The paper introduces the concept of polar decision rules and establishes that majority rules are polar rules. We identify second best rules and penultimate rules in cases that majority rules are optimal or the most inferior, respectively. We especially specify the almost expert rule and the almost majority rule as the secondary rules of the expert and majority rules, respectively.

Bird’s Ultimate Argument sought to show that Armstrong’s N relationships involving categorical universals can’t entail nomic regularities. In N’s place Bird offered the non-categorical SR relation. Two kinds of objection have been raised: either Bird’s own alternative metaphysics fails in just the same way as Armstrong’s or the target of Bird’s argument may anyway have a way out of the problem. My aim is to reclaim the victory for Bird. I argue that the responses in defence of Armstong’s N (...) relationships fail to acknowledge that Bird was explicitly concerned with Armstrong’s commitment to a categoricalist view of universals. Moreover, Bird’s alternative account does not suffer the same problem since his metaphysics of properties is essentialist. Nevertheless, Bird’s account does need elaborating on to explainwhySR relationships entail their regularities. I offer Schaffer’s Axiomatic Solution as a candidate for this purpose. (shrink)

A proof-theoretical analysis of elementary theories of order relations is effected through the formulation of order axioms as mathematical rules added to contraction-free sequent calculus. Among the results obtained are proof-theoretical formulations of conservativity theorems corresponding to Szpilrajn’s theorem on the extension of a partial order into a linear one. Decidability of the theories of partial and linear order for quantifier-free sequents is shown by giving terminating methods of proof-search.

This paper first discusses how Russell and Hochberg have addressed some phenomena of relatedness, notably relational order, in a similarly ‘positionalist’ way, yet by appealing to different sorts of formal relations: “positions” in Russell's case and “ordering relations” in Hochberg's. After pointing out some shortcomings of both approaches, the paper then proposes an alternative view based on ‘o-roles’, which are, roughly speaking, ontological counterparts of the thematic roles postulated in linguistics. It is argued that o-roles are sort of middle-of-the-road (...) entities in the sense that they have the virtues of positions and those of ordering relations, without having their respective vices. Some tentative ideas on which o-roles should be acknowledged are also put forward. (shrink)

We prove that if ≼ is an analytic partial order then either ≼ can be extended to a Δ 2 1 linear order similar to an antichain in 2 ω 1 , ordered lexicographically, or a certain Borel partial order ⩽ 0 embeds in ≼. Similar linearization results are presented, for κ -bi-Souslin partial orders and real-ordinal definable orders in the Solovay model. A corollary for analytic equivalence relations says that any Σ 1 1 equivalence relation (...) E , such that E 0 does not embed in E , is fully determined by intersections with E -invariant Borel sets coded in L. (shrink)

Data on toddler language acquisition and use support the idea of a cognitive that can resolve contradictory claims about human-animal similarities. Examples of imagination, aesthetic evaluation, theory of mind (ToM), and language learning reveal higher-order, relational, abstract capabilities early on. Although language itself may be a consequence of exercising this supermodule, it enables further cognitive operations on indirect experience to go far beyond animal accomplishments.

Using a variation of the rainbow construction and various pebble and colouring games, we prove that RRA, the class of all representable relation algebras, cannot be axiomatised by any first-order relation algebra theory of bounded quantifier depth. We also prove that the class At(RRA) of atom structures of representable, atomic relation algebras cannot be defined by any set of sentences in the language of RA atom structures that uses only a finite number of variables.

This paper states two sets of axioms sufficient for extensive measurement. The first set, like previously published axioms, requires that each of the objects measured must be classifiable as either greater than, or less than, or indifferent to each other object. The second set, however, requires only that any two objects be classifiable as either indifferent or different, and does not need any information about which object is greater. Each set of axioms produces an extensive scale with the usual properties (...) of additivity and uniqueness except for unit. Moreover, the axioms imply Weber's Law: whether two objects are indifferent depends only upon the ratio of their scale values. (shrink)

Contraction-free sequent calculi for intuitionistic theories of apartness and order are given and cut-elimination for the calculi proved. Among the consequences of the result is the disjunction property for these theories. Through methods of proof analysis and permutation of rules, we establish conservativity of the theory of apartness over the theory of equality defined as the negation of apartness, for sequents in which all atomic formulas appear negated. The proof extends to conservativity results for the theories of constructive (...) class='Hi'>order over the usual theories of order. (shrink)

The representation of quantification over relations in monadic third-order logic is discussed; it is shown to be possible in numerous special cases of foundational interest, but not in general unless something akin to the Axiom of Choice is assumed.

