Single and double meter simultaneous measurements of a harmonic oscillator are reviewed and compared. Naimark extensions are constructed and relevant projection properties are exhibited for both cases. The theory is extended to the simultaneous measurement of squeezed position and momentum measurements.

Measurements involving spin-1/2 meters which result in the simultaneous measurement of spin components are described. The spin analoge of the Arthur-Kelly experiment is contrasted with a simultaneous measurement which interacts with the system. Naimark extensions are constructed and Bloch state projection properties are discussed for each case. The theory is extended to squeezed angular momentum measurement.

Two different characterizations of POV measures with commutative range are compared using a representation of some stochastic operators by (weak) Markov kernels. A representation by Choquet theorem is obtained as an integral over functions of a sharp observable appearing in one of the characterizations. A Naimark extension is constructed.

The language of manuals may be used to discuss inference in measurement in a general experimental context. Specializing to the context of the frame manual for Hilbert space, this inference leads to state dominance of the inferred state from partial measurements; this in turn, by Sakai's theorem, determines observables which are described by positive operator-valued measures. Symmetries are then introduced, showing that systems of covariance, rather than systems of imprimitivity, are natural objects to study in quantum mechanics. Experiments measuring different (...) polarization components simultaneously are reexamined in this language. Finally, implications of the Naimark extension theorem for the manual approach are investigated. (shrink)

Extensive measurement is the standard measurement-theoretic approach for constructing a ratio scale. It involves the comparison of objects that can be concatenated in an additively representable way. This paper studies the implications of extensively measurable welfare for social choice theory. We do this in two frameworks: an Arrovian framework with a fixed population and no interpersonal comparisons, and a generalized framework with variable populations and full interpersonal comparability. In each framework we use extensive measurement to introduce novel domain restrictions, independence (...) conditions, and constraints on social evaluation. We prove a welfarism theorem for these domains and characterize the social welfare functions that satisfy the axioms of extensive measurement at both the individual and social levels. The main results are simple axiomatizations of strong dictatorship in the Arrovian framework and classical utilitarianism in the generalized framework. (shrink)

In proof-theoretic semantics, meaning is based on inference. It may be seen as the mathematical expression of the inferentialist interpretation of logic. Much recent work has focused on base-extension semantics, in which the validity of formulas is given by an inductive definition generated by provability in a ‘base’ of atomic rules. Base-extension semantics for classical and intuitionistic propositional logic have been explored by several authors. In this paper, we develop base-extension semantics for the classical propositional modal systems (...) K, KT , K4, and S4, with □ as the primary modal operator. We establish appropriate soundness and completeness theorems and establish the duality between □ and a natural presentation of ♢. We also show that our semantics is in its current form not complete with respect to euclidean modal logics. Our formulation makes essential use of relational structures on bases. (shrink)

Classical logic has proved inadequate in various areas of computer science, artificial intelligence, mathematics, philosopy and linguistics. This is an introduction to extensions of first-order logic, based on the principle that many-sorted logic (MSL) provides a unifying framework in which to place, for example, second-order logic, type theory, modal and dynamic logics and MSL itself. The aim is two fold: only one theorem-prover is needed; proofs of the metaproperties of the different existing calculi can be avoided by borrowing them from (...) MSL. To make the book accessible to readers from different disciplines, whilst maintaining precision, the author has supplied detailed step-by-step proofs, avoiding difficult arguments, and continually motivating the material with examples. Consequently this can be used as a reference, for self-teaching or for first-year graduate courses. (shrink)

In this paper, we generalize the soft set to the hypersoft set by transforming the function F into a multi-attribute function. Then we introduce the hybrids of Crisp, Fuzzy, Intuitionistic Fuzzy, Neutrosophic, and Plithogenic Hypersoft Set.

Domain extension in mathematics occurs whenever a given mathematical domain is augmented so as to include new elements. Manders argues that the advantages of important cases of domain extension are captured by the model-theoretic notions of existential closure and model completion. In the specific case of domain extension via ideal elements, I argue, Manders’s proposed explanation does not suffice. I then develop and formalize a different approach to domain extension based on Dedekind’s Habilitationsrede, to which Manders’s (...) account is compared. I conclude with an examination of three possible stances towards extensions via ideal elements. (shrink)

