This book presents an English translation of a classic Russian text on duality theory for Heyting algebras. Written by Georgian mathematician Leo Esakia, the text proved popular among Russian-speaking logicians. This translation helps make the ideas accessible to a wider audience and pays tribute to an influential mind in mathematical logic. The book discusses the theory of Heyting algebras and closure algebras, as well as the corresponding intuitionistic and modal logics. The author introduces the key notion of (...) a hybrid that “crossbreeds” topology and order, resulting in the structures now known as Esakia spaces. The main theorems include a duality between the categories of closure algebras and of hybrids, and a duality between the categories of Heyting algebras and of so-called strict hybrids. Esakia’s book was originally published in 1985. It was the first of a planned two-volume monograph on Heyting algebras. But after the collapse of the Soviet Union, the publishing house closed and the project died with it. Fortunately, this important work now lives on in this accessible translation. The Appendix of the book discusses the planned contents of the lost second volume. (shrink)

The duality theory established by Halmos in [2] for boolean hemimorphism applies of course to the diagonalizable algebra, because ντν is an hemimorphism. For commodity in working on diagonalizable algebras we recall the basic facts and give the characteristic conditions on the dual of ντν.

We start from Marra–Spada duality between semisimple MV-algebras and Tychonoff spaces, and we consider the particular cases when the \-skeleta of the MV-algebras are restricted in some way. In particular we consider antiskeletal MV-algebras, that is, the ones whose \-skeleton is trivial.

In this work we present some new contributions towards two different directions in the study of modal logic. First we employ tense logics to provide a temporal interpretation of intuitionistic quantifiers as “always in the future” and “sometime in the past.” This is achieved by modifying the Gödel translation and resolves an asymmetry between the standard interpretation of intuitionistic quantifiers.Then we generalize the classic Gelfand–Naimark–Stone duality between compact Hausdorff spaces and uniformly complete bounded archimedean $\ell $ -algebras to a (...) duality encompassing compact Hausdorff spaces with continuous relations. This leads to the notion of modal operators on bounded archimedean $\ell $ -algebras and in particular on rings of continuous real-valued functions on compact Hausdorff spaces. This new duality is also a generalization of the classic Jónsson-Tarski duality in modal logic.Abstract taken directly from the thesis.E-mail: [email protected]: https://www.proquest.com/openview/5d284dbfb954383da9364149fa312b6f/1?pq-origsite=gscholar&cbl=18750& diss=y. (shrink)

In this paper we continue the investigation of monadic Heyting algebras which we started in [2]. Here we present the representation theorem for monadic Heyting algebras and develop the duality theory for them. As a result we obtain an adequate topological semantics for intuitionistic modal logics over MIPC along with a Kripke-type semantics for them. It is also shown the importance and the effectiveness of the duality theory for further investigation of monadic Heyting algebras and logics (...) over MIPC. (shrink)

Both syntactic and semantic solutions are given for the entailment problem of duality theory. The test algebra theorem provides both a syntactic solution to the entailment problem in terms of primitive positive formulae and a new derivation of the corresponding result in clone theory, viz. the syntactic description of $\operatorname{Inv(Pol}(R))$ for a given set R of finitary relations on a finite set. The semantic solution to the entailment problem follows from the syntactic one, or can be given (...) in the form of an algorithm. It shows, in the special case of a purely relational type, that duality-theoretic entailment is describable in terms of five constructs, namely trivial relations, intersection, repetition removal, product, and retractive projection. All except the last are concrete, in the sense that they are described by a quantifier-free formula. It is proved that if the finite algebra $\underline{M}$ generates a congruence-distributive variety and all subalgebras of $\underline{M}$ are subdirectly irreducible, then concrete constructs suffice to describe entailment. The concept of entailment appropriate to strong dualities is also introduced, and described in terms of coordinate projections, restriction of domains, and composition of partial functions. (shrink)

