We introduce a class of models of the bounded arithmetic theoryPV n . These models, which are generated by their tally part, have a curious feature: they have end-extensions or satisfyB∑ n b only in case they are closed under exponentiation. As an application, we show that if then the polynomial hierarchy does not collapse.

In 1941, Tarski published an abstract, finitely axiomatized version of the theory of binary relations, called the theory of relation algebras, He asked whether every model of his abstract theory could be represented as a concrete algebra of binary relations. He and Jonsson obtained some initial, positive results for special classes of abstract relation algebras. But Lyndon showed, in 1950, that in general the answer to Tarski's question is negative. Monk proved later that the answer remains negative even if one (...) adjoins finitely many new axioms to Tarski's system. In this paper we describe a far-reaching generalization of the positive results of Jonsson and Tarski, as well as of some later, related results of Maddux. We construct a class of concrete models of Tarski's axioms-called coset relation algebras-that are very close in spirit to algebras of binary relations, but are built using systems of groups and cosets instead of elements of a base set. The models include all algebras of binary relations, and many non-representable relation algebras as well, We prove that every atomic relation algebra satisfying a certain measurability condition-a condition generalizing the conditions imposed by Jonsson and Tarski-is essentially isomorphic to a coset relation algebra. The theorem raises the possibility of providing a positive solution to Tarski's problem by using coset relation algebras instead of the standard algebras of binary relations. (shrink)

We discuss some new properties of the natural Galois connection among set relation algebras, permutation groups, and first order logic. In particular, we exhibit infinitely many permutational relation algebras without a Galois closed representation, and we also show that every relation algebra on a set with at most six elements is Galois closed and essentially unique. Thus, we obtain the surprising result that on such sets, logic with three variables is as powerful in expression as full first order logic.

We introduce a variety of algebras in the language of Boolean algebras with an extra implication, namely the variety of pseudo-subordination algebras, which is closely related to subordination algebras. We believe it provides a minimal general algebraic framework where to place and systematise the research on classes of algebras related to several kinds of subordination algebras. We also consider the subvariety of pseudo-contact algebras, related to contact algebras, and the subvariety of the strict implication algebras introduced in Bezhanishvili et al. (...) [(2019). A strict implication calculus for compact Hausdorff spaces. Annals of Pure and Applied Logic, 170, 102714]. The variety of pseudo-subordination algebras is term equivalent to the variety of Boolean algebras with a binary modal operator. We exploit this fact in our study. In particular, to obtain a topological duality from which we derive the known topological dualities for subordination algebras and contact algebras. (shrink)

The ordering tendency in binary f -Ti solid solution containing 3sp or 4sp simple-metal or 3d transition-metal solute is investigated systematically by the linear muffin-tin orbital method within the atomic sphere approximation . We demonstrated that the effective pairwise interaction energy in a solid solution is equal to half the solute-solute interaction energies and can be evaluated by a supercell total energy approach. The calculations of EPI energy both with and without volume relaxation of the supercells and local density of (...) states show that the EPI energies of Ti-SM and Ti-TM solutions are dominated by different factors. For Ti-SM solutions, the EPI energies are of large absolute values with a negative sign, indicating strong ordering tendency in these solutions. The volume relaxation does not alter the EPI energy substantially. The calculated LDOS shows that the ordering tendency in Ti-SM solutions may be related to the hybridization between the electrons of the SM atoms when they are close to each other. For most Ti-TM solutions, if calculated without relaxation, the absolute EPI energies are very small; however, if calculated with relaxation, they are of relatively large positive values, indicating a clustering tendency in these solutions. By combining the calculated EPI energy and Flinn's model for short-range order strengthening, the increase in critical shear stress ‰ sro due to SRO is estimated for Ti-SM alloys, and the results qualitatively agree with experiment. (shrink)

A monosexual configuration of sexuality assumes that sexual desire is directed at either men or women. Bisexuality resists a choice between oppositional categories and is often theorised as having a transgressive potential to destabilise binary logic, not only in relation to sexuality but also to gender. There is, however, a lack of empirical work exploring how this potential might be realised in the accounts of bisexual individuals. Drawing on interviews with South African bisexual women, we use a narrative-discursive lens to (...) examine the discursive resources employed by participants to trouble or resist hetero-gendered norms. Our findings demonstrate how resistance to the gender binary hinges on citational politics that are fundamentally gendered and linked to sexuality. Instead of entirely destabilising hetero-gendered norms, participants draw on gendered scripts that simultaneously expand norms to accommodate their sexual difference and, through processes of othering, function to reiterate hetero-gendered norms. While complete subversion of gender binaries is not possible in participants’ discursive contexts, what does occur is a ‘slow bending’ of norms. Theorising bisexuality as transgressing oppositional categories closes off opportunities to interrogate the pervasive influence of gender binaries in contexts that remain marked by pervasive heteronormativity and heterosexism. Significantly, it also obscures more modest improvisations of gender scripts that hold potential for destabilising gender binaries. (shrink)

A new reduction class is presented for the satisfiability problem for well-formed formulas of the first-order predicate calculus. The members of this class are closed prenex formulas of the form ∀ x∀ yC. The matrix C is in conjunctive normal form and has no disjuncts with more than three literals, in fact all but one conjunct is unary. Furthermore C contains but one predicate symbol, that being unary, and one function symbol which symbol is binary.

We prove that there exists no sentence F of the language of rings with an extra binary predicat I2 satisfying the following property: for every definable set X ⊆ ℂ2, X is connected if and only if ⊧ F, where I2 is interpreted by X. We conjecture that the same result holds for closed subset of ℂ2. We prove some results motivated by this conjecture.

We investigate topological properties of subsets S of the real plane, expressed by first-order logic sentences in the language of the reals augmented with a binary relation symbol for S. Two sets are called topologically elementary equivalent if they have the same such first-order topological properties. The contribution of this paper is a natural and effective characterization of topological elementary equivalence of closed semi-algebraic sets.

This paper addresses how multiple individual credences on logically related issues should be aggregated into collective binary beliefs. We call this binarizing belief aggregation. It is vulnerable to dilemmas such as the discursive dilemma or the lottery paradox: proposition-wise independent aggregation can generate inconsistent or not deductively closed collective judgments. Addressing this challenge using the familiar axiomatic approach, we introduce general conditions on a binarizing belief aggregation rule, including rationality conditions on individual inputs and collective outputs, and determine which rules (...) (if any) satisfy different combinations of these conditions. Furthermore, we analyze similarities and differences between our proofs and other related proofs in the literature and conclude that the problem of binarizing belief aggregation is a free-standing aggregation problem not reducible to judgment aggregation or probabilistic opinion pooling. (shrink)

This thesis is a sceptical investigation into the notion that the metaphorical wave-particle binary of Virginia Woolf's To the Lighthouse is related to quantum physics. Indeed, the field of literature and science has employed conceptual similarities as the main means of connecting quantum concepts to novels, however, this has led to a host of scholarly difficulties, prompting the need for a re-examination of analogical linkages. Woolf is the model candidate for such a re-examination, given her historical and philosophical proximity with (...) the developments of quantum mechanics. To the Lighthouse, in particular, was written between 1925 and 1927: precisely when quantum physicists were attempting to resolve the wave-particle duality, leading to Niels Bohr's complementarity. This parallel has been noted by researchers as relevant to the novel, as it too displays a general binary that can be read as wave-particle-like, which the author also attempts to resolve along similar philosophical lines. Nevertheless, other than proximity and similarity, there are no reasons to affirm that the science is related to the novel; hence, it is an ideal case study to examine in order to ascertain the value of conceptual similarities in literature and science. To do so, To the Lighthouse's binary and its resolution are interpreted in a thesis-long close reading, in order to compare aspects of it firstly to Woolf's personal thought, and secondly to various non-quantum intellectual contexts that preceded and surrounded her. In doing so, it becomes clear not only that invoking quantum physics serves no clear purpose in better understanding the novel, but also that the notion of resolving wave- particle-like binaries was a widespread philosophical procedure at the turn of the century, decades before Bohr's concept. This negative conclusion interrogates what the scholarly value of interpreting similarities is, though a hypothetical solution can be found in conceptual metaphor theory. (shrink)

This paper explores issues of abstraction and space in Sande Zeig's movie The Girl (2001), based on a novella by Monique Wittig, who also co-wrote the script. It argues that, with this movie, Zeig and Wittig strive to re-materialize the lesbian body abstracted by the ‘Straight Mind’ as defined by Wittig in her 1980 essay. The plot revolves around the love affair of two women, the narrator and the Girl (a lesbian painter and a straight B-grade jazz singer), under the (...) oppressive scrutiny of the Man (the Girl's boss). Each part of this paper corresponds to one space where the characters interact: first, it shows how the nightclub stands as a space of transgression for the lesbian subject and how it opens a reflection on temporality as represented by space; then the analysis of the hotel room where the Girl lives and meets the narrator demonstrates how, notably through the use of close-ups in erotic scenes, Zeig redefines abstraction as a new way to envisage the materiality of the lesbian body; finally, in the third part, the discussion on abstraction moves on to the narrator's studio where, through her work as a painter, she also explores abstract representations of the female body. This approach reconciles Wittigian theory with post-modern queer theory, and confirms that the former still offers paths to positive reformulations of modes of relationality through the de-spatialization of gender – de-genderization; the rematerialization of the body outside of heteronormative conventions; and a reconfiguration of human relationships that refuses the inevitability of social and amorous hierarchy, and that shifts the representation of the erotic from possessiveness to disinterested reciprocity. (shrink)

This essay delves into Alan Turing's "Computing Machinery and Intelligence", examining the entrenched the binary opposition between human and machine. Through deconstruction, the essay seeks to reveal and analyze the deeper insights inherent in Turing's work. Of particular significance is Turing's equivalence between language and thought, particularly in the context of the imitation game. Within the grammatological realm of Turing's text, writing and mechanism are nearly interchangeable, representing the intersection of humans and machines. By adopting a deconstructive perspective, the essay (...) illuminates the philosophical origins of the artificial intelligence debates and their implications for our understanding of human existence. (shrink)

In "Doing Well Enough: Toward a Logic for Common Sense Morality", Paul McNamara sets out a semantics for a deontic logic which contains the operator It is supererogatory that. As well as having a binary accessibility relation on worlds, that semantics contains a relative ordering relation, . For worlds u, v and w, we say that u w v when v is at least as good as u according to the standards of w. In this paper we axiomatize logics complete (...) over three versions of the semantics. We call the strongest of these logics DWE for Doing Well Enough. (shrink)

Algebraic structures on linguistic sets associated with a linguistic variable are introduced. The linguistics with single closed binary operations are only semigroups and monoids. We describe the new notion of linguistic semirings, linguistic semifields, linguistic semivector spaces and linguistic semilinear algebras defined over linguistic semifields. We also define algebraic structures on linguistic subsets of a linguistic set associated with a linguistic variable. We define the notion of linguistic subset semigroups, linguistic subset monoids and their respective substructures. We also define as (...) in case of deals in classical semigroups, linguistic ideals in linguistic semigroups and linguistic monoids. This concept of linguistic ideals is extended to the case of linguistic subset semigroups and linguistic subset monoids. We also define linguistic substructures. (shrink)

Technologies of communication condition human sense-making. They do so by creating the social environment we inhabit and extending their structural biases and logics through human use. As such, this essay inquires into the prevailing habits of mind in the digital era. Employing a media ecology of communication, I argue that digital computers and microprocessors are defined by three structural properties and, hence, underlying logics: digitization (binary code), algorithmic execution (input/output), and efficiency (machine logic). Repeated exposure to these logics cultivates a (...) digital mind, a model of thinking, communicating, and sense-making characterized by intransigence, impertinence, and impulsivity. I conclude the essay by exploring the broader implications of a digital mind, paying particular attention to the challenges it poses to democratic politics. (shrink)

The binary divide between traditional cognitivist and enactivist paradigms is tied to their respective commitments to understanding cognition as based on knowing that as opposed to knowing how. Using O’Regan’s and No¨e’s landmark sensorimotor contingency theory of perceptual experience as a foil, I demonstrate how easy it is to fall into conservative thinking. Although their account is advertised as decidedly ‘skill-based’, on close inspection it shows itself to be riddled with suppositions threatening to reduce it to a rules-and-representations approach. (...) To remain properly enactivist it must be purged of such commitments and indeed all commitment to mediating knowledge: it must embrace a more radical enactivism. (shrink)

This binary also speaks more generally to how we tend to conceive of the relation between place and space: As many scholars have pointed out, place often comes valued as "the sphere of the everyday, or real and valued practices," or "a locus of denial" trailing ideas of the "closed, coherent, integrated as authentic, as 'home', a secure retreat" (Massey 2005, 5-6). Doreen Massey's descriptions speak to our perceptions of a center, of a core toward which energy flows, and in (...) so doing marking, in Michel de Certeau's words, "an instantaneous configuration of positions. (shrink)

The algebra of all binary relations on a given set is the most important example of a relation algebra . In this note we will examine the possible isomorphisms within some subclasses of a closely related class ; A is a relation set algebra with base U if its Boolean reduct is a field of sets with unit element 2 U, its universe A contains the identity relation on U and it is closed under the operations −1 and |, where (...) R |S = { : ∈ R, ∈ S for some z} and R−1 = { : ∈ R} for any R, S ⊆ 2 U. The class of all relation set algebras is denoted by Rs. (shrink)

