A classic result in the foundations of Yang-Mills theory, due to J. W. Barrett ["Holonomy and Path Structures in General Relativity and Yang-Mills Theory." Int. J. Th. Phys. 30, ], establishes that given a "generalized" holonomy map from the space of piece-wise smooth, closed curves based at some point of a manifold to a Lie group, there exists a principal bundle with that group as structure group and a principal connection on that bundle such that the holonomy map corresponds to (...) the holonomies of that connection. Barrett also provided one sense in which this "recovery theorem" yields a unique bundle, up to isomorphism. Here we show that something stronger is true: with an appropriate definition of isomorphism between generalized holonomy maps, there is an equivalence of categories between the category whose objects are generalized holonomy maps on a smooth, connected manifold and whose arrows are holonomy isomorphisms, and the category whose objects are principal connections on principal bundles over a smooth, connected manifold. This result clarifies, and somewhat improves upon, the sense of "unique recovery" in Barrett's theorems; it also makes precise a sense in which there is no loss of structure involved in moving from a principal bundle formulation of Yang-Mills theory to a holonomy, or "loop", formulation. (shrink)

In his ontological works Kurt Grelling tries to give a rigorous analysis of the foundations of the so-called Gestalt-psychology. Gestalten are peculiar emergent qualities, ontologically dependent on their foundations, but nonetheless non reducible to them. Grelling shows that this concept, as used in psychology and ontology, is often ambiguous. He distinguishes two important meanings in which the word “Gestalt” is used: Gestalten as structural aspects available to transposition and Gestalten as causally self-regulating wholes. Gestalten in the first meaning are, according (...) to Grelling, “equivalence classes of correspondences”, while Gestalten as self-regulating wholes have more to do with relations of ontological dependence. Grelling’s clarification of the concept of Gestalt is doubtless an excellent piece of philosophical analysis, but at the end of the day it turns out that his analysis captures at best only a part of intuitions traditionally connected with the notion of Gestalt. (shrink)

The equivalence connective in ukasiewicz logic has its algebraic counterpart which is the distance function d(x,y) =|x–y| of a positive cone of a commutative -group. We make some observations on logically motivated algebraic structures involving the distance function.

. The connection presented in this paper mirror-links two metamathematical structures, the finitary closure operators, and the compact consistency properties, in such a way that a specification of one structure induces a provably equivalent specification of the other.

Conversation analytic research has advanced understanding of the psychotherapeutic process by understanding how psychotherapy is organised over time in and through interaction between clients and therapists. This study progresses knowledge in this area by examining how psychological accounts of experience are progressively developed across a range of helping relationships. Data include: (1) approximately 30 h of psychotherapy sessions involving trainee therapists; (2) approximately 15 h of psychotherapy demonstration sessions involving expert therapists; and (3) approximately 30 h of everyday conversations involving (...) close friends or family members. This article reports an analysis of techniques that are used to bring together two experiences that were discussed separately, to proffer a candidate connection between them. This proffering of candidate connections was recurrently used in psychotherapy. If confirmed by a client, a proffered connection could be used to develop a psychological account of a client’s experiences, which could then warrant some psychological intervention. In contrast, the proffering of connections was observed in only one of the everyday conversations included in the current study, where it was used to develop psychological accounts of experience. This shows that although proffering candidate connections is an everyday interactional practice, it appears to be used with greater frequency in psychotherapy, to advance its specific institutional aims. (shrink)

A cornerstone of physics, Maxwell‘s theory of electromagnetism, apparently contains a fatal flaw. The standard expressions for the electromagnetic field energy and the self-mass of an electron of finite extension do not obey Einstein‘s famous equation, \, but instead fulfill this relation with a factor 4/3 on the left-hand side. Furthermore, the energy and momentum of the electromagnetic field associated with the charge fail to transform as a four-vector. Many famous physicists have contributed to the debate of this so-called 4/3-problem (...) without arriving at a complete solution. It has generally been assumed that, as originally suggested by Poincaré, the problems are connected to the question of stability of the charge distribution, and that relativistic equivalence between energy and self-mass can only be restored by inclusion of stabilizing forces. Alternative solutions to the problems have also been proposed. Nearly a century ago Fermi suggested a covariant definition of the electromagnetic energy and momentum, and sixty years later Kalckar et al. argued that the 4/3 problem is caused by omission of a relativistic correction in the standard evaluation of the self-force from Coulomb self-interaction. However, the relation between these suggestions has not been clear. We show that the relativistic correction implies that the mechanical momentum of an accelerated rigid body must be defined as the sum of the momenta of its parts for fixed time in the momentary rest frame of the body. For the total momentum of particles and field to be conserved, the total energy–momentum tensor must be divergence free, and this then requires that the momentum of the associated electromagnetic field be defined in the same way, consistent with the suggestion by Fermi. This comprehensive solution of the 4/3-problem demonstrates that there is no conflict of Maxwell‘s theory with special relativity and the questions of equivalence of electromagnetic energy and self-mass and of stability of a classical charge distribution are independent. In appendices we discuss the relations of our treatment with Fermi‘s seminal paper and with a classic paper by Dirac where he evaluated the damping self-force on a point electron from transport of energy and momentum in the electromagnetic field. (shrink)

Given a Polish group G, let $E(G)$ be the right coset equivalence relation $G^{\omega }/c(G)$, where $c(G)$ is the group of all convergent sequences in G. The connected component of the identity of a Polish group G is denoted by $G_0$. Let $G,H$ be locally compact abelian Polish groups. If $E(G)\leq _B E(H)$, then there is a continuous homomorphism $S:G_0\rightarrow H_0$ such that $\ker (S)$ is non-archimedean. The converse is also true when G is connected and compact. For $n\in (...) {\mathbb {N}}^+$, the partially ordered set $P(\omega )/\mbox {Fin}$ can be embedded into Borel equivalence relations between $E({\mathbb {R}}^n)$ and $E({\mathbb {T}}^n)$. (shrink)

A symbol is defined as a species of sign. The concept of a sign coincides with Skinner's (1938) concept of a discriminative stimulus. Symbols differ from other signs in five respects: (1) They are stimuli which the organism can both respond to and produce, either as a self-directed stimulus (as in thinking) or as a stimulus for another individual with a predictably similar response from the recipient in each case. (2) they act as discriminative stimuli for the same kind of (...) object for all members of the verbal community within which they function as a symbols; (3) they acquire their properties by virtue of arbitrary social convention rather than any natural and intrinsic connection between the sign and what it is a sign of; (4) competent members of the verbal community can both produce the appropriate symbol in response to a naturally occurring sign of the presence of the object or a sample of the kind of object which the symbol stands for and select the appropriate object when presented with the symbol; (5) they form stimulus equivalence classes of the kind demonstrated in the matching-to-sample task (Sidman, 1971; Sidman and Tailby, 1982) both with other symbols having the same meaning and, more important, with the naturally-occurring non-symbolic signs of the presence of the object or kind of object which the symbol stands for. (shrink)

In this paper, we present a prenex form theorem for a version of Independence Friendly logic, a logic with imperfect information. Lifting classical results to such logics turns out not to be straightforward, because independence conditions make the formulas sensitive to signalling phenomena. In particular, nested quantification over the same variable is shown to cause problems. For instance, renaming of bound variables may change the interpretations of a formula, there are only restricted quantifier extraction theorems, and slashed connectives cannot be (...) so easily removed. Thus we correct some claims from Hintikka [8], Caicedo & Krynicki [3] and Hodges [11]. We refine definitions, in particular the notion of equivalence, and sharpen preconditions, allowing us to restore those claims, including the prenex form theorem of Caicedo & Krynicki [3], and, as a side result, we obtain an application to Skolem forms of classical formulas. It is a known fact that a complete calculus for IF-logic is impossible, but with our results we establish several quantifier rules that form a partial calculus of equivalence for a general version of IF-logic reflecting general properties of information flow in games. (shrink)

The properties, theoretical magnitudes, and natural kinds which science seeks to characterize, and not the sense or meanings which parts of speech may possess, are the subject of this paper. Many philosophers (e.g., Putnam [1971] and Achinstein [1974]) have agreed that two predicates of different meaning may pick out the same property, but they usually hold that that logically equivalent predicates must pick out the same properties. I propose to deny this thesis. My argument is by way of an example (...) and by appeal to a principle. The principle connects propertyhood with causal efficacy. The example situation I will describe in what follows is intended to show that the property of being a closed straight-sided figure having three sides is not the same property as that of being a closed straight-sided figure having three angles (a triangle). (shrink)

In this paper we investigate some basic semantic and syntactic conditions characterizing the equivalence connective. In particular we define three basic classes of algebras: the class of weak equivalential algebras, the class of equivalential algebras and the class of regular equivalential algebras.Weak equivalential algebras can be used to study purely equivalential fragments of relevant logics and strict equivalential fragments of some modal logics. Equivalential algebras are suitable to study purely equivalential fragment of BCI and BCK logic. A subclass (...) of the class of regular equivalential algebras is suitable to study equivalential fragments of ukasiewicz logics. Some subvarieties of the class of regular equivalential algebras provide natural semantics for equivalential fragments of the intuitionistic prepositional logic and various intermediate logics. (shrink)

Given a Polish group G, let $E(G)$ be the right coset equivalence relation $G^\omega /c(G)$, where $c(G)$ is the group of all convergent sequences in G. We first established two results: (1) Let $G,H$ be two Polish groups. If H is TSI but G is not, then $E(G)\not \le _BE(H)$. (2) Let G be a Polish group. Then the following are equivalent: (a) G is TSI non-archimedean; (b) $E(G)\leq _B E_0^\omega $ ; and (c) $E(G)\leq _B {\mathbb {R}}^\omega /c_0$. (...) In particular, $E(G)\sim _B E_0^\omega $ iff G is TSI uncountable non-archimedean. A critical theorem presented in this article is as follows: Let G be a TSI Polish group, and let H be a closed subgroup of the product of a sequence of TSI strongly NSS Polish groups. If $E(G)\le _BE(H)$, then there exists a continuous homomorphism $S:G_0\rightarrow H$ such that $\ker (S)$ is non-archimedean, where $G_0$ is the connected component of the identity of G. The converse holds if G is connected, $S(G)$ is closed in H, and the interval $[0,1]$ can be embedded into H. As its applications, we prove several Rigid theorems for TSI Lie groups, locally compact Polish groups, separable Banach spaces, and separable Fréchet spaces, respectively. (shrink)

In this essay I argue that even though egalitarianism and prioritarianism are different theories of social welfare, they can use the same social welfare measures. I present six different arguments for this thesis. The first argument is that conceptual connections between egalitarianism and prioritarianism ensure that any measure that works for either theory works for both. The second argument is that conditions necessary and sufficient to identify egalitarian and prioritarian measures, respectively, are equivalent. The third argument is that both egalitarianism (...) and prioritarianism can use two standard measures, typically proposed for only one theory. The fourth to sixth arguments contend that four properties that have been proposed as distinctive of either egalitarian or prioritarian measures cannot distinguish between them. I conclude that any egalitarian measure is also a prioritarian measure, and vice versa. (shrink)

Say that a property is topological if and only if it is invariant under homeomorphism. Homeomorphism would be a successful criterion for the equivalence of logical systems only if every logically significant property of every logical system were topological. Alas, homeomorphisms are sometimes insensitive to distinctions that logicians value: properties such as functional completeness are not topological. So logics are not just devices for exploring closure topologies. One still wonders, though, how much of logic is topological. This essay examines (...) some logically significant properties that are topological (or are topological in some important class). In the process, we learn something about the conditions under which the meaning of a connective can be "given by the connective's role in inference.". (shrink)

Normative systems have been advocated as an effective tool to regulate interaction in multi-agent systems. The use of deontic operators and the ability to represent defeasible information are known to be two fundamental ingredients to represent and reason about normative systems. In this paper, after introducing a framework that combines standard deontic logic and non-monotonic logic programming, deontic logic programs (DLP), we tackle the fundamental problem of equivalence between normative systems using a deontic extension of David Pearce’s Equilibrium Logic (...) and its monotonic basis, the logic of Here-and-There. We also show how deontic logic programs can be used to represent and reason about normative systems, and establish a strong connection with input-output logic. (shrink)

We present a new notion of game equivalence that captures basic powers of interacting players. We provide a representation theorem, a complete logic, and a new game algebra for basic powers. In doing so, we establish connections with imperfect information games and epistemic logic. We also identify some new open problems concerning logic and games.

In this article I connect two debates in the philosophy of science: the questions of scientific representation and both model and theoretical equivalence. I argue that by paying attention to how a model is used to draw inferences about its target system, we can define a notion of theoretical equivalence that turns on whether models license the same claims about the same target systems. I briefly consider the implications of this for two questions that have recently been discussed (...) in the context of the formal philosophy of science. (shrink)

The "space" of Lascar strong types, on some sort and relative to a given complete theory T, is in general not a compact Hausdorff topological space. We have at least three aims in this paper. The first is to show that spaces of Lascar strong types, as well as other related spaces and objects such as the Lascar group Gal L of T, have well-defined Borel cardinalities. The second is to compute the Borel cardinalities of the known examples as well (...) as of some new examples that we give. The third is to explore notions of definable map, embedding, and isomorphism, between these and related quotient objects. We also make some conjectures, the main one being roughly "smooth if and only if trivial". The possibility of a descriptive set-theoretic account of the complexity of spaces of Lascar strong types was touched on in the paper [E. Casanovas, D. Lascar, A. Pillay and M. Ziegler, Galois groups of first order theories, J. Math. Logic1 305–319], where the first example of a "non-G-compact theory" was given. The motivation for writing this paper is partly the discovery of new examples via definable groups, in [A. Conversano and A. Pillay, Connected components of definable groups and o-minimality I, Adv. Math.231 605–623; Connected components of definable groups and o-minimality II, to appear in Ann. Pure Appl. Logic] and the generalizations in [J. Gismatullin and K. Krupiński, On model-theoretic connected components in some group extensions, preprint, arXiv:1201.5221v1]. (shrink)

We study the complexity of isomorphism of classes of metric structures using methods from infinitary continuous logic. For Borel classes of locally compact structures, we prove that if the equivalence relation of isomorphism is potentially $\mathbf {\Sigma }^0_2$, then it is essentially countable. We also provide an equivalent model-theoretic condition that is easy to check in practice. This theorem is a common generalization of a result of Hjorth about pseudo-connected metric spaces and a result of Hjorth–Kechris about discrete structures. (...) As a different application, we also give a new proof of Kechris’s theorem that orbit equivalence relations of actions of Polish locally compact groups are essentially countable. (shrink)

The present paper studies two approaches to the expressiveness of propositional modal logics based on first-degree entailment logic, FDE. We first consider the basic FDE-based modal logic BK and certain systems in its vicinity, and then turn to some FDE-based modal logics in a richer vocabulary, including modal bilattice logic, MBL. On the one hand, model-theoretic proofs of the definability of connectives along the lines of [McCullough, “Logical connectives for intuitionistic propositional logic”, Journal of Symbolic Logic 36, 1 (1971): 15–20. (...) DOI: 10.2307/2271511] and [[17] Wansing, “Logical connectives for constructive modal logic”, Synthese 150, 3 (2006): 459–482. DOI: 10.1007/s11229-005-5518-5] are given for various FDE-based modal logics. On the other hand, building on [Odintsov and Wansing, “Disentangling FDE-based paraconsistent modal logics, Studia Logica 105, 6 (2017): 1221–1254. DOI: 10.1007/s11225-017-9753-9], expressibility is considered in terms of mutual faithful embeddability of one logic into another logic. A distinction is drawn between definitional equivalence, which is defined with respect to a pair of structural translations between two languages, and weak definitional equivalence, which is defined with respect to a weaker notion of translations. Moreover, the definitional equivalence of some FDE-based modal logics is proven, especially the definitional equivalence of MBL and a conservative extension of the logic BK□×BK□, which underlines the central role played by BK among FDE-based modal logics. (shrink)

Chs. 8 and 9 convert the two basic forms of slingshot argument—one used by Alonzo Church, W. V. Quine, and Donald Davidson, the other by Kurt Gödel—into knock‐down deductive proofs that Donald Davidson's and Richard Rorty's cases against facts and the representation of facts are unfounded, and their slingshot arguments for discrediting the existence of facts unsatisfactory. The proofs are agnostic on key semantic issues; in particular, they assume no particular account of reference and do not even assume that sentences (...) have references. Using this procedure, Ch. 8 shows that cleaned‐up versions of the slingshot arguments used by Davidson, Church, and Quine demonstrate theses weaker than those their authors were seeking—the impossibility of facts and non‐truth‐functional connectives. This is primarily because the arguments depend upon logical equivalences and theories of definite descriptions that must satisfy certain semantic conditions if the aforementioned logical equivalences are to obtain. The four sections of the chapter are: Introductory Remarks; The Quinean Strategy; An Incomplete Connective Proof; and A Complete Connective Proof. (shrink)