According to an influential idea in the philosophy of set theory, certain mathematical concepts, such as the notion of a well-order and set, are indefinitely extensible. Following Parsons (1983), this has often been cashed out in modal terms. This paper explores instead an extensional articulation of the idea, formulated in higher-order logic, that flat-footedly formalizes some remarks of Zermelo. The resulting picture is incompatible with the idea that the entire universe can be well-ordered, but entirely consistent with the idea that (...) the sets of any set-theoretic universe can be. (shrink)

The Cooperative Extension System serves as the conduit through which scientific knowledge generated by the 130 land-grant colleges and universities in the United States is translated and delivered directly to its constituents. Since its inception over 100 years ago, Extension has been integral in developing, delivering, and applying cutting-edge knowledge in agriculture and natural resources, youth development, family and consumer sciences, and community and rural development. Today, more than ever, Extension will need to lead the way in (...) building and maintaining sustainable partnerships across disciplines and with organizations at the local, state, and national levels to tackle complex issues considering diminishing resources. Bringing together leading and emerging scholars, this volume discusses how Extension is addressing issues and opportunities relevant to children, youth, families, and communities across the country both now and in the future. Topics include Extension's role in supporting childcare, social media use, entrepreneurship, rural communities, and underserved audiences. (shrink)

We introduce the notion of τ-like partial order, where τ is one of the linear order types ω, ω*, ω + ω*, and ζ. For example, being ω-like means that every element has finitely many predecessors, while being ζ-like means that every interval is finite. We consider statements of the form “any τ-like partial order has a τ-like linear extension” and “any τ-like partial order is embeddable into τ” . Working in the framework of reverse mathematics, we show that (...) these statements are equivalent either to equation image or to equation image over the usual base system equation image. (shrink)

In recent years, many people writing on set theory have invoked the notion of an indefinitely extensible concept. The notion, it is usually claimed, plays an important role in solving the paradoxes of absolute infinity. It is not clear, however, how the notion should be formulated in a coherent way, since it appears to run into a number of problems concerning, for example, unrestricted quantification. In fact, the notion makes perfectly good sense if one endorses a dialetheic solution to the (...) paradoxes. It then morphs from a supposed solution to the paradoxes into a diagnosis of their structure. In this paper I show how. (shrink)

Anglophone philosophers have shown a surprising interest in Alexander Gottlieb Baumgarten’s aesthetics in recent years. At the same time, new approaches to aesthetics have been proposed that come very close to the original conception of aesthetics that Baumgarten introduced in the middle of the eighteenth century. In light of these developments, this article undertakes a critical examination of a central concept in Baumgarten’s poetics and aesthetics—extensive clarity. It argues that historians of philosophy and contemporary aestheticians should be wary of this (...) concept for two reasons. First, in Baumgarten’s poetics, the extensive clarity of sensible representations constitutes a dubious standard with which to determine whether those representations are “poetic.” Second, in aesthetics, using extensive clarity as an alternative standard with which to determine the perfection of sensible cognition undermines the “marriage of reason and experience” that characterized the Leibnizian-Wolffian philosophy and raises the specter of dualism that Kant tried, with questionable success, to address in the ‘Schematism’ chapter of the Critique of Pure Reason. The article concludes that historians of philosophy should acknowledge the philosophical shortcomings of Baumgarten’s conception of extensive clarity and contemporary aesthetics should not reproduce them. (shrink)

On the ethics of extending human life: healthy people have a right to carry on livingHumanity has long demonstrated a paradoxical ambivalence concerning the extension of a healthy human lifespan. Modest health extension has been universally sought, whereas extreme health extension has been regarded as a snare and delusion—a dream beyond all others at first blush, but actually something we are better off without. The prevailing pace of biotechnological progress is bringing ever closer the day when humanity (...) will be able to act on the latter view by rejecting a clear and present opportunity for much longer healthy lives. Indeed, some biogerontologists contend that that day has already arrived, to the extent that our hesitation in embarking on a vigorous “war on ageing” is already delaying the point at which a cure for ageing will be developed. Here I consider whether our present caution concerning the wisdom of truly curing ageing is likely to survive the increased scrutiny that it will receive in coming years as a result of these technical advances. I conclude that it will not, because of its irreconcilability with values that are more deeply held by the large majority of humanity than any values that argue against the quest for a cure. I further conclude that all the major current reasons given for not curing ageing are mere crutches to help us cope with the immutability of ageing that we have been brought up to accept. Our failure to set aside such irrationality is already shortening potential longevity—quite probably of those already alive today—to a staggering degree. Once we realise this, our determination to consign human ageing to history will be second …. (shrink)

We analyze extensive games as interactive process models, using modallanguages plus matching notions of bisimulation as varieties of gameequivalences. Our technical results show how to fit existing modalnotions into this new setting.

