relation between the claim that a and b are equal, and the claim that the quantity of a is identical to the quantity of b? I argue that we can find a positive answer to this problem in Leibniz. (shrink)

We may define words or concepts, and we may also, as Aristotle and others have thought, define the things for which words stand and of which concepts are concepts. Definitions of words or concepts may be explicit or implicit, and may seek to report preexisting synonymies, as Quine put it, but they may instead be wholly or partly stipulative. Definition by abstraction, of which Hume’s principle is a much discussed example, seek to define a term-forming operator, such as (...) the number operator, by fixing the truth-conditions of identity-statements featuring terms formed by means of that operator. Such definitions are a species of implicit definition. They are typically at least partly stipulative. Definitions of things, or real definitions, are, by contrast, typically conceived as true or false statements about the nature or essence of their definienda, and so not stipulative. There thus appears to be an obvious and head-on clash between taking Hume’s principle as an implicit and at least partly stipultative definition of the number operator and taking it as a real definition, stating the nature or essence of cardinal numbers. This paper argues that this apparent tension can be resolved, and that resolving it sheds light on part of the epistemology or essence and necessity, showing how some of our knowledge of essence and necessity can be a priori. (shrink)

I offer in this paper a contextual analysis of Frege's Grundlagen, section 64. It is surprising that with so much ink spilled on that section, the sources of Frege's discussion of definitions by abstraction have remained elusive. I hope to have filled this gap by providing textual evidence coming from, among other sources, Grassmann, Schlömilch, and the tradition of textbooks in geometry for secondary schools . In addition, I put Frege's considerations in the context of a widespread debate (...) in Germany on ‘directions’ as a central notion in the theory of parallels. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Obesity as Disease:Definition by DesperationJeremy ShermakI hated removing my shirt. Each visit to my doctor’s office, following a blood pressure and temperature check, the nurse would instruct me to take off my shirt so the doctor could examine me further. She would then leave the room. I remained perched atop the exam table, now half exposed, and a mirror on the wall would not leave me alone. In the (...) reflection, I saw my oversized breasts and “fat roll” oozing out from my pants, hiding my belt. I tried to straighten my back—breathe in, no wait, breathe out—to resculpt the appearance, but it did nothing. I hated myself. Soon, a gentle knock at the door would interrupt the loathing and I’d shake hands with my doctor.I have been obese for the majority of my life—from childhood until I turned 34, when shortly thereafter, I had gastric bypass surgery. Since that time, I have lost 155 pounds and now write this at a comfortable, healthy 180 pounds. That said, my mind remains “obese”. My thinking remains shrouded deep within the obese personality that I embodied for all of those years. I still automatically walk to the big–and–tall section of clothing stores; I don’t recognize the 34–waist pants when they come [End Page 114] out of the dryer; I’m not used to seeing my ribs; and I’m still terrified of that exam room mirror.Upon hearing the news that the American Medical Association (AMA) now classifies obesity as a disease, my initial reaction was that of relief. My recent success was due in large part to the discovery and treatment of hindering endocrinological issues. My history—like repeatedly slamming head–long into a brick wall—suggests that these medically verified roadblocks have existed for some time. Despite my efforts, I didn’t stand a chance.If anyone were a “poster child” for obesity as a disease, it would be me. The lifelong sparring with my weight had its ups and downs; I worked very hard to maintain or lose weight with mixed results. I lost 85 pounds my freshman year of college by running and fervently watching my portions. I kept that weight off for a good five years before the summer of 2003, when I started gaining weight very quickly. I gained 25 pounds in each month of June and July that year, despite maintaining my usual eating habits and running schedule. I soon lost faith. I felt as though I was up against forces that I could not control and I began to forget about losing weight. I didn’t binge, but I didn’t eat well. By 2008, my weight had reached 375 pounds—an all–time personal high.Following a bad relationship and seeking rejuvenation, I embarked on a diet that summer like I’ve never tried before. I cooked everything naturally—from regular dishes like stuffed peppers all the way to condiments like ketchup. It was a militant effort on my part. I lost 50 pounds that summer and felt that I was well on my way back to the body that once felt so comfortable. It was at that point where I hit a wall. I could not get past 321. I can still see that number staring back at me on the scale. It was frustrating. I felt I had done all I could. That’s when I visited a doctor who ran some tests that revealed problems with my hormones and metabolic systems.As it turned out, I had a tumor on my pituitary gland. This may have been the issue all along. That led to low testosterone, which had a really negative impact on my weight loss efforts. After thorough consideration, I decided to have gastric bypass surgery to “reset” my hormones and get a “boost” in my weight loss efforts. It worked.In my particular circumstance, I could view obesity as a disease because its causes, for the most part, were not a result of conscious lifestyle choices. I had no control over the pituitary tumor. However, I know that I could have done more to limit my... (shrink)

In this article, I try to shed new light on Frege’s envisaged definitional introduction of real and complex numbers in _Die Grundlagen der Arithmetik_ (1884) and the status of cross-sortal identity claims with side glances at _Grundgesetze der Arithmetik_ (vol. I 1893, vol. II 1903). As far as I can see, this topic has not yet been discussed in the context of _Grundlagen_. I show why Frege’s strategy in the case of the projected definitions of real and complex numbers (...) in _Grundlagen_ is modelled on his definitional introduction of cardinal numbers in two steps, tentatively via a contextual definition and finally and definitively via an explicit definition. I argue that the strategy leaves a few important questions open, in particular one relating to the status of the envisioned abstraction principles for the real and complex numbers and another concerning the proper handling of cross-sortal identity claims. (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)