In the first exposition of the doctrine of indeterminacy of translation, Quine asserted that the individuation and translation of truth-functional sentential connectives like 'and', 'or', 'not' are not indeterminate. He changed his mind later on, conjecturing that some sentential connectives might be interpreted in different non-equivalent ways. This issue has not been debated much by Quine, or in the subsequent literature, it is, as it were, an unsolved problem, not well understood. For the sake of the argument, I will adopt (...) Quine's background assumption that all the semantic features of a language can be reduced to the speakers' dispositions toward assent and dissent, as far as only the truth-conditional core of the meaning of sentences is concerned. I will put forward an argument to the effect that the speech dispositions of most, if not all, English (French, Italian, etc.) speakers constrain a unique translation of their connectives. This argument crucially relies on an empirical conjecture concerning the behaviour of these operators. (shrink)

This rare publication continues an exploratory journey in relational biology, a study of biology in terms of the organization of networked connections in living systems. It builds on the author's two earlier monographs which looked at the epistemology of life and the ontogeny of life. Here the emphasis is on the intangibility of life, that the real nature of living systems is conveyed not by their tangible material basis but by their intangible inherent processes. Relational biology is the approach that (...) hails 'function dictates structure'; it is mathematics decoded into biological realizations. Therefore, the work begins with a concise introduction to category theory, equiping the reader with the mathematical metalanguage of relation biology. The book is organized around three parts: Part I is a comprehensive study of the most important functor in relational biology, the power set functor. The author lays the set-theoretic foundations of the functorial connections in relational biology, exploring relations, mappings, and set-valued mappings. In Part II, Natural Law receives a new mathematical formulation founded on two axioms: 'Everything is a set.' and 'Every process is a set-valued mapping.' The reader sees how Metabolism-Repair networks, equipped with set-valued processors, expand their role from models of biological entities to generic models of all natural systems. Part III expounds the various shades of invertibility in general, and the inversion of encoding to decoding in particular. A plethora of mathematical and biological examples illustrate the category-theoretic concepts of equivalence and adjunction. This book's algebraic approach to biological models will appeal to researchers and graduate students in mathematics, biology, and the philosophy of science. (shrink)

We construct a notion of teleparallelization for Newton-Cartan theory, and show that the teleparallel equivalent of this theory is Newtonian gravity; furthermore, we show that this result is consistent with teleparallelization in general relativity, and can be obtained by null-reducing the teleparallel equivalent of a five-dimensional gravitational wave solution. This work thus strengthens substantially the connections between four theories: Newton-Cartan theory, Newtonian gravitation theory, general relativity, and teleparallel gravity.

We investigate diagonal actions of Polish groups and the related intersection operator on closed subgroups of the acting group. The Borelness of the diagonal orbit equivalence relation is characterized and is shown to be connected with the Borelness of the intersection operator. We also consider relatively tame Polish groups and give a characterization of them in the class of countable products of countable abelian groups. Finally an example of a logic action is considered and its complexity in the Borel (...) reducbility hierarchy determined. (shrink)

The objective of the present study was to show that the use of adversative and conclusive connectives to mark off the prototypical schema of argumentative text begins to set in at approximately the age of 10 or 11. Based on Adam's (1992) proposals, we constituted an argumentative text with two blocks of arguments separated by an adversative instruction (the connective but or an equivalent) and followed by a conclusion introduced by a conclusive instruction (the connective thus or an (...) equivalent). Four revising tasks (insertion or substitution with or without five connectives) have been used to asses children's knowledge of the argumentative schema and the use of connectives that punctuate them. The study of good and erroneous locations (concerning the placement of but and thus) showed that there was some regularity in the choices made by the children. The main result of this study shows that argumentative connectives are used differently by children aged 9 and children aged 10 or 11. The argumentative schema is used more consciously by 11-year-olds to guide revising tasks than by 10-year-olds. (shrink)

We present a new notion of game equivalence that captures basic powers of interacting players. We provide a representation theorem, a complete logic, and a new game algebra for basic powers. In doing so, we establish connections with imperfect information games and epistemic logic. We also identify some new open problems concerning logic and games.