This paper explores relationships between many-valued logic and fuzzy topology from the viewpoint of duality theory. We first show a fuzzy topological duality for the algebras of Łukasiewicz n -valued logic with truth constants, which generalizes Stone duality for Boolean algebras to the n -valued case via fuzzy topology. Then, based on this duality, we show a fuzzy topological duality for the algebras of modal Łukasiewicz n -valued logic with truth constants, which generalizes Jónsson-Tarski (...) duality for modal algebras to the n -valued case via fuzzy topology. We emphasize that fuzzy topological spaces naturally arise as spectrums of algebras of many-valued logics. (shrink)

The main aim of this paper is to make a remark about the relation between dualities between theories, as `duality' is understood in physics and equivalence of theories, as `equivalence' is understood in logic and philosophy. The remark is that in physics, two theories can be dual, and accordingly get called `the same theory', though we interpret them as disagreeing---so that they are certainly not equivalent, as `equivalent' is normally understood. So the remark is simple: but, I shall (...) argue, worth stressing---since often neglected. My argument for this is based on the account of duality developed by De Haro: which is illustrated here with several examples, from both elementary physics and string theory. Thus I argue that in some examples, including in string theory, two dual theories disagree in their claims about the world. I also spell out how this remark implies a limitation of proposals to understand theoretical equivalence as either logical equivalence or a weakening of it. (shrink)

Some philosophers of science have suggested that contemporary science should be the source of inspiration to the new analytic metaphysics (A. Chakra vartty, C. Callender, S. French, J. Ladyman, T. Maudlin, etc.). This paper explores the prospect of a string metaphysics: a research program in analytic metaphysics based on string theory. Different forms of fundamentalism and pluralism are discussed in this context. The paper focus on string metaphysics with S-dualities (a relation between models of string theory at different (...) coupling regimes) and argues that fundamentality and compositionality have to be reconceptualized. String metaphysics with dualities is better couched in terms of metaphysical pluralism. Grounding, as well as a sketch of a string modality, are briefly discussed. The paper concludes with a suggestion for future work: the metaphysician may find a productive ground for research in discussing other dualities (especially the T-dualities, or the AdS/CFT duality), the emergence of spacetime, the concept of time in string theory, the multiverse etc. (shrink)

The article investigates one of the key contributions to modern structural mathematics, namely Hilbert’sFoundations of Geometry and its mathematical roots in nineteenth-century projective geometry. A central innovation of Hilbert’s book was to provide semantically minded independence proofs for various fragments of Euclidean geometry, thereby contributing to the development of the model-theoretic point of view in logical theory. Though it is generally acknowledged that the development of model theory is intimately bound up with innovations in 19th century geometry, so (...) far, little has been said about how exactly model-theoretic concepts grew out of methodological investigations within projective geometry. This article is supposed to fill this lacuna and investigates this geometrical prehistory of modern model theory, eventually leading up to Hilbert’sFoundations. (shrink)

String theory promises to be able to provide us with a working theory of quantum gravity and a unified description of all fundamental forces. In string theory there are so called ‘dualities’; i.e. different theoretical formulations that are physically equivalent. In this article these dualities are investigated from a philosophical point of view. Semantic and epistemic questions relating to the problem of underdetermination of theories by data and the debate on realism concerning scientific theories are discussed. Depending (...) on ones views on semantic issues and realism different interpretations are possible of the dualities. (shrink)

: Frege’s remarks about the first-person pronoun in Der Gedanke have elicited numerous commentaries, but his insight has not been fully appreciated or developed. Commentators have overlooked Frege’s reasons for claiming that there are two distinct first-person senses, and failed to realize that his remarks easily generalize to all indexicals. I present a perspectival theory of indexicals inspired by Frege’s claim that all indexical types have a dual meaning which, in turn, leads to a duality of senses expressed (...) by indexical tokens. Keywords : Indexicals; First Person; Perspective; Senses La doppia natura degli indessicali: una posizione fregeana Riassunto : Le osservazioni di Frege sul pronome di prima persona contenute in Der Gedanke hanno sollevato numerosi commenti, ma le sue intuizioni non sono state pienamente comprese o sviluppate. I commentatori di Frege hanno trascurato le ragioni per le quali egli sosteneva che ci sono due distinti sensi della prima persona e non hanno colto come queste sue osservazioni possono essere facilmente estese a tutti gli indessicali. Intendo presentare qui una posizione prospettivista sugli indessicali, ispirata dall’affermazione di Frege per cui tutti i tipi di indessicali hanno un doppio significato che, a sua volta, porta a una doppia natura dei sensi espressi dalle occorrenze indessicali. Parole chiave : Indessicali; Prima persona; Prospettiva; Sensi. (shrink)