"Philosophical discussion of the notion of existence, or being, has centered on two main problems which have not always been very clearly distinguished. First, there is the problem of what we are to say about the existence of fictitious objects, such as centaurs, dragons, and Pegasus; second, there is the problem of what we are t o say about the existence of abstract objects, such as qualities, relations, and numbers. Both problems have tempted philosophers to say that there are inferior (...) sorts of existence a s well as the ordinary straightforward sort, and they therefore often suggest that we use the word "being" to cover both kinds but restrict "existence" to "being" of the common, non-fictitious, non-abstract sort. (Sometimes the term "reality" is proposed for "existence" or for "being.") T he problems of fiction and abstraction are different, how ever, for there are both real and fictitious abstractions. For example, the integer between two and four is real, but the integer between two and three is fictitious. On the other hand, there are both concrete and abstract fictions; for example, the winged horse of Bellerophon and the integer between two and three. Accordingly, philosophers have often dealt with the two problems in quite different ways and perhaps ought to do so. While these are the two main problems, there are others, f or example, that of what we are to say of the being of objects which have not yet begun, or have now ceased, to exist. The history of this subject, moreover, has been tangled with theological issues, to which it will be necessary to refer at certain points.". (shrink)

Paolo Mancosu provides an original investigation of historical and systematic aspects of the notions of abstraction and infinity and their interaction. A familiar way of introducing concepts in mathematics rests on so-called definitions by abstraction. An example of this is Hume's Principle, which introduces the concept of number by stating that two concepts have the same number if and only if the objects falling under each one of them can be put in one-one correspondence. This principle is (...) at the core of neo-logicism. In the first two chapters of the book, Mancosu provides a historical analysis of the mathematical uses and foundational discussion of definitions by abstraction up to Frege, Peano, and Russell. Chapter one shows that abstraction principles were quite widespread in the mathematical practice that preceded Frege's discussion of them and the second chapter provides the first contextual analysis of Frege's discussion of abstraction principles in section 64 of the Grundlagen. In the second part of the book, Mancosu discusses a novel approach to measuring the size of infinite sets known as the theory of numerosities and shows how this new development leads to deep mathematical, historical, and philosophical problems. The final chapter of the book explore how this theory of numerosities can be exploited to provide surprisingly novel perspectives on neo-logicism. (shrink)

Play behaviour is notorious for constituting a much debated, yet little clarified field of research. In this article, attempts are made to reach conclusions on the relation between human play and the play of other animals (especially cat play), as well as on the very character of play. The concept of Umwelt is reviewed, as are definitions of animal play, categorization of animal play and the role of meta-communication in playful behaviour. For some, play is a symbol of everythingthat (...) is good. The author of the current article does not deny that social morality may have originated from play behaviour, but stresses the existence of cruelty play, which leads to additional assumptions. Another notion that is treated in some detail is perceptual play, which proves to demonstrate complex semiotic play that is related first of all to signification. At the end of the article an alternative categorization of animal play is suggested, in which the fundamental role of mind games is emphasized. Throughout the text, examples of play behaviour are offered by the two domestic cats Muki and Maluca. (shrink)

The neo-Fregean program in the philosophy of mathematics seeks a foundation for a substantial part of mathematics in abstraction principles—for example, Hume’s Principle: The number of Fs D the number of Gs iff the Fs and Gs correspond one-one—which can be regarded as implicitly definitional of fundamental mathematical concepts—for example, cardinal number. This paper considers what kind of abstraction principle might serve as the basis for a neo- Fregean set theory. Following a brief review of the main difficulties (...) confronting the most widely discussed proposal to date—replacing Frege’s inconsistent Basic Law V by Boolos’s New V which restricts concepts whose extensions obey the principle of extensionality to those which are small in the sense of being smaller than the universe—the paper canvasses an alternative way of implementing the limitation of size idea and explores the kind of restrictions which would be required for it to avoid collapse. (shrink)

Magical ersatzism is the view that possible worlds are primitive abstract entities. In On the Plurality of Worlds, David Lewis presented what appeared to many to be a devastating argument against magical ersatzism. In this paper, I show that Lewis’ central argument does not succeed. Magical ersatzism remains a viable theory of possible worlds.