In this paper I analyze three artificial examples of empirical equivalence: van Fraassen’s alternative formulations of Newton’s theory, the Poincaré-Reichenbach argument for the conventionality of geometry; and predictively equivalent ‘systems of the world’. These examples have received attention in the philosophy of science literature because they are supposed to illustrate the connection between predictive equivalence and underdetermination of theory choice. I conclude that this view is wrong. These examples of empirical equivalence are harmless with respect to the (...) problem of underdetermination. (shrink)

If a parsimony criterion may be used to choose between theories that make different predictions, may the same criterion be used to choose between theories that are predictively equivalent? The work of the statistician H. Akaike (1973) is discussed in connection with this question. The results are applied to two examples in which parsimony has been invoked to choose between philosophical theories-Shoemaker's (1969) discussion of the possibility of time without change and the discussion by Smart (1959) and Brandt and Kim (...) (1967) of mind/body dualism and the identity theory. (shrink)

An extensions by new axioms and rules of an algebraizable logic in the sense of Blok and Pigozzi is not necessarily algebraizable if it involves new connective symbols, or it may be algebraizable in an essentially different way than the original logic. However, extension whose axioms and rules define implicitly the new connectives are algebraizable, via the same equivalence formulas and defining equations of the original logic, by enriched algebras of its equivalente quasivariety semantics. For certain strongly algebraizable (...) logics, all connectives defined implicitly by axiomatic extensions of the logic are explicitly definable. (shrink)

We introduce a new framework for classifying logics on finite structures and studying their expressive power. This framework is based on the concept of almost everywhere equivalence of logics, that is to say, two logics having the same expressive power on a class of asymptotic measure 1. More precisely, if L, L ′ are two logics and μ is an asymptotic measure on finite structures, then $\scr{L}\equiv _{\text{a.e.}}\scr{L}^{\prime}(\mu)$ means that there is a class C of finite structures with μ (...) (C)=1 and such that L and L ′ define the same queries on C. We carry out a systematic investigation of $\equiv _{\text{a.e.}}$ with respect to the uniform measure and analyze the $\equiv _{\text{a.e.}}$ -equivalence classes of several logics that have been studied extensively in finite model theory. Moreover, we explore connections with descriptive complexity theory and examine the status of certain classical results of model theory in the context of this new framework. (shrink)

We study the computational content of the Brouwer Fixed Point Theorem in the Weihrauch lattice. Connected choice is the operation that finds a point in a non-empty connected closed set given by negative information. One of our main results is that for any fixed dimension the Brouwer Fixed Point Theorem of that dimension is computably equivalent to connected choice of the Euclidean unit cube of the same dimension. Another main result is that connected choice is complete for dimension greater than (...) or equal to two in the sense that it is computably equivalent to Weak Kőnig’s Lemma. While we can present two independent proofs for dimension three and upward that are either based on a simple geometric construction or a combinatorial argument, the proof for dimension two is based on a more involved inverse limit construction. The connected choice operation in dimension one is known to be equivalent to the Intermediate Value Theorem; we prove that this problem is not idempotent in contrast to the case of dimension two and upward. We also prove that Lipschitz continuity with Lipschitz constants strictly larger than one does not simplify finding fixed points. Finally, we prove that finding a connectedness component of a closed subset of the Euclidean unit cube of any dimension greater than or equal to one is equivalent to Weak Kőnig’s Lemma. In order to describe these results, we introduce a representation of closed subsets of the unit cube by trees of rational complexes. (shrink)

One of the basic theorems in universal algebra is Birkhoff's variety theorem: the smallest equationally axiomatizable class containing a class K of algebras coincides with the class obtained by taking homomorphic images of subalgebras of direct products of elements of K. G. Gratzer asked whether the variety theorem is equivalent to the Axiom of Choice. In 1980, two of the present authors proved that Birkhoff's theorem can already be derived in ZF. Surprisingly, the Axiom of Foundation plays a crucial role (...) here: we show that Birkhoff's theorem cannot be derived in ZF + AC {Foundation}. even if we add Foundation for Finite Sets. We also prove that the variety theorem is equivalent to a purely set-theoretical statement, the Collection Principle. This principle is independent of $ZF \{\operatorname{Foundation}$ . The second part of the paper deals with further connections between axioms of ZF-set theory and theorems of universal algebra. (shrink)

We follow the ideas given by Chen and Grätzer to represent Stone algebras and adapt them for the case of Stonean residuated lattices. Given a Stonean residuated lattice, we consider the triple formed by its Boolean skeleton, its algebra of dense elements and a connecting map. We define a category whose objects are these triples and suitably defined morphisms, and prove that we have a categorical equivalence between this category and that of Stonean residuated lattices. We compare our results (...) with other works and show some applications of the equivalence. (shrink)