Science does not exist in vacuum; it arises and works in context. Ground-breaking achievements transforming the scientific landscape often stem from philosophical thought, just as symbolic logic and computer science were born from the early analytic philosophy, and for the very reason they impact our global worldview as a coherent whole as well as local knowledge production in different specialised domains. Here we take first steps in elucidating rich philosophical contexts in which Samson Abramsky’s far-reaching work centring around categorical science (...) as a new kind of science may be placed, explicated, and articulated. We argue, inter alia, that Abramsky’s work, as a whole, may be construed as demonstrating the categorical unity of science, or rather the sciences, in a mathematically rigorous, down-to-earth manner, which has been a salient feature of his work. At the same time we trace his intellectual history, leading from duality, to intensionality, and to contextuality, and place it in a broader context of philosophy beyond the analytic-continental divide, namely towards the reintegration of them as in the post-analytic tradition. Besides, we address issues in philosophy of category theory, such as the foundational autonomy of category theory and the (presumably two) dogmas of set-theoretical foundationalism, which Abramsky actually touch upon in one of his few philosophically inclined works. As to philosophy of category theory, we also address categorical structuralism as higher-order structuralism, categorical epistemology as elucidating higher-order meta-laws, and categorical ontology as allowing for reduction of ontological commitment via structural realism, the structuralist resolution of Benaceraff’s dilemma, and the pluralistic multiverse view of science as opposed to the set-theoretical reductionist ‘universe’ view. We conclude by speculating about the existence of the Oxford School of (Pluralistic) Unified Science as opposed to the Vienna Circle of (Monistic) Unified Science and to the Stanford School of (Pluralistic) Disunified Science; Categorical Unified Science may potentially allow us to reconcile the two camps on the unity and disunity of science whilst doing justice to both of them. Categorical unity arguably allows for unification via epistemological and ontological networking, and via knowledge transfer thus enabled, rather than unification via the reduction of everything and every truth to a single foundationalist framework, whilst taking at face value disunity, plurality, and diversity, and their significance in science and in human civilisation as a whole. (shrink)

Methods developed in a previous paper are employed to define an exact correspondence between the states of a deterministic cellular automaton in 1+1 dimensions and those of a bosonic quantum field theory. The result may be used to argue that quantum field theories may be much closer related to deterministic automata than what is usually thought possible.

– The paper explores first the postulational basis and significance of‘measures of information’in current information theory and their possible relations to physical entropy and Brillouin's‘negentropy’regarded as the negative of entropy. For some purposes, the same pattern or formal structure may be abstracted from both‘entropy’and‘information’. The paper analyzes, in the second place, the mathematical analogies which have been traced between information theory and quantum mechanics and argues that the analogies have but a limited value when we come to grips (...) with the deeper, and yet unsolved, problems of quantum theory. Lastly it is urged that the indiscriminate extensions of statistical and information‐theoretic methods, and of quantum‐mechanical analogies, e.g.‘linguistic duality’, to psycholinguistics, literary and aesthetic Gestalten, can result only in crude exaggerations. Dialectical methodology requires the continual reformulation of all scientific descriptions which have to reckon perpetually with the richness of human materials. (shrink)

