Let ( * X, * T) be the nonstandard extension of a Hausdorff space (X, T). After Wattenberg [6], the monad m(x) of a near-standard point x in * X is defined as m(x) = μ T (st(x)). Consider the relation $R_{\mathrm{ns}} = \{\langle x, y \rangle \mid x, y \in \mathrm{ns} (^\ast X) \text{and} y \in m(x)\}.$ Frank Wattenberg in [6] and [7] investigated the possibilities of extending the domain of R ns to the whole of * X. Wattenberg's (...) extensions of R ns were required to be equivalence relations, among other things. Because the nontrivial ways of constructing such extensions usually produce monadic relations, the said condition practically limits (to completely regular spaces) the class of spaces for which such extensions are possible. Since symmetry and transitivity are not, after all, characteristics of the kind of nearness that is obtained in a general topological space, it may be expected that if these two requirements are relaxed, then a monadic extension of R ns to * X should be possible in any topological space. A study of such extensions of R ns is the purpose of the present paper. We call a binary relation $W \subseteq ^\ast X \times ^\ast X$ an infinitesimal on * X if it is monadic and reflexive on * X. We prove, among other things, that the existence of an infinitesimal on * X that extends R ns is equivalent to the condition that the space (X, T) be regular. (shrink)

In this article we establish the undecidability of representability and of finite representability as algebras of binary relations in a wide range of signatures. In particular, representability and finite representability are undecidable for Boolean monoids and lattice ordered monoids, while representability is undecidable for Jónsson's relation algebra. We also establish a number of undecidability results for representability as algebras of injective functions.

The properties of antisymmetry and linearity are easily seen to be sufficient for a recursively enumerable binary relation to be recursively isomorphic to a recursive relation. Removing either condition allows for the existence of a structure where no recursive isomorph exists, and natural examples of such structures are surveyed.

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 shall prove here that any binary relation on a base E with cardinality n > 6 is reconstructible from its restrictions of cardinality 2, 3, 4 and . This proof needs results of part I of this paper where we characterize any pair of relations R, R' which are 2-, 3- and 4-hypomorphic. As a corollary we obtain that any binary relation is -reconstructible.

A characterization of a property of binary relations is of finite type if it is stated in terms of ordered T-tuples of alternatives for some positive integer T. The concept was introduced informally by Knoblauch (2005). We give a clear, complete definition below. We prove that a characterization of finite type can be used to determine in polynomial time whether a binary relation over a finite set has the property characterized. We also prove a simple but useful (...) nonexistence theorem and apply it to three examples. (shrink)

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)

The transitive closure of a binary relation R can be thought of as the best possible approximation of R "from above" by a transitive relation. We consider the question of approximating a relation from below by transitive relations. Our main result is that every thick relation (a relation whose complement contains no infinite chain) on a countable set has a transitive thick subrelation. This allows for a solution to a problem arising from previous work by the author and (...) Alan Taylor. We also exhibit a thick relation on an uncountable set with no transitive thick subrelation. (shrink)

Let S be a signature of operations and relations definable in relation algebra, let R be the class of all S-structures isomorphic to concrete algebras of binary relations with concrete interpretations for symbols in S, and let F be the class of S-structures isomorphic to concrete algebras of binary relations over a finite base. To prove that membership of R or F for finite S-structures is undecidable, we reduce from a known undecidable problem—here we use (...) the tiling problem, the partial group embedding problem and the partial group finite embedding problem to prove undecidability of finite membership of R or F for various signatures S. It follows that the equational theory of R is undecidable whenever S includes the boolean operators and composition. We give an exposition of the reduction from the tiling problem and the reduction from the group embedding problem, and summarize what we know about the undecidability of finite membership of R and of F for different signatures S. (shrink)

This paper presents a metatheorem with the following property: Given a proven axiom-free lemma, which interrelates some of the elementary properties of a binary relation, the metatheorem mechanically transforms this lemma into its uniquely corresponding complementary lemma. Therefore, the metatheorem nearly doubles our knowledge about the elementary properties of binary relations, for application to statistical decision theory. At present we do not know whether there exists a nontrivial axiom-free lemma that is its own complementary lemma.

We prove that algebras of binary relations whose similarity type includes intersection, union, and one of the residuals of relation composition form a nonfinitely axiomatizable quasivariety and that the equational theory is not finitely based. We apply this result to the problem of the completeness of the positive fragment of relevance logic with respect to binary relations.

This paper is concerned with the problem of extending an antisymmetric binary relation on a set to a linear order on the power set. A necessary and sufficient condition is offered.

(2012). Modal logics for reasoning about infinite unions and intersections of binary relations. Journal of Applied Non-Classical Logics: Vol. 22, No. 4, pp. 275-294. doi: 10.1080/11663081.2012.705960.

We characterize the existence of semicontinuous weak utilities in a general framework, where the axioms of transitivity and acyclicity are relaxed to that of consistency in the sense of Suzumura (Economica 43:381–390, 1976). This kind of representations allow us to transfer the problem of the existence of the ${{\mathcal{G}}{\mathcal{O}}{\mathcal{C}}{\mathcal{H}}{\mathcal{A}}}$ set of a binary relation to the easier problem of getting maxima of a real function. Finally, we show that the maxima of these representations correspond to the different levels of (...) satiation that each of individual has (an individual reaches his or her level of satiation when an increase of consuming an alternative product/service brings no increase in utility). (shrink)

The topological structures imposed on the collection of binary relations on a given set by the symmetric difference metric and the Hausdorff metric provide opportunities for learning about how collections of binary relations with various properties fit into the collection of all binary relations. For example, there is some agreement and some disagreement between conclusions drawn about the rarity of certain properties of binary relations using first the symmetric difference metric and then (...) the Hausdorff metric. (shrink)

Alternative food networks (AFNs) have become increasingly important in response to growing concerns about industrial animal agriculture’s harmful impacts on animals. Alternative animal agriculture seeks to address problems with industrial animal agriculture given its purported emphases on animal welfare and, more controversially, “happy” meat. Debates in critical food studies and animal studies literature, however, caution against the glorification of “alterity” and how welfare claims provide an ethical façade for violence towards farm animals. These debates, while critically important, leave little room (...) for exploring the complexities of alternative animal agriculture, human-farm animal relations, and care. In this article, I build upon the insights of critical food studies scholarship that takes a reflexive and hybrid approach to studying AFNs and literature that explores the complex dynamics of human-farm animal relations. Drawing on qualitative interviews with grass-fed beef farmers in Ontario, Canada, I present an analysis aimed at unsettling divisive thinking about food practices and relations. First, I reflect on how farmers understand their work as an ethical alternative to industrial beef production, benefiting animals, people, and the environment. I then detail the emotional dynamics that shape cows’ commodification and relations with farmers, adding to scholarship that demands more attention to affective dimensions of care for animals in AFNs. Lastly, I discuss grass-fed beef farming challenges and contradictions, empirically exploring, for example, how grass-fed beef farming operates as a niche market whose accessibility is shaped by socio-economic privilege. Collectively, these insights contribute to discussions about the complexities of grass-fed beef farming as an alternative food network with benefits and problems, impacting the wellbeing of animals, people, and the environment. (shrink)

Agents are often assumed to have degrees of belief (“credences”) and also binary beliefs (“beliefs simpliciter”). How are these related to each other? A much-discussed answer asserts that it is rational to believe a proposition if and only if one has a high enough degree of belief in it. But this answer runs into the “lottery paradox”: the set of believed propositions may violate the key rationality conditions of consistency and deductive closure. In earlier work, we showed that this (...) problem generalizes: there exists no local function from degrees of belief to binary beliefs that satisfies some minimal conditions of rationality and non-triviality. “Locality” means that the binary belief in each proposition depends only on the degree of belief in that proposition, not on the degrees of belief in others. One might think that the impossibility can be avoided by dropping the assumption that binary beliefs are a function of degrees of belief. We prove that, even if we drop the “functionality” restriction, there still exists no local relation between degrees of belief and binary beliefs that satisfies some minimal conditions. Thus functionality is not the source of the impossibility; its source is the condition of locality. If there is any non-trivial relation between degrees of belief and binary beliefs at all, it must be a “holistic” one. We explore several concrete forms this “holistic” relation could take. (shrink)

This article asks whether justice is binary, whether matters are either-or with respect to it. This question has been inexplicably neglected, and the elementary conceptual work has not been done. We consider this question through exploring the implications of free-will-related justice. We see that there are actually two questions of very different scope here, and that two distinct notions of binarity need to be distinguished. In the process, the plausibility of considering justice as a binary notion is evaluated.

Contrastivism is the claim that the knowledge relation is ternary, it relates three relata: a subject, a proposition, and a class of contrastive propositions. The present paper is a discussion of Jonathan Schaffer’s arguments in favour of contrastivism. The case is made that these are unconvincing: the traditional binary account of knowledge can handle the phenomena that ternarity is claimed to handle in a superior way.

Despite a recent wave in global recognition of the rights of transgender and gender-diverse populations, referred to in this text by the umbrella label of trans*, international law continues to presume a cisgender binary definition of gender — dismissing the lived realities of trans* individuals throughout the world. This gap in international legal recognition and protection has fundamental implications for health, where trans* persons have been and continue to be subjected to widespread discrimination in health care, longstanding neglect of (...) health needs, and significant violations of bodily autonomy. (shrink)

Let S2S [WS2S] espectively be the storn [weak] monadic second order theory of the binary tree T in the language of two successor functions. An S2S-formula whose free variables are just individual variables defines a relation on T (rather than on the power set of T). We show that S2S and WS2S define the same relations on T, and we give a simple characterization of these relations.

We compute the domination monoid in the theory of dense meet‐trees. In order to show that this monoid is well‐defined, we prove weak binarity of and, more generally, of certain expansions of it by binary relations on sets of open cones, a special case being the theory from [7]. We then describe the domination monoids of such expansions in terms of those of the expanding relations.

(1) This paper uses the following binary relations: > (is preferred to); ⩽ (is not preferred to); y ⇔ y are complementary relations onA: [x y]. (3) Pratt, Raiffa and Schlaifer used primitive ⩽, postulated to be transitive onL (the set of lotteries), to define the others with a different expression for are not necessarily complementary onL; since ⩽ is not postulated to be connected onL, but connected ⩽ is necessary and sufficient for such complementarity. Since the (...) restriction of ⩽ to the subsetA ofL is connected, ⩽ and > are complementary onA. (4) Fishburn used primitive are complementary, since asymmetric are asymmetric, negatively transitive, antisymmetric, irreflexive, and transitive; but only ~ is symmetric. (shrink)

Saul Kripke’s work on the semantics of modal logics is well known, unlike his work on Anderson and Belnap’s system E of Entailment (a modal relevance logic), which included his proof of the decidability of its implicational fragment E_>, and also a counterexample to the conjecture of Belnap that E_> is the intersection of the implicational fragments of the relevance logic R and the modal logic S4. This led to Storrs McCall’s suggesting that the “mingle” axiom might be added to (...) E_>, and that this system might be used instead of S4_>. We give a counterexample to this conjecture. We also disprove the conjecture Anderson and Belnap attribute to McCall, in which “restricted mingle” is added to E_>. We use a modal extension of the lattice M_0 and follow ideas of Meyer. In the 1970s, Dunn showed that RM (R with the mingle axiom) can be modeled using a binary accessibility relation, like the accessibility relation in Kripke’s semantics for normal modal logics and his semantics for intuitionistic logic. Dunn also gave a variant interpretation for EM (E with the mingle axiom). We explore some other variations of these binary relational semantics. (shrink)

Traditionally the study of chemical elements has been limited to well-known concepts like the periodic properties and chemical families. However, current information shows a new and rich language that allows us to observe relations in the elements that are not limited to their positions in the table. These relations are evident when reactions are represented through networks, as in the case of similar reactivity of organic compounds sharing functional groups. For the past two decades, it has been argued (...) that network reactions may be considered the core of chemistry. This network, which constitutes the basic set of chemical knowledge, provides the basis for classification and delivers routes to obtain new and known substances. In order to provide an example, we constructed and analyzed a network of chemical elements from the formation of stoichiometric binary compounds, providing to the chemistry, a formal structure of extracting the chemical knowledge. Thus, we explored all possible presence of relationships among elements. Concepts like degree centrality and centralization, the relationships among metals, semimetal and nonmetal classes, blocks, and chemical families were analyzed. We observed that the network structure had a small core set of elements of high reactivity and also a peripheral set of elements of low reactivity. The classes, blocks and families show the following increasing order of reactivity: nonmetals > semimetals > metals; p > s > d > f; and families: boron, carbon, pnictogens, chalcogens, halogens, lanthanoids > other families. This example shows, from the network perspective, that the elements and their classifications exhibit properties such as the reactivity, order, and similarity. (shrink)

This paper deals with two sorts of binary Kripke-style semantics, i.e., algebraic and non-algebraic semantics, for three-valued logic. We first introduce three systems, their corresponding algebraic structures, and associated algebraic completeness results. We next introduce various types of algebraic and non-algebraic binary relational Kripke-style semantics. © 2016 Elsevier B.V., All rights reserved.

Causation is a topic of perennial philosophical concern. As well as being of intrinsic interest, almost all philosophical concepts — such as knowledge, beauty, and moral responsibility — involve a causal dimension. Nonetheless, attempts to provide a satisfactory account of the nature of causation have typically led to barrages of counterexamples. I hope to show that a number of the difficulties plaguing theories of causation have a common source.Most philosophical theories of causation describe a binary relation between cause and (...) effect, or at any rate, a relation that reduces to such a binary relation when certain background information is held fixed. Indeed, most theories provide the same general account of when this relation holds: in order to evaluate whether C causes E, we must make a comparison between two cases, which we may neutrally label as C and ∼C. Where theories of causation differ, of course, is in precisely what is being so compared. Regularity theories of causation require a comparison between what actually happens whenever C occurs, and what actually happens, elsewhere and elsewhen, when C does not occur. (shrink)

The article deals with linguistic relativism in the decisions of the Austrian Verfassungsgerichtshof and the Czech Constitutional Court. It focuses on the Courts argumentation in which the state of language and its limiting perception of the word gender about non-binary and intersex people were used. The article conducts an in-depth analysis of two judgments. The first is the ruling of the Constitutional Court of the Czech Republic Case No. Pl. ÚS 2/20, in which the Court argued the absence of (...) gender-neutral names in Czech as one of the reasons for rejecting the constitutional complaint. The second is the judgment of the Austrian Verfassungsgerichtshof Case No. G 77/2018-9, in which the Court concluded as part of the constitutional review that the language can be adapted to the societal situation. The paper analyses selected parts of both decisions, interpreted using fundamental theses of linguistic relativism (and determinism). I argue that the Czech Constitutional Court has adopted a conservative approach without considering the current societal situation. This opened space for the unequal status of people with an alternative gender identity in society based on ignoring inclusive language, which is the exact opposite of the position of the Austrian Verfassungsgerichtshof. (shrink)