An infinite binary sequence X is Kolmogorov–Loveland random if there is no computable non-monotonic betting strategy that succeeds on X in the sense of having an unbounded gain in the limit while betting successively on bits of X. A sequence X is KL-stochastic if there is no computable non-monotonic selection rule that selects from X an infinite, biased sequence.One of the major open problems in the field of effective randomness is whether Martin-Löf randomness is the same as KL-randomness. Our first (...) main result states that KL-random sequences are close to Martin-Löf random sequences in so far as every KL-random sequence has arbitrarily dense subsequences that are Martin-Löf random. A key lemma in the proof of this result is that for every effective split of a KL-random sequence at least one of the halves is Martin-Löf random. However, this splitting property does not characterize KL-randomness; we construct a sequence that is not even computably random such that every effective split yields two subsequences that are 2-random. Furthermore, we show for any KL-random sequence A that is computable in the halting problem that, first, for any effective split of A both halves are Martin-Löf random and, second, for any computable, nondecreasing, and unbounded function g and almost all n, the prefix of A of length n has prefix-free Kolmogorov complexity at least n−g. Again, the latter property does not characterize KL-randomness, even when restricted to left-r.e. sequences; we construct a left-r.e. sequence that has this property but is not KL-stochastic and, in fact, is not even Mises–Wald–Church stochastic.Turning our attention to KL-stochasticity, we construct a non-empty class of KL-stochastic sequences that are not weakly 1-random; by the usual basis theorems we obtain such sequences that in addition are left-r.e., are low, or are of hyperimmune-free degree.Our second main result asserts that every KL-stochastic sequence has effective dimension 1, or equivalently, a sequence cannot be KL-stochastic if it has infinitely many prefixes that can be compressed by a factor of α<1. This improves on a result by Muchnik, who has shown that were they to exist, such compressible prefixes could not be found effectively. (shrink)

The science of operations research is one of the modern sciences that have made a great revolution in all areas of life through the methods provided by it, suitable and appropriate to solve most of the problems that were facing researchers, scholars and those interested in the development of societies, and the most beneficiaries of this science were companies and institutions that are looking for scientific methods that help them manage their work so that they achieve the greatest profit and (...) the lowest cost, and one of the important methods that have been used in the management of companies we offer in this research two methods, Dynamic programming method. This method has been used in many practical matters and helped decision-makers in companies to achieve a maximum profit and less cost by formulating the reality of the state of the company and the data provided by decision-makers with a dynamic mathematical model that is solved using methods of solving dynamic models and we will provide in this research an example of this through the issue of choosing the optimal investment for the budget of a company so that it achieves a maximum profit, and the method of programming with integers: the method that provided these companies with solutions with integer values suitable for the nature of its work, through the use of the binary integer in the formulation of the appropriate mathematical model on the one hand, and on the other hand, the use of the binary integer variable helped to convert some nonlinear models that lead to some practical problems into linear models, and it should be noted here that in the previous two methods there is something indeterminable because we must make a decision in choosing or not choosing something, but the optimal solution that we will get remains A specific value because we are building the mathematical model for any realistic issue through the data provided by those responsible for the work and these data are calculated quantities and therefore they are uncertain values because their validity depends on the circumstances surrounding the work environment, they may be exposed to increase or decrease, and therefore the optimal solution on which the company will base its decision is suitable for specific values and any change in them can cause the company an uncalculated loss, so in this research we will use the concepts of neutrosophic science, the branch of science founded by the American scientist Florentin Smarandache in 1995 based on his belief that there is no absolute truth, a science that is interested in the study of ideas and concepts that are neither true nor false, but just in between, and we will take the data (calculated quantities) neutrosophic values that are specified or unspecified values are any set close to the calculated quantities, then the resulting mathematical model is a neutrosophic model and the optimal solution has neutrosophic values and thanks to the indefinite uncertainty that these values have, companies from the development of appropriate plans for all circumstances and thus achieve the greatest profit and the lowest cost, and we will clarify the above through two issues, the issue of optimal designation of a warehouse site, which we will formulate the mathematical model of using the neutrosophic integer programming method - and the issue of capital budget, which we will present in two different forms, we use in the first form the neutrosophic integer programming method and in the second the neutrosophic dynamic programming method. (shrink)

We introduce and study a class ofbetweenness algebras—Boolean algebras with binary operators, closely related to ternary frames with a betweenness relation. From various axioms for betweenness, we chose those that are most common, which makes our work applicable to a wide range of betweenness structures studied in the literature. On the algebraic side, we work with two operators ofpossibilityand ofsufficiency.

This paper is about a substitution-failure in attitude ascriptions, but not the one you think. A standard view about the semantic shape of ‘that’-clause attitude ascriptions is that they are fundamentally relational. The attitude verb expresses a binary relation whose extension, if not empty, is a collection of pairs each of which consists in an individual and a proposition, while the ‘that’-clause is a term for a proposition. One interesting problem this view faces is that, within the scope of many (...) attitude verbs, ‘that’-clauses are not interchange able with certain other terms which stand for the same propositions as the clauses are supposed to. For example, (1a) below may be true, but (1c) is probably not: (1) a. Holmes {fears/suspects} that Moriarty has returned. b. That Moriarty has returned is the proposition that Moriarty has re turned. c. Holmes {fears/suspects} the proposition that Moriarty has returned. For only the exceptionally timorous fear propositions, and, excluding contexts in which we speak of a ‘suspect premise’, only the unusually paranoid suspect them. The truth-conditional change eVected by substituting propositional descrip tion for ‘that’- clause illustrated in (1) occurs with a wide range of attitude verbs. If you understand that the window of opportunity is closing, perhaps you should act before it is too late, but if you mere-. (shrink)

We investigate the connections between computability theory and Nonstandard Analysis. In particular, we investigate the two following topics and show that they are intimately related. A basic property of Cantor space$2^ $ is Heine–Borel compactness: for any open covering of $2^ $, there is a finite subcovering. A natural question is: How hard is it to compute such a finite subcovering? We make this precise by analysing the complexity of so-called fan functionals that given any $G:2^ \to $, output a (...) finite sequence $\langle f_0, \ldots,f_n \rangle $ in $2^ $ such that the neighbourhoods defined from $\overline {f_i } G\left$ for $i \le n$ form a covering of $2^ $. A basic property of Cantor space in Nonstandard Analysis is Abraham Robinson’s nonstandard compactness, i.e., that every binary sequence is “infinitely close” to a standard binary sequence. We analyse the strength of this nonstandard compactness property of Cantor space, compared to the other axioms of Nonstandard Analysis and usual mathematics.Our study of yields exotic objects in computability theory, while leads to surprising results in Reverse Mathematics. We stress that and are highly intertwined, i.e., our study is holistic in nature in that results in computability theory yield results in Nonstandard Analysis and vice versa. (shrink)