Moral language displays a characteristic duality. On the one hand, moral claims seem to be similar to descriptive claims: To say that an act is right seems to be a matter of making an assertion, thus indicating that the speaker has a moral belief about which she can be correct or mistaken. On the other hand, moral claims seem to be different from descriptive claims: There is a sense in which, by claiming that an act is right, a speaker (...) indicates that she is inclined to perform the act, to want others to perform it, and to react with praise towards those who do it. Hybrid theories maintain that utterances of moral sentences have both a descriptive and a practical linguistic function. In this thesis, I investigate how a hybrid theory can best accommodate the descriptive and the practical aspects of moral language. I evaluate and criticize the way that existing hybrid theories have proposed to explain the duality of moral language. The main arguments against these views are that they either: fail to accommodate the way that the practical or attitudinal import of moral claims vary across different contexts and linguistic environments; or that they fail to show how the practical or attitudinal import is reliably inferred by interlocutors in virtue of general communicative principles. I develop and defend a new hybrid theory called ‘dynamic descriptivism’. This theory accommodates the duality of moral language by appeal to the way that moral utterances affect a dimension of communication which consists of the shared assumptions that interlocutors mutually assume to mutually accept. Moreover, I discuss a problem which has affected a wide range of theories of moral language – including existing hybrid theories – which is the problem of explaining what it is that interlocutors disagree about when they disagree about matters of morality. I argue that that dynamic descriptivism offers new resources for solving this problem. (shrink)

A ‘duality’ is a formal mapping between the spaces of solutions of two empirically equivalent theories. In recent times, dualities have been found to be pervasive in string theory and quantum field theory. Naïvely interpreted, duality-related theories appear to make very different ontological claims about the world—differing in e.g. space-time structure, fundamental ontology, and mereological structure. In light of this, duality-related theories raise questions familiar from discussions of underdetermination in the philosophy of science: in the (...) presence of dual theories, what is one to say about the ontology of the world? In this paper, we undertake a comprehensive and non-technical survey of the landscape of possible ontological interpretations of duality-related theories. We provide a significantly enriched and clarified taxonomy of options—several of which are novel to the literature. (shrink)

In any probabilistic theory, we say that a bipartite state ω on a composite system AB steers its marginal state ω B if, for any decomposition of ω B as a mixture ω B =∑ i p i β i of states β i on B, there exists an observable {a i } on A such that the conditional states $\omega_{B|a_{i}}$ are exactly the states β i . This is always so for pure bipartite states in quantum mechanics, a (...) fact first observed by Schrödinger in 1935. Here, we show that, for weakly self-dual state spaces (those isomorphic, but perhaps not canonically isomorphic, to their dual spaces), the assumption that every state of a system A is steered by some bipartite state of a composite AA consisting of two copies of A, amounts to the homogeneity of the state cone. If the state space is actually self-dual, and not just weakly so, this implies (via the Koecher-Vinberg Theorem) that it is the self-adjoint part of a formally real Jordan algebra, and hence, quite close to being quantum mechanical. (shrink)

We develop a Priestley-style duality theory for different classes of algebras having a bilattice reduct. A similar investigation has already been realized by B. Mobasher, D. Pigozzi, G. Slutzki and G. Voutsadakis, but only from an abstract category-theoretic point of view. In the present work we are instead interested in a concrete study of the topological spaces that correspond to bilattices and some related algebras that are obtained through expansions of the algebraic language.

Karl Popper developed a theory of deductive logic in the late 1940s. In his approach, logic is a metalinguistic theory of deducibility relations that are based on certain purely structural rules. Logical constants are then characterized in terms of deducibility relations. Characterizations of this kind are also called inferential definitions by Popper. In this paper, we expound his theory and elaborate some of his ideas and results that in some cases were only sketched by him. Our focus (...) is on Popper's notion of duality, his theory of modalities, and his treatment of different kinds of negation. This allows us to show how his works on logic anticipate some later developments and discussions in philosophical logic, pertaining to trivializing connectives, the duality of logical constants, dual-intuitionistic logic, the conservativeness of language extensions, the existence of a bi-intuitionistic logic, the non-logicality of minimal negation, and to the problem of logicality in general. (shrink)

By creating an unbounded topological reduction theory for complex Hilbert spaces over Stonean spaces, we can give a category-theoretic duality between Boolean valued analysis and topological reduction theory for complex Hilbert spaces. MSC: 03C90, 03E40, 06E15, 46M99.