The contribution of this paper lies with providing a systematically specified and intuitive interpretation pattern and delineating a class of relational structures and models providing a natural interpretation of logical operators on an underlying propositional calculus of Positive Lattice Logic and subsequently proving a generic completeness theorem for the related class of logics, sometimes collectively referred to as Generalized Galois Logics.

I ate my broccoli, though my broccoli did not eat me. The eating relation, like many other relations, differentiates between its arguments. The fact that eating holds between a and b does not entail that it holds between b and a. How are we to make sense of this? The standard view is that relations are sensitive to the order of their arguments. As natural as this view is, it has been the target of a powerful objection from (...) Kit Fine. I examine Fine’s objection and defend the standard view. (shrink)

Our instinctive tendency is to believe that the relations of succession can be adequately symbolized by geometrical relations. The persistence of this belief has had disastrous influence through the centuries on philosophical and theological thought, and upon physical theories as well.

Boolos has suggested a plural interpretation of second-order logic for two purposes: to escape Quine’s allegation that second-order logic is set theory in disguise, and to avoid the paradoxes arising if the second-order variables are given a set-theoretic interpretation in second-order set theory. Since the plural interpretation accounts only for monadic second-order logic, Rayo and Yablo suggest an new interpretation for polyadic second-order logic in a Boolosian spirit. The present paper argues that Rayo and (...) Yablo’s interpretation does not achieve the goal. (shrink)

States of affairs involving a non-symmetric relation such as loving are said to have a relational order, something that distinguishes, for instance, Romeo’s loving Juliet from Juliet’s loving Romeo. Relational order can be properly understood by appealing to o-roles, i.e., ontological counterparts of what linguists call thematic roles, e.g., agent, patient, instrument, and the like. This move allows us to meet the appropriate desiderata for a theory of relational order. In contrast, the main theories that try (...) to do without o-roles, proposed by philosophers such as Russell, Hochberg, and Fine, are in trouble with one or another of these desiderata. After discussing some alternatives, it is proposed that o-roles are best viewed as very generic properties characterizable as ways in which objects jointly exemplify a relation. This makes for exemplification relations understood as complex entities having o-roles as building blocks. (shrink)

`Authority', `competence' and other related concepts are determined on the basis of the concept of law as a dynamic order of norms. The norms which regulate the processes of norm creation establish empowerments (Ermächtigungen). The material domain of validity of the empowering norm is called `competence'. The concept of `person' in relation to empowering norms yields the concepts of `organ' and `authority'. The spatial domain of the validity of these norms is the spatial or territorial jurisdiction. This paper (...) analyses the basic norm and its legal functions; it considers the irregularity of legal acts and norms, as well as the legal consequences thereof, namely nullity and annulment. Additionally, the Kelsenian `Tacit Alternative Clause' is criticized and a possible solution to the problem of irregular norms is offered through new definitions of the existence, validity and legitimacy of norms. (shrink)

We show how to interpret the language of first-order set theory in an elementary topos endowed with, as extra structure, a directed structural system of inclusions . As our main result, we obtain a complete axiomatization of the intuitionistic set theory validated by all such interpretations. Since every elementary topos is equivalent to one carrying a dssi, we thus obtain a first-order set theory whose associated categories of sets are exactly the elementary toposes. In addition, we show that (...) the full axiom of Separation is validated whenever the dssi is superdirected. This gives a uniform explanation for the known facts that cocomplete and realizability toposes provide models for Intuitionistic Zermelo-Fraenkel set theory. (shrink)

Until the end of the last century, media sociology was synonymous with the investigation of mass media as a social domain. Today, media sociology needs to address a much higher level of complexity, that is, a deeply mediatized world in which all human practices, social relations, and social order are entangled with digital media and their infrastructures. This article discusses this shift from a sociology of mass communication to the sociology of a deeply mediatized world. The principal aim of (...) the article is to outline a new media-sociological imagination: media sociology as a cross-sectional sociology, a sociology of entanglement, and a new critical sociology of technological deep structures. (shrink)

The relationship between international order and justice has long been central to the study and practice of international relations. For most of the twentieth century, states and international society gave priority to a view of order that focused on the minimum conditions for coexistence in a pluralist, conflictual world. Justice was seen either as secondary or sometimes even as a challenge to order. Recent developments have forced a reassessment of this position. This book sets current concerns within (...) a broad historical and theoretical context; explores the depth and scope of this presumed solidarism amidst the difficulties of acting on the basis of a more strongly articulated liberal position; and underscores the complexity and abiding tensions inherent in the relationship between order and justice. Chapters examine a wide range of state and transnational perspectives on order and justice, including those from China, India, Russia, the United States, and the Islamic world. Other chapters investigate how the order-justice relationship is mediated within major international institutions, including the United Nations, the World Trade Organization and the global financial institutions. (shrink)

