Pareto principles are at the core of ethics and decision theory. The Strong Pareto principle says that if one thing is better than another for someone and at least as good for everyone else, then the one is overall better than the other. But a host of famous figures express it differently, with ‘not worse’ in place of ‘at least as good.’ In the presence of parity (or incommensurability), this results in a strictly stronger Pareto principle, which I call (...) Super‐Strong Pareto. Super‐Strong Pareto, however, yields cyclic betterness and is therefore false. I point out a number of influential arguments—concerning population ethics, collective action problems, and decision‐making in the face of parity and uncertainty—that crucially rely on Super‐Strong Pareto and are therefore unsound. I then turn to the most influential argument against the possibility of parity—Broome's collapsing argument—and argue that it likewise relies on Super‐Strong Pareto reasoning and is therefore question‐begging. Finally, I turn to the much‐neglected question of how to justify Strong Pareto. The answer I arrive at, which emphasizes tie‐breaking, yields a striking insight, namely that Super‐Strong Pareto amounts to the denial of insensitivity to mild sweetening. That is what makes it problematic in the presence of parity. (shrink)

The (moral) permissibility of an act is determined by the relative weights of reasons, or so I assume. But how many weights does a reason have? Weight Monism is the idea that reasons have a single weight value. There is just the weight of reasons. The simplest versions hold that the weight of each reason is either weightier than, less weighty than, or equal to every other reason. We’ll see that this simple view leads to paradox in at least two (...) ways. We must complicate the picture somehow. I consider two candidate complications. The first, Parity Monism, is inspired by Ruth Chang’s suggestion that parity is a fourth comparative beyond the traditional three (>, <, =). This view complicates the single weight relation by allowing that the weights of reasons can be on a par. Unfortunately, Parity Monism resolves only one of the two paradoxes that afflict simple versions of Weight Monism. To resolve both paradoxes, we need our second candidate complication, Weight Pluralism. This view holds that reasons have at least two weight values (e.g., justifying weight and requiring weight) and these two values aren’t always equivalent. Parity is no substitute for Pluralism. (shrink)

Crispin Wright and Filippo Ferrari have accused relativism of not accounting for “parity” – the idea that, when we argue over matters of taste, we take our opponents’ opinions to be “as good as ours” from our own, committed perspective. In this paper, I show that i) explaining parity has not been taken to be a desideratum by relativists and thus they cannot be accused of failing to fulfil a promise; ii) Wright’s and Ferrari’s reasons for claiming that (...) parity should be a desideratum are unconvincing. (shrink)

This paper begins with a response to Josh Gert’s challenge that ‘on a par with’ is not a sui generis fourth value relation beyond ‘better than’, ‘worse than’, and ‘equally good’. It then explores two further questions: can parity be modeled by an interval representation of value? And what should one rationally do when faced with items on a par? I argue that an interval representation of value is incompatible with the possibility that items are on a par (a (...) mathematical proof is given in the appendix). I also suggest that there are three senses of ‘rationally permissible’ which, once distinguished, show that parity does distinctive practical work that cannot be done by the usual trichotomy of relations or by incomparability. In this way, we have an additional argument for parity from the workings of practical reason. (shrink)

This article explores the main similarities and differences between Derek Parfit’s notion of imprecise comparability and a related notion I have proposed of parity. I argue that the main difference between imprecise comparability and parity can be understood by reference to ‘the standard view’. The standard view claims that 1) differences between cardinally ranked items can always be measured by a scale of units of the relevant value, and 2) all rankings proceed in terms of the trichotomy of (...) ‘better than’, ‘worse than’, and ‘equally good’. Imprecise comparability, which can be understood in terms of the more familiar notions of cardinality and incommensurability, rejects only the first claim while parity rejects both claims of the standard view. -/- I then argue that insofar as those attracted to imprecise comparability assume that all rankings are trichotomous, as Parfit appears to, the view should be rejected. This is because imprecise equality is not a form of equality but is a sui generis ‘fourth’ basic way in which items can be ranked. We should, I argue, understand imprecise equality as parity, and imprecise comparability as entailing ‘tetrachotomy’ – that if two items are comparable, one must better than, worse than, equal to, or on a par with the other. Thus those attracted to the idea that cardinality can be imprecise should abandon trichotomy and accept parity and tetrachotomy instead. -/- Finally, I illustrate the difference between Parfit’s trichotomous notion of imprecise comparability and parity by examining how each notion might be employed in different solutions to the problem posed by the Repugnant Conclusion in population ethics. I suggest that parity provides the arguably more ecumenical solution to the problem. (shrink)

One Berkeleyan case for idealism, recently developed by Robert M. Adams, relies on a seeming disparity between our concepts of matter and mind. Thomas Reid’s critique of idealism directly challenges the alleged disparity. After highlighting the role of the disparity thesis in Adams’s updated Berkeleyan argument for idealism, this chapter offers an updated version of Reid’s challenge, and assesses its strength. What emerges from this historico-philosophical investigation is that a contemporary Reidian has much work to do to transpose her objections (...) to Berkeley into good objections to Adams’s argument. (shrink)

We can often permissibly choose a worse self-interested option over a better altruistic alternative. For example, it is permissible to eat out rather than donate the money to feed five hungry children for a single meal. If we eat out, we do something permissibly partial toward ourselves. If we donate, we go beyond the call of moral duty and do something supererogatory. Such phenomena aren’t easy to explain, and they rule out otherwise promising moral theories. Incommensurability and Ruth Chang’s notion (...) of parity can explain certain small improvement puzzles, but they can’t explain permissible partiality and supererogation. On the other hand, Josh Gert’s distinction between justifying and requiring weight can explain all three phenomena: permissible partiality, supererogation, and the relevant small improvement puzzle. Indeed, this chapter provides a reason to endorse the justifying/requiring weight distinction by showing that it provides the only extant explanation of all three phenomena. (shrink)

This article discusses the possibility of a rationally justified choice between two options neither of which is better than the other while they are not equally good either (‘3NT’). Joseph Raz regards such options as incomparable and argues that reason cannot guide the choice between them. Ruth Chang, by contrast, tries to show that many cases of putative incomparability are instead cases of parity—a fourth value relation of comparability, in addition to the three standard value relations ‘better than’, ‘worse (...) than’ and ‘equally good as’. It follows, she argues, that many choice situations in which rationally justified choice seems precluded are in fact situations within the reach of practical reason. This article has three aims: (1) it challenges Chang’s argument for the possibility of parity; (2) it demonstrates that, even if parity would exist, its problematic implications for practical reason would not differ from those of Raz’s incomparability; (3) it discusses the underlying cause of hard cases of comparison: the fact that none of the three standard value relations applies (‘3NT’). It will be shown that the problematic implications for the rational justification of the choice are due to 3NT itself, irrespective of whether 3NT is explained as incomparability or parity. (shrink)

Let’s say that you regard two things as on a par when you don’t prefer one to other and aren’t indifferent between them. What does rationality require of you when choosing between risky options whose outcomes you regard as on a par? According to Prospectism, you are required to choose the option with the best prospects, where an option’s prospects is a probability-distribution over its potential outcomes. In this paper, I argue that Prospectism violates a dominance principle—which I call The (...) Principle of Predominance—because it sometimes requires you to do something that’s no better than the alternatives and might be worse. I argue that this undermines the strongest argument that’s been given in favor of Prospectism. (shrink)

Ruth Chang has argued for the existence of a fourth positive value relation, distinct from betterness, worseness and equality, which she calls “parity.” In an earlier article I seemed to criticize Chang's suggestion by offering an interval model for the values of items that I claimed could accommodate all the phenomena characteristic of parity. Wlodek Rabinowicz, offering his own model of value relations, endorsed one central feature of my proposal: the need to distinguish permissible preferences from required ones. (...) But he, along with Chang, correctly pointed out that my own model suffered from serious formal problems. The present article offers an improved interval model, but the main point is not the model itself. Rather, it is the derivation of the model from a response-dependent account of practical rationality. This derivation helps explain why we have a concept for which Rabinowicz's model is appropriate, and therefore supplements Chang's existence arguments. My conclusion is that we can take Chang's view, Rabinowicz's and mine as providing a reinforcing set of views. (shrink)

Ruth Chang has defended a concept of "parity", implying that two items may be evaluatively comparable even though neither item is better than or equally good as the other. This article takes no stand on whether there actually are cases of parity. Its aim is only to make the hitherto somewhat obscure notion of parity more precise, by defining it in terms of the standard value relations. Given certain plausible assumptions, the suggested definiens is shown to state (...) a necessary and sufficient condition for parity, as this relation is envisaged by Chang. (shrink)

