Non-Duality Questions, Non-Duality Answers is an organic, interactive exploration of non-duality--the understanding that in life there is no separation, only seamless Oneness. Designed to answer questions as they arise, this book invites readers to participate in the argument for non-separation, inquire about the nature of the self and the experience of liberation, and understand how non-dual awareness can impact our lives. With a unique blend of patience, compassion, respect, provocation, and humor, author Richard Sylvester helps frustrated spiritual (...) seekers gain clarity on what non-duality is--and what it isn't. (shrink)

NON-DUALISM, DUALISM AND MONISM Q.1 What do the following terms, often used by the Vedanta: dualism, monism, monotheism and non-dualism, mean? A. Every philosophical or cosmological vision which affirms two opposing and irreducible...

Inquiry into the felt dimension of experience can be deeply transformative, but more than promoting transcendence it can help us make contact with the intricacy of our open and embodied self-….

In lieu of an abstract, here is a brief excerpt of the content:Duality and Non-Duality in Christian Practice:Reflections on the Benefits of Buddhist-Christian Dialogue for Constructive TheologyWendy FarleyThe question before us is the desirability of Buddhist-Christian dialogue in the work of (what Christians call) constructive theology. As a feminist theologian whose work is ever more deeply shaped by such a dialogue, my immediate answer is an unequivocal yes.1 This dialogue fits a general pattern over two thousand years in (...) which theologians have drawn from the wisdom of other traditions or cultures to better understand the mysteries of human experience. Without neo-Platonism and Aristotle there could hardly have been anything we recognize as Christian theology. Without pre-Christian Irish religion, there would be no Saint Brigit or Celtic Christianity. Without phenomenology, process, and Marxist philosophy the greatest works of twentieth-century Christian thought would not exist. Christianity, like all religions, is a moving river changed by time and culture and nearly as interiorly diverse as its members. The dialogue with Buddhism might be seen as a contemporary example of Christianity's openness to exchange with other ways of thinking that broadens and deepens its best insights. It is perhaps analogous to Origen's reliance on neo-Platonism or Thomas Aquinas's inspiration by the rediscovery of Aristotle.2The porous boundaries between Christianity and its culture open it to wisdom beyond itself but they also open it to cruelty and ignorance. Nothing protects religions from participating in the human condition and thus the prejudices and power structures that mutilate human society. Consumer culture and oppressions by class, ethnicity, and gender are seamlessly incorporated into Christian identity. The tacit synthesis of a culture's values with those of a religion neutralizes possible critique and can elevate a society's oppressions to a sacred duty. For this reason, too, dialogue with Buddhism is important to call Christianity back to its ethical moorings in the ideal of universal love and compassion.Religions are porous to the world around them, but they tend to conceal this from themselves and often imagine themselves to be not only untainted but also uniquely [End Page 135] qualified to mediate salvation. Religions provide human beings with language and practice through which dimensions of reality that cannot simply be read off of nature or experience are (in some sense) encountered. This is their great gift. The difficulty of religions is that tradition can become an end in itself. Because a tradition really does offer meaning and tools for transformation it is easy to slip into the existential posture of so idealizing a tradition that it seems in some direct and exclusive way to correlate to ultimate truth. The curvature of the ego onto itself, so attached to its own pain and pleasure that everything else seems a mere shadow, moves from the microcosm of the self to the macrocosm of religious tradition. The particular religion—or that element within it with which I identify—becomes identical with sacred reality. In the hostility to other religions characteristic of many Christians we see the illusion of egocentrism in its religious form. Paul Tillich described this tendency to absolutize itself as the demonic distortion of religion.3 Religious dialogue is a potent medicine against this overattachment to one's own tradition.These different senses in which a religion integrates into itself the wisdom and illusions of its time and place create enormous diversity within a religious tradition. Root concepts within Christianity concerning soteriology, divinity, and suffering do not have any agreed upon meaning. That is, there is no "Christian" view of salvation. Gregory of Nyssa's ideal of theosis, Luther's emphasis on salvation by grace through faith, an evangelical's experience of being born again, a liberal Protestant's commitment to social justice are so diverse that they seem to represent almost different religions.In light of the malleability and interior diversity of traditions, dialogue may not be between two self-sufficient religions but between participants who inhabit their tradition in a variety of ways.4 With this in mind, I would like to experiment with practice as a basis for some kinds of dialogue. Practice is concretely located in a religious tradition and yet reflects... (shrink)

This essay considers some meta-ethical questions that emerge from a consideration of the phenomena of terrorism in the context of Buddhist metaphysics: what, in the Buddhist view, ultimately causes terrorism (and its subsidiary effects)? What resources do the Buddhist metaphysical claims of no-self, karma, emptiness and related concepts bring to a meta-ethical understanding of terrorism and its effects?

A fascinating compilation of the life stories of perfectly ordinary people, all of whom have been through extraordinary experiences leading to amazing new perceptions. Some have been seekers from a very young age, whilst others had previously never even thought in terms of seeking. Nevertheless, all shared a common sense of dissatisfaction with their lives. This collection of stories explores the remarkable means by which each individual went about achieving an end to their suffering.

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)

We agree with Josipovic that a fundamental differentiating feature of meditation techniques is whether they remain within the dualistic subject–object cognitive structure, or they transcend this structure to reveal an underlying level of non-dual awareness. Further discussion is needed to delineate the basic non-dual experience in meditation, where all phenomenal content is absent, from the more advanced experience of non-duality in daily life, where phenomenal content is obviously present as well. In this discussion, it is important to recognize that (...) the experiencer–object relation makes the experience dual or non-dual, rather than the nature of the object experienced. (shrink)

Inquiry into object happiness -- What is enlightenment? -- The means of knowledge -- Qualifications -- The self -- Obstructions -- Inquiry into karma and dharma -- Inquiry into practice -- Love -- The assimilation of experience -- Lifestyle -- Knowledge yoga -- Meditation -- After enlightenment -- The teachings of Ramana Maharshi -- Neo-Advaita.

This paper describes the contemporary phenomenological understanding of subjectivity as non dual awareness.

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)

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)

This article problematizes the way of thinking schooling in discourse of sign system, which involves opposition, and double gesture of inclusion/exclusion. Drawing on two fundamental texts of Taoism, the Tao Te Ching and Chuang Tzu, this article puts forward the seemingly passive Non-Word and Non-Self to resist the hierarchy ordering of conceptions and man, and to undo duality of binary opposition. It links the history of assessment and PISA to the rethinking of evidence and sign in contemporary movements. The (...) second section proposes teaching as wandering (you, 游), the transcendence into Non-Self with response to men’s natural inquiry, but not artificially manipulating and shaping man by privileging certain knowledge, value, and thoughts over other. Instead of making categories of race, gender, and normal/abnormal, Taoist provides a way of reconsider the self operating on assessing, standardizing, normalizing, correcting and predicting others in school accountability. It is built on the ontology of considering individual as belonging to an ultimate sameness, the oneness. The article concludes with underlying assumption that students are subjected to change, and their difference must be eliminated through standard test. It further questions tactics of reason that educational reform face in the current regime of school accountability. (shrink)

Recently, a number of epistemologists have argued that there are no non-conceptual elements in representational content. On their view, the only sort of non-conceptual elements are components of sub-personal organic hardware that, because they enjoy no veridical role, must be construed epistemologically irrelevant. By reviewing a 35-year-old debate initiated by Dagfinn F.

This article problematizes the way of thinking schooling in discourse of sign system, which involves opposition, and double gesture of inclusion/exclusion. Drawing on two fundamental texts of Taois...

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...

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)

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)

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)

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)

