This article offers an overview of philosophical methodologies. In an attempt to avoid a certain circularity, the article itself tries to avoid consciously or solely deploying and engaging with any current standard notion of what constitutes a philosophical method or philosophy itself. It hopes to find some of the possible places in which philosophy occurs, and this turns out to include such endeavours as literature, art, poetry, and linguistics. From here it considers how almost anything—for example, conversation, everyday life, and (...) love—can also be philosophy. An attempt is made to identify some characteristic feature of philosophy as it occurs in all such forms. In the end, the simultaneous enacting of a peculiar optimism and particular humility both, in the search for knowledge, is put forward as potentially sufficient to at least begin to identify philosophy across its many guises. (shrink)

Richard L. Epstein with Carolyn Kerbrger, Critical Thinking, 3rd Edition, Advanced Reasoning Forum, 2012, 464 pp., ISBN-10: 1938421000, ISBN-13: 978-1938421006.Richard L. Epstein, The Pocket Guide to Critical Thinking, 4th Edition, Advanced Reasoning Forum, 2011, 162 pp., paperback ISBN-13: 9780981550770, ISBN-10: 0981550770, ebook ISBN-13: 9780981550787.

Richard L. Epstein, Reasoning in Science and Mathematics: Essayson Logic as The Art of Reasoning Well, Advanced Reasoning Forum,2011, 134 pp., ISBN-13: 978-0983452126, ISBN-10: 0983452121.

This paper explores the question of what logic is not. It argues against the wide spread assumptions that logic is: a model of reason; a model of correct reason; the laws of thought, or indeed is related to reason at all such that the essential nature of the two are crucially or essentially co-illustrative. I note that due to such assumptions, our current understanding of the nature of logic itself is thoroughly entangled with the nature of reason. I show that (...) most arguments for the presence of any sort of essential re- lationship between logic and reason face intractable problems and demands, and fall well short of addressing them. These arguments include those for the notion that logic is normative for reason (or that logic and correct reason are in some way the same thing), that logic is some sort of description of correct reason and that logic is an abstracted or idealised version of correct reason. A strong version of logical realism is put forward as an alternative view, and is briefly explored. (shrink)

I argue that one in particular of CrispinWright’s attempts to capture our common or intuitive concepts of objectivity, warrant, and other associated notions, relies on an ambiguity between a given constructivist reading of the concepts and at least one other, arguablymore ‘ordinary’, version of the notions he tries to accommodate. I do this by focusing on one case in point, and concluding with a brief argument showing how this case generalises. I demonstrate why this ambiguity is unacceptable and also that (...) its resolution undermines the aim it serves: to account for and accommodate our ordinary conception of (at least) objectivity, warrant (or justication) and truth. (shrink)

Featuring fourteen new essays from an international team of renowned contributors, this volume explores the key issues, debates and questions in the metaphysics of logic. The book is structured in three parts, looking first at the main positions in the nature of logic, such as realism, pluralism, relativism, objectivity, nihilism, conceptualism, and conventionalism, then focusing on historical topics such as the medieval Aristotelian view of logic, the problem of universals, and Bolzano's logical realism. The final section tackles specific issues such (...) as glutty theories, contradiction, the metaphysical conception of logical truth, and the possible revision of logic. The volume will provide readers with a rich and wide-ranging survey, a valuable digest of the many views in this area, and a long overdue investigation of logic's relationship to us and the world. It will be of interest to a wide range of scholars and students of philosophy, logic, and mathematics. (shrink)

This paper seeks to identify and defend an approach to inquiry dubbed ‘metaphysical optimism’, particularly as it is evidenced at crisis points in the fields of physics, mathematics and logic. That the practice of metaphysical optimism at such moments, wherein it has appeared that there is no clear way to proceed or understand where we have arrived, is both reasonable and useful suggests it is to be taken seriously as capable of progressing fields and increasing knowledge. Given this, the paper (...) then looks in more depth at what such an approach involves and why it might be useful both as a methodological approach in general and to help clarify positions along the realism/anti-realism spectrum in philosophy. From here, the paper arrives at a possible argument in defence of the realist attitude to transcendence. (shrink)

