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)

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. (shrink)

We integrate dualistic conceptions of the real with Peirce's perspectives about reality and abduction, emphasizing the concept of reason underlying Peirce's thoughts. Peirce's abduction is related to the notions of retrogression and grounding in Hegel, later re-encountered in Hansonian-abduction. Abduction in turn is considered in relation to abstraction acquiring its fullest sense as a stage in the process of producing a theory. The process is iterative and self improving, it incorporates ``turbid thinking'' making it increasingly ``clear'' at successive iterations that (...) incorporate the lessons taught by failed predictions, i.e., refutations. The cycle of thoughts promoted by doubts comes to rest when belief is reached. We discuss how this coming to rest depends on a criterion for cessation of doubts. The observation is illustrated with two different criteria, one proposed by Mach that only demands analogy and the criteria of dualists such as Goethe and Whewell that inspire the present work. Hence, it is possible to produce, and socially accept, imperfect theories unless we demand the highest level of rationality, avoiding any leftover of the turbid thoughts that have been used in the early developments. This work rests upon the existence of some objective form of reason. Influenced by a constructivist, Piagetian, perspective of science, we propose and discuss a small number of conditions that we identify as characteristics of rational abduction: rules for the rational construction of theories. We show how a classical example of belief that satisfies today's most common definition of abduction does not match the standards of rational retroduction. We further show how the same rules indicate the partial detachment of Special Relativity from the observable world, a fact actually known to Einstein. We close arguing that there is an urgent need to develop a critical epistemology incorporating dualistic perspectives. (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...

In its classical formulation as found in Svātmārāma’s Haṭhapradīpikā, haṭhayoga is a Śaiva appropriation of an older extra-Vedic soteriological method. But this appropriation was not accompanied by an imposition of Śaiva philosophy. In general, the texts of haṭhayoga reveal, if not a disdain for, at least an insouciance towards metaphysics. Yoga is a soteriology that works regardless of the yogin’s philosophy. But the various texts that were used to compile the Haṭhapradīpikā (a table identifying these borrowings is given at the (...) end of the article) were not composed in metaphysical vacua. Analysis of their allusions to doctrine shows that the texts from which Svātmārāma borrowed most were products of a Vedantic milieu—bearing testament to Vedānta’s newfound interest in yoga as a complement to jñāna—but that many others were Śaiva non-dual works. Because of the lack of importance given to the niceties of philosophy in haṭhayogic works, these two non-dualities were able to combine happily and thus the Śaiva tenets incorporated within haṭhayoga survived the demise of Śaivism as part of what was to become in the medieval period the dominant soteriological method in scholarly religious discourse in India. (shrink)

The Duality Theory of the Single Universe (DT) proposes that reality emerges from the perpetual interaction of two primordial, irreducible forces: F1, the centripetal, integrative, and aggregative force, and F2, the centrifugal, dispersive, and differentiating force. Rejecting monism, pure randomness, and rigid binarism, DT argues that the observable universe arises from the dynamic tension and occasional synergy between these co-present forces. Drawing on religious traditions (Asha-Drugh, light-darkness, Yin-Yang), mathematics, social and legal systems, biology, physics, and cosmology, the theory reveals recurring (...) dual structures that reflect a deeper F1-F2 architecture. Foundational principles - sequential gradualism, duality interval, duality gap, alteration, and duality synergy - are offered to explain the emergence of ordered complexity. DT further posits that duality is a permanent feature of the created world and the primary epistemological lens through which human cognition interprets reality. DT offers an internally coherent model that reconciles ancient wisdom with modern inquiry, though it is not uniformly compatible with the standard model of particle physics. DT generates testable predictions and provides a generative compass for fields including artificial intelligence, legal philosophy, and institutional design. (shrink)