In this paper we investigate several extensions of the first order-language with finitely many binary relations. The most interesting of the studied extensions appears to be the monadic second-order one. We show that the extended languages have the same expressive power as the first-order language over the class of all relational structures of equivalence relations in local agreement by providing appropriate translation of formulae. The decidability of the considered extensions over the above mentioned class of structures is (...) also shown. (shrink)

In this paper I contribute a further element to the case for admitting non-symmetric relations by dismantling the case against them. Armstrong and Dorr have both argued (1) that asymmetric relations give rise to ‘brute necessities’, whilst Dorr further argues (2) that admitting non-symmetric relations generates spurious possibilities and (3) that exploiting work of Goodman and Hazen, we can do without non-symmetric relations anyway. Against (1) I argue that neither Armstrong nor Dorr succeed in avoiding brute necessities themselves. Against (2) (...) and (3) I argue that admitting non-symmetric relations doesn’t give rise to spurious possibilities and Goodman and Hazen’s work cannot be used to establish that non-symmetric relations are dispensable to science and mathematics. (shrink)

We present a relational proof system in the style of dual tableaux for a multimodal propositional logic for order of magnitude qualitative reasoning to deal with relations of negligibility, non-closeness, and distance. This logic enables us to introduce the operation of qualitative sum for some classes of numbers. A relational formalization of the modal logic in question is introduced in this paper, i.e., we show how to construct a relational logic associated with the logic for order-of-magnitude reasoning and (...) its dual tableau system which is a validity checker for the modal logic. For that purpose, we define a validity preserving translation of the modal language into relational language. Then we prove that the system is sound and complete with respect to the relational logic defined as well as with respect to the logic for order of magnitude reasoning. Finally, we show that in fact relational dual tableau does more. It can be used for performing the four major reasoning tasks: verification of validity, proving entailment of a formula from a finite set of formulas, model checking, and verification of satisfaction of a formula in a finite model by a given object. (shrink)

We show how to interpret the language of first-order set theory in an elementary topos endowed with, as extra structure, a directed structural system of inclusions (dssi). As our main result, we obtain a complete axiomatization of the intuitionistic set theory validated by all such interpretations. Since every elementary topos is equivalent to one carrying a dssi, we thus obtain a first-order set theory whose associated categories of sets are exactly the elementary toposes. In addition, we show that (...) the full axiom of Separation is validated whenever the dssi is superdirected. This gives a uniform explanation for the known facts that cocomplete and realizability toposes provide models for Intuitionistic Zermelo—Fraenkel set theory (IZF). (shrink)

John Sealey; Education as a Second-order Form of Experience and its Relation to Religion, Journal of Philosophy of Education, Volume 13, Issue 1, 30 May 2006, P.

Technology and its interfaces with nursing care, patients and carers, and the home are many and varied. To date, healthcare services research has generally focussed on pragmatic issues such access to and the optimization of technology, while philosophical inquiry has tended to focus on the ethics of how technology makes the home more hospital like. However, the ethical implications of the ways in which technology shapes the subjectivities of patients and carers have not been explored. In order to explore (...) this, poststructural theory, in particular the work of Butler, Foucault, and Deleuze, is used to theorize the relationship between subjectivity and materiality as ethically mandated on producing rather than precluding the development of subjectivities in novel ways. This theoretical understanding is then utilized through a process of ‘plugged in’ as described by Jackson and Massie that aims to link empirical data, research, and philosophical inquiry. Through this process, it is suggested that power, which the empirical data demonstrate, is frequently exercised through medical discourses and restricts patients' and carers' ability to shape the material environment of the home as a place to live and be cared for in palliative stages of illness. Alternative discourses are suggested both from the empirical data as well as other research, which may offer patients and carers the possibility of reclaiming power over the home and their subjectivities. Finally, the dichotomy between the home and hospital, mediated via technology, is posited as being problematic. It is argued the dichotomy is false and should be moved away from in order to allow an ethical embrace of technology in palliative care. (shrink)