This Element looks at the problem of inter-translation between mathematical realism and anti-realism and argues that so far as realism is inter-translatable with anti-realism, there is a burden on the realist to show how her posited reality differs from that of the anti-realist. It also argues that an effective defence of just such a difference needs a commitment to the independence of mathematical reality, which in turn involves a commitment to the ontological access problem – the problem of how knowable (...) mathematical truths are identifiable with a reality independent of us as knowers. Specifically, if the only access problem acknowledged is the epistemological problem – i.e. the problem of how we come to know mathematical truths – then nothing is gained by the realist notion of an independent reality and in effect, nothing distinguishes realism from anti-realism in mathematics. (shrink)

Four Basic Logical Issues: The paper addresses what we see as the four major issues in logic. The overriding issue is that of the choice of logic. We start with some discussion of the preliminary issue of whether there is such a 'one true logic,' but we reserve the main discussion for the first issue of 'classical logic versus nonclassical logic.' Here, we discuss the role of meaning and truth, the relation between classical logic and classical negation, and whether and, (...) if so, how classical logic should reside at the base world. Given the argument in favor of an overall use of nonclassical logic, the second issue is that of the choice of nonclassical logic. Brady's logic MC of meaning containment is argued for, with some comparison made with other relevant logics. For the remaining two issues, we make a case for relevant deduction, in comparison with classical deduction, and we explore possibilities for the appropriate metalogic, comparing classical and nonclassical approaches. (shrink)

ABSTRACTGrounded in Emotional Cascade Theory, we explored whether rumination and multisensory imagery-based cognitions moderated the relationships between affect and both odds of non-suicidal self-injury, and frequency of the behaviour. A sample of 393 university students completed self-report questionnaires assessing the constructs of interest. Contrary to expectations, rumination did not emerge as a significant moderator of the affect-NSSI relationship. However, the relationship between affect and frequency of NSSI was moderated by the use of imagery. Further, the relationship between negative affect and (...) NSSI was moderated by positive affect, underscoring the need to consider both negative and positive affect in models of NSSI. Most youth who self-injured reported thinking in images while the urge to self-injure was strong, with 53% thinking in images at least half the time. Future work is needed to explore how positive and negative affect work in concert to govern NSSI, and how imagery might eit... (shrink)

Sleep enhances integration across multiple stimuli, abstraction of general rules, insight into hidden solutions and false memory formation. Newly learned information is better assimilated if compatible with an existing cognitive framework or schema. This article proposes a mechanism by which the reactivation of newly learned memories during sleep could actively underpin both schema formation and the addition of new knowledge to existing schemata. Under this model, the overlapping replay of related memories selectively strengthens shared elements. Repeated reactivation of memories in (...) different combinations progressively builds schematic representations of the relationships between stimuli. We argue that this selective strengthening forms the basis of cognitive abstraction, and explain how it facilitates insight and false memory formation. (shrink)

A patient's experience unfolds through a nurse's personal conversation with herself. Conveyed through three voices, the nurse's dialogue highlights her many internal struggles; those with her conscience on what she understands to be best practice, those important to her as a person, those of an ethical nature that profoundly affect one's search for meaning, and those in the personal–professional realm driven in part by institutional culture. These multivoiced knowledges are confronted in ways that foreground language and understanding as performative acts. (...) At the same time, another journey is co‐constructed with the reader, one that weaves in‐between the symbolic and the real, engaging the imaginary in (inter)play. The nurse's response to the inner conversation with her ‘self/selves’ problematizes practice, illuminates the patient's perspective while highlighting the nurse's sense of her marginal position. Insight into reified and hegemonic assumptions, strategies of how control is maintained through organizational surveillance, trust and moral agency help to foreground personal expectations as the nurse begins to grapple with her own feelings of betrayal. Tackling these insights offers opportunities to rethink oppressive practices in the provision of care. It also enables an alternative appreciation of the everyday dilemmas confronting nurses and offers new meaning to practice. (shrink)

Fifty years on from its first publication, The Concept of Law is still the starting point for the study of legal philosophy and is widely heralded as a classic work of modern philosophy. This third edition features a new introduction by Leslie Green, looking at Hart's work from the perspective of modern jurisprudence.