Not long after the completion of Michael Mann’s “quadrilogy” on The Sources of Social Power (1986–2012), social scientists began to interrogate the meaning of his concepts of “despotic” and “infrastructural” power. While we know that the former is the most evident sign of danger in times of war, less well understood is the role of infrastructural power in state/civil society relations. Most important is the ambiguous relationship between the two types of power and the possibility that—especially in times of war—infrastructural (...) power can become the vehicle for despotic ends. But infrastructural power is also reciprocal, offering firms and civil society groups channels with which to contest the state’s projects. In this article, I first explicate the different meanings that Mann gave to his concept of infrastructural power. In the second section, I turn to how the concept has been “received” in political science and historical sociology. In the third part, I argue that the main danger to American democracy in wartime lies not in its becoming a despotic state, but in the use of the state’s infrastructural channels for the exercise of despotic ends. The fourth part illustrates the complexities of infrastructural power in business/government/civil society relations in cybersecurity, which Mann—for understandable reasons—did not examine in his encompassing work. (shrink)

Both for Kant and for Nietzsche, aesthetics must not be considered as a systematic science based merely on logical premises but rather as a set of intuitively attained artistic ideas that constitute or reconstitute the sensible perceptions and supersensible representations into a new whole. Kantian and Nietzschean aesthetics are both aiming to see beyond the forms of objects to provide explanations for the nobility and sublimity of human art and life. We can safely say that Kant and Nietzsche used the (...) dualities of the beautiful/sublime and Apollonian/Dionysian to advocate their general philosophical worldview, and that the initial formation (in Observations and The Birth of Tragedy) and final dissolution (in the Critique of Judgment and Zarathustra and other later works) of these dualities are determined by the gradually established telos of their philosophical endeavor. Therefore, by observing the evolution of these so-called dualities, Kaplama gathers important clues as to how Kant’s and Nietzsche’s aesthetics transformed into different ways to affirm human art and life. On the one hand, Kaplama argues, the Dionysian came to be the heart and soul of Nietzschean aesthetics and ethics, and the Apollonian (or the formal drive of individuation) was reduced into a mere aesthetic criterion. On the other, Kant treats the sublime (which is originally an idea-producing feeling and/or judgment) as a mere appendix to his Critique of Judgment and aesthetic theory teleologically reducing it into its possible moral consequences. This is why Schopenhauer calls the sublime “by far the most excellent thing in the Critique of Judgment” which touches on the real problem of aesthetics very closely but does not provide a real solution for it. Kant’s forced teleological move is to make his theory of aesthetic judgment stand as a ‘reaffirmation’ of the earlier ethical justification he believed to have accomplished in the first two Critiques and the Groundwork where he defends an affirmation of human life through a teleological morality centered on the principle of free-will. In contrast, Nietzsche’s aesthetics (particularly the Dionysian) guides his ethics and metaphysics again through defining an ideal human nature without which ethos would be static and meaningless, lacking the ability to move and change and represent the tragic pathos of human life. (shrink)

Dualities offer new possibilities for relating fundamentality and emergence. In particular, as this chapter aims to show, it may happen that the relations of fundamentality and emergence between dual theories are inverted. In other words, the direction of emergence typically found in these cases is opposite to the direction of emergence followed in the standard accounts: that is, while the standard emergence direction is that of decreasing fundamentality—in that there is emergence of less fundamental, high-level entities, out of more (...) fundamental, low-level entities—in these cases of duality, on the contrary, a more fundamental entity can emerge out of a less fundamental one. In fact, this possibility can be traced back to the existence of different classical limits in quantum field theories and string theories. (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.

We show that in 4-spacetime modified at very short distances due to the weakening of classical logic, the higher dimensions emerge. We analyse the case of some smooth topoi, and the case of some class of pointless topoi. The pointless topoi raise the dimensionality due to the forcing adding “string” objects and thus replacing classical points in spacetime. Turning to strings would be something fundamental and connected with set theoretical forcing. The field theory/strings dualities originate at the set theoretical (...) level of the theories. It is argued that this fundamental level can help solving some difficulties of the physical dualities. (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.

Priestley duality can be used to study subalgebras of Heyting algebras and related structures. The dual concept is that of congruence on the dual space and the congruence lattice of a Heyting space is dually isomorphic to the subalgebra lattice of the dual algebra. In this paper we continue our investigation of the congruence lattice of a Heyting space that was undertaken in [10], [8] and [12]. Our main result is a characterization of the modularity of this lattice (Theorem 2.12). (...) Partial results about its complementedness are also given, and among other things a characterization of those finite Heyting algebras with a complemented subalgebra lattice (Theorem 3.5). (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.

We present two discrete dualities for double Stone algebras. Each of these dualities involves a different class of frames and a different definition of a complex algebra. We discuss relationships between these classes of frames and show that one of them is a weakening of the other. We propose a logic based on double Stone algebras.

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)

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)

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.

This article explores a duality inherent in care between the treatment of disease and the care of the person; or, as Winnicott might put it, between the dimensions of cure and care. It shows how these two dimensions are distinct from one another, despite their entanglement in care practices, and argues that their effect in practice varies sharply according to the situation. Taking as a starting point Nathalie Zaccaï-Reyners’s analysis of the care of dependent older adults in nursing homes (EHPADs), (...) I argue that the care–cure dimensions may be understood as a particular case of the more general duality of our relation to the world and to others. I then consider Hartmut Rosa’s account of this duality, between an instrumental relation and a relation of resonance. According to Plessner’s philosophical anthropology, this duality in our relation to the world arises from a distinctly human property: the capacity to sustain a double relation both to the environment and to oneself, at once immediate and mediated through the perspectives of others. Plessner characterizes this condition as the human being’s ‘eccentric’ or decentered positionality. Understanding this duality at the core of ordinary human experience – and the continual adjustments required to preserve its equilibrium – may shed light on the duality inherent in care, where balance must likewise be ceaselessly recalibrated, in response to circumstance, between its different dimensions. (shrink)

Natural dualities are developed for varieties ofn-valued ukasiewicz algebras with and without negation. These dualities are based on hom-functors, and parallel Stone duality for Boolean algebras. A translation is described which relates the natural dualities to the corresponding restricted Priestley dualities. This enables a unified approach to free algebras to be presented, whence R. Cignoli's characterisations of the finitely generated free algebras are elucidated and new descriptions of arbitrary free algebras obtained. Finally it is shown how (...) dualities for subvarieties encode equational bases. An alternative approach is thereby provided to results of M.E. Adams and R. Cignoli. (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)

This paper develops a categorical distinction between two kinds of informational object: compressed descriptions, which specify a full admissible configuration space without selecting any particular element, and resolved descriptions, which contain selection information identifying one particular element as actual in a structural sense. -/- The distinction is anchored in the Kolmogorov structure function decomposition, which independently identifies model and selection components as formally separable individual-level facts in individual strings, and motivates the extension to descriptions in general by analogy. On this (...) basis, the paper develops and argues for the Compression--Resolution Duality: no single informational object can simultaneously be compressed and resolved with respect to the same admissible configuration space. This is the paper's central formal result. -/- The paper then develops two structural consequences. The Hierarchical Asymmetry argues that elimination across levels is local while resolution is perspectivally global: from the perspective of any resolved description, the containing descriptions are presupposed as resolved from within that description, whereas compressed descriptions generate no such chain. -/- The Boundedness interpretation, motivated by the Hierarchical Asymmetry, proposes that resolved descriptions are self-limiting---unable to represent the containing structure without access to compressed information they lack. -/- These structural claims are explored in application to the relationship between the universal wavefunction and any particular experienced quantum outcome: from within a resolved description of an outcome, other outcomes are not absent from what the wavefunction encodes---they are inaccessible from within that descriptive environment. -/- The paper does not adjudicate between interpretations of quantum mechanics. The analysis is structural and informational throughout, making no claims about physical dynamics, temporal processes, or empirical matters. (shrink)