We shall address from a conceptual perspective the duality between algebra and geometry in the framework of the refoundation of algebraic geometry associated to Grothendieck’s theory of schemes. To do so, we shall revisit scheme theory from the standpoint provided by the problem of recovering a mathematical structure A from its representations \ into other similar structures B. This vantage point will allow us to analyze the relationship between the algebra-geometry duality and the structure-semiotics duality. (...) Whereas in classical algebraic geometry a certain kind of rings can be recovered by considering their representations with respect to a unique codomain B, Grothendieck’s theory of schemes permits to reconstruct general rings by considering representations with respect to a category of codomains. The strategy to reconstruct the object from its representations remains the same in both frameworks: the elements of the ring A can be realized—by means of what we shall generally call Gelfand transform—as quantities on a topological space that parameterizes the relevant representations of A. As we shall argue, important dualities in different areas of mathematics can be understood as particular cases of this general pattern. In the wake of Majid’s analysis of the Pontryagin duality, we shall propose a Kantian-oriented interpretation of this pattern. We shall use this conceptual framework to argue that Grothendieck’s notion of functor of points can be understood as a “relativization of the a priori” that generalizes the relativization already conveyed by the notion of domain extension to more general variations of the corresponding domains. (shrink)

There exists a common view that for theories related by a ‘duality’, dual models typically may be taken ab initio to represent the same physical state of affairs, i.e. to correspond to the same possible world. We question this view, by drawing a parallel with the distinction between ‘interpretational’ and ‘motivational’ approaches to symmetries.

This paper defines a category of bounded distributive lattice-ordered grupoids with a left-residual operation that corresponds to a weak system in the family of relevant logics. Algebras corresponding to stronger systems are obtained by adding further postulates. A duality theoey piggy-backed on the Priestley duality theory for distributive lattices is developed for these algebras. The duality theory is then applied in providing characterizations of the dual spaces corresponding to stronger relevant logics.

This paper focuses on natural dualities for varieties of bilattice-based algebras. Such varieties have been widely studied as semantic models in situations where information is incomplete or inconsistent. The most popular tool for studying bilattices-based algebras is product representation. The authors recently set up a widely applicable algebraic framework which enabled product representations over a base variety to be derived in a uniform and categorical manner. By combining this methodology with that of natural duality theory, we demonstrate how (...) to build a natural duality for any bilattice-based variety which has a suitable product representation over a dualisable base variety. This procedure allows us systematically to present economical natural dualities for many bilattice-based varieties, for most of which no dual representation has previously been given. Among our results we highlight that for bilattices with a generalised conflation operation. Here both the associated product representation and the duality are new. Finally we outline analogous procedures for pre-bilattice-based algebras. (shrink)

Duality has often been described as a means of extending our knowledge with a minimal additional outlay of investigative resources. I consider possible arguments for this view. Major elements of this argument are out of keeping with certain widely held views concerning the nature of axiomatic theories (both in projective geometry and elsewhere). They also require a special form of consistency requirement.

In this paper I develop a framework for relating dualities and emergence: two notions that are close to each other but also exclude one another. I adopt the conception of duality as 'isomorphism', from the physics literature, cashing it out in terms of three conditions. These three conditions prompt two conceptually different ways in which a duality can be modified to make room for emergence; and I argue that this exhausts the possibilities for combining dualities and emergence. I (...) apply this framework to gauge/gravity dualities, considering in detail three examples: AdS/CFT, Verlinde's scheme, and black holes. My main point about gauge/gravity dualities is that the theories involved, qua theories of gravity, must be background-independent. I distinguish two senses of background-independence: minimalistic and extended. I argue that the former is sufficiently strong to allow for a consistent theory of quantum gravity; and that AdS/CFT is background-independent on this account; while Verlinde's scheme best fits the extended sense of background-independence. I argue that this extended sense should be applied with some caution: on pain of throwing the baby out with the bath-water. Nevertheless, it is an interesting and potentially fruitful heuristic principle for quantum gravity theory construction. It suggests some directions for possible generalisations of gauge/gravity dualities. The interpretation of dualities is discussed; and the so-called 'internal' vs. 'external' viewpoints are articulated in terms of: epistemic and metaphysical commitments; parts vs. wholes. I then analyse the emergence of gravity in gauge/gravity dualities in terms of the two available conceptualisations of emergence; and I show how emergence in AdS/CFT and in Verlinde's scenario differ from each other. Finally, I give a novel derivation of the Bekenstein-Hawking black hole entropy formula based on Verlinde's scheme; the derivation sheds light on several aspects of Verlinde's scheme and how it compares to Bekenstein's original calculation. (shrink)