Can a person be harmed or wronged by being brought into existence? Can a person be benefited by being brought into existence? Following David Heyd, I refer to these questions as “genethical questions”. This chapter examines three broad approaches to genethics: the no-faults model, the dual-benchmark model, and the parity model. The no-faults model holds that coming into existence is not properly subject to moral evaluation, at least so far as the interests of the person that is to be (...) brought into existence are concerned. The dual benchmark model allows that coming into existence can be subject to moral evaluation, but holds that our judgments about the kinds of lives worth starting ought not be aligned with our judgments about the kinds of lives worth sustaining. I argue against both the no-faults and dual benchmark models. In their place, I argue for a parity approach to genethics, according to which our judgments about the conditions under which life is worth creating ought to be constrained by our judgments about the conditions under which life is worth continuing. (shrink)

I discuss the general form of arguments that profess to prove that the view that things endure in tensed time through causally produced change (the dynamic view) must be false because it involves contradictions. I argue that these arguments implicitly presuppose what has been called the temporal parity thesis, i.e. that all moments of time are equally existent and real, and that this thesis must be understood as the denial of the dynamic view. When this implicit premise is made (...) explicit, the arguments turn out to be either circular, they presuppose what they profess to prove, or mere demonstrations of the fact that the dynamic view is incompatible with its own negation. Furthermore, I discuss the metaphysical consequences of accepting the temporal parity thesis, arguing that it deprives us of the means to provide natural explanations to empirical phenomena. (shrink)

On parity views of mere addition if someone is added to the world at a range of well-being levels – or ‘neutral range’ – leaving existing people unaffected, addition is on a par with the initial situation. Two distinct parity views – ‘rough equality’ and fittingattitudes views – defend the ‘intuition of neutrality’. The first can be interpreted or adjusted so that it can rebut John Broome’s objection that the neutral range is wide. The two views respond in (...) distinct ways to two of Broome’s other objections. Both views can, nonetheless, be plausibly defended against these objections. (shrink)

The principles of ontological parity and ordinality have distinct functions in Buchler's ontology. Ontological parity could be independently subscribed to, whereas ordinality signals the positive conception of the nature of reality as irreducibly complex or indefinitely related, which Buchler's metaphysical system seeks to articulate. Both principles inform Buchler's system, but each has a distinctive function. They are not, I suggest, necessarily at odds with one another, as some critics claim. I do identify several difficulties that follow from (1) (...) the level of generality claimed by Buchler and (2) the claim of irreducible complexity or indefinite relatedness. (shrink)

According to Ruth Chang the three standard positive value relations: “better than”, “worse than” and “equally good” do not fully exhaust the conceptual space for positive value relations. According to her, there is room for a fourth positive value relation, which she calls “parity”. Her argument for parity comes in three parts. First, she argues that there are items that are not related by the standard three value relations. Second, that these items are not incomparable, and third, that (...) the phenomena she has focused on are not due to the vagueness of the comparative predicates. This paper focuses on the second part of the argument and an objection is presented. By assuming the Small Unidimensional Difference Principle, which is a key premise for the second part of the argument, Chang’s argument could be accused of begging the question. More so, by assuming this principle, the space for incomparability gets severely limited. If these worries are justified, then Chang’s argument for parity as a fourth form of comparability is unsuccessful. (shrink)

In the first part of this paper a relational account of incongruent counterparts is defended against an argument due to Kant. I then consider a more recent attack on such an account, due to John Earman, which alleges that the relationalist cannot account for the lawlike left--right asymmetry manifested in parity-violating phenomena. I review Hoefer's, Huggett's and Saunders' responses to Earman's argument and argue that, while a relationalist account of parity-violating laws is possible, it comes at the cost (...) of non-locality. (shrink)

According to the moral error theory, all moral judgments are false. Until lately, most error theorists were local error theorists; they targeted moral judgments specifically and were less skeptical of other normative areas. These error theorists now face so-called “prudential parity objections”, according to which whatever evidence there is in favor of the moral error theory is also evidence for a prudential error theory. The present paper rejects three prudential parity objections: one based on the alleged irreducible normativity (...) of prudential reasons; another on the lack of a story about the normativity of hypothetical reasons; yet another on the very nature of reasons generally. I argue that these objections leave an important variant of a local moral error theory intact. (shrink)