We consider a sequence of successively more restrictive definitions of abstraction for causal models, starting with a notion introduced by Rubenstein et al. (2017) called exact transformation that applies to probabilistic causal models, moving to a notion of uniform transformation that applies to deterministic causal models and does not allow differences to be hidden by the "right" choice of distribution, and then to abstraction, where the interventions of interest are determined by the map from low-level states to (...) high-level states, and strong abstraction, which takes more seriously all potential interventions in a model, not just the allowed interventions. We show that procedures for combining micro-variables into macro-variables are instances of our notion of strong abstraction, as are all the examples considered by Rubenstein et al. (shrink)

Abstract A lot of science has passed under the bridge since the classic definition of evolution as a change in gene frequencies in a population became common. Much knowledge has accumulated since then about evolution, heredity, ecology, development, phenotypic plasticity, niche construction and genetic drift. Building on Van Valen’s description of evolution as “the control of development by ecology,” it is suggested that the classic definition be replaced by a updated ‘eco‐evo-evo’ definition of evolution by natural selection which acknowledges this (...) accumulation. (shrink)

ABSTRACT This essay considers definitions of racism which emphasise its behavioural, motivational, and cognitive features. The behavioural definition (‘the failure to give equal consideration, based on the fact of race alone’) is rejected, primarily due to its inability to distinguish between ‘true’and ‘ordinary’racism. It is the former which is morally most objectionable — and which identifies the essence of the racist attitude and belief. The central part of the essay argues in favour of the motivational approach to the definition (...) (‘the infliction of unequal consideration, motivated by the desire to dominate, based on race alone’) and clarifies the way in which racism, thus understood, conflicts with the principle of human equality. Finally, the cognitivist definition (‘unequal consideration, out of a belief in the inferiority of another race’) is also rejected, despite its intuitive appeal. The overall discussion has important implications for moral philosophy. It is shown both that the principle of human equality does not strictly imply equal consideration, and that one may violate the principle of equal perception, yet not deny the principle which, from the perspective of this argument, is the more fundamental element in the principle of human equality, the principle of equal human worth. (shrink)

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

In lieu of an abstract, here is a brief excerpt of the content:Philosophy, Psychiatry, & Psychology 10.3 (2003) 265-266 [Access article in PDF] Definition Is Limited and Values Inescapable Richard Mullen THIS IS A welcome paper that lays bare some of the presumptions of those who seek to determine the status of psychiatric disorder. At different times debate on the subject reflects stigma, prejudice, needs for coherent categorization, and occasionally just antipsychiatric resentment. As Pickering hints, much philosophical argument may be (...) selected to support a position already chosen.The article illustrates that the likeness argument is insufficient to demonstrate that mental disorders are or are not disorders in any other more profound sense. Similarly perhaps, if a debate were conducted about whether or not a bony fracture, hypertension, obesity, or bacterial infection were real illnesses, the likeness argument might be unfruitful. This is perhaps unsurprising; notions of health and illness, mental or physical, are largely abstractions. In psychiatry, so-called symptoms are themselves abstract classifications of incompletely articulated mental phenomena far more diverse than the definitions and criteria to be found in a diagnostic manual.Argument by analogy is always limited. If my car has a flat tire, the car will be dysfunctional or disordered. Its loss of performance will be the symptoms of the flat tire, which is the lesion. It is a problem quickly diagnosed and treated. If convenient, I suppose my car could have been said to be ill, but this would not lead to any particular insights into the nature of cars, mechanics, health, or anything else.A variety of concepts can be contained within terms such as illness,disease, and disorder. They include the clinicopathological approach (the disease is the organic lesion), the disease is deviance approach, disease as distress, and approaches that consider disease—particularly mental disorder—as largely socially constructed. None may resolve the questions that Pickering raises, but recognizing that our terms are not always precise or used consistently is part of formulating the problem more clearly. For the most part clinical psychiatry (and the rest of medicine) has no difficulty in properly identifying most of our patients as disordered or ill. The fact that there are some borderline instances, difficult to classify either way, does not invalidate the classification.Consideration of whether or not the features of a disorder can in themselves determine whether or not a person is disordered is timely for two reasons. As stigma about (some) psychiatric disorder lessens, and opportunities for complaint and compensation expand, an era of self-definition as mentally disordered has arrived. How are mental health practitioners to respond to someone whose mental distress appears not to be clinically significant, but who demands access to the sick role? Also of current concern for clinical psychiatry is widespread pressure to define as disordered those persons whose antisocial behavior brings them into conflict with society. Professional recognition that definitions of mental disorder are inadequate, or rather incomplete, [End Page 265] may be valid. Worryingly, there are others who are willing to define it for us.Both major international systems of classification of mental disorder offer definitions of mental disorder that turn on what is to be meant by clinical significance or clinically recognizable. In the current Diagnostic and Statistical Manual, mental disorder implies "clinically significant impairment or distress," and criteria for specific disorders mostly including a comment about "clinically significant distress or impairment in social, occupational, or other important areas of functioning" (APA 2000, 8). In the International Classification of Disease, disorder is used to imply "existence of a clinically recognizable set of symptoms or behavior associated in most cases with distress and with interference with personal function" (WHO 1992, 5). It seems that no definition of mental disorder is adequate for all purposes. Seeking to improve existing definitions by further defining the terms used quickly becomes an exercise in pedantry, far removed from practical utility.Pickering mentions as an example the "symptom" of delusions. Available definitions are widely recognized as incomplete or otherwise unsatisfactory (David 1999). This leads to debate about what the individual elements within the definition precisely mean. However in practice delusions are identified with remarkably high reliability (WHO 1973... (shrink)