Aims: The experience of a compromised sense of ‘self’, termed ego-dissolution, is a key feature of the psychedelic experience and acute psychosis. This study aimed to validate the Ego-Dissolution Inventory (EDI), a new 8-item self-report scale designed to measure ego-dissolution. Additionally, we aimed to investigate the specificity of the relationship between psychedelics and ego-dissolution. Method: Sixteen items relating to altered ego-consciousness were included in an internet questionnaire; 8 relating to the experience of ego-dissolution (comprising the EDI), and 8 relating to (...) the antithetical experience of increased self-assuredness. Items were rated using a visual analogue scale. Participants answered the questionnaire for experiences with classical psychedelic drugs, cocaine or alcohol. They also answered the 7 questions from the Mystical Experiences Questionnaire (MEQ) relating to the experience of unity with one’s surroundings. Results: 691 participants completed the questionnaire, providing data for 1828 drug experiences (1043 psychedelics. 377 cocaine. 408 alcohol). Exploratory factor analysis demonstrated that the 8 EDI items loaded exclusively onto a single common factor, which was orthogonal to a second factor comprised of the items relating to increased self-assuredness (rho= -.110), demonstrating discriminant validity. The EDI correlated strongly with our measure of unitive experience (rho =.735), demonstrating convergent validity. EDI internal consistency was excellent (Cronbach’s alpha 0.93). Three analyses confirmed the specificity of ego-dissolution for experiences occasioned by psychedelic drugs. Firstly, EDI score correlated with drug-dose for psychedelic drugs (rho=.371), but not for cocaine (rho=.115) or alcohol (rho=-0.055). Secondly, the linear regression line relating the subjective intensity of the experience to EDI was significantly steeper for psychedelics (unstandardized B coefficient= 0.701) compared with cocaine (0.135) or alcohol (0.144). Finally, a binary support vector machine classifier identified experiences occasioned by psychedelic drugs vs. cocaine or alcohol with over 85% accuracy using ratings of ego-dissolution and ego-inflation alone. Conclusions: Our results demonstrate the psychometric structure, internal consistency and construct validity of the EDI. Moreover, we demonstrate the close relationship between ego-dissolution and the psychedelic experience. The EDI will facilitate the study of the neuronal correlates of ego-dissolution, which is relevant for psychedelic-assisted psychotherapy and our understanding of psychosis. (shrink)

Computationalism about the brain is the view that the brain literally performs computations. For the view to be interesting, we need an account of computation. The most well-developed account of computation is Turing Machine computation, the account provided by theoretical computer science which provides the basis for contemporary digital computers. Some have thought that, given the seemingly-close analogy between the all-or-nothing nature of neural spikes in brains and the binary nature of digital logic, neural computation could be a species (...) of digital computation. A few recent authors have offered arguments against this idea; here, I review recent findings in neuroscience that further cement the implausibility of this view. However, I argue that we can retain the view that the brain is a computer if we expand what we mean by “computation” to include analog computation. I articulate an account of analog computation as the manipulation of analog representations based on previous work on the difference between analog and non-analog representations, extending a view originally articulated in Shagrir :271–279, 2010). Given that analog computation constitutes a significant chapter in the history of computation, this revision of computationalism to include analog computation is not an ad hoc addition. Brains may well be computers, but of the analog kind, rather than the digital kind. (shrink)

A model of judgment aggregation is presented in which judgments on propositions are not binary but come in degrees. The primitives are a set of propositions, an entailment relation, and a “triangular norm” which establishes a lower bound on the degree to which a proposition is true whenever it is entailed by a set of propositions. Under standard assumptions, we identify a necessary and sufficient condition for the collective judgments to be both deductively closed and free from veto power. This (...) condition says that the triangular norm used to establish the lower bound must contain a zero divisor. (shrink)

Reuven Leigh provides the first in-depth introduction to the pioneering philosophy of Rabbi Shalom Schneersohn. Bringing him into dialogue with key continental philosophers Emmanuel Levinas, Jacques Derrida and Julia Kristeva, this book reveals how Schneersohn's views anticipated many prominent themes in 20th-century thought. Shalom Schneersohn (1860-1920) was the fifth Rebbe of the Habad-Lubavitch dynasty. He was a traditional, kabbalistic thinker and yet, beyond mysticism, he wrote extensively on speech, gender and the body. So why is he not better known? Leigh (...) begins by uncovering and contesting numerous scholarly assumptions that have operated to exclude traditional rabbinic thinkers from contemporary philosophical debates. Seeking to correct this, this book offers a close reading of Schneersohn's 1898 discourses. With the disruption of traditional binary structures being the dominant theme pervading Schneersohn's work, Leigh challenges Levinas' controversial ideas on the feminine. Examining Schneersohn on language, too, he highlights how Derridean deconstruction involves a more positive approach to presence that was already anticipated in the writings of Schneersohn. And from the disruption of the hierarchy of signification to the semiotic aspect of language and the maternal body, this book demonstrates how Schneersohn foreshadows a number of Kristeva's central philosophical concerns. A wide-ranging and inclusive volume, The Philosophy of Rabbi Shalom Schneersohn demonstrates not only how forward-thinking Schneersohn's ideas were a century ago, but how relevant they still are today. (shrink)

We introduce the notions of stationarily ordered types and theories; the latter generalizes weak o-minimality and the former is a relaxed version of weak o-minimality localized at the locus of a single type. We show that forking, as a binary relation on elements realizing stationarily ordered types, is an equivalence relation and that each stationarily ordered type in a model determines some order-type as an invariant of the model. We study weak and forking non-orthogonality of stationarily ordered types, show that (...) they are equivalence relations and prove that invariants of non-orthogonal types are closely related. The techniques developed are applied to prove that in the case of a binary, stationarily ordered theory with fewer than countable models, the isomorphism type of a countable model is determined by a certain sequence of invariants of the model. In particular, we confirm Vaught's conjecture for binary, stationarily ordered theories. (shrink)

Category Theory has developed rapidly. This book aims to present those ideas and methods which can now be effectively used by Mathe­ maticians working in a variety of other fields of Mathematical research. This occurs at several levels. On the first level, categories provide a convenient conceptual language, based on the notions of category, functor, natural transformation, contravariance, and functor category. These notions are presented, with appropriate examples, in Chapters I and II. Next comes the fundamental idea of an adjoint (...) pair of functors. This appears in many substantially equivalent forms: That of universal construction, that of direct and inverse limit, and that of pairs offunctors with a natural isomorphism between corresponding sets of arrows. All these forms, with their interrelations, are examined in Chapters III to V. The slogan is "Adjoint functors arise everywhere". Alternatively, the fundamental notion of category theory is that of a monoid -a set with a binary operation of multiplication which is associative and which has a unit; a category itself can be regarded as a sort of general­ ized monoid. Chapters VI and VII explore this notion and its generaliza­ tions. Its close connection to pairs of adjoint functors illuminates the ideas of universal algebra and culminates in Beck's theorem characterizing categories of algebras; on the other hand, categories with a monoidal structure lead inter alia to the study of more convenient categories of topological spaces. (shrink)

Abstract.Formal topologies are today an established topic in the development of constructive mathematics. One of the main tools in formal topology is inductive generation since it allows to introduce inductive methods in topology. The problem of inductively generating formal topologies with a cover relation and a unary positivity predicate has been solved in [CSSV]. However, to deal both with open and closed subsets, a binary positivity predicate has to be considered. In this paper we will show how to adapt to (...) this framework the method used to generate inductively formal topologies with a unary positivity predicate; the main problem that one has to face in such a new setting is that, as a consequence of the lack of a complete formalization, both the cover relation and the positivity predicate can have proper axioms. (shrink)