With catastrophic events of «nature» like global warming, arguments emerge that insinuate an equivalence of vulnerability, responsibility or being affected by these catastrophes. Such an alleged equivalence when facing climate catastrophe is already visible, for example, in the notion of the «Anthropocene» itself, which obscures both causes and various vulnerabilities in a homogenized as well as universalized concept of humanity (anthropos). Taking such narratives as a starting point, the paper explores questions about the connection between catastrophe, temporality, and (...) history, following mainly Walter Benjamin, Dipesh Chakrabarty and Jean-Luc Nancy with the goal to provide (1) a critique of the concept «Anthropocene» on the basis of nonequivalence while retaining its key features to still grasp the catastrophic present, (2) an analysis of anthropocenic time and the chronical structure of catastrophe, (3) philosophical considerations on the intersection of catastrophe and history. (shrink)

Suppose λ is a singular cardinal of uncountable cofinality κ. For a model M of cardinality λ, let No (M) denote the number of isomorphism types of models N of cardinality λ which are L ∞λ - equivalent to M. In [7] Shelah considered inverse κ- systems A of abelian groups and their certain kind of quotient limits Gr(A)/ Fact(A). In particular Shelah proved in [7, Fact 3.10] that for every cardinal μ there exists an inverse κ-system A such that (...) A consists of abelian groups having cardinality at most μ κ and card(Gr(A)/Fact(A)) = μ. Later in [8, Theorem 3.3] Shelah showed a strict connection between inverse κ-systems and possible values of No (under the assumption that $\theta^\kappa for every $\theta ): if A is an inverse κ- system of abelian groups having cardinality $\theta^\kappa for every $\theta ): for every nonzero $\mu or μ = λ κ there is a model M μ of cardinality λ with No(M μ ) = μ. In this paper we show: for every nonzero μ ≤ λ κ there is an inverse κ-system A of abelian groups having cardinality $2^\kappa and $\theta^{ for all $\theta when $\mu > \lambda$ ), with the obvious new consequence concerning the possible value of No. Specifically, the case No(M) = λ is possible when $\theta^\kappa for every $\theta. (shrink)

Chaos is often explained in terms of random behaviour; and having positive Kolmogorov–Sinai entropy (KSE) is taken to be indicative of randomness. Although seemly plausible, the association of positive KSE with random behaviour needs justification since the definition of the KSE does not make reference to any notion that is connected to randomness. A common way of justifying this use of the KSE is to draw parallels between the KSE and ShannonÕs information theoretic entropy. However, as it stands this no (...) more than a heuristic point, because no rigorous connection between the KSE and ShannonÕs entropy has been established yet. This paper fills this gap by proving that the KSE of a Hamiltonian dynamical system is equivalent to a generalized version of ShannonÕs information theoretic entropy under certain plausible assumptions. Ó 2005 Elsevier Ltd. All rights reserved. (shrink)

I argue that the best spacetime setting for Newtonian gravitation (NG) is the curved spacetime setting associated with geometrized Newtonian gravitation (GNG). Appreciation of the ‘Newtonian equivalence principle’ leads us to conclude that the gravitational field in NG itself is a gauge quantity, and that the freely falling frames are naturally identified with inertial frames. In this context, the spacetime structure of NG is represented not by the flat neo-Newtonian connection usually made explicit in formulations, but by the sum (...) of the flat connection and the gravitational field. 1 Introduction2 Newtonian Gravity: The Orthodox Approach3 Newtonian Gravity: Additional Symmetries4 Cosmological Considerations5 A Newtonian Equivalence Principle: Inertial Frames in Newtonian Gravitation6 Theory Equivalence?7 Conclusion. (shrink)

Originating from work in operations research the cutting plane refutation systemCP is an extension of resolution, where unsatisfiable propositional logic formulas in conjunctive normal form are recognized by showing the non-existence of boolean solutions to associated families of linear inequalities. Polynomial sizeCP proofs are given for the undirecteds-t connectivity principle. The subsystemsCP q ofCP, forq≥2, are shown to be polynomially equivalent toCP, thus answering problem 19 from the list of open problems of [8]. We present a normal form theorem forCP (...) 2-proofs and thereby for arbitraryCP-proofs. As a corollary, we show that the coefficients and constant terms in arbitrary cutting plane proofs may be exponentially bounded by the number of steps in the proof, at the cost of an at most polynomial increase in the number of steps in the proof. The extensionCPLE +, introduced in [9] and there shown top-simulate Frege systems, is proved to be polynomially equivalent to Frege systems. Lastly, since linear inequalities are related to threshold gates, we introduce a new threshold logic and prove a completeness theorem. (shrink)