Sentences containing definite descriptions, expressions of the form ‘The F’, can be formalised using a binary quantifier ι that forms a formula out of two predicates, where ιx[F, G] is read as ‘The F is G’. This is an innovation over the usual formalisation of definite descriptions with a term forming operator. The present paper compares the two approaches. After a brief overview of the system INFι of intuitionist negative free logic extended by such a quantifier, which was presented in (...) (Kürbis 2019), INFι is first compared to a system of Tennant’s and an axiomatic treatment of a term forming ι operator within intuitionist negative free logic. Both systems are shown to be equivalent to the subsystem of INFι in which the G of ιx[F, G] is restricted to identity. INFι is then compared to an intuitionist version of a system of Lambert’s which in addition to the term forming operator has an operator for predicate abstraction for indicating scope distinctions. The two systems will be shown to be equivalent through a translation between their respective languages. Advantages of the present approach over the alternatives are indicated in the discussion. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Rhetorical Definition:A French InitiativeNancy S. StrueverRhetoric as TheoryIl y a quelque chose de démesuré et de prématuré à entreprendre une histoire de la rhétorique dans I'Europe moderne(Fumaroli 1999).When in his preface to the Histoire de la rhétorique Marc Fumaroli states that the project itself is overambitious and premature, he proceeds to justify his judgment by listing the complications of rhetorical definition: rhetoric is Protean in nature, and in this (...) it simply matches the fluidity of the phenomena of speech, parole ; it is oxymoronic as the science of the uncertain, trying to ally incertitude of topic with intelligibility of explanation; it intricates theoria and ergon, the concept of parole with an engagement with parole in act. Long a part of early education, it generates grace in adult expression, finesse. It is as well a primary source for reflection on communication phenomena. It is manifest in works of genius, yet, it is democratic in its accessibility; it is perdurable, remarkably persistent through change; and, it recognizes that the sharing and abusing of parole are intimately bound together. Finally rhetoric stimulates, for it may illumine the contemporary "exile" of literature and the arts, an exile that poses for us the issue of, perhaps intractable, cultural decline (Fumaroli 1999, 2–3). [End Page 401]These disparate elements of rhetoric, I would argue, by their very complexity justify rhetoric as mode of inquiry, and prompt our continuing consideration of the investigative protocols of rhetoric as hermeneutic. Roman rhetoric, subdivided into three communicative tasks—docere, delectare, movere —sponsors three investigative tasks: instructing requires command of the strategies of argument; delighting sponsors an ecology of desire, and moving requires a socio-pathology of community assent and dissent. The rhetorical interest in discursive effect, then, sponsors a second-order inquiry, the task of reviewing the basic assumptions of the communicative community: the beliefs, doxa, the shared opinions, endoxa, habits, dispositions. Rhetoric collects assumptions; thus Aristotle's Rhetoric has been characterised as a compendium of the collective wisdom of the polis. Rhetorical topoi, commonplaces, are the rubrics that organize common sense, sensus communis. Rhetoric has the task of understanding the means of asserting, rejecting, altering beliefs, the transfer of theory into practice, dealing with the transformations of idea into ideology.Rhetorical interests, in the Habermasian sense of motives that color knowledge initiatives, can correct careers of inquiry; a shift to a rhetorical investigative focus is pragmatic, to a most distinctive engagement with practice. And thus Martin Heidegger's aphorism: "Rhetoric functions inside politics." Rhetoric is framed, compromised by politics defined in the generous sense of discursive negotiations within the communal domain of beliefs, shared opinions, habits of action, dispositions. Rhetorical inquiry has to identify the beliefs and habits of action of a community in its efforts to preserve and extend its identity, and rhetoric must analyze the discursive negotiations for affirmation and alteration of beliefs vital to the continued existence of the polity.The work of Marc Fumaroli is of intrinsic interest to the historian of rhetoric as an instantiation of rhetorical interests driving an entire career, a practice shaped by rhetorical tasks and habits of action in inquiry; the effect is that of a kind of alchemy, of the transformation, at every turn, of the history of culture into the history of cultural politics. "Cultural politics" does not denote, of course, an exclusive focus on state policy, dirigisme in culture. Rather, we have a combination of a compendious notion of culture as all our inventions, artifices—political, theological, affective, aesthetic—with a generous sense of politics as all our discursive negotiations of differences of public import, a combination that sets a topic that Fumaroli prefers to call a "politics of the spirit" (Fumaroli 1992a, 80–81).1 If we distinguish rhetoric as discipline, oratory as product, eloquence as value, we [End Page 402] see Fumaroli's career is exemplary of a modern initiative that links rhetoric as disciplined inquiry to the politics of value. Thus, he will give us a rhetorical definition of the Académie Française's communal, historical duty to French "eloquence," to a pursuit of a national excellence (Fumaroli 1996a, 304, 1992b, 78). And... (shrink)

An important lesson that philosophy can learn from the Turing Test and computer science more generally concerns the careful use of the method of Levels of Abstraction (LoA). In this paper, the method is first briefly summarised. The constituents of the method are “observables”, collected together and moderated by predicates restraining their “behaviour”. The resulting collection of sets of observables is called a “gradient of abstractions” and it formalises the minimum consistency conditions that the chosen abstractions must satisfy. Two (...) useful kinds of gradient of abstraction – disjoint and nested – are identified. It is then argued that in any discrete (as distinct from analogue) domain of discourse, a complex phenomenon may be explicated in terms of simple approximations organised together in a gradient of abstractions. Thus, the method replaces, for discrete disciplines, the differential and integral calculus, which form the basis for understanding the complex analogue phenomena of science and engineering. The result formalises an approach that is rather common in computer science but has hitherto found little application in philosophy. So the philosophical value of the method is demonstrated by showing how making the LoA of discourse explicit can be fruitful for phenomenological and conceptual analysis. To this end, the method is applied to the Turing Test, the concept of agenthood, the definition of emergence, the notion of artificial life, quantum observation and decidable observation. It is hoped that this treatment will promote the use of the method in certain areas of the humanities and especially in philosophy. (shrink)