This article analyzes U.S. court cases involving reproductive technologies in terms of their implications for reproductive choice, mothers' versus fathers' rights, definitions and evaluations of parenting, and the nuclear family structure. The analysis reveals that the courts have tended not to recognize how social conditions shape women's reproductive choices, to promote fathers' rights more than mothers' rights, to ignore the social relationships that constitute childbearing and child rearing and value men's over women's biological contribution to these processes, to reflect certain (...) assumptions about the proper roles of mothers and fathers, and to privilege the nuclear family. The implications of developing reproductive technology policy for an understanding of the relationships among gender, reproductive technologies, and the state are considered, and recommendations for the equitable regulation of these technologies are offered. (shrink)

IntroductionAcademic integrity policy that is inaccessible, ambiguous or confusing is likely to result in inconsistent policy enactment. Additionally, policy analysis and development are often undertaken as top down processes requiring passive acceptance by users of policy that has been developed outside the context in which it is enacted. Both these factors can result in poor policy uptake, particularly where policy users are overworked, intellectually critical and capable, not prone to passive acceptance and hold valuable grass roots intelligence about policy enactment.Case (...) descriptionThe case study presented in this paper describes the actions of a community of practice at a regional Australian university to deconstruct and translate ambiguous academic integrity policy into a suite of accessible academic integrity resources that were intelligible to staff and students, and which assisted academic staff to consistently enact policy. The paper narrates the formation of the CoP, the tangible and intangible value it created, the social learning practices enacted by its members, its grassroots policy work and the material resources produced from that work.Discussion and evaluationAn evaluation of the CoP was conducted using a value creation framework to explore its immediate value, potential value, applied value, realised value, and reframing value. These values were considered at each stages of the CoP’s lifespan. The evaluation was a useful process that demonstrated the wide-ranging value created by the CoP. Six insights were drawn from the evaluation which promote understanding of the value created for a university by a CoP, particularly in contributing to academic integrity culture over a sustained period of time.ConclusionsThis paper contributes to a research gap on specific examples of discretion within rule-based systems. It illustrates how academics and members of the CoP used their discretion to interpret and enact academic integrity policy within a higher education setting. Drawing from the evaluation of the CoP we argue for greater understanding of the grass-roots contribution of academic and professional staff to academic integrity policy translation and enactment. (shrink)

One diagnosis of Fitch’s paradox of knowability is that it hinges on the factivity of knowledge: that which is known is true. Yet the apparent role of factivity and non-factive analogues in related paradoxes of justified belief can be shown to depend on familiar consistency and positive introspection principles. Rejecting arguments that the paradox hangs on an implausible consistency principle, this paper argues instead that the Fitch phenomenon is generated both in epistemic logic and logics of justification by the interaction (...) of analogues of the knowability principle and positive introspection theses that are characteristic of, even if not entailed by, epistemic internalism. (shrink)

We first consider the entailment logic MC, based on meaning containment, which contains neither the Law of Excluded Middle (LEM) nor the Disjunctive Syllogism (DS). We then argue that the DS may be assumed at least on a similar basis as the assumption of the LEM, which is then justified over a finite domain or for a recursive property over an infinite domain. In the latter case, use is made of Mathematical Induction. We then show that an instance of the (...) LEM is intrumental in the proof of Cantor's Theorem, and we then argue that this is based on a more general form than can be reasonably justified. We briefly consider the impact of our approach on arithmetic and naive set theory, and compare it with intuitionist mathematics and briefly with recursive mathematics. Our "Four Basic Logical Issues" paper would provide useful background, the current paper being an application of the some of the ideas in it. (shrink)

When in May 1930, the Council of Trinity College, Cambridge, had to decide whether to renew Wittgenstein's research grant, it turned to Bertrand Russell for an assessment of the work Wittgenstein had been doing over the past year. His verdict: "The theories contained in this new work... are novel, very original and indubitably important. Whether they are true, I do not know. As a logician who likes simplicity, I should like to think that they are not, but from what I (...) have read of them I am quite sure that he ought to have an opportunity to work them out, since, when completed, they may easily prove to constitute a whole new philosophy." "[Philosophical Remarks] contains the seeds of Wittgenstein's later philosophy of mind and of mathematics. Principally, he here discusses the role of indispensable in language, criticizing Russell's The Analysis of Mind. He modifies the Tractatus's picture theory of meaning by stressing that the connection between the proposition and reality is not found in the picture itself. He analyzes generality in and out of mathematics, and the notions of proof and experiment. He formulates a pain/private-language argument and discusses both behaviorism and the verifiability principle. The work is difficult but important, and it belongs in every philosophy collection."—Robert Hoffman, Philosophy "Any serious student of Wittgenstein's work will want to study his Philosophical Remarks as a transitional book between his two great masterpieces. The Remarks is thus indispensible for anyone who seeks a complete understanding of Wittgenstein's philosophy."—Leonard Linsky, American Philosophical Association. (shrink)

Many philosophers claim to be naturalists, but there is no common understanding of what naturalism is. Maddy proposes an austere form of naturalism called 'Second Philosophy', using the persona of an idealized inquirer, and she puts this method into practice in illuminating reflections on logical truth, philosophy of mathematics, and metaphysics.

