We generalize Priestley duality for distributive lattices to a duality for distributive meet-semilattices. On the one hand, our generalized Priestley spaces are easier to work with than Celani’s DS-spaces, and are similar to Hansