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)

The aim of this article is to discuss the extent to which certain substructural logics are related through the phenomenon of duality. Roughly speaking, metainferences are inferences between collect...

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.

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)

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)

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)

We establish a duality between the category of involutive bisemilattices and the category of semilattice inverse systems of Stone spaces, using Stone duality from one side and the representation of involutive bisemilattices as Płonka sum of Boolean algebras, from the other. Furthermore, we show that the dual space of an involutive bisemilattice can be viewed as a GR space with involution, a generalization of the spaces introduced by Gierz and Romanowska equipped with an involution as additional operation.

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.

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)

The final version of the paper is published pp. 117-166 in: Myrdene Anderson and Floyd Merrell (eds.): On Semiotic Modeling . Mouton de Gruyter, Berlin and New York, 1991.

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.

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.

Contrary to received opinion among philosophers, psychologists, and neuroscientists, conscious duality as a principle of brain organization is neither incoherent nor demonstrably false. The present paper begins by reviewing the history of the theory and its anatomical basis and defending it against the claim that it rests upon an arbitrary decision as to what constitutes the biological substratum of mind or person.

Synthesizing situated cognition, reinforcement learning, and hybrid connectionist modeling, a generic cognitive architecture focused on situated involvement and interaction with the world is developed in this book. The architecture notably incorporates the distinction of implicit and explicit processes.

This is the first of two papers concerned with the philosophical significance of dualities as applied in recent fundamental physics. The general idea is that, for its peculiarity, this ‘new’ ingredient in theory construction can open unexpected perspectives in the current philosophical reflection on contemporary physics. In particular, today’s physical dualities represent an unusual type of intertheory relation, the meaning of which deserves to be investigated. The aim is to show how discussing this point brings into play, at the same (...) time, what is intended by a 'theory’ and in which sense dualities are to be considered 'symmetries'. This paper is introductory and focusses on the first form of duality explicitly applied in twentieth century physics, that is electromagnetic duality as discussed in Dirac’s theory of magnetic poles. The extension of electromagnetic duality in the context of quantum field theory and string theory is explored in a forthcoming companion paper. (shrink)

Płonka sums consist of an algebraic construction similar, in some sense, to direct limits, which allows to represent classes of algebras defined by means of regular identities. Recently, Płonka sums have been connected to logic, as they provide algebraic semantics to logics obtained by imposing a syntactic filter to given logics. In this paper, I present a very general topological duality for classes of algebras admitting a Płonka sum representation in terms of dualisable algebras.

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)

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)

Duality in Logic and Language [draft--do not cite this article] Duality phenomena occur in nearly all mathematically formalized disciplines, such as algebra, geometry, logic and natural language semantics. However, many of these disciplines use the term ‘duality’ in vastly different senses, and while some of these senses are intimately connected to each other, others seem to be entirely … Continue reading Duality in Logic and Language →.

This paper presents duality results between categories of neighbourhood frames for modal logic and categories of modal algebras (i.e. Boolean algebras with an additional unary operation). These results extend results of Goldblatt and Thomason about categories of relational frames for modal logic.

We present an epistemological schema of natural sciences inspired by Peirce's pragmaticist view, stressing the role of the \emph{phenomenological map}, that connects reality and our ideas about it. The schema has a recognisable mathematical/logical structure which allows to explore some of its consequences. We show that seemingly independent principles as the requirement of reproducibility of experiments and the Principle of Sufficient Reason are both implied by the schema, as well as Popper's concept of falsifiability. We show that the schema has (...) some power in demarcating science by first comparing with an alternative schema advanced during the first part of the 20th century which has its roots in Hertz and has been developed by Einstein and Popper. Further, the identified differences allow us to focus in the construction of Special Relativity, showing that it uses an intuited concept of velocity that does not satisfy the requirements of reality in Peirce. While the main mathematical observation connected with this issue has been known for more than a century, it has not been investigated from an epistemological point of view. A probable reason could be that the socially dominating epistemology in physics does not encourage such line of work. We briefly discuss the relation of the abduction process presented in this work with discussions regarding ``abduction'' in the literature and its relation with ``analogy''. (shrink)

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)

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)

We prove that the unification type of Łukasiewicz logic and of its equivalent algebraic semantics, the variety of MV-algebras, is nullary. The proof rests upon Ghilardiʼs algebraic characterisation of unification types in terms of projective objects, recent progress by Cabrer and Mundici in the investigation of projective MV-algebras, the categorical duality between finitely presented MV-algebras and rational polyhedra, and, finally, a homotopy-theoretic argument that exploits lifts of continuous maps to the universal covering space of the circle. We discuss the (...) background to such diverse tools. In particular, we offer a detailed proof of the duality theorem for finitely presented MV-algebras and rational polyhedra—a fundamental result that, albeit known to specialists, seems to appear in print here for the first time. (shrink)

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.