Background Military medics, who are both professional soldiers and qualified nurses, can face situations where their training and moral ethos conflict in the performance of duty. Their role has intrinsic duality. They are both a soldier, thus a member of a military organisation as well as well as a healer, a nurse with a corresponding duty of care. Both roles have ethical, legal and professional responsibilities, codes of conduct and moral codes. Both also are roles which have strong cultural images (...) and distinct expectations from individuals and those around them. This can lead to moral dilemmas, moral injury and long-term psychological illness. In the 2024 Royal Commission into Defence and Veteran Suicide in Australia, moral injury was cited as a relatively new, but not widely acknowledged, risk factor that may lead to suicide. In the context of defence, moral injury can be described as being experienced by a person who participates in, or witnesses, actions that go against their moral code or moral beliefs. Research design The research presented is a component of a larger study into the role duality of the Australian Army medic in warzones. It focuses on interviews from twelve currently serving army medics who have deployed overseas in recent conflicts with the research question: How does the intrinsic duality of the role of the medic impact the individual performing this specialised role. Ethical considerations This research has ethics approval from both the Australian Defence Human Research Ethics Committee #778–15 and the Human Research Ethics Committee #2015/024. Participants have given formal consent for their reflections to be used in publications. Conclusion For this research the psychological impact of these ethical conflicts on Australian Army Medics who have served in this dual role has been investigated, with specific focus on moral injury. (shrink)

The symmetries between points and lines in planar projective geometry and between points and planes in solid projective geometry are striking features of these geometries that were extensively discussed during the nineteenth century under the labels “duality” or “reciprocity.” The aims of this article are, first, to provide a systematic analysis of duality from a modern point of view, and, second, based on this, to give a historical overview of how discussions about duality evolved during the nineteenth century. Specifically, we (...) want to see in which ways geometers’ preoccupation with duality was shaped by developments that lead to modern logic towards the end of the nineteenth century, and how these developments in turn might have been influenced by reflections on 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 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)

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.

As Kantorowicz acknowledged in his seminal work on the theological groundings of medieval political theory, the legal doctrine of the king’s two bodies was not a medieval legal doctrine but an Elizabethan one. It was developed by Plowden, an English jurist, in the second half of the sixteenth century. With Plowden’s theory of the king’s two bodies – one natural and the other politic – the medieval notion of the Crown as a corporation aggregate of many was progressively abandoned. The (...) king’s body politic was instead confused with the concept of office of kingship, which could only vest in the king. Building on Plowden’s doctrine, Coke wrote of the king’s body politic in the early seventeenth century that it was a corporation sole. With time, the Crown conceived as a corporation sole and as a corporation aggregate understood as the government became an important feature of English legal thought. The Crown as a corporation aggregate – comprising the whole realm – is more apt to reflect the constitutional arrangement reached after the Glorious Revolution, characterised by the sovereignty of Parliament and, later, responsible government. (shrink)

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)

Many results in logic and mathematics rely on techniques that allow for concrete, often visual, representations of abstract concepts. A primary example of this phenomenon in logic is the distinction between syntax and semantics, itself an example of the more general duality in mathematics between algebra and geometry. Such representations, however, often rely on the existence of certain maximal objects having particular properties such as points, possible worlds or Tarskian first-order structures. -/- This dissertation explores an alternative to such representations (...) known as possibility semantics. Its core idea is to replace maximal objects with ordered systems of partial approximations. Although it originates in the semantics of modal logic and the representation of abstract ordered structures, I argue that it has far-reaching mathematical, foundational and philosophical significance, especially in the context of semiconstructive mathematics, a foundational framework that does not assume any fragment of the Axiom of Choice beyond the Axiom of Dependent Choices. -/- The dissertation is divided in two main parts. The first part explores various applications of the mathematical framework underlying possibility semantics to lattice theory and non-classical propositional logics. A major theme is the development of constructive dualities for various categories of lattices, which are related to standard non-constructive dualities via Vietoris constructions. -/- The second part of the dissertation explores the alternative foundational setting of semi-constructive mathematics, focusing on three applications of possibility semantics for classical first-order logic to the philosophy of the mathematical infinite. In particular, we introduce generic powers, a semi-constructive analogue of ultrapowers in classical model theory, and we explore the merits of these structures from a foundational, conceptual and historical viewpoint. (shrink)

Kumar and Banerjee (Some algebras and logics from quasiorder-generated covering-based approximation spaces. J Appl Non-classical Log 2024;34:248-68) characterized a subclass of quasiorder-generated covering-based approximation spaces for which the algebra of definable sets forms a Stone algebra. In this paper, we characterize those subclasses for which the definable sets form dual Stone, regular double Stone, linear Heyting, and well-connected Heyting algebras. As a consequence, discrete dualities of the aforementioned algebras are also obtained. Furthermore, we provide representation theorems of the aforementioned (...) algebras in terms of rough sets determined by a quasiorder. (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 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 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)

A growing consensus in the literature on agentive modals has it that ability modals like ‘can’ or ‘able to’ have a _dual_, i.e. interpretations of ‘must’ or ‘cannot but’ which stand to _necessity_ as ability stands to _possibility_. We argue that this thesis (which we call ‘Agentive Duality’) is much more controversial than meets the eye. While Agentive Duality follows from the orthodox possibility analysis of ability given natural assumptions, it sits uneasily with a wide range of alternative proposals which (...) are unified by the idea that ability requires _control_. In particular, we show that against the background of a control requirement on ability, Agentive Duality can be used to derive absurd predictions featuring this dual. Far from being a purely definitional thesis, Agentive Duality thus affords a new lens through which to assess the long-standing debate between possibility analyses of ability and their discontents. (shrink)

The categorical dualities presented are: (first) for the category of bi-algebraic lattices that belong to the variety generated by the smallest non-modular lattice with complete (0,1)-lattice homomorphisms as morphisms, and (second) for the category of non-trivial (0,1)-lattices belonging to the same variety with (0,1)-lattice homomorphisms as morphisms. Although the two categories coincide on their finite objects, the presented dualities essentially differ mostly but not only by the fact that the duality for the second category uses topology. Using the (...) presented dualities and some known in the literature results we prove that the Q-lattice of any non-trivial variety of (0,1)-lattices is either a 2-element chain or is uncountable and non-distributive. (shrink)

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.

Theoretical equivalence and duality are two closely related notions: but their interconnection has so far not been well understood. In this paper I explicate the contribution of a recent schema for duality to discussions of theoretical equivalence. I argue that duality suggests a construal of theoretical equivalence in the physical sciences. The construal is in terms of the isomorphism of models, as defined by the schema. This construal gives interpretative constraints that should be useful for discussions of theoretical equivalence more (...) generally. I illustrate the construal in various formulations of Maxwell’s electromagnetic theory. (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)

Clans are categorical representations of generalized algebraic theories that contain more information than the finite-limit categories associated to the locally finitely presentable categories of models via Gabriel–Ulmer duality. Extending Gabriel–Ulmer duality to account for this additional information, we present a duality theory between clans and locally finitely presentable categories equipped with a weak factorization system of a certain kind.

We prove that the opposite of the category of coalgebras for the Vietoris endofunctor on the category of compact Hausdorff spaces is monadic over $\mathsf {Set}$. We deliver an analogous result for the upper, lower, and convex Vietoris endofunctors acting on the category of stably compact spaces. We provide axiomatizations of the associated (infinitary) varieties. This can be seen as a version of Jónsson–Tarski duality for modal algebras beyond the zero-dimensional setting.

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)