In Telling Flesh: the Substance 0f the C0rporeul, Vicki Kirby suggests, among other things, that it is not in the interests of feminism to propound what she describes as an ‘inessentialist’ position in regards to embodiment. While she objects to undifferentiating biological givens that might, for example, attempt to construe women as confined to a nurturing role, she also does not want to simplistically insist that embodiment has nothing to do with subjectivity. To pose the problem in terms more closely (...) aligned with her own, Kirby is wary of the tendency to simply reverse binary oppositions, to swap nature for culture, reality for representation, and originary cause for interpretive effect. According to her, themes like ‘textuality’, and linguistic ideality have all but replaced the notion of ‘reality’. As arguably the pre-eminent ‘continental’ philosopher of our generation, the work of Derrida is invariably associated with this reversal of binary oppositions that seem to prohibit recourse to questions concerning embodiment. Several critics have even suggested that deconstruction is nothing but semiological reductionism in disguise. However, Kirby’s thesis, via an extended meditation upon Derrida’s claim that "there is nothing outside of the text," constitutes an important attempt to redeem him from such criticism. Rather than eschewing any and every reference to the body, she wants to insist that deconstruction cannot be contained within such a framework, and that it makes sense, within the logic of Of Grummutology (and she also pays cursory attention to Derrida’s ""Eating Well," or the Calculation of the Subject"), to conceive of embodiment in deconstructive terms. Examining the coherence of this claim will be the main focus of this paper, though in order to facilitate this task, this paper will also compare the notion of embodiment that Kirby espouses, to a curiously similar conception of the body that Merleau-Ponty theorizes in his unfinished text The Wsible und the Invisible.. (shrink)

A key issue in the philosophy of biology is evolutionary contingency, the degree to which evolutionary outcomes could have been different. Contingency is typically contrasted with evolutionary convergence, where different evolutionary pathways result in the same or similar outcomes. Convergences are given as evidence against the hypothesis that evolutionary outcomes are highly contingent. But the best available treatments of contingency do not, when read closely, produce the desired contrast with convergence. Rather, they produce a picture in which any degree of (...) contingency is compatible with any degree of convergence. This is because convergence is the repeated production of a given outcome from different starting points, and contingency has been defined without reference to the size of the space of possible outcomes. In small spaces of possibilities, the production of repeated outcomes is almost assured. This paper presents a definition of contingency which includes this modal dimension in a way that does not reduce it to the binary notion of contingency found in standard modal logic. The result is a conception of contingency which properly contrasts with convergence, given some domain of possibilities and a measure defined over it. We should therefore not ask whether evolution is contingent or convergent simpliciter, but rather about the degree to which it is contingent or convergent in various domains, as measured in various ways. (shrink)

This essay argues that Zizek’s post-Lacanian critique of contemporary culture stays within the logic of the discourse of the university and often functions to repress psychoanalysis and the unconscious. By looking at how Zizek divides Lacan work into a bad early Symbolic stage and a good late period that promotes the Real, enjoyment, and the death drive, I reveal how this binary and linear reading functions to efface important connections and differences concerning the key concepts of psychoanalysis. In fact, Zizek’s (...) work utilizes the obsessional processes of negation, splitting, fragmenting, and projection in order to absorb analytic practice into a self-consuming Hegelian negative dialectic. Moreover, his use of repetition in his own writings replicates the short-circuit of the death drive that finds enjoyment by returning to the subject in an act of self-negation. In looking closely at how Zizek reads Lacan, I place Zizek’s work within the contemporary backlash against progressive social movements and the social construction of knowledge. Thus, by creating a closed circle between the subject of autonomous repetition and the universal discourse of global nihilism, he represses social Symbolic mediation and reiterates a fundamentalist logic of obscene enjoyment. In fact, we can read his use of humor and popular culture as an effort to bond with his audience by providing enjoyment in exchange for the suspension of criticism and responsibility. In short, I argue that Zizek’s work is structured like a joke. (shrink)

We propose a logic for reasoning about metric spaces with the induced topologies. It combines the 'qualitative' interior and closure operators with 'quantitative' operators 'somewhere in the sphere of radius r.' including or excluding the boundary. We supply the logic with both the intended metric space semantics and a natural relational semantics, and show that the latter (i) provides finite partial representations of (in general) infinite metric models and (ii) reduces the standard '∈-definitions' of closure and interior to simple constraints (...) on relations. These features of the relational semantics suggest a finite axiomatisation of the logic and provide means to prove its EXPTIME-completeness (even if the rational numerical parameters are coded in binary). An extension with metric variables satisfying linear rational (in)equalities is proved to be decidable as well. Our logic can be regarded as a 'well-behaved' common denominator of logical systems constructed in temporal, spatial, and similarity-based quantitative and qualitative representation and reasoning. Interpreted on the real line (with its Euclidean metric), it is a natural fragment of decidable temporal logics for specification and verification of real-time systems. On the real plane, it is closely related to quantitative and qualitative formalisms for spatial representation and reasoning, but this time the logic becomes undecidable. (shrink)

Blockage contraction is an operation of belief contraction that acts directly on the outcome set, i.e. the set of logically closed subsets of the original belief set K that are potential contraction outcomes. Blocking is represented by a binary relation on the outcome set. If a potential outcome X blocks another potential outcome Y, and X does not imply the sentence p to be contracted, then Y ≠ K ÷ p. The contraction outcome K ÷ p is equal to the (...) (unique) inclusion-maximal unblocked element of the outcome set that does not imply p. Conditions on the blocking relation are specified that ensure the existence of such a unique inclusion-maximal set for all sentences p. Blockage contraction is axiomatically characterized and its relations to AGM-style operations are investigated. In a finite-based framework, every transitively relational partial meet contraction is also a blockage contraction. (shrink)

We study the partial orderings of the form ⟨P,⊂⟩\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\langle \mathbb{P}, \subset\rangle}$$\end{document}, where X\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb{X}}$$\end{document} is a binary relational structure with the connectivity components isomorphic to a strongly connected structure Y\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb{Y}}$$\end{document} and P\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb{P} }$$\end{document} is the set of substructures of X\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} (...) \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {X}}$$\end{document} isomorphic to X\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb{X}}$$\end{document}. We show that, for example, for a countable X\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb{X}}$$\end{document}, the poset ⟨P,⊂⟩\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\langle \mathbb {P}, \subset\rangle}$$\end{document} is either isomorphic to a finite power of P\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb{P} }$$\end{document} or forcing equivalent to a separative atomless σ-closed poset and, consistently, to P/fin. In particular, this holds for each ultrahomogeneous structure X\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb{X}}$$\end{document} such that X\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb{X}}$$\end{document} or Xc\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb{X}^{c}}$$\end{document} is a disconnected structure and in this case Y\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb{Y}}$$\end{document} can be replaced by an ultrahomogeneous connected digraph. (shrink)