The purpose of this paper is to examine how any domain of Meinongian objects can be structured by a special kind of mereology. The basic definition of this mereology is the following: an object is part of another iff every characteristic property of the former is also a characteristic property of the latter. I will show that this kind of mereology ends up being very powerful for dealing with Meinongian objects. Mereological sums and products are not restricted in any way (...) in a domain of Meinongian objects: there is a sum and a product for any pair of Meinongian objects. With the mereological operations of sum, product and complement, and two special Meinongian objects, we can define a full boolean algebra on Meinongian objects. Moreover, this kind of mereology is atomic and extensional: an atom is a Meinongian object having just one characteristic property and two objects are identical iff the same atoms are parts of both of them. A Meinongian object can finally be defined in mereological terms as the sum of the atoms of its characteristic properties. (shrink)

We continue [2], developing simplicity in the framework of compact abstract theories. Due to the generality of the context we need to introduce definitions which differ somewhat from the ones use in first order theories. With these modified tools we obtain more or less classical behaviour: simplicity is characterized by the existence of a certain notion of independence, stability is characterized by simplicity and bounded multiplicity, and hyperimaginary canonical bases exist.

The Logic of Causation: Definition, Induction and Deduction of Deterministic Causality is a treatise of formal logic and of aetiology. It is an original and wide-ranging investigation of the definition of causation (deterministic causality) in all its forms, and of the deduction and induction of such forms. The work was carried out in three phases over a dozen years (1998-2010), each phase introducing more sophisticated methods than the previous to solve outstanding problems. This study was intended as part of a (...) larger work on causal logic, which additionally treats volition and allied cause-effect relations (2004). The Logic of Causation deals with the main technicalities relating to reasoning about causation. Once all the deductive characteristics of causation in all its forms have been treated, and we have gained an understanding as to how it is induced, we are able to discuss more intelligently its epistemological and ontological status. In this context, past theories of causation are reviewed and evaluated (although some of the issues involved here can only be fully dealt with in a larger perspective, taking volition and other aspects of causality into consideration, as done in Volition and Allied Causal Concepts). Phase I: Macroanalysis. Starting with the paradigm of causation, its most obvious and strongest form, we can by abstraction of its defining components distinguish four genera of causation, or generic determinations, namely: complete, partial, necessary and contingent causation. When these genera and their negations are combined together in every which way, and tested for consistency, it is found that only four species of causation, or specific determinations, remain conceivable. The concept of causation thus gives rise to a number of positive and negative propositional forms, which can be studied in detail with relative ease because they are compounds of conjunctive and conditional propositions whose properties are already well known to logicians. The logical relations (oppositions) between the various determinations (and their negations) are investigated, as well as their respective implications (eductions). Thereafter, their interactions (in syllogistic reasoning) are treated in the most rigorous manner. The main question we try to answer here is: is (or when is) the cause of a cause of something itself a cause of that thing, and if so to what degree? The figures and moods of positive causative syllogism are listed exhaustively; and the resulting arguments validated or invalidated, as the case may be. In this context, a general and sure method of evaluation called ‘matricial analysis’ (macroanalysis) is introduced. Because this (initial) method is cumbersome, it is used as little as possible – the remaining cases being evaluated by means of reduction. Phase II: Microanalysis. Seeing various difficulties encountered in the first phase, and the fact that some issues were left unresolved in it, a more precise method is developed in the second phase, capable of systematically answering most outstanding questions. This improved matricial analysis (microanalysis) is based on tabular prediction of all logically conceivable combinations and permutations of conjunctions between two or more items and their negations (grand matrices). Each such possible combination is called a ‘modus’ and is assigned a permanent number within the framework concerned (for 2, 3, or more items). This allows us to identify each distinct (causative or other, positive or negative) propositional form with a number of alternative moduses. This technique greatly facilitates all work with causative and related forms, allowing us to systematically consider their eductions, oppositions, and syllogistic combinations. In fact, it constitutes a most radical approach not only to causative propositions and their derivatives, but perhaps more importantly to their constituent conditional propositions. Moreover, it is not limited to logical conditioning and causation, but is equally applicable to other modes of modality, including extensional, natural, temporal and spatial conditioning and causation. From the results obtained, we are able to settle with formal certainty most of the historically controversial issues relating to causation. Phase III: Software Assisted Analysis. The approach in the second phase was very ‘manual’ and time consuming; the third phase is intended to ‘mechanize’ much of the work involved by means of spreadsheets (to begin with). This increases reliability of calculations (though no errors were found, in fact) – but also allows for a wider scope. Indeed, we are now able to produce a larger, 4-item grand matrix, and on its basis find the moduses of causative and other forms needed to investigate 4-item syllogism. As well, now each modus can be interpreted with greater precision and causation can be more precisely defined and treated. In this latest phase, the research is brought to a successful finish! Its main ambition, to obtain a complete and reliable listing of all 3-item and 4-item causative syllogisms, being truly fulfilled. This was made technically feasible, in spite of limitations in computer software and hardware, by cutting up problems into smaller pieces. For every mood of the syllogism, it was thus possible to scan for conclusions ‘mechanically’ (using spreadsheets), testing all forms of causative and preventive conclusions. Until now, this job could only be done ‘manually’, and therefore not exhaustively and with certainty. It took over 72’000 pages of spreadsheets to generate the sought for conclusions. This is a historic breakthrough for causal logic and logic in general. Of course, not all conceivable issues are resolved. There is still some work that needs doing, notably with regard to 5-item causative syllogism. But what has been achieved solves the core problem. The method for the resolution of all outstanding issues has definitely now been found and proven. The only obstacle to solving most of them is the amount of labor needed to produce the remaining (less important) tables. As for 5-item syllogism, bigger computer resources are also needed. (shrink)