C. Karp has shown that if α is an ordinal with ω α = α and A is a linear ordering with a smallest element, then α and $\alpha \bigotimes A$ are equivalent in L ∞ω up to quantifer rank α. This result can be expressed in terms of Ehrenfeucht-Fraïssé games where player ∀ has to make additional moves by choosing elements of a descending sequence in α. Our aim in this paper is to prove a similar result for Ehrenfeucht-Fraïssé (...) games of length ω 1 . One implication of such a result will be that a certain infinite quantifier language cannot say that a linear ordering has no descending ω 1 -sequences (when the alphabet contains only one binary relation symbol). Connected work is done by Hyttinen and Oikkonen in [H] and [O]. (shrink)

The meaning of a formula built out of proof-functional connectives depends in an essential way upon the intensional aspect of the proofs of the component subformulas. We study three such connectives, strong equivalence (where the two directions of the equivalence are established by mutually inverse maps), strong conjunction (where the two components of the conjunction are established by the same proof) and relevant implication (where the implication is established by an identity map). For each of these connectives we (...) give a type assignment system, a realizability semantics, and a completeness theorem. This form of completeness implies the semantic completeness of the type assignment system. (shrink)

This article examines a series of Greek myths which establish a structural equivalence between two motifs, stoning and blinding; the two penalties either substitute for one another in alternative versions of a single story, or appear in sequence as repayments in kind. After reviewing other theories concerning the motives behind blinding and lapidation, I argue that both punishments-together with petrifaction and live imprisonment, which frequently figure alongside the other motifs-are directed against individuals whose crimes generate pollution. This miasma affects (...) not only the perpetrator of the deed, but risks spreading to the community at large, and prompts measures aimed at containing the source of the disease. Both blinding and lapidation are designed to cordon off the contaminant by removing him from all visual and tactile contact with other men. But it is not only the nature of the crimes that explains the kinship between the two penalties. I further argue that the attributes Greek thinking assigned to stones, repeatedly characterized as unseeing, mute, immobile, and dry, and symbolic of the condition of the dead, elucidate the connections and clarify the antagonism that myth suggests between lapidation and sight. Stoning, blinding, imprisonment, and petrifaction all consign the criminal to an existence exactly parallel to that of the stone, stripping him of the properties that distinguish the living from the dead, and making him both unseeing and unseen. Three examples drawn from archaic and classical literature provide examples of these interactions between stones, blindness, invisibility, and death: the snake portent sent by Zeus in Book 2 of the Iliad, the Perseus myth, and Hermes' activity in both the Homeric Hymn to Hermes and Aeschylus' Choephoroe. (shrink)

In this paper we show that some versions of Dung’s abstract argumentation frames are equivalent to classical propositional logic. In fact, Dung’s attack relation is none other than the generalised Peirce–Quine dagger connective of classical logic which can generate the other connectives ${\neg, \wedge, \vee, \to}$ of classical logic. After establishing the above correspondence we offer variations of the Dung argumentation frames in parallel to variations of classical logic, such as resource logics, predicate logic, etc., etc., and create resource (...) argumentation frames, predicate argumentation frames, etc., etc. We also offer the notion of logic proof as a geometrical walk along the nodes of a Dung network and thus we are able to offer a geometrical abstraction of the notion of inference based argumentation. Thus our paper is also a contribution to the question: “What is a logical system” in as much as it integrates logic with abstract argumentation networks. (shrink)

The concept of constructive negation we refer to in this paper is (minimally) intuitionistic in character (see [1]). The idea is to understand the negation of a proposition A as equivalent to A implying a falsity constant of some sort. Then, negation is introduced either by means of this falsity constant or, as in this paper, by means of a propositional connective defined with the constant. But, unlike intuitionisitc logic, the type of negation we develop here is, of course, (...) devoid of paradoxes of relevance. (shrink)

The minimal ordinal-connection axiom $$MOC$$ was introduced by the first author in R. Freire. (South Am. J. Log. 2:347–359, 2016). We observe that $$MOC$$ is equivalent to a number of statements on the existence of certain hierarchies on the universe, and that under global choice, $$MOC$$ is in fact equivalent to the $${{\,\mathrm{GCH}\,}}$$. Our main results then show that $$MOC$$ corresponds to a weak version of global choice in models of the $${{\,\mathrm{GCH}\,}}$$ : it can fail in models of the (...) $${{\,\mathrm{GCH}\,}}$$ without global choice, but also global choice can fail in models of $$MOC$$. (shrink)