There are increasing concerns that research regulatory requirements exceed those required to manage risks, particularly for low- and negligible-risk research projects. In particular, inconsistent documentation requirements across research sites can delay the conduct of multi-site projects. For a one-year, negligible-risk project examining biobank operations conducted at three separate Australian institutions, we found that the researcher time required to meet regulatory requirements was eight times greater than that required for the approved research activity. In total, 76 business days were required to (...) obtain the necessary approvals, and site-specific processes required twice as long as primary Human Research Ethics Committee and Research Governance Office processes. We describe the impact of this administrative load on the conduct of a one-year, externally-... (shrink)

Mathematicians tend to think of themselves as scientists investigating the features of real mathematical things, and the wildly successful application of mathematics in the physical sciences reinforces this picture of mathematics as an objective study. For philosophers, however, this realism about mathematics raises serious questions: What are mathematical things? Where are they? How do we know about them? Offering a scrupulously fair treatment of both mathematical and philosophical concerns, Penelope Maddy here delineates and defends a novel version of mathematical (...) realism. She answers the traditional questions and poses a challenging new one, refocusing philosophical attention on the pressing foundational issues of contemporary mathematics. (shrink)

This unique book applies concepts from the field of anthropology to clinical settings to result in a powerful and dynamic model/theory of clinical anthropology.

I provide a characterization of weakly pseudo-rationalizable choice functions---that is, choice functions rationalizable by a set of acyclic relations---in terms of hyper-relations satisfying certain properties. For those hyper-relations Nehring calls extended preference relations, the central characterizing condition is weaker than (hyper-relation) transitivity but stronger than (hyper-relation) acyclicity. Furthermore, the relevant type of hyper-relation can be represented as the intersection of a certain class of its extensions. These results generalize known, analogous results for path independent choice functions.

There is a theorem that shows that it is impossible for an algorithm to jointly satisfy the statistical fairness criteria of Calibration and Equalised Odds non-trivially. But what about the recently advocated alternative to Calibration, Base Rate Tracking? Here, we show that Base Rate Tracking is strictly weaker than Calibration, and then take up the question of whether it is possible to jointly satisfy Base Rate Tracking and Equalised Odds in non-trivial scenarios. We show that it is not, thereby establishing (...) an even more general impossibility theorem. (shrink)

We adopt and extend the concept of ‘noncooperative space’ to analyze how (aspirant) black women intellectual activists attempt to sustain their efforts within settings that publicly endorse racial equality, while, in practice, the contexts remain deeply racist. Noncooperative spaces reflect institutional, organizational, and social environments portrayed by powerful white agents as conducive to anti-racism work and promoting racial equality but, indeed, constrain individuals who challenge racism. Our work, which is grounded in intersectionality, draws on an autoethnographic account of racially motivated (...) domestic violence suffered by our lead author. Our analysis suggests that (aspirant) black women intellectual activists must develop courage to sustain their ‘voice’ within noncooperative spaces. However, the three interlinked dimensions of noncooperative spaces—namely, deceiving design, hegemonic actors’ indifference to racism, and (some assimilated gatekeepers’) false equivalence—may gradually erode a black female scholar’s courage. This forces her ‘voice’ to vanish temporarily, or even permanently. Courage is thus fragile and depletable. Yet, courage can be regenerated, resulting in regaining voice. Consequently, we propose courageous collective action by white allies and black and brown individuals who voluntarily and officially cooperate within and across various spaces to achieve racial equality. (shrink)