In Kant’s logical texts the reference of the form S is P to an “unknown = x” is well known, but its understanding still remains controversial. Due to the universality of all concepts, the subject as much as the predicate is regarded as predicate of the x, which, in turn, is regarded as the subject of the judgment. In the CPR, this Kantian interpretation of the S-P relationship leads to the question about the relations between intuition and concept in judgment. (...) In contrast to intuition, if no concept, due to its universality, refers immediately to an object, how should one understand the relations of S and P to one another, as well as their relations to intuition, which corresponds to the possible individuality of the object in general = x? To answer this question, it is necessary to understand Kant’s notion of extension, and to prove its irreducibility to the Port-Royal notion of extension as well as to the Fregean one. (shrink)

Extension is probably the most general natural property. Is it a fundamental property? Leibniz claimed the answer was no, and that the structureless intuition of extension concealed more fundamental properties and relations. This paper follows Leibniz's program through Herbart and Riemann to Grassmann and uses Grassmann's algebra of points to build up levels of extensions algebraically. Finally, the connection between extension and measurement is considered.

The worst possible way to resolve this issue is to leave it up to individual choice. There is no known social good coming from the conquest of death (Bailey, 1999). - Daniel Callahan Dramatically extending the human lifespan seems increasingly possible. Many bioethicists object that life-extension will have Malthusian consequences as new Methuselahs accumulate, generation by generation. I argue for a Life-Years Response to the Malthusian Objection. If even a minority of each generation chooses life-extension, denying it to (...) them deprives them of many years of extra life, and their total extra life-years are likely to exceed the total life-years of a majority who do not want life-extension. This is a greater harm to those who want extended life than the Malthusian harms to those who refuse extended life, both because losing an extra year of life is worse than enduring a year of Malthusian conditions, and because the would-be Methuselahs have more life-years at stake. Therefore, even if life-extension seems likely to cause severe overcrowding and resource shortages, that threat is not sufficient to justify society in restricting the development or availability of life-extension. (shrink)

When we remember a scene, the scene’s boundaries are wider than the boundaries of the scene we saw. This phenomenon is called boundary extension. The most important philosophical question about boundary extension is whether it is a form of perceptual adjustment or adjustment during memory encoding. The aim of this paper is to propose a third explanatory scheme, according to which the extended boundary of the original scene is represented by means of mental imagery. And given the similarities (...) between perception and mental imagery, the memory system encodes both the part of the scene that is represented perceptually and the part of the scene that is represented by means of mental imagery. (shrink)

This article introduces and motivates the notion of a “properly extensive” quantity by means of a puzzle about the reliability of certain canonical length measurements. An account of these measurements’ success, I argue, requires a modally robust connection between quantitative structure and mereology that is not mediated by the dynamics and is stronger than the constraints imposed by “mere additivity.” I outline what it means to say that length is not just extensive but properly so and then briefly sketch an (...) application of proper extensiveness to the project of providing a reductive ground for metric quantitative structure. (shrink)

The paper considers certain extensions of the system LC introduced in Dunn & Meyer 1997. LC is a structurally free system , but it has combinators as formulas in the place of structural rules. We consider two ways to extend LC with conjunction and disjunction depending on whether they distribute over each other or not. We prove the elimination theorem for the systems. At the end of the paper we give a Routley-Meyer style semantics for the distributive extension, including (...) some new definitions and an algorithm which are used in proving a representation theorem in general, i.e., for arbitrary combinators. (shrink)

We give alternative proofs of results due to Paris and Wilkie concerning the existence of end extensions of countable models of $B\Sigma_{1}$, that is, the theory of $\Sigma_{1}$ collection.

Extended simple regions have been the focus of recent developments in philosophical logic, metaphysics, and philosophy of physics. However, only a handful of works provides a rigorous characterization of an extended simple region. In particular, a recent paper in this journal defends a definition based on an extrinsic notion of least distance. Call it the Least Distance proposal. This paper provides the first assessment of it. It argues that Least Distance faces difficulties and drawbacks. The paper then goes on to (...) suggest a different proposal, the Diameter proposal that is able to handle such drawbacks and difficulties. (shrink)

This article explains some of the technical terms used in semantic theory and philosophy of language.