Andy Clark and David Chalmers (1998) argue that certain mental states and processes can be partially constituted by objects located beyond one’s brain and body: this is their extended mind thesis (EM). But they maintain that consciousness relies on processing that is too high in speed and bandwidth to be realized outside the body (see Chalmers, 2008, and Clark, 2009). I evaluate Clark’s and Chalmers’ reason for denying that consciousness extends while still supporting unconscious state extension. I argue that their (...) reason is not well grounded and does not hold up against foreseeable advances in technology. I conclude that their current position needs re-evaluation. If their original parity argument works as a defence of EM, they have yet to identify a good reason why it does not also work as a defence of extended consciousness. I end by advancing a parity argument for extended consciousness and consider some possible replies. (shrink)

It is often supposed that, given two potential objects of choice X and Y, a specific set of circumstances, and a specific choosing agent, one of the following must be true: (1) opting for X is a better choice than opting for Y, (2) opting for Y is a better choice than opting for X, or (3) opting for X and opting for Y are exactly equally good choices. My aim in this paper is to show how some philosophical insights (...) concerning color perception can illuminate the possibility of two options, X and Y, being such that, even given a specific set of circumstances and a specific choosing agent, none of (1), (2), or (3) holds, but X and Y are positively related to one another as “on a par.”. (shrink)

In other work I have argued that items can be on a par, where being on a par is a fourth, basic, sui generis value relation beyond the usual trichotomy of ‘better than’, ‘worse than’, and ‘equally good’. In this paper, I aim to marshal non-technical, intuitive arguments for this view. First, I try to cast doubt on the leading source of intuitive resistance to parity, the conviction that if two items are comparable, one must be better than the (...) other, worse than it, or they must be equally good. Second, I explain how parity can arise by appealing to an uncontroversial distinction between quantity and quality of value. I propose both sufficient conditions for parity and a nontechnical model of the notion. My overall aim is to bring into view a simple and intuitive picture of value – and more generally of normativity – in which parity plays a significant role. (shrink)

This article advocates a new interpretation ofinner experience– the experience that one has of one’s empirical-psychological features ‘from within’ – in Kant. It argues that for Kant inner experience is the empirical cognition of mental states, but not that of a persistent mental substance. The schema of persistence is thereby substituted with the regulative idea of the soul. This view is shown to be superior to two opposed interpretations: the parity view that regards inner experience as empirical cognition of (...) a mental object on a par with outer experience and the disparity view that denies altogether that inner experience is empirical cognition. (shrink)

Some philosophers believe that two objects of value can be ‘roughly equal’, or ‘on a par’, or belong to the same ‘clump’ of value in a sense that is fundamentally different from that in which some objects are ‘better than’, ‘worse than’, or ‘equally as good as’ others. This article shows that if two objects are on a par, or belong to the same clump, then an agent accepting a few plausible premises can be exploited in a money-pump. The central (...) premise of the argument is that value is choice-guiding. If one object is more valuable than another, then it is not permitted to choose the less valuable object; and if two objects are equally valuable it is permitted to choose either of them; and if two objects are on a par or belong to the same clump it is also permitted to choose either of them. (shrink)

In this paper I briefly set out Susan Moller Okin’s liberal feminist position and then rehearse a number of criticisms of Okin which together suggest that dismantling the gender system and adopting the principle of androgyny would not be compatible with liberalism. This incompatibility appears to vindicate an extreme feminist critique of liberalism. I argue that nevertheless a liberal feminism is possible. The liberal feminist ought to adopt the principle of parity, that is, guaranteed equal representation of both sexes (...) in parliament, rather than the requirement of androgyny. Parity follows from a conception of procedural justice, for it provides a mechanism which guarantees that the interests of both sexes are fairly represented in the legislature. Parity may also go some way to alleviating the tensions which exist between feminism and multiculturalism. (shrink)