It is clear now that open education is much more than a binary consideration of open versus closed but also includes "opening." This book maps a range of different theoretical and practice-oriented approaches and proposals to (re)considering open education.

continent. 1.4 (2011): 286—310. This mad play of writing —Stéphane Mallarmé Somewhere in between mathematics and theory, light and dark, physicality and projection, oscillates the poetry of Alessandro De Francesco. The texts hold no periods or commas, not even a capital letter for reference. Each piece stands as an individual construction, and yet the poetry flows in and out of the frame. Images resurface from one poem to the next, haunting the reader with reincarnations of an object lost in the (...) grass or a representation of a pear in a Dutch still life, embedded in cycles of cinematic close ups and multiple dimensions. As a whole, De Francesco’s oeuvre suggests, to use the title of one of his works, a redefinition —linguistic, epistemological, personal. The poem The End (un’agenda) (The End (an agenda)) opens with a quote by literary theorist Maurice Blanchot, which may be translated as follows: Speaking is not seeing. Speaking frees thought from this optical imperative that in the Western tradition, for thousands of years, has subjugated out approach to things, and induced us to think under the guaranty of light or under the threat of its absence.[1] In writing, De Francesco seeks to redefine our approach to things. His poems, like Blanchot’s writings, speak to both illuminate and to obscure. The reader hangs in an absence of reality within language that necessitates an attempt at interpretation, yet renders such a search ultimately futile. De Francesco draws the anonymous images that appear in this poem from the city of Paris, his current home. The fragmented narrative depicts an inner doubt of realness, calling for a confirmation of a physical existence beyond verbalization and cogitation. Cyclical references to film throw into question the reliability of the reality he creates. The light, sometimes red and safe, other times absent and murky, cannot necessarily be trusted as a guaranteed means of illumination. Blanchot’s distinction between language and visualization, and the relationship of each to reality, though linked specifically to The End (un’agenda) , can be a lens through which to read any of the three works that follow. assente obliqua (absent oblique) is composed of descriptions of six still life paintings, ranging from Dutch masterpieces of the seventeenth century to 1960s minimalism. The actual images of the paintings are absent, and the text is the sole source and object of a virtual reality created around the artwork. “Absent oblique” refers to the light source placed outside of the frame, which both informs and distracts viewing of the scene. Although mere description can be unreliable and unsatisfying in its necessary limitation, the representational act of description is shown as more than just a human want. It is a need, a means by which to grapple with an unsure reality. As the illustrations progress, they descend into abstraction and chaos. The precariously placed plate triggers the movement: the bottle bursts and the objects collapse into the movement of a tracking film camera. corpo estraneo in moto ascensionale (foreign body in ascending motion) is prose poetry in action. Its objective tone of description is skewed by the subjectivity of the narrative as horrific images are relayed in fragmented but matter-of-fact manner. A “foreign body” is both a mass of cancerous cells, and a corporeal description, and the poem takes the reader on a journey back and forth between the two, destination unclear. Beyond a questioning of the real, corpo estraneo in moto ascensionale is an examination of the possible. De Francesco’s poetry becomes a camera obscura through which to observe the constant flux of light and dark that blurs our reality. His work is colored by the idea of apo koinou , a Greek term meaning “from the common” that involves a construction of two clauses with a shared word or phrase. Through such melting of meaning, De Francesco creates a fluid ambiguity that is nonetheless grounded in bodily concepts. In assente obliqua , the reader is confronted with a description of a still life that rapidly transforms into a more chilling scene: the membrane of the mushroom in the basket emerges behind the white cloth a body could hide itself that asks for help In the realm of pure description, there is a mushroom that peeks out from behind a white cloth in a basket. Behind the white cloth, too, there could be a body, hiding, asking for help. Such an elision elicits disorienting mistrust in temporality, but also expands the planar possibilities of syntactical meaning. A white cloth is not merely a construct in still life painting, but veil that both shields and reveals potential mysteries. It is a proposed alternative to a traditional Western binary of subjectivity and objectivity. The apo koinou construction is rare in modern English, having slipped out of use after its resurgence in Old English poetry. De Francesco reintroduces this blur of object and subject relations into a modern light. He draws inspiration from a device used by Gruppo 63 founder and avant-garde poet Antonio Porta tended toward repetition of phrases in third person perspective. In Italian, it is perfectly grammatically correct to omit the subject, including only a verb conjugation. Gender, even humanity unknown, the subject dissolves into the action, thus giving to the poetry the power to change the reader’s perception of who or what is actually acting. To further push the relative boundaries of experience, De Francesco expands his written poetry into what he calls “reading environments.” In these sound experiments, De Francesco’s voice reading his poetry is manipulated by real time digital voice processing. The written is transformed to the spoken, and then injected with currents of electric sound. “At each activation of word in the darkness,” (from The End (un’agenda) ) language is permutated into a nonphysical yet all-encompassing being. In these projects, realized all over Europe, De Francesco continues his examination of the word’s potency beyond the page and into an “n-dimensional space,” where the lines between nature and affect are intertwined. To publish excerpts of De Francesco’s oeuvre is to show screen shots of full-length film, to read pull quotes from a philosophy treatise. In these poems, there are questions, not to be answered, but to be questioned again. The words will oscillate in and out of the square construction found in his poems, out of the borders of the page, out of the rectangle of the laptop screen. NOTES (1) Blanchot, Maurice. The Infinite Conversation . Trans. Susan Hanson. Minneapolis: University of Minnesota Press, 1993. 27. EDITORS' NOTE The .pdf above contains about twenty pages of translation and the original Italian poetry. Below we provide only a brief account. The End (an agenda) Parler, ce n’est pas voir. Parler libère la pensée de cette exigence optique qui, dans la tradition occidentale, soumet depuis des millénaires notre approche des choses et nous invite à penser sous la garantie de la lumière ou sous la menace de l’absence de lumière. —Maurice Blanchot in this n-dimensional space i could spy on myself from the cracks while going back and forth from the summer to the wardrobe from the night to the parking lot but something filters in through the half-closed shutters becomes an event enunciation we stay without explanation like parallel curves on the back of a solid growing older she enters the garden always repeats certain phrases the elevator to be redone the neighbor’s music the steps the pavement furls like the leaves of a fern when touched she is framed from behind while she moves rocking she covers distances slowly a bag overflows from the wardrobe something stirs inside but then the suitcase comes open not even by my hand it’s the objects that are starting to filter out the little red haired monster watches idling the appointments written on one of your agendas two years old still seem urgent each letter holds itself upright on the page with flourishes and arabesques each angle has its own geography i’m trying to describe the path the fingers trace on the carpet the zipper takes the opposite route we are suspended on the stairs above the water in the center is the summer seen from above the darkness of the city passes from one headlight to the next the surfaces of our arms cohere and are shiny underneath the blinking of a sign each pore is an open expanse the body dreams the hair gives form to the possible some daring swimmers were throwing themselves into the seine they swam upstream up to the first quay look there could exist behind the screen a room where even when the lamp is out and the curtains drawn even when the suitcase wasn’t closed everything seemed red everything calm construction of a four-dimensional tetrahedron by R. Courant – H. Robbins, What is Mathematics? Oxford University Press, 1941  . (shrink)