Rush Rhees (1905-1989) was a philosopher, and a pupil and close friend of Ludwig Wittgenstein. While some of Rhees's own published papers became classics, most of his work remained unpublished during his lifetime. After his death, his papers were found to comprise sixteen thousand pages of manuscript on every aspect of philosophy, from philosophical logic to Simone Weil. This collection of unpublished papers, edited by D. Z. Phillips, includes Rhees's outstanding work on philosophy and religion. Written over an academic (...) lifetime, some of the papers are sympathetic to religion while others are not. It is Rhees's ability to interweave the personal and philosophical, and his integrity and intellectual honesty, which make this one of the most impressive books in twentieth-century philosophy of religion. (shrink)

Many philosophers these days consider themselves naturalists, but it's doubtful any two of them intend the same position by the term. In Second Philosophy, Penelope Maddy describes and practices a particularly austere form of naturalism called "Second Philosophy". Without a definitive criterion for what counts as "science" and what doesn't, Second Philosophy can't be specified directly ("trust only the methods of science" for example), so Maddy proceeds instead by illustrating the behaviors of an idealized inquirer she calls the "Second (...) Philosopher". mhis Second Philosopher begins from perceptual common sense experimentation, theory formation and testing, working all the while to asses, correct and improve her methods as she goes. Second Philosophy is then the result of the Second Philosopher's investigations. Maddy delineates the Second Philosopher's approach by tracing her reactions to various familiar skeptical and transcendental views (Descartes, Kant, Carnap, late Putnam, van Fraassen), comparing her methods to those of other self-described naturalists (especially Quine), and examining a prominent contemporary debate (between disquotationalists and correspondence theorists in the theory of truth) to extract a properly second-philosophical line of thought. She then undertakes to practice Second Philosophy in her reflections on the ground of logical truth, the methodology, ontology and epistemology of mathematics, and the general prospects for metaphysics naturalized. (shrink)

The question of how the probabilistic opinions of different individuals should be aggregated to form a group opinion is controversial. But one assumption seems to be pretty much common ground: for a group of Bayesians, the representation of group opinion should itself be a unique probability distribution, 410–414, [45]; Bordley Management Science, 28, 1137–1148, [5]; Genest et al. The Annals of Statistics, 487–501, [21]; Genest and Zidek Statistical Science, 114–135, [23]; Mongin Journal of Economic Theory, 66, 313–351, [46]; Clemen and (...) Winkler Risk Analysis, 19, 187–203, [7]; Dietrich and List [14]; Herzberg Theory and Decision, 1–19, [28]). We argue that this assumption is not always in order. We show how to extend the canonical mathematical framework for pooling to cover pooling with imprecise probabilities by employing set-valued pooling functions and generalizing common pooling axioms accordingly. As a proof of concept, we then show that one IP construction satisfies a number of central pooling axioms that are not jointly satisfied by any of the standard pooling recipes on pain of triviality. Following Levi, 3–11, [39]), we also argue that IP models admit of a much better philosophical motivation as a model of rational consensus. (shrink)

Ludwig Wittgenstein gave priority to aesthetics over other disciplines due to its invaluable capacities for revealing certain aspects of the nature of human understanding and for guiding our actions toward an ethical life. Although Wittgenstein did not focus on these issues in a systematic way, these worries were present in his philosophy during his lifetime. That is why I use a very wide range of his writings, from the Tractatus to letters and diaries. Aesthetic inquiries can throw light upon the (...) existence of an intransitive understanding–that is, a kind of knowledge that cannot be expressed by language shared by all competent users of a language. I also discuss Wittgenstein’s conception of ethics in order to explain the close relationship that he proposed between ethics and aesthetics. I hope this will clarify why he believed there was so much to learn from paintings, films, music, poems, or novels. I compare works of art and religious belief and how the former can help us to guide our lives as the latter does. I do this while extending the scope and claiming the importance of aesthetic education, given that it can help us to deal with our nature and with our history. (shrink)