Duality has often been described as a means of extending our knowledge with a minimal additional outlay of investigative resources. I attempt to construct a serious argument for this view. Certain major elements of this argument are then considered at length. They’re found to be out of keeping with certain widely held views concerning the nature of axiomatic theories (both in projective geometry and elsewhere). They’re also found to require a special form of consistency requirement.

(1980). The development of attitudes to the wave-particle duality of light and quantum theory, 1900–1920. Annals of Science: Vol. 37, No. 1, pp. 59-79.

Dualities are a pervasive phenomenon in contemporary physics, in which two physical theories are empirically equivalent, yet prima facie make different ontological claims about the world (potentially very different claims—differing in e.g. the number and radius of dimensions of the universe). Dualities thus present a particular instantiation of the well-known notion of underdetermination of theory by evidence. Many different philosophical proposals have been made for how such putative underdetermination might be resolved—this continues to be a programme of active research.

This is a chapter of the planned monograph "Out of Nowhere: The Emergence of Spacetime in Quantum Theories of Gravity", co-authored by Nick Huggett and Christian Wüthrich and under contract with Oxford University Press. (More information at www<dot>beyondspacetime<dot>net.) This chapter investigates the meaning and significance of string theoretic dualities, arguing they reveal a surprising physical indeterminateness to spacetime.

. Relational semantics for nonclassical logics lead straightforwardly to topological representation theorems of their algebras. Ortholattices and De Morgan lattices are reducts of the algebras of various nonclassical logics. We define three new classes of topological spaces so that the lattice categories and the corresponding categories of topological spaces turn out to be dually isomorphic. A key feature of all these topological spaces is that they are ordered relational or ordered product topologies.

We discuss some aspects of the relation between dualities and gauge symmetries. Both of these ideas are of course multi-faceted, and we confine ourselves to making two points. Both points are about dualities in string theory, and both have the ‘flavour’ that two dual theories are ‘closer in content’ than you might think. For both points, we adopt a simple conception of a duality as an ‘isomorphism’ between theories: more precisely, as appropriate bijections between the two theories’ sets (...) of states and sets of quantities. The first point is that this conception of duality meshes with two dual theories being ‘gauge related’ in the general philosophical sense of being physically equivalent. For a string duality, such as T-duality and gauge/gravity duality, this means taking such features as the radius of a compact dimension, and the dimensionality of spacetime, to be ‘gauge’. The second point is much more specific. We give a result about gauge/gravity duality that shows its relation to gauge symmetries to be subtler than you might expect. For gauge theories, you might expect that the duality bijections relate only gauge-invariant quantities and states, in the sense that gauge symmetries in one theory will be unrelated to any symmetries in the other theory. This may be so in general; and indeed, it is suggested by discussions of Polchinski and Horowitz. But we show that in gauge/gravity duality, each of a certain class of gauge symmetries in the gravity/bulk theory, viz. diffeomorphisms, is related by the duality to a position-dependent symmetry of the gauge/boundary theory. (shrink)

String dualities establish empirical equivalence between theories that often look entirely different with respect to their basic ontology and physical structure. Therefore, they represent a particularly interesting example of empirical equivalence in physics. However, the status of duality relations in string physics differs substantially from the traditional understanding of the role played by empirical equivalence. The paper specifies three important differences and argues that they are related to a substantially altered view on the underdetermination of theory building.