Recent developments in theories for responsible innovation have focused on the importance of actively accounting for values in our technological designs. Leading among these theories is that of Value Sensitive Design which attempts to guide the design process on the basis of evaluative analysis. However, values often come into conflict and VSD has been criticized for not providing a proper method to resolve such inevitable conflicts. This paper examines three such methods and argues that although each has its merits, they (...) all fail to account for a common source of value conflicts known as value incommensurability. Drawing on literature from the field of axiology, this paper argues that by incorporating the evaluative relation of ‘parity’ each of these three methods, and the VSD framework in general, will be able to properly understand the relation which holds between conflicting design options stemming from the incommensurable of values and be able to guide designers in making rational decision in the face of such conflicts. (shrink)

Over the past decade, Nancy Fraser has developed a sophisticated theory of social justice. At its heart lies the principle of parity of participation, according to which all adult members of society must be in a position to interact with one another as peers. This article examines some obstacles to the implementation of that principle. Concentrating on the contemporary status order, it asks two specific questions. Is it possible to produce a precise account of how the status order might (...) need to be ordered for parity of participation to be realized? And is it possible to derive a detailed and coherent political strategy capable of achieving such parity within the status order? The argument of this article is that, while Fraser has recognized the difficulties that the complexity of the contemporary status order poses for achieving parity of participation, she has nevertheless underestimated those difficulties. If parity of participation requires status equality, important and difficult work remains to be done in delineating the nature, and demands, of equality in the contemporary status order. (shrink)

Why are hard decisions hard? According to the incomparabilists, hard choices are hard because the options cannot be compared. Proponents of parity hold that hard choices are hard because the options can be compared but only in terms of a fourth value relation—parity—in addition to the three standard relations: better, worse, and equally good. Others claim that hard choices are hard because it is vague (or indeterminate) how the options relate in terms of the three standard relations. Lastly, (...) there is the epistemicist. For the epistemicist, hard choices are hard because one is irresolvably ignorant about how the options compare. In the debate about hard choices, epistemicism is often mentioned but rarely defended. The present paper is a contribution to closing this gap. (shrink)

This paper argues for the existence of a fourth positive generic value relation that can hold between two items beyond ‘better than’, ‘worse than’, and ‘equally good’: namely ‘on a par’.

Larry Temkin challenged what seems to be an analytic truth about comparatives: if A is Φ-er than B and B is Φ-er than C, then, A is Φ-er than C. Ruth Chang denies a related claim: if A is Φ-er than B and C is not Φ-er than B, but is Φ to a certain degree, then A is Φ-er than C. In this paper I advance a context-sensitive semantics of gradability according to which the data uncovered by Temkin and (...) Chang leave both statements intact. (shrink)

This article begins from the premise that, in contemporary conditions of immense cultural, social and ethnic diversity, a just and stable political order must be one in which all citizens are able to enjoy due recognition. In order to determine what form such a politics of recognition must take in practice, the article focuses on the case of Northern Ireland. More specifically, it examines the principle of ‘parity of esteem’ which forms the keystone of the Belfast Agreement of 1998. (...) A detailed examination of this idea is conducted by developing a contrast between it and an alternative principle of ‘due recognition’. It is contended that, despite initial impressions, parity of esteem and due recognition are not direct rivals since they aim to occupy different places in a politics of recognition. To be specific, while the former describes a vertical relationship between state and citizen, the latter depicts a horizontal relationship between citizens themselves. Building on this analysis, it is claimed that this examination of the principle of parity of esteem reveals something interesting and important both about the specific case of Northern Ireland and about the necessary features of a politics of recognition in general. (shrink)

A diagrammatic representation is given of the 24 rays of Peres that makes it easy to pick out all the 512 parity proofs of the Kochen-Specker theorem contained in them. The origin of this representation in the four-dimensional geometry of the rays is pointed out.

It is shown how the 300 rays associated with the antipodal pairs of vertices of a 120-cell (a four-dimensional regular polytope) can be used to give numerous “parity proofs” of the Kochen–Specker theorem ruling out the existence of noncontextual hidden variables theories. The symmetries of the 120-cell are exploited to give a simple construction of its Kochen–Specker diagram, which is exhibited in the form of a “basis table” showing all the orthogonalities between its rays. The basis table consists of (...) 675 bases (a basis being a set of four mutually orthogonal rays), but all the bases can be written down from the few listed in this paper using some simple rules. The basis table is shown to contain a wide variety of parity proofs, ranging from 19 bases (or contexts) at the low end to 41 bases at the high end. Some explicit examples of these proofs are given, and their implications are discussed. (shrink)