For much of its history, philosophy was not merely a theoretical discipline but a way of life, an "art of living." This practical aspect of philosophy has been much less dominant in modernity than it was in ancient Greece and Rome, when philosophers of all stripes kept returning to Socrates as a model for living. The idea of philosophy as an art of living has survived in the works of such major modern authors as Montaigne, Nietzsche, and Foucault. Each of (...) these writers has used philosophical discussion as a means of establishing what a person is and how a worthwhile life is to be lived. In this wide-ranging, brilliantly written account, Alexander Nehamas provides an incisive reevaluation of Socrates' place in the Western philosophical tradition and shows the importance of Socrates for Montaigne, Nietzsche, and Foucault. Why does each of these philosophers—each fundamentally concerned with his own originality—return to Socrates as a model? The answer lies in the irony that characterizes the Socrates we know from the Platonic dialogues. Socratic irony creates a mask that prevents a view of what lies behind. How Socrates led the life he did, what enabled or inspired him, is never made evident. No tenets are proposed. Socrates remains a silent and ambiguous character, forcing readers to come to their own conclusions about the art of life. This, Nehamas shows, is what allowed Montaigne, Nietzsche, and Foucault to return to Socrates as a model without thereby compelling them to imitate him. This highly readable, erudite study argues for the importance of the tradition within Western philosophy that is best described as "the art of living" and casts Montaigne, Nietzsche, and Foucault as the three major modern representatives of this tradition. Full of original ideas and challenging associations, this work will offer new ways of thinking about the philosophers Nehamas discusses and about the discipline of philosophy itself. (shrink)

“I can’t read. Show me” is a student’s cry heard by teachers of the arts in all kinds of classes. Demonstrating a particular process one on one is a very effective way to learn, but sometimes teachers need a way for students to take notes or follow a guide to aid in remembering a complex technique. Notation systems have developed as the educational solution to this need.1 Adela Bay, a private piano teacher, relates in her book’s dedication the reason she (...) developed a notation system for her music students, saying they were “intimidated by traditional music instruction.”2 In the arts, written notations are available from places and times that most people cannot visit in order to learn a particular technique.3 .. (shrink)

In these discussions, Rush Rhees, who was a student and close friend of Wittgenstein, works out what he has learned from Wittgenstein's personal teaching and from study of his published and (at the time) unpublished writings. Some are review articles of books on Wittgenstein, and these are devoted to exposition of Wittgenstein's views. Others are independent discussions of special points in Wittgenstein's philosophy. The longest article, here published for the first time, is an account or record of what Wittgenstein (...) said about continuity in private discussion in 1938. (shrink)

"First Published in 1991, Routledge is an imprint of Taylor & Francis, an informa company.".

Essays offer a glimpse of the Vienna-born philosopher's personality, character, and life's work.

Four years after the publication of Wittgenstein's Investigations, Rush Rhees began writing critical reflections on the masterpiece he had helped to edit. In this edited collection of his previously unpublished writings, Rhees argues, contra Wittgenstein, that although language lacks the unity of a calculus it is not simply a family of language games. The unity of language is found in its dialogical character. It is in this context that we say something, and grow in understanding: notions not captured in (...) Wittgenstein's emphasis on language games, following rules, and using language. Rhees develops Wittgenstein's notion that to imagine a language is to image a form of life, without suggesting that we are all engaged in an all-inclusive conversation. The result is not only a major contribution to Wittgenstein scholarship, but an original discussion of central philosophical questions concerning the possibility of discourse. (shrink)

In many assessment problems—aptitude testing, hiring decisions, appraisals of the risk of recidivism, evaluation of the credibility of testimonial sources, and so on—the fair treatment of different groups of individuals is an important goal. But individuals can be legitimately grouped in many different ways. Using a framework and fairness constraints explored in research on algorithmic fairness, I show that eliminating certain forms of bias across groups for one way of classifying individuals can make it impossible to eliminate such bias across (...) groups for another way of dividing people up. And this point generalizes if we require merely that assessments be approximately bias-free. Moreover, even if the fairness constraints are satisfied for some given partitions of the population, the constraints can fail for the coarsest common refinement, that is, the partition generated by taking intersections of the elements of these coarser partitions. This shows that these prominent fairness constraints admit the possibility of forms of intersectional bias. (shrink)

Our aim here is to present a result that connects some approaches to justifying countable additivity. This result allows us to better understand the force of a recent argument for countable additivity due to Easwaran. We have two main points. First, Easwaran’s argument in favour of countable additivity should have little persuasive force on those permissive probabilists who have already made their peace with violations of conglomerability. As our result shows, Easwaran’s main premiss – the comparative principle – is strictly (...) stronger than conglomerability. Second, with the connections between the comparative principle and other probabilistic concepts clearly in view, we point out that opponents of countable additivity can still make a case that countable additivity is an arbitrary stopping point between finite and full additivity. (shrink)