A well-established historiography has pointed out the distinction between first philosophy and theology, proposed by Benet Perera in his De communibus, as the “birth” of modern ontology. Ontology is often defined as an independent or neutral science by modern authors as well as contemporary scholars.This paper aims to show the way in which Perera comes to this distinction, after a long reflection on the status of mathematics and abstractions of theoretical sciences matured during his lectures at the Collegio Romano.Interestingly, it (...) appears that after the subject of ontology is removed from its relationship with God and rational theology, ontology does not seem to acquire the status of neutral and universal science that Perera sought. Ontology becomes, in fact, a science dependent on the definitions provided by other disciplines and sciences, such as logic and mathematics.Following these recent findings, the paper also provides a renewed confrontation b.. (shrink)

The Quantified argument calculus (Quarc) has received a lot of attention recently as an interesting system of quantified logic which eschews the use of variables and unrestricted quantification, but nonetheless achieves results similar to the Predicate calculus (PC) by employing quantifiers applied directly to predicates instead. Despite this noted similarity, the issue of the relationship between Quarc and PC has so far not been definitively resolved. We address this question in the present paper, and then expand upon that result. Utilizing (...) recent developments in structural proof theory, we develop a G3-style sequent calculus for Quarc and briefly demonstrate its structural properties. We put these properties to use immediately to construct direct proofs of the meta-theoretical properties of the system. We then incorporate an abstract (and, as we shall see, logical) predicate into the system in a way that preserves all the structural properties. This allows us to identify a system of Quarc which is deductively equivalent to PC, and also yields a constructive method of demonstrating the Craig interpolation theorem (which speaks in favor of the aforementioned predicate being logical). We further generalize this extension to develop a bivalent system of Quarc with defining clauses that still maintains all the desirable properties of a good proof system. (shrink)

In recent discussions concerning the definition of argument, it has been maintained that the word ‘argument’ exhibits the process-product ambiguity, or an act/object ambigu-ity. Drawing on literature on lexical ambiguity we argue that ‘argument’ is not ambiguous. The term ‘argu-ment’ refers to an object, not to a speech act. We also examine some of the important implications of our argument by considering the question: what sort of abstract objects are arguments?

We provide a version of first-order hybrid tense logic with predicate abstracts and definite descriptions as the only non-rigid terms. It is formalised by means of a tableau calculus working on sat-formulas. A particular theory of DD exploited here is essentially based on the approach of Russell, but with descriptions treated as genuine terms. However, the reductionist aspect of the Russellian approach is retained in several ways. Moreover, a special form of tense definite descriptions is formally developed. A constructive proof (...) of the interpolation theorem for this calculus is given, which is an extension of the result provided by Blackburn and Marx. (shrink)

The first part of the article demonstrates how the concept of mental resilience has developed from the 1970s to the 2010s, spreading from the field of developmental psychopathology to a wide range of psychological and psychotherapeutic disciplines. Today, there are many varied definitions of mental resilience, ranging from a relation to the concept of mental vulnerability to a related concept of mental strength. The second part of the article demonstrates how the current popularisation of the idea of individual resilience (...) is associated with a social and cultural problematisation of the vulnerable life as well as with a psychopolitical and governmental focus on enhancing the mental capacity of citizens. (shrink)

The words “mind” and “matter” are used glibly, both by ordinary people and by philosophers, without any adequate attempt at definition. Philosophers are much to blame for this. My own feeling is that there is not a sharp line, but a difference of degree; an oyster is less mental than a man, but not wholly un-mental. And I think “ mental ” is a character, like “ harmonious ” or “ complicated,” that cannot belong to a single entity, but only (...) to a system of entities. But before defending this view, I wish to spend some time on traditional theories. (shrink)

The essay on abstraction provides an historical review of the notion of abstraction with an attempt being made to show that there is a basic similarity between the doctrines of Aristotle and Aquinas, on the one hand, and Locke on the other. The conclusion that is then drawn is that the nominalistic critique initiated by Berkeley and refined by Hume in direct answer to the Lockean theory of general ideas is effective against all doctrines of abstraction which (...) hope to end up in a moderate realism. Other concerns are 1) to show the basic affinities between Berkeleyean and Humean nominalism, and late medieval nominalism, in particular that of Ockham, 2) to show that such a nominalism is inadequate and must be replaced by, what has been termed in contemporary analytical circles, psychological nominalism. The essay on relation shows the continuity in the history of the doctrine of relations from the time of Melissus to the Rusellian critique of the substance-accident ontology, critique which Weinberg thinks definitively replaced the unilateral-attribute doctrine of relations. The essay on induction attempts to show how the medieval commentators modified Aristotle's intuitionistic and vestigally [[sic]] Platonic doctrine of induction into a doctrine of covert inference, and anticipated, in part, Bacon's and Mill's theories.—E. A. R. (shrink)

In lieu of an abstract, here is a brief excerpt of the content: Against Definition. Eric Matthews. -/- Keywords: definition, mental illness, ideology -/- Varelius rightly says, in the very first sentence of his article, that 'The nature of mental disorder is a controversial issue.' But I do not think he recognizes just how deeply controversial it is. He hopes to be able to find a way of resolving the controversy by his chosen method, based on a consideration of 'the (...) goals that we have in classifying certain persons as mentally disordered.' However, not only does the adoption of that method presuppose that the controversy is resoluble, it is also somewhat circular. For to be able to set out the goals that we have in classifying some people as mentally disordered, we already have to know which persons we so classify, and so what is meant by calling them 'mentally disordered.' There are several appeals to alleged 'intuitions' and one to a 'paradigm case' in Varelius's article; but, if we had such intuitions, they would be all we needed to define the concept of mental disorder, and there would be no need for the further reference to our goals in demarcating this class of human conditions.I want to argue that any alleged 'intuitions' we might have in regard to mental disorder are philosophically useless, and that the controversy about the nature of mental disorder is intrinsically irresoluble. Any claimed intuition can always be met by a rival intuition that is irreconcilable with the first. It is this, indeed, that makes the dispute about mental disorder essentially philosophical, rather than scientific, in character. Philosophical positions have to be based on, and defended by, rational argument, but there necessarily comes a point in any philosophical dispute at which no rational argument can settle the matter one way or the other. Varelius derives his methodology of considering goals of demarcation from some work done using this method on the concept of science. But people have different goals for demarcating 'science' from 'non-science' or 'pseudo-science.' The demarcation is not made, as Varelius suggests, on the basis of such criteria as testability of hypotheses or clarity of concepts; or rather, the application of these criteria themselves is determined by the goals of demarcation. The debate about what is to count as a 'science' is irresoluble in part because what we classify as such has evaluative and practical consequences. To be a 'science,' in our culture, is to be a valued activity, to have a status to be aspired to. This makes arguments about what it is to be scientific what I would like to call 'ideological' in character. Similar considerations apply to the dispute about the nature of mental disorder. Human societies have always distinguished madness from reason, disturbed or disordered behavior, thoughts, and emotions from those that are not disordered. But the purpose of making the distinction, and thus the character of the distinction made, has varied from society to society and from one historical period to another. The historian Roy Porter, in his short work Madness: A Brief History (2002) describes some of the different images of madness which have prevailed at different times in Western culture—as a divine punishment for hubris, as [End Page 53] a source of poetic inspiration, as the inevitable result of moral excess, as diabolical possession, and so on. The modern idea of mental disorder as mental illness is only one possible way of classifying together certain sorts of deviations from normality. Varelius's method of arriving at a definition of mental illness involves 'considering the goals that we have in classifying certain persons as mentally disordered.' But we can ask who are the 'we' in question, and whether 'we' have only one set of goals in classifying people in this way. The history referred to in the preceding paragraph suggests that 'we' means those members of society who do not regard themselves as disordered—the 'normal' people whose behavior can be readily understood by other normal people. Classifying someone as mentally disordered is distinguishing them as 'not one of us,' and this can serve various purposes, for instance, treating them as objects of entertainment in a human zoo, as in Bedlam; or treating them as moral. (shrink)

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

The first part of the article demonstrates how the concept of mental resilience has developed from the 1970s to the 2010s, spreading from the field of developmental psychopathology to a wide range of psychological and psychotherapeutic disciplines. Today, there are many varied definitions of mental resilience, ranging from a relation to the concept of mental vulnerability to a related concept of mental strength. The second part of the article demonstrates how the current popularisation of the idea of individual resilience (...) is associated with a social and cultural problematisation of the vulnerable life as well as with a psychopolitical and governmental focus on enhancing the mental capacity of citizens. (shrink)

Most of the arguments usually appealed to in order to support the view that some abstraction principles are analytic depend on ascribing to them some sort of existential parsimony or ontological neutrality, whereas the opposite arguments, aiming to deny this view, contend this ascription. As a result, other virtues that these principles might have are often overlooked. Among them, there is an epistemic virtue which I take these principles to have, when regarded in the appropriate settings, and which I (...) suggest to call ‘epistemic economy’. My purpose is to isolate and clarify this notion by appealing to some examples concerning the definition of natural and real numbers. (shrink)

I survey several counterexamples (by Raz and MacCormick) to Hohfeld's conjecture that a claim-right is correlative to a directed duty and (by Cornell and Frick) to Bentham's suggestion that a claim-right is correlative to a wronging. We can vindicate these claims of correlativity if we acknowledge that entitlements like claim-rights and directed duties admit of degrees of abstraction: that they may be general rather than specific, unspecified rather than specified, or indefinite rather than definite. I provide an error theory (...) consisting in linguistic and practical reasons for why we articulate normative incidents in ways that threaten correlativity. And I deny that abstraction imposes a heavy metaphysical cost on rights theory, though I leave open whether abstraction excludes certain explanatory accounts of rights such as the interest theory or will theory. (shrink)