In a paper published in 1939, Ernest Nagel described the role that projective duality had played in the reformulation of mathematical understanding through the turn of the nineteenth century, claiming that the discovery of the principle of duality had freed mathematicians from the belief that their task was to describe intuitive elements. While instances of duality in mathematics have increased enormously through the twentieth century, philosophers since Nagel have paid little attention to the phenomenon. In this paper (...) I will argue that a reassessment is overdue. Something beyond doubt is that category theory has an enormous amount to say on the subject, for example, in terms of arrow reversal, dualising objects and adjunctions. These developments have coincided with changes in our understanding of identity and structure within mathematics. While it transpires that physicists have employed the term ‘duality’ in ways which do not always coincide with those of mathematicians, analysis of the latter should still prove very useful to philosophers of physics. Consequently, category theory presents itself as an extremely important language for the philosophy of physics. (shrink)

[About the book] This book explores the idea that we have two minds - automatic, unconscious, and fast, the other controlled, conscious, and slow. In recent years there has been great interest in so-called dual-process theories of reasoning and rationality. According to such theories, there are two distinct systems underlying human reasoning - an evolutionarily old system that is associative, automatic, unconscious, parallel, and fast, and a more recent, distinctively human system that is rule-based, controlled, conscious, serial, and slow. Within (...) the former, processes the former, processes are held to be innate and to use heuristics that evolved to solve specific adaptive problems. In the latter, processes are taken to be learned, flexible, and responsive to rational norms. Despite the attention these theories are attracting, there is still poor communication between dual-process theorists themselves, and the substantial bodies of work on dual processes in cognitive psychology and social psychology remain isolated from each other. This book brings together leading researchers on dual processes to summarize the state-of-the-art, highlight key issues, present different perspectives, explore implications, and provide a stimulus to further work. It includes new ideas about the human mind both by contemporary philosophers interested in broad theoretical questions about mental architecture and by psychologists specialising in traditionally distinct and isolated fields. For all those in the cognitive sciences, this is a book that will advance dual-process theorizing, promote interdisciplinary communication, and encourage further applications of dual-process approaches. (shrink)

The AdS/CFT duality has been a source of several strong conceptual claims in the physics literature that have yet to be explored by philosophers. In this paper I focus on one of these: the extent to which spacetime geometry and locality can be said to emerge from this duality, so that neither is fundamental. I argue: that the kind of emergence in question is relatively weak, involving one kind of spacetime emerging from another kind of spacetime; inasmuch as (...) there is something conceptually interesting to say about the emergence of spacetime and locality , it is no different from that already well known to those within canonical quantum gravity; that at the core of AdS/CFT is an issue of representation and redundancy in representation. (shrink)

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.

The great variety of meditation techniques found in different contemplative traditions presents a challenge when attempting to create taxonomies based on the constructs of contemporary cognitive sciences. In the current issue of Consciousness and Cognition, Travis and Shear add ‘automatic self-transcending’ to the previously proposed categories of ‘focused attention’ and ‘open monitoring’, and suggest characteristic EEG bands as the defining criteria for each of the three categories. Accuracy of current taxonomies and potential limitations of EEG measurements as classifying criteria are (...) discussed. (shrink)

The main goal of this paper is to explain the link between the algebraic and the Kripke-style models for certain classes of propositional logics. We start by presenting a Priestley-type duality for distributive lattices endowed with a general class of well-behaved operators. We then show that finitely-generated varieties of distributive lattices with operators are closed under canonical embedding algebras. The results are used in the second part of the paper to construct topological and non-topological Kripke-style models for logics that (...) are sound and complete with respect to varieties of distributive lattices with operators in the above-mentioned classes. (shrink)

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)

In this article I bring the recent philosophical literature on theoretical equivalence to bear on dualities in physics. Focusing on electromagnetic duality, which is a simple example of S-duality i...

We present dualities for implicative and residuated lattices. In combination with our recent article on a discrete duality for lattices with unary modal operators, the present article contributes in filling in a gap in the development of Orłowska and Rewitzky’s research program of discrete dualities, which seemed to have stumbled on the case of non-distributive lattices with operators. We discuss dualities via truth, which are essential in relating the non-distributive logic of two-sorted frames with their sorted, residuated modal logic, (...) as well as full Stone duality for residuated lattices. Our results have immediate applications to the semantics of related substructural logical calculi. (shrink)

The variety of N4? -lattices provides an algebraic semantics for the logic N4?, a version of Nelson 's logic combining paraconsistent strong negation and explosive intuitionistic negation. In this paper we construct the Priestley duality for the category of N4?-lattices and their homomorphisms. The obtained duality naturally extends the Priestley duality for Nelson algebras constructed by R. Cignoli and A. Sendlewski.