LetEbe a computably enumerable equivalence relation on the setωof natural numbers. We say that the quotient set$\omega /E$realizesa linearly ordered set${\cal L}$if there exists a c.e. relation ⊴ respectingEsuch that the induced structure is isomorphic to${\cal L}$. Thus, one can consider the class of all linearly ordered sets that are realized by$\omega /E$; formally,${\cal K}\left = \left\{ {{\cal L}\,|\,{\rm{the}}\,{\rm{order}}\, - \,{\rm{type}}\,{\cal L}\,{\rm{is}}\,{\rm{realized}}\,{\rm{by}}\,E} \right\}$. In this paper we study the relationship between computability-theoretic properties ofEand algebraic properties of linearly (...) ordered sets realized byE. One can also define the following pre-order$ \le _{lo} $on the class of all c.e. equivalence relations:$E_1 \le _{lo} E_2 $if every linear order realized byE1is also realized byE2. Following the tradition of computability theory, thelo-degrees are the classes of equivalence relations induced by the pre-order$ \le _{lo} $. We study the partially ordered set oflo-degrees. For instance, we construct various chains and anti-chains and show the existence of a maximal element among thelo-degrees. (shrink)

I examine notions of equivalence between logics (understood as languages interpreted model-theoretically) and develop two new ones that invoke not only the algebraic but also the string-theoretic structure of the underlying language. As an application, I show how to construe modal operator languages as what might be called typographical notational variants of _bona fide_ first-order languages.

The goal of this short note is to interest set theorists in the order type ω*ω1, and to encourage them to work on the question of whether or not the Continuum Hypothesis decides the partition relation τ→2, for τ=ω*ω1 and for τ=ω1ω+2.

A relation on a linearly ordered structure is called semi-bounded if it is definable in an expansion of the structure by bounded relations. We study ultimate behavior of semi-bounded relations in an ordered module M over an ordered commutative ring R such that M/rM is finite for all nonzero r $\epsilon$ R. We consider M as a structure in the language of ordered R-modules augmented by relation symbols for the submodules rM, and prove several quantifier elimination results for (...) semi-bounded relations and functions in M. We show that these quantifier elimination results essentially characterize the ordered modules M with finite indices of the submodules rM. It is proven that (1) any semi-bounded k-ary relation on M is equal, outside a finite union of k-strips, to a k-ary relation quantifier-free definable in M, (2) any semibounded function from $M^{k}$ to M is equal, outside a finite union of k-strips, to a piecewise linear function, and (3) any semi-bounded in M endomorphism of the additive group of M is of the form x $\mapsto \sigmax$ , for some $\mapsto \sigma$ from the field of fractions of R. (shrink)

This article discusses the emergence of the concept of ‘transnational feminisms’ as a differentiated notion from ‘global sisterhood’ within feminist postcolonial criticism. This is done in order to examine its usefulness for interrogating the globalization of reproductive technologies and women’s right to selfdetermination over their own bodies by using these technologies. In particular, women’s use of technologies for assisted conception, and the local and global transactions in reproductive body parts form a testing ground for transnational feminisms. Does the construction (...) of individual reproductive rights still leave some ground for women’s collective struggles? It is proposed that, if at all, transnational solidarity on this issue is possible, it will have to be built on the concept of universal ethical norms regarding human dignity. (shrink)

The block relation B(x,y) in a linear order is satisfied by elements that are finitely far apart; a block is an equivalence class under this relation. We show that every computable linear order with dense condensation-type (i.e., a dense collection of blocks) but no infinite, strongly η-like interval (i.e., with all blocks of size less than some fixed, finite k ) has a computable copy with the nonblock relation ¬ B(x,y) computably enumerable. This implies that (...) every computable linear order has a computable copy with a computable nontrivial self-embedding and that the long-standing conjecture characterizing those computable linear orders every computable copy of which has a computable nontrivial self-embedding (as precisely those that contain an infinite, strongly η-like interval) holds for all linear orders with dense condensation-type. (shrink)

In this paper, we review Keith Lehrer’s account of the basing relation, with particular attention to the two cases he offered in support of his theory, Raco (Lehrer, Theory of knowledge, 1990; Theory of knowledge, (2nd ed.), 2000) and the earlier case of the superstitious lawyer (Lehrer, The Journal of Philosophy, 68, 311–313, 1971). We show that Lehrer’s examples succeed in making his case that beliefs need not be based on the evidence, in order to be justified. These (...) cases show that it is the justification (rather than the belief) that must be based in the evidence. We compare Lehrer’s account of basing with some alternative accounts that have been offered, and show why Lehrer’s own account is more plausible. (shrink)