A key working hypothesis in neuroscience is ‘materialistic reductionism’, i.e., the assumption whereby all physiological, behavioral or cognitive phenomena is produced by localized neurochemical brain activation (but not vice versa). However, analysis of sub-threshold Weber’s psychophysical stimulation indicates its computational irreducibility to the direct interaction between psychophysical stimulation and any neuron/s. This is because the materialistic-reductionistic working hypothesis assumes that the determination of the existence or non-existence of any psychophysical stimulation [s] may only be determined through its direct interaction [di1] (...) with a given neuron/s [N] that together forms the ‘neural registry’ computational level [NR/di1]. But, this implies that in cases of (initial) sub-threshold (sensory-specific) psychophysical stimulation which is increased above the sensory-specific threshold but below Weber’s psychophysical ‘dv’—the psychophysical computational processing [PCP] produces an apparently ‘computationally indeterminate’ output. This is because materialistic reductionism asserts the contingency of PCP upon the existence of a direct interaction between ‘s’ and ‘N’ within the NR/di1 level, but in the special case of Weber’s sub-threshold psychophysical stimulation the same PCP/di1 also asserts the non-existence of ‘s’ (as demanded by Weber’s psychophysical law). However, given robust empirical evidence indicating the capability of PCP to determine whether (or not) ‘s’ exists, we must conclude that PCP may not be carried out from within NR’s direct interaction between a particular psychophysical stimulation and any set of neuron/s in the brain. Hence, the Duality Principle asserts the conceptual irreducibility of sub-threshold psychophysical stimulation to any direct NR/di1: s-N interaction, thereby challenging the current materialistic-reductionistic assumption. (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 many philosophical traditions, it is held that reality is non-dual. Of course, to be non-dual, as opposed to dual, is itself to partake of a certain duality. If reality really is non-dual, it must transcend this duality too. But what could this mean? Can one make coherent sense of it? To keep the discussion focussed, I will locate it in one specific tradition: the Mahāyāna Buddhist tradition. The idea that ultimate reality is non-dual goes back to the (...) earliest Mahāyāna sūtras at the turn of the Common Era. Thereafter, the question of what it means to transcend duality plays a central role in Buddhist philosophy. The point that reality must transcend even the duality between duality and non-duality plays a significant role in the Chinese Sanlun philosopher Jizang (, 549–623). His discussion points the way to an answer to our problem which may be articulated with the techniques of contemporary paraconsistent logic, as we will see. (shrink)

We report on the simultaneous determination of complementary wave and particle aspects of light in a double-slit type “welcher-weg” experiment beyond the limitations set by Bohr’s Principle of Complementarity. Applying classical logic, we verify the presence of sharp interference in the single photon regime, while reliably maintaining the information about the particular pinhole through which each individual photon had passed. This experiment poses interesting questions on the validity of Complementarity in cases where measurements techniques that avoid Heisenberg’s uncertainty principle and (...) quantum entanglement are employed. We further argue that the application of classical concepts of waves and particles as embodied in Complementarity leads to a logical inconsistency in the interpretation of this experiment. (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)

There are two foundational, but not fully developed, ideas in paraconsistency, namely, the duality between paraconsistent and intuitionistic paradigms, and the introduction of logical operators that express meta-logical notions in the object language. The aim of this paper is to show how these two ideas can be adequately accomplished by the Logics of Formal Inconsistency (LFIs) and by the Logics of Formal Undeterminedness (LFUs). LFIs recover the validity of the principle of explosion in a paraconsistent scenario, while LFUs recover (...) the validity of the principle of excluded middle in a paracomplete scenario. We introduce definitions of duality between inference rules and connectives that allow comparing rules and connectives that belong to different logics. Two formal systems are studied, the logics mbC and mbD, that display the duality between paraconsistency and paracompleteness as a duality between inference rules added to a common core– in the case studied here, this common core is classical positive propositional logic (CPL + ). The logics mbC and mbD are equipped with recovery operators that restore classical logic for, respectively, consistent and determined propositions. These two logics are then combined obtaining a pair of logics of formal inconsistency and undeterminedness (LFIUs), namely, mbCD and mbCDE. The logic mbCDE exhibits some nice duality properties. Besides, it is simultaneously paraconsistent and paracomplete, and able to recover the principles of excluded middle and explosion at once. The last sections offer an algebraic account for such logics by adapting the swap-structures semantics framework of the LFIs the LFUs. This semantics highlights some subtle aspects of these logics, and allows us to prove decidability by means of finite non-deterministic matrices. (shrink)

I review at a non-technical level the use of the gauge-string duality to study aspects of heavy ion collisions, with special emphasis on the trailing string calculation of heavy quark energy loss. I include some brief speculations on how variants of the trailing string construction could provide a toy model of black hole formation and evaporation. This essay is an invited contribution to “Forty Years of String Theory” and is aimed at philosophers and historians of science as well as (...) physicists. (shrink)

Dualistic approaches to the mind-body relationship are commonplace; however, the adoption of dualistic thinking can often obscure aspects of the way the organism functions as a whole biological entity. Future versions of the emulation theory will, it is hoped, address some of these issues, including the nature of process noise, how distinct iterations can occur, and how to deal with non-emulated aspects of motor control.