Since their introduction by Partee and Rooth (1983) into linguistic theory, type shifting principles have been extensively employed in various linguistic domains, including nominal predicates (Partee 1987), kind denoting NPs (Chierchia 1998), interrogatives (Groenendijk and Stokhof 1989), scrambled definites (De Hoop and Van der Does 1998) and plurals (Winter 2001,2002). Most of the accounts that use type shifting principles employ them as ``last resort'' mechanisms, which apply only when other compositional mechanisms fail. This failure is often sloppily referred to as (...) type mismatch . The motivation for introducing type mismatch into the compositional mechanism is twofold: on the one hand it allows lexical items to be assigned the minimal types that are needed for describing their denotation; on the other hand, it has been argued that the ``last resort'' strategy of type shifting prevents derivation of undesired meanings. The first goal of this paper is to define a simple notion of type mismatch, which will rather closely follow Partee and Rooth's original proposal but will be expressed within more familiar terms of categorial semantics. After introducing this implementation of traditional type mismatch, it will be argued that in fact, it covers only one possible kind of trigger for type shifting principles. Partee and Rooth's notion of mismatch is ``external'' in that the type of an expression is changed only when it combines with another type to which it cannot compose using the ``normal'' compositional mechanism. It will be argued that, within an appropriate type system, another notion of mismatch is also useful. This is the kind of mismatch in which the semantic type of an expression does not match its syntactic category. Two such cases will be explored: mismatch between morpho-syntactic number (singular or plural) and semantic number (a denotation ranging over atoms or sets), and mismatch between syntactic category (noun, DP, adjective etc.) and semantic role (predicate, quantifier, predicate modifier etc.).. (shrink)

Cantor's abstractionist account of cardinal numbers has been criticized by Frege as a psychological theory of numbers which leads to contradiction. The aim of the paper is to meet these objections by proposing a reassessment of Cantor's proposal based upon the set theoretic framework of Bourbaki - called BK - which is a First-order set theory extended with Hilbert's ε-operator. Moreover, it is argued that the BK system and the ε-operator provide a faithful reconstruction of Cantor's insights on cardinal numbers. (...) I will introduce first the axiomatic setting of BK and the definition of cardinal numbers by means of the ε-operator. Then, after presenting Cantor's abstractionist theory, I will point out two assumptions concerning the definition of cardinal numbers that are deeply rooted in Cantor’s work. I will claim that these assumptions are supported as well by the BK definition of cardinal numbers, which will be compared to those of Zermelo-von Neumann and Frege-Russell. On the basis of these similarities, I will make use of the BK framework in meeting Frege's objections to Cantor's proposal. A key ingredient in the defence of Cantorian abstraction will be played by the role of representative sets, which are arbitrarily denoted by the ε-operator in the BK definition of cardinal numbers. (shrink)

In the following I take issue with the allegation that liberalism must inevitably be guilty of ‘abstract individualism’. I treat Michael Sandel’s well-known claim that there are ‘loyalties and convictions whose moral force consists partly in the fact that living by them is inseparable from understanding ourselves as the particular persons we are’ as representative of this widely held view. Specifically, I argue: (i) that Sandel’s account of the manner in which ‘constitutive’ loyalties function as reasons for action presupposes the (...) possibility of there being (what I call) ‘underivable particular obligations’, but that such obligations are, in fact, a logical impossibility; and (ii) that Sandel’s account of the self as necessarily ‘encumbered’ presupposes an account of personal identity which confuses identification with definition, and which is, therefore, fundamentally flawed. (shrink)

This thesis is concerned with explaining how a subject can acquire a priori knowledge of arithmetic. Every account for arithmetical, and in general mathematical knowledge faces Benacerraf's well-known challenge, i.e. how to reconcile the truths of mathematics with what can be known by ordinary human thinkers. I suggest four requirements that jointly make up this challenge and discuss and reject four distinct solutions to it. This will motivate a broadly Fregean approach to our knowledge of arithmetic and mathematics in general. (...) Pursuing this strategy appeals to the context principle which, it is proposed, underwrites a form of Platonism and explains how reference to and object-directed thought about abstract entities is, in principle, possible. I discuss this principle and defend it against different criticisms as put forth in recent literature. Moreover, I will offer a general framework for implicit definitions by means of which - without an appeal to a faculty of intuition or purely pragmatic considerations - a priori and non-inferential knowledge of basic mathematical principles can be acquired. In the course of this discussion, I will argue against various types of opposition to this general approach. Also, I will highlight crucial shortcomings in the explanation of how implicit definitions may underwrite a priori knowledge of basic principles in broadly similar conceptions. In the final part, I will offer a general account of how non-inferential mathematical knowledge resulting from implicit definitions is best conceived which avoids these shortcomings. (shrink)

The question of what should be meant by a measurement is tackled from a mathematical perspective whose physical interpretation is that a measurement is a fundamental process via which a finite amount of classical information is produced. This translates into an algebraic and topological definition of measurement space that caters for the distinction between quantum and classical measurements and allows a notion of observer to be derived.