Weak/strong duality is usually accompanied by what seems a puzzling ontological feature: the fact that under this kind of duality what is viewed as 'elementary' in one description gets mapped to what is viewed as 'composite' in the dual description. This paper investigates the meaning of this apparent 'particle democracy', as it has been called, by adopting an historical approach. The aim is to clarify the nature of the correspondence between 'dual particles' in the light of an historical (...) analysis of the developments of the idea of weak/strong duality, starting with Dirac's electric-magnetic duality and its successive generalizations in the context of field theory, to arrive at its first extension to string theory. This analysis is then used as evidential basis for discussing the 'elementary/composite' divide and, after taking another historical detour by analysing an instructive analogy case, drawing some conclusions on the particle-democracy issue. (shrink)

In this paper, we examine the Dirac monopole in the framework of Off-Shell Electromagnetism, the five-dimensional U(1) gauge theory associated with Stueckelberg–Schrodinger relativistic quantum theory. After reviewing the Dirac model in four dimensions, we show that the structure of the five-dimensional theory prevents a natural generaliza tion of the Dirac monopole, since the theory is not symmetric under duality transforma tions. It is shown that the duality symmetry can be restored by generalizing the electromagnetic (...) field strength to an element of a Clifford algebra. Nevertheless, the generalized framework does not permit us to recover the phenomenological (or conventional) absence of magnetic monopoles. (shrink)

Part I of this paper is developed in the tradition of Stone-type dualities, where we present a new topological representation for general lattices (influenced by and abstracting over both Goldblatt's [17] and Urquhart's [46]), identifying them as the lattices of stable compact-opens of their dual Stone spaces (stability refering to a closure operator on subsets). The representation is functorial and is extended to a full duality.In part II, we consider lattice-ordered algebras (lattices with additional operators), extending the Jónsson and (...) Tarski representation results [30] for Boolean algebras with Operators. Our work can be seen as developing, and indeed completing, Dunn's project of gaggle theory [13, 14]. We consider general lattices (rather than Boolean algebras), with a broad class of operators, which we dubb normal, and which includes the Jónsson-Tarski additive operators. Representation of l-algebras is extended to full duality. (shrink)

In the wake of profound social changes, which have been accelerated due to a global pandemic, educators reconsider the role and goals of education, and subsequently, how its pragmatic expression should look like in a VUCA-world. We address this challenge by offering basic tenets of education and principles that are tailored to the current reality. We concentrate primarily on the merits of embracing dualities, dilemmas and tensions, for engaging in deep learning and personal development. Jon Wergin’s theory of ‘deep (...) learning’ and aspects of Etienne Wenger’s theory of ‘communities of practice’ provide a fertile framework for this endeavor. Arnett’s concept of ‘emerging adulthood’ provides an additional theoretical resource for connecting principles of education to a timely perception of personal development. Based on the notion of ‘emerging adulthood’, we offer four principles of education, articulated in the form of dualities, along with examples of how to implement these principles at college-level education. (shrink)

In this paper, we will analyse the superloop space formalism for a four dimensional supersymmetric Yang–Mills theory in deformed superspace. We will deform the \ superspace by imposing imposing non-anticommutativity. This non-anticommutative deformation of the superspace will break half the supersymmetry of the original theory. So, this theory will have \ supersymmetry. We will analyse the superloop space duality for this deformed supersymmetric Yang–Mills theory using the \ superspace formalism. We will demonstrate that the sources (...) in the original theory will become monopoles in the dual theory, and the monopoles in the original theory will become sources in the dual theory. (shrink)

We argue that dualities offer new possibilities for relating fundamentality, levels, and emergence. Namely, dualities often relate two theories whose hierarchies of levels are inverted relative to each other, and so allow for new fundamentality relations, as well as for epistemic emergence. We find that the direction of emergence typically found in these cases is opposite to the direction of emergence followed in the standard accounts. Namely, the standard emergence direction is that of decreasing fundamentality: there is emergence of less (...) fundamental, high-level entities, out of more fundamental, low-level entities. But in cases of duality, a more fundamental entity can emerge out of a less fundamental one. This possibility can be traced back to the existence of different classical limits in quantum field theories and string theories. (shrink)