Radical enactive and embodied approaches to cognitive science oppose the received view in the sciences of the mind in denying that cognition fundamentally involves contentful mental representation. This paper argues that the fate of representationalism in cognitive science matters significantly to how best to understand the extent of cognition. It seeks to establish that any move away from representationalism toward pure, empirical functionalism fails to provide a substantive “mark of the cognitive” and is bereft of other adequate means for individuating (...) cognitive activity. It also argues that giving proper attention to the way the folk use their psychological concepts requires questioning the legitimacy of commonsense functionalism. In place of extended functionalism—empirical or commonsensical—we promote the fortunes of extensive enactivism, clarifying in which ways it is distinct from notions of extended mind and distributed cognition. (shrink)

Extensive social choice theory is used to study the problem of measuring group fitness in a two-level biological hierarchy. Both fixed and variable group size are considered. Axioms are identified that imply that the group measure satisfies a form of consequentialism in which group fitness only depends on the viabilities and fecundities of the individuals at the lower level in the hierarchy. This kind of consequentialism can take account of the group fitness advantages of germ-soma specialization, which is not possible (...) with an alternative social choice framework proposed by Okasha, but which is an essential feature of the index of group fitness for a multicellular organism introduced by Michod, Viossat, Solari, Hurand, and Nedelcu to analyze the unicellular-multicellular evolutionary transition. The new framework is also used to analyze the fitness decoupling between levels that takes place during an evolutionary transition. (shrink)

The contour of a family of filters along a filter is a set-theoretic lower limit. Topologicity and regularity of convergences can be characterized with the aid of the contour operation. Contour inversion is studied, in particular, for iterated contours of sequential cascades. A related problem of continuous extension of maps between maximal elements of sequential cascades to full subcascades is solved in full generality.

0. Introductory remarks. I assume that every serious theory of language must give some explicit account of the relationship between expressions in the language described and expressions in some interpreted language which spells out the semantics of the language.1 Let's call this relationship the translation relation. Theories differ as to how this relation is specified. In the Aspects theory of syntax, taken together with a Katz-Postal view of "semantic rules", it was assumed that the relation was defined on deep structures. (...) Serious problems led to modifications of this view in several directions. Chomsky's latest papers assume that a modified 'intermediate structure' is transformed by rules of interpretation to a level of representation called 'logical form'. Common to all of these approaches is the assumption that the translation rules are defined initially on syntactic structures of one sort or another. Let us call this assumption the configurational hypothesis: the translation rules all have the form: [183]. (shrink)

It seems to be a widespread opinion that increasing the length of existing happy lives is better than creating new happy lives and that it may be better even when the total welfare is lower in the outcome with extended lives. The chapter discusses two interesting suggestions that seem to support this idea. The first is critical level utilitarianism (CLU) and the other is view comparativism. The chapter describes the pure case of life extension versus life replacement and then (...) presents some different views about the value of life extension, indicating some of the arguments in favor and against life extension fail. Then, it turns to the implications of critical level utilitarianism and comparativism in regards to life extension versus replacement, the main topic of this chapter. A case is presented to explain that there is a conflict between intuitions regarding life extension and comparativism. (shrink)

The paper investigates completions in the context of finitely generated lattice-based varieties of algebras. In particular the structure of canonical extensions in such a variety $${\mathcal {A}}$$ is explored, and the role of the natural extension in providing a realisation of the canonical extension is discussed. The completions considered are Boolean topological algebras with respect to the interval topology, and consequences of this feature for their structure are revealed. In addition, we call on recent results from duality theory (...) to show that topological and discrete dualities for $${\mathcal {A}}$$ exist in partnership. (shrink)

Institute of Theoretical Astronomy, University of Cambridge, Cambridge, UK 3fS. received 28th August 1970, in final revised form 1st July 1971..

Separated and immediate extensions of valued fields. The notion of separated extension of valued fields was introduced by Baur. He showed that extensions of maximal fields are separated. We prove that, when $(K, v)$ is Henselian with residual characteristic 0, then $(K, v) \subset (L, w)$ is separated iff $L$ is linearly disjoint over $K$ from each immediate extension of $K$.

Extension of the system that includes the key substrates for sensation, perception, emotion, volition, and cognition, and all representational sources for cognition, supports the view that there is an extended mind and an extended body. These intellectual views can be made practical in a humanist system based on extensions and in religious systems based on extensions. Independently, there is also an institutional extension of secularism. Hence, I maintain, there are five principal forms of extension.

This paper investigates the role of a pre-existing body-model that is an enabling constraint for the incorporation of objects into the body. This body-model is also a basis for the distinction between body extensions (e.g., in the case of tool-use) and incorporation (e.g., in the case of successful prosthesis use). It is argued that, in the case of incorporation, changes in the sense of body-ownership involve a reorganization of the body-model, whereas extension of the body with tools does not (...) involve changes in the sense of body-ownership. (shrink)

Following considerable debate, the practice of euthanasia was legalized in Belgium in 2002, thereby making Belgium one of the few places in the world where this practice is legal. In 2014 the law was amended for the first time. The 2014 amendment makes euthanasia legally possible for all minors who repeatedly and voluntarily request euthanasia and who are judged to possess “capacity of discernment”, as well as fulfil a number of other criteria of due care. This extension of the (...) 2002 euthanasia law generated a lot of national and international debate and has been applauded by many and heavily criticized by others. This evolution is clearly of interest to end-of-life debates in the entire world. This paper will therefore describe how this amendment came to get passed using official documents from Belgium's Senate and Chamber of Representatives where this amendment was discussed and subsequently passed. Next, some of the most commonly given arguments in favour of the law are identified, as well as the arguments most often voiced against the amendment. All these arguments will be expanded upon and it will be examined whether they hold up to ethical scrutiny. Analysing the official documents and identifying the most commonly voiced arguments gives valuable insight into how Belgium came to amend its euthanasia law and why it did so in 2014. It also becomes clear that although the current amendment is often seen as far-reaching, more radical ideas were proposed during the drafting of the law. Also, in analysing those arguments in favour of the amendment and those against, it is clear that the validity of some of these is questionable. (shrink)

This paper investigates the role of a pre-existing body-model that is an enabling constraint for the incorporation of objects into the body. This body-model is also a basis for the distinction between body extensions (e.g., in the case of tool-use) and incorporation (e.g., in the case of successful prosthesis use). It is argued that, in the case of incorporation, changes in the sense of body-ownership involve a reorganization of the body-model, whereas extension of the body with tools does not (...) involve changes in the sense of body-ownership. (shrink)

It is shown by Parsons [2] that the first-order fragment of Frege's logical system in the Grundgesetze der Arithmetic is consistent. In this note we formulate and prove a stronger version of this result for arbitrary first-order theories. We also show that a natural attempt to further strengthen our result runs afoul of Tarski's theorem on the undefinability of truth.

Dummett's account of the semantic paradoxes in terms of his theory of indefinitely extensible concepts is compared with Bürge's account in terms of indexicality. Dummett's appeal to intuitionistic logic does not block the paradoxes but Bürge's attempt to avoid the Strengthened Liar is unconvincing. It is argued that in order to avoid the Strengthened Liar and other semantic paradoxes involving nonindexical expressions (constants), one must postulate that when we reflect on the paradoxes there are slight shifts in the meaning (not (...) just reference) we ascribe to metalinguistic expressions (in particular 'say', and derivatively 'true' and 'false'). Consideration of metaphor and gradual linguistic change suggests that such semantic shifts are consistent with language-learning and communication. On this account there is no threat to classical logic, bivalence or the fundamental principles governing 'true' and 'false'. (shrink)

Contrary to what is stated in Lemma 7.1 of [8], it is shown that some Boolean algebras of finitary logic admit finitely additive probabilities that are not σ-additive. Consequences of Lemma 7.1 are reconsidered. The concept of a C-σ-additive probability on B (where B and C are Boolean algebras, and $\mathscr{B} \subseteq \mathscr{C}$ ) is introduced, and a generalization of Hahn's extension theorem is proved. This and other results are employed to show that every S̄(L)-σ-additive probability on s̄(L) can (...) be extended (uniquely, under some conditions) to a σ-additive probability on S̄(L), where L belongs to a quite extensive family of first order languages, and S̄(L) and s̄(L) are, respectively, the Boolean algebras of sentences and quantifier free sentences of L. (shrink)

During the past decade, the so-called “hypothesis of cognitive extension,” according to which the material vehicles of some cognitive processes are spatially distributed over the brain and the extracranial parts of the body and the world, has received lots of attention, both favourable and unfavourable. The debate has largely focussed on three related issues: (1) the role of parity considerations, (2) the role of functionalism, and (3) the importance of a mark of the cognitive. This paper critically assesses these (...) issues and their interconnections. Section 1 provides a brief introduction. Section 2 argues that some of the most prominent objections against the appeal to parity considerations fail. Section 3 shows that such considerations are nevertheless unsuitable as an argument for cognitive extension. First, the actual argumentative burden is carried by an underlying commitment to functionalism, not by the parity considerations themselves. Second, in the absence of an independently motivated mark of the cognitive, the argument based on parity considerations does not get off the ground, but given such a mark, it is superfluous. Section 4 argues that a similar dilemma arises for the attempt to defend cognitive extension by a general appeal to functionalism. Unless it can be independently settled what it is for a process to be cognitive, functionalism itself will be undermined by the possibility of cognitive extension. Like parity considerations, functionalism is thus either unable to support cognitive extension or superfluous. Hence, nothing short of the specification of an appropriate mark of the cognitive that can be fulfilled not only by intracranial but also by extended processes will do as an argument for cognitive extension. (shrink)

This paper brings together several strands of thought from both the analytic and phenomenological traditions in order to critically examine accounts of cognitive enhancement that rely on the idea of cognitive extension. First, I explain the idea of cognitive extension, the metaphysics of mind on which it depends, and how it has figured in recent discussions of cognitive enhancement. Then, I develop ideas from Husserl that emphasize the agential character of thought and the distinctive way that conscious thoughts (...) are related to one another. I argue that these considerations are necessary for understanding why forms of cognitive extension may diminish our cognitive lives in different ways. This does not lead to a categorical rejection of cognitive enhancement as unethical or bad for human flourishing, but does warrant a conservative approach to the design and implementation of cognitive artifacts. (shrink)

There is growing public interest in developing biomedical technologies capable of extending the human lifespan. This chapter discusses the desirability of a substantial life extension with regard to its implications for personal identity over time. The possibility of significant lifespan extension involves two different personal‐identity questions. First, it raises the question of whether one is identifiable as the very same person throughout a lifetime; secondly, it challenges our self‐conception. The chapter describes the way in which personal identity as (...) “sameness” might be affected by the prospect of substantial life extension. It then focuses on the significance of the conception of personal identity as “selfhood” for radically extended lifespans. Finally, the chapter argues that a substantial life extension raises important concerns about the continuity of one's identity over time and is not straightforwardly desirable for the individual. (shrink)

The object of this paper is to examine half and full connexive extensions of the basic regular conditional logic CR. Extensions of this system are of interest because it is among the strongest well-known systems of conditional logic that can be augmented with connexive theses without inconsistency resulting. These connexive extensions are characterized axiomatically and their relations to one another are examined proof-theoretically. Subsequently, algebraic semantics are given and soundness, completeness, and decidability are proved for each system. The semantics is (...) also used to establish independence results. Finally, a deontic interpretation of one of the systems is examined and defended. (shrink)

It seems to be a widespread opinion that increasing the length of existing happy lives is better than creating new happy lives although the total welfare is the same in both cases, and that it may be better even when the total welfare is lower in the outcome with extended lives. I shall discuss two interesting suggestions that seem to support this idea. Firstly, the idea there is a positive level of well-being above which a life has to reach to (...) have positive contributive value to a population. This view is usually called critical level utilitarianism. Secondly, the viewthat it makes an outcome worse if people are worse off than they otherwise could have been. I shall call this view comparativism. Firstly, I shall describe what I call the pure case of life extension versus replacement. Then I shall very briefly describe some different views about the value of life extension and indicate why I think some of the arguments in favor and against life extension fail. I shall then turn to the implications of critical level utilitarianism and comparativism in regards to life extension versus replacement, which is the main topic of this chapter. (shrink)

In the present paper we prove that the poset of all extensions of the logic defined by a class of matrices whose sets of distinguished values are equationally definable by their algebra reducts is the retract, under a Galois connection, of the poset of all subprevarieties of the prevariety generated by the class of the algebra reducts of the matrices involved. We apply this general result to the problem of finding and studying all extensions of the logic of paradox. In (...) order to solve this problem, we first study the structure of prevarieties of Kleene lattices. Then, we show that the poset of extensions of the logic of paradox forms a four-element chain, all the extensions being finitely many-valued and finitely-axiomatizable logics. There are just two proper consistent extensions of the logic of paradox. The first is the classical logic that is relatively axiomatized by the Modus ponens rule for the material implication. The second extension, being intermediate between the logic of paradox and the classical logic, is the one relatively axiomatized by the Ex Contradictione Quodlibet rule. (shrink)