Within debates concerning responsibility for ignorance the distinction between moral and factual ignorance is often treated as crucial. Many prominent accounts hold that while factual ignorance routinely exculpates, moral ignorance never does so. The view that there is an in-principle distinction between moral and factual ignorance has been referred to as the “Asymmetry Thesis.” This view stands in opposition to the “Parity Thesis,” which holds that moral and factual ignorance are in-principle similar. The Parity Thesis has been closely (...) aligned with volitionist accounts of moral responsibility, whereas the Asymmetry Thesis has been closely aligned with Quality of Will accounts. Two central questions are at work here: how ignorance excuses, and whether it excuses in the same way for both moral and factual ignorance. I will argue that these questions have often been confused in the present debate, and once we have distinguished more clearly between them, it seems that Quality of Will accounts are compatible with the Parity Thesis. And more generally: that the distinction between moral and factual ignorance is far less important in debates about responsibility for ignorance than it has often appeared. (shrink)

Some recent authors suggest that the extended view fails because it does not follow from functionalism. For although functionalism can tell us whether a system is cognitive, it does not show whether such a newly identified cognitive system can be attributed to the very same subject. I argue that Clark and Chalmers can dodge this attack by claiming that the Parity Principle is essentially an analogy. In their crucial thought experiment, it can be argued that Otto’s notebook is similar (...) to Inga’s biological memory in that they are functionally equivalent, and it seems that the only relevant difference between them is concerned with their being located inside/outside the skull/skin. Provided that Inga’s biological memory is part of Inga’s cognition, analogously, Otto’s notebook should also be regarded as part of Otto’s cognition. However, I argue that this alleged analogy does not hold because the location is not the only difference that matters. Otto’s notebook and Inga’s biological memory are taken as part of a whole for different reasons, and because of this, they actually belong to different kinds of wholes. Otto’s notebook is part of a whole because such a whole functions as a cognitive system, but Inga’s biological memory is part of a whole because it is within a “proper whole” whose boundary is determined by reproduction. As a result, the analogy does not really work. (shrink)

In this paper I will focus on the case of the discovery of parity nonconservation in weak interactions from the period spanning 1947–1957, and the lessons this episode provides for successful theory construction in HEP. I aim to (a) summarize the history into a coherent story for philosophers of science, and (b) use the history as a case study for the epistemological evolution of the understanding of weak interactions in HEP. I conclude with some philosophical lessons regarding theory construction (...) in physics. (shrink)

Sam Buss produced the first polynomial size Frege proof of thepigeonhole principle. We introduce a variation of that problem and producea simpler proof based on parity. The proof appearing here has an upperbound that is quadratic in the size of the input formula.

This commentary describes limitations of mental health parity requirements in ensuring access to insurance coverage for mental health treatment and surveys regulatory options employed by states in Medicaid managed care programs as supplements to parity that can further reduce the risk of inappropriate denials of coverage.

Causal selection is the task of picking out, from a field of known causally relevant factors, some factors as elements of an explanation. The Causal Parity Thesis in the philosophy of biology challenges the usual ways of making such selections among different causes operating in a developing organism. The main target of this thesis is usually gene centrism, the doctrine that genes play some special role in ontogeny, which is often described in terms of information-bearing or programming. This paper (...) is concerned with the attempt of confronting the challenge coming from the Causal Parity Thesis by offering principles of causal selection that are spelled out in terms of an explicit philosophical account of causation, namely an interventionist account. I show that two such accounts that have been developed, although they contain important insights about causation in biology, nonetheless fail to provide an adequate reply to the Causal Parity challenge: Ken Waters's account of actual-difference making and Jim Woodward's account of causal specificity. A combination of the two also doesn't do the trick, nor does Laura Franklin-Hall's account of explanation (in this volume). We need additional conceptual resources. I argue that the resources we need consist in a special class of counterfactual conditionals, namely counterfactuals the antecedents of which describe biologically normal interventions. (shrink)

One response to Derek Parfit's invokes the relation of . Since parity is a form of in John Broome's terms, three doubts which Broome raises about accounts involving incommensurateness in Weighing Lives pose a challenge for this response. I discuss two of these. They emerge from a discussion of various intuitions about . I argue that an account based on parity may be no less consistent with Broome's intuitions than is his own vagueness view.

The set of 60 real rays in four dimensions derived from the vertices of a 600-cell is shown to possess numerous subsets of rays and bases that provide basis-critical parity proofs of the Bell-Kochen-Specker (BKS) theorem (a basis-critical proof is one that fails if even a single basis is deleted from it). The proofs vary considerably in size, with the smallest having 26 rays and 13 bases and the largest 60 rays and 41 bases. There are at least 90 (...) basic types of proofs, with each coming in a number of geometrically distinct varieties. The replicas of all the proofs under the symmetries of the 600-cell yield a total of almost a hundred million parity proofs of the BKS theorem. The proofs are all very transparent and take no more than simple counting to verify. A few of the proofs are exhibited, both in tabular form as well as in the form of MMP hypergraphs that assist in their visualization. A survey of the proofs is given, simple procedures for generating some of them are described and their applications are discussed. It is shown that all four-dimensional parity proofs of the BKS theorem can be turned into experimental disproofs of noncontextuality. (shrink)

The Routley star, an involutive function between possible worlds or set-ups against which negation is evaluated, is a hallmark feature of Richard Sylvan and Val Plumwood's set-up semantics for the logic of first-degree entailment. Less frequently acknowledged is the weaker mate function described by Sylvan and his collaborators, which results from stripping the requirement of involutivity from the Routley star. Between the mate function and the Routley star, however, lies an broad field of intermediate semantical conditions characterizing an infinite number (...) of consequence relations closely related to first-degree entailment. In this paper, we consider the semantics and proof theory for deductive systems corresponding to set-up models in which the mate function is cyclical. We describe modifications to Anderson and Belnap's consecution calculus LE_fde2 that correspond to these constraints, for which we prove soundness and completeness with respect to the set-up semantics. Finally, we show that a number of familiar metalogical properties are coordinated with the parity of a mate function's period, including refined versions of the variable-sharing property and the property of gentle explosiveness. (shrink)

Uninformed learning mechanisms will not discover “type- 2” regularities in their inputs, except fortuitously. Clark & Thornton argue that error back-propagation only learns the classical parity problem – which is “always pure type-2” – because of restrictive assumptions implicit in the learning algorithm and network employed. Empirical analysis showing that back-propagation fails to generalise on the parity problem is cited to support their position. The reason for failure, however, is that generalisation is simply not a relevant issue. Nothing (...) can be gleaned about back-propagation in particular, or learning in general, from this failure. (shrink)

Clark & Thornton take issue with my claim that parity is not a generalisation problem, and that nothing can be inferred about back-propagation in particular, or learning in general, from failures of parity generalisation. They advance arguments to support their contention that generalisation is a relevant issue. In this continuing commentary, I examine generalisation more closely in order to refute these arguments. Different learning algorithms will have different patterns of failure: back-propagation has no special status in this respect. (...) This is not to deny that a particular algorithm might fortuitously happen to produce the “intended” function in an parity-generalisation task. (shrink)

This article shows that introducing the positive comparative relation parity only helps one block so‐called “Spectrum Arguments” in order to avoid their unsavoury implications if one specifies parity in a specific way with respect to its persistence. The article illustrates how parity must both admit of persistency and be weakly non‐persistent for parity to block Spectrum Arguments, and identifies some consequences of that discovery for the general debate on Spectrum Arguments, value theory and comparability problems.

Agacinski's articulation of the theory of "parity" helped inspire a law that went into effect in May 2000 requiring France's political parties to fill 50 percent of the candidacies in every race with women. Agacinski's model of parity does not strive for the nebulous ideal of "equality" between the sexes; instead, it demands a concrete formula for political contests: an equal number of female and male candidates in every election. This theory that has sparked impassioned debate across France (...) and around the globe. (shrink)

By forming the intersections of the parity and time reversal equivalence classes of physical entities that are represented by differential forms and differential form densities, a number of subsets of discrete symmetry classes for electromagnetic systems can be generated. Only one of these subsets is consistent with elementary thermodynamic arguments for dissipative systems and at the same time yields the notion that both charge and mass are spacetime scalars. This subset is not in correspondence with the two self-consistent presentations (...) that now are implied in the literature. (shrink)