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)

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)

The basic notions in Prior’s Ockhamist and Peircean logics of branching-time are the notion of moment and that of history (or course of events). In the tree semantics, histories are defined as maximal linearly ordered sets of moments. In the geometrical approach, both moments and histories are primitive entities and there is no set theoretical (and ontological) dependency of the latter on the former. In the topological approach, moments can be defined as the elements of a rank 1 base of (...) a non-Archimedean topology on the set of histories. In this paper, it will be shown that the topological approach, and hence the other approaches, can be reconstructed in a framework in which the basic notions are those of history and of relative closeness relation among histories. (shrink)

The basic notions in Prior's Ockhamist and Peircean logics of branching-time are the notion of moment and that of history. In the tree semantics, histories are defined as maximal linearly ordered sets of moments. In the geometrical approach, both moments and histories are primitive entities and there is no set theoretical dependency of the latter on the former. In the topological approach, moments can be defined as the elements of a rank 1 base of a non-Archimedean topology on the set (...) of histories. In this paper, it will be shown that the topological approach, and hence the other approaches, can be reconstructed in a framework in which the basic notions are those of history and of relative closeness relation among histories. (shrink)

_ Source: _Volume 11, Issue 2, pp 117 - 148 The paper examines the problem of self-knowledge from the perspectives of Plotinus and the Indian Advaita school. Analyzing the subject-object relation, I show that according to both Plotinus and Advaita thinkers, full self-knowledge demands complete absence of otherness. Plotinus argues that if self-consciousness is divided into subject-object relation then one will know oneself as contemplated but not as contemplating and no real self-knowledge obtains in this case. Śaṅkara, who constitutes an (...) important representative of Advaita thought, points out that the self cannot know itself as an object because what is called an object to be known becomes established when it is separated from the self, the subject. I argue that at the level of the One, similar to the state of _ātman_ consciousness in Advaita framework, the soul experiences itself in expansive non-dual consciousness. Lastly, I examine the role of non-duality as the foundation of knowledge. (shrink)

Apartness spaces were introduced as a constructive counterpart to proximity spaces which, in turn, aimed to model the concept of nearness of sets in a metric or topological environment. In this paper we introduce apartness algebras and apartness frames intended to be abstract counterparts to the apartness spaces of (Bridges et al., 2003), and we prove a discrete duality for them.

We define a family of intuitionistic non-normal modal logics; they can be seen as intuitionistic counterparts of classical ones. We first consider monomodal logics, which contain only Necessity or Possibility. We then consider the more important case of bimodal logics, which contain both modal operators. In this case we define several interactions between Necessity and Possibility of increasing strength, although weaker than duality. We thereby obtain a lattice of 24 distinct bimodal logics. For all logics we provide both a (...) Hilbert axiomatisation and a cut-free sequent calculus, on its basis we also prove their decidability. We then define a semantic characterisation of our logics in terms of neighbourhood models containing two distinct neighbourhood functions corresponding to the two modalities. Our semantic framework captures modularly not only our systems but also already known intuitionistic non-normal modal logics such as Constructive K and the propositional fragment of Wijesekera’s Constructive Concurrent Dynamic Logic. (shrink)

A recent paper by Jakl, Jung and Pultr succeeded for the first time in establishing a very natural link between bilattice logic and the duality theory of d-frames and bitopological spaces. In this paper we further exploit, extend and investigate this link from an algebraic and a logical point of view. In particular, we introduce classes of algebras that extend bilattices, d-frames and N4-lattices to a setting in which the negation is not necessarily involutive, and we study corresponding logics. (...) We provide product representation theorems for these algebras, as well as completeness, algebraizability results for the corresponding logics. (shrink)