Matthew Spinks [35] introduces implicative BCSK-algebras, expanding implicative BCK-algebras with an additional binary operation. Subdirectly irreducible implicative BCSK-algebras can be viewed as flat posets with two operations coinciding only in the 1- and 2-element cases, each, in the latter case, giving the two-valued implication truth-function. We introduce the resulting logic (for the general case) in terms of matrix methodology in §1, showing how to reformulate the matrix semantics as a Kripke-style possible worlds semantics, thereby displaying the distinction between the two (...) implications in the more familiar language of modal logic. In §§2 and 3 we study, from this perspective, the fragments obtained by taking the two implications separately, and – after a digression (in §4) on the intuitionistic analogue of the material in §3 – consider them together in §5, closing with a discussion in §6 of issues in the theory of logical rules. Some material is treated in three appendices to prevent §§1–6 from becoming overly distended. (shrink)

In the author's Relative lawlessness in intuitionistic analysis [this JOURNAL. vol. 52 (1987). pp. 68-88] and An intuitionistic theory of lawlike, choice and lawless sequences [Logic Colloquium '90. Springer-Verlag. Berlin. 1993. pp. 191-209] a notion of lawless ness relative to a countable information base was developed for classical and intuitionistic analysis. Here we simplify the predictability property characterizing relatively lawless sequences and derive it from the new axiom of closed data (classically equivalent to open data) together with a natural principle (...) of invariance under finite translation. We characterize relative lawlessness in terms of a notion of forcing. Finally, we study relative lawlessness on an arbitrary fan and show that the collection of lawless binary sequences (which is comeager in the sense of Baire) has probability measure zero. The reasoning is predominantly constructive. (shrink)

The paper studies how the localic notion of sublocale transfers to formal topology. For any formal topology (not necessarily with positivity predicate) we define a sublocale to be a cover relation that includes that of the formal topology. The family of sublocales has set-indexed joins. For each set of base elements there are corresponding open and closed sublocales, boolean complements of each other. They generate a boolean algebra amongst the sublocales. In the case of an inductively generated formal topology, the (...) collection of inductively generated sublocales has coframe structure. Overt sublocales and weakly closed sublocales are described, and related via a new notion of "rest closed" sublocale to the binary positivity predicate. Overt, weakly closed sublocales of an inductively generated formal topology are in bijection with "lower powerpoints", arising from the impredicative theory of the lower powerlocale. Compact sublocales and fitted sublocales are described. Compact fitted sublocales of an inductively generated formal topology are in bijection with "upper powerpoints", arising from the impredicative theory of the upper powerlocale. (shrink)

In the framework of the theory of the semiosphere elaborated by Juri Lotman in the 1980s, the notion of translation acquires a new, broadened meaning and is used to describe a general mechanism of cultural dynamics. This is a direct consequence of the understanding of the semiosphere as a “continuum of semiotic systems” of which heterogeneity and polyglotism are constitutive features. If the “smallest functioning semiotic mechanism” is not an isolated system, but always a (at least) binary system, translation will (...) play a central role in communication not only between different cultures, but also within every single culture.The article focuses on the different places of translation within the semiosphere as a system of systems and its relations to the notion of border. It argues that in Lotman’s theory, translation can function both as a homogenizing and a heterogenizing semiotic force, depending on its position at the center or the periphery of a given semiotic system. The first part of the article develops a theoretical argument on the relation between translation processes and cultural borders, which acquire different meanings and functions within Lotman’s theory of the semiosphere corresponding to differently oriented kinds of translation. The second part of the article develops a close analysis of Giacomo Leopardi’s poem “L’Infinito,” where the tension and dialogue between the different functions of the Lotmanian borders and the different kinds of cultural translation become devices for the generation of new meanings. Finally, the article briefly considers the significance of Lotman’s semiospheric understanding of the relation between borders and translation for contemporary translation and border studies. (shrink)

This paper aims to show that genders are enacted, by providing an account of how an individual can be said to enact a gender and explaining how, consequently, genders can be fluid. On the enactive-ecological view we defend, individuals first and foremost perceive the world as fields of affordances, that is, structured sets of action possibilities. Fields of natural affordances offer action possibilities because of the natural properties of organisms and environments. Handles offer graspability to humans because of physical-structural properties (...) of handles and the anatomical-physiological properties of humans. Although humans live in fields of bodily, action, and cultural affordances, our work focuses on cultural affordances, where action possibilities are offered to individuals because of the normative responses of individuals in that culture. Knocking on a door affords entrance because knocking provides cultured individuals on the other side of the door an affordance to which they themselves behave normatively. Usually, behaving normatively in response to cultural affordances brings about sequences of perception-action loops, which we will call “scripts”: for instance, closed doors afford knocking, which affords the individual inside opening the door, which affords an interpersonal meeting, which afford entrance. Although the notion of script has a strong cognitivist flavor, one of the aims of the paper to provide an ecological account of scripts, to show that what cognitivists viewed as representations are in fact environmentally structured perception-action loops. On our account of gender, gendered cultures build and maintain gendered cultural affordance landscapes, that is, landscapes in which the action possibilities individuals face are normed according to a specific body type or situation; most often biological sex. Individuals enact a given gender when they come to perceive the affordances reserved for one gender by their culture and respond in the culturally normative way, thus enacting gendered sequences of perception-action loops. With the shifting landscapes of cultural affordances brought about by several recent social, technological, and epistemic developments in some cultures, the gendered landscapes of affordances offered to individuals in these cultures have become more varied and less rigid, thus increasing the variety and flexibility of scripts individuals can enact. This entails that individuals in such cultures have an increased possibility for gender fluidity, which may in part explain the increasing number of people currently identifying outside the binary. (shrink)

A structure A for the language L, which is the first-order language (without equality) whose only nonlogical symbol is the binary predicate symbol , is called a quasi -struoture iff (a) the universe A of A consists of sets and (b) a b is true in A ([p) a = {p } & p b] for every a and b in A, where a(b) is the name of a (b). A quasi -structure A is called an -structure iff (c) {p (...) } A whenever p a A. Then a closed formula in L is derivable from Leniewski's axiom x, y[x y u (u x) u; v(u, v x u v) u(u x u y)] (from the axiom x, y(x y x x) x, y, z(x y z y x z)) iff is true in every -structure (in every quasi -structure). (shrink)

One of the issues with introducing animal rights law is whether the problem is quantitative or qualitative, whether it can be achieved by working within existing legal paradigms or whether it requires a new set of paradigms. The answer is fundamental: a quantitative problem can be solved by applying more of the same solutions, while a qualitative problem requires completely different solutions. The qualitative camp can be represented by, say, Professor Gary Francione, demanding not only rights for animals but that (...) they have personhood and are no longer considered “property.” The quantitative approach can be finely represented by Professor David Favre, who has argued for many years that we can make progress by developing existing legal principles to give some rights to some animals, and he does so eloquently in his new book. He is clear in his view that progress is, and can continue to be, made with a series of steps, as he puts it, “climbing the slippery slope toward personhood” (p. viii).His case is supported by looking backward—progress made in establishing the rules governing humans in their relationships with animals—as well as forward to developments that will support continuing improvement in the lives of animals, not least the potential for technology (for example, through cell- and plant-based meat). At the same time, he also rightly rejects the concepts of animal welfare and animal rights being at odds, let alone either of them being a panacea for better lives for animals.One aspect of animal law that Favre believes does need to change is that of property status. As he says, the status of animals as personal property is “critical to the future of animal law” (p. 25). In general, the solutions have been largely binary—that animals simply cannot be property for them to have meaningful rights or that animals can be property and still have rights. The former makes intuitive good sense (and is consistent with the objectives of the human abolitionist movements) but would have enormous effects on economic activity throughout the world. The latter is appealing because it does not change the current paradigm, so can be developed more easily.Favre's premise is that we are moving toward rights for animals, so we need to find a way of integrating animals into the current legal system rather than having to destroy and reconstruct the legal system. His ingenious solution is to propose a fourth category of property, alongside real, personal, and intellectual property: “living property.”Within this category, he envisages using the common law concept of the trust to provide equitable “self-ownership,” where an animal has equitable “ownership” while a human retains legal “ownership.” He then connects this concept to adaptations in other aspects of law that start to recognize the selfhood of animals, creating a platform onto which broader and deeper rights for animals can develop, such as the status of “animalhood” (p. 72) and the corresponding legal right of consideration before decisions are made by the state affecting an animal's fundamental interests (such as continuing to be alive). By putting animals in a separate class—rather than uncomfortably leaving them in personal property—he creates a possible route for animal rights that seems less paradigm-busting than removing them from being property altogether.Drawing on evidence from criminal law, tort law, equity (trusts), property law, and family law, he shows how in many places—with a focus on states in the United States—existing laws are being adapted (“green shoots” p. 81) to recognize that animals are more than personal property, in ways that cannot be explained simply through the growth of regulations. For example, family law is starting to recognize that in the division of matrimonial assets, companion animals cannot be treated simply like any other personal property such as a table or a car based on who paid for them. Because animals are “living property,” the courts should take account of other aspects, such as who cared for the animals and who looked after them. Similarly, the concept of extending trust law so that an animal can be the beneficiary of a trust—in the same way as a human can be but inanimate personal property cannot—shows that the law can understand that companion animals can play a large emotional role in our lives (as well as we in theirs), such that we want to ensure proper care for them after we die.The book raises important issues. First, it is intentionally about domestic animals, with their close links with humans, so the potential for laws to engage is much easier than with, say, farmed animals. The danger with solutions that focus on domestic or charismatic animals (e.g., dogs, chimpanzees, elephants) is that the arguments may be pertinent essentially only to those species and therefore hard to apply to animals of less charismatic species, say in a factory farm (but who are present in much greater numbers and generally badly treated in comparison). He “leaves for another day” (p. 32) the discussion about other animals. Hopefully that day will come, as it would be interesting to see how Favre could extend his concept to animals for whom the public has less affection and less appreciation of their capacities (and for whom the prospect of death is much more real and imminent).Secondly, public support for these ideas about living property may simply not be present. If there is not a sufficient groundswell of support for changing the status of animals, then whether one proposes changing it to not-property or “self-owned” property doesn't matter (if someone doesn't like needles, it doesn't matter whether the needle you propose to use is big or small). This may be too pessimistic, of course (people can be helped to overcome a fear of needles, and one way to start is with small needles). Perhaps the law should start with small changes to recognize at least a few animals—their sentience, their capacities for pleasure and pain, their emotions and preferences, their potential for autonomy—and build public support from there.There is also the underlying and more radical point about whether these ideas—the trust for a beloved companion set up by the human, the divorce settlement that doesn't just give the companion to the party who paid for them—are actually bringing about changes for animals as such or are simply adapting laws to help humans, recognizing human needs, and do not actually change the status of animals. The ends are still humans, and the means are still animals.This is not intended to paint a bleak picture: The more bluntand uncompromising part of the animal rights movement that has been focused on the qualitative problem has hardly been successful at getting animal rights laws passed anywhere in the world, so everything that moves us closer to a better, richer, more thoughtful view of animals will help. Favre's approach has ingenuity, shows good sense, and has the potential to bring about real animal rights. The themes and cases in this book that show how courts and legislatures are becoming more aware of the sentience of at least some animals, as well as proposing a middle path in the debate about animals as property, are important in the journey toward recognition of animal rights. Even more important, they may actually work. (shrink)

In this article I explore how a philosophical conception of love may be used to draw debate on the death of God beyond the binary opposition between theology and philosophy through a comparative study of the work of Jean-Luc Marion and Jean-Luc Nancy. Although Marion’s reading of love—in both its theological and phenomenological guises—proposes an innovative phrasing of a non-metaphysical notion of divinity, I argue that it is ultimately unable to maintain its coherence in nominal discourse due to Marion’s insistence (...) on keeping love and being separate. In contrast, through his unique thinking on ontology, Nancy’s philosophy allows us to reposition love at the heart of being, from whence it may serve as a means of disrupting the very principle governing the atheist and theist world-views. Far from dismantling Marion’s core proposition, I conclude that the two thinkers may in fact be more closely aligned than previous scholarship has acknowledged, coming together in the need to affirm an experience of the impossible that takes place in the present rather than in any eschatological fulfillment to come. At this intersection between two nominally very different thinkers, love forms an ontological limit-point for thought that poses a challenge to both theology and philosophy alike as they continue to grapple with the question of religion in the twenty-first century. (shrink)

In this erudite study, Anna Becker employs the lens of gender to explore the political thought of nine Renaissance writers. She argues that political thinkers in the Italian and French Renaissance perceived the domestic realm to be essentially political, particularly within relationships of marriage and parenting. She demonstrates that "The Great Dichotomy"—the assumed binary opposition between a public-civic realm that is political and male, and a private-domestic realm that is apolitical and female—did not exist in canonical Renaissance political thought.The first (...) chapter closely examines how three Italian commentators, Leonardo Bruni, Donato Acciaioli, and Bernardo Segni... (shrink)

A Π01 class is an effectively closed set of reals. We study properties of these classes determined by cardinality, measure and category as well as by the complexity of the members of a class P. Given an effective enumeration {Pe:e < ω} of the Π01 classes, the index set I for a certain property is the set of indices e such that Pe has the property. For example, the index set of binary Π01 classes of positive measure is Σ02 complete. (...) Various notions of boundedness are discussed and classified. For example, the index set of the recursively bounded classes is Σ03 complete and the index set of the recursively bounded classes which have infinitely many recursive members is Π04 complete. Consideration of the Cantor-Bendixson derivative leads to index sets in the transfinite levels of the hyperarithmetic hierarchy. (shrink)