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 essay, we consider the formal and ontological implications of one specific and intensely contested dialectical context from which Deleuze’s thinking about structural ideal genesis visibly arises. This is the formal/ontological dualism between the principles, ἀρχαί, of the One (ἕν) and the Indefinite/Unlimited Dyad (ἀόριστος δυάς), which is arguably the culminating achievement of the later Plato’s development of a mathematical dialectic.3 Following commentators including Lautman, Oskar Becker, and Kenneth M. Sayre, we argue that the duality of the One (...) and the Indefinite Dyad provides, in the later Plato, a unitary theoretical formalism accounting, by means of an iterated mixing without synthesis, for the structural origin and genesis of both supersensible Ideas and the sensible particulars which participate in them. As these commentators also argue, this duality furthermore provides a maximally general answer to the problem of temporal becoming that runs through Plato’s corpus: that of the relationship of the flux of sensory experiences to the fixity and order of what is thinkable in itself. Additionally, it provides a basis for understanding some of the famously puzzling claims about forms, numbers, and the principled genesis of both attributed to Plato by Aristotle in the Metaphysics, and plausibly underlies the late Plato’s deep considerations of the structural paradoxes of temporal change and becoming in the Parmenides, the Sophist, and the Philebus. After extracting this structure of duality and developing some of its formal, ontological, and metalogical features, we consider some of its specific implications for a thinking of time and ideality that follows Deleuze in a formally unitary genetic understanding of structural difference. These implications of Plato’s duality include not only those of the constitution of specific theoretical domains and problematics, but also implicate the reflexive problematic of the ideal determinants of the form of a unitary theory as such. We argue that the consequences of the underlying duality on the level of content are ultimately such as to raise, on the level of form, the broader reflexive problem of the basis for its own formal or meta-theoretical employment. We conclude by arguing for the decisive and substantive presence of a proper “Platonism” of the Idea in Deleuze, and weighing the potential for a substantive recuperation of Plato’s duality in the context of a dialectical affirmation of what Deleuze recognizes as the “only” ontological proposition that has ever been uttered. This is the proposition of the univocity of Being, whereby “being is said in the same sense, everywhere and always,” but is said (both problematically and decisively) of difference itself. (shrink)

The main goal of this paper is to explain the link between the algebraic models and the Kripke-style models for certain classes of propositional non-classical logics. We consider logics that are sound and complete with respect to varieties of distributive lattices with certain classes of well-behaved operators for which a Priestley-style duality holds, and present a way of constructing topological and non-topological Kripke-style models for these types of logics. Moreover, we show that, under certain additional assumptions on the variety (...) of the algerabic models of the given logics, soundness and completeness with respect to these classes of Kripke-style models follows by using entirely algebraical arguments from the soundness and completeness of the logic with respect to its algebraic models. (shrink)

In this paper we show a duality between two approaches to represent quantum structures abstractly and to model the logic and dynamics therein. One approach puts forward a “quantum dynamic frame” :2267–2282, 2005), a labelled transition system whose transition relations are intended to represent projections and unitaries on a Hilbert space. The other approach considers a “Piron lattice”, which characterizes the algebra of closed linear subspaces of a Hilbert space. We define categories of these two sorts of structures and (...) show a duality between them. This result establishes, on one direction of the duality, that quantum dynamic frames represent quantum structures correctly; on the other direction, it gives rise to a representation of dynamics on a Piron lattice. (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)

In this paper, we develop a duality for the varieties of a Łukasiewicz n + 1-valued modal System. This duality is an extension of Stone duality for modal algebras. Some logical consequences (such as completeness results, correspondence theory...) are then derived and we propose some ideas for future research.

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)

A method of defining semantics of logics based on not necessarily distributive lattices is presented. The key elements of the method are representation theorems for lattices and duality between classes of lattices and classes of some relational systems . We suggest a type of duality referred to as a duality via truth which leads to Kripke-style semantics and three-valued semantics in the style of Allwein-Dunn. We develop two new representation theorems for lattices which, together with the existing (...) theorems by Urquhart and Bimbo-Dunn, constitute a complete, in a sense, representation theory for lattices. As observed by Dunn and Hardegree, variations of Urquhart's duality arise by varying his disjointness assumption on the canonical frame. Four possible assumptions – disjoint, exhaustive, non-disjoint and non-exhaustive – are discussed in the paper. Each of the four corresponding representation theorems is expanded to a duality via truth. Based on these dualities we suggest four corresponding types of semantics for lattice-based logics. We also discuss a new topological representation of lattices. (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)

In this paper we present a schema for describing dualities between physical theories, and illustrate it in detail with the example of bosonization: a boson-fermion duality in two-dimensional quantum field theory. The schema develops proposals in De Haro : these proposals include construals of notions related to duality, like representation, model, symmetry and interpretation. The aim of the schema is to give a more precise criterion for duality than has so far been considered. The bosonization example, or (...) boson-fermion duality, has the feature of being simple, yet rich enough, to illustrate the most relevant aspects of our schema, which also apply to more sophisticated dualities. The richness of the example consists, mainly, in its concern with two non-trivial quantum field theories: including massive Thirring-sine-Gordon duality, and non-abelian bosonization. This prompts two comparisons with the recent philosophical literature on dualities:--- Unlike the standard cases of duality in quantum field theory and string theory, where only specific simplifying limits of the theories are explicitly known, the boson-fermion duality is known to hold {\it exactly}. This exactness can be exhibited explicitly. The bosonization example illustrates both the cases of isomorphic and {\it non-isomorphic} models: which we believe the literature on dualities has not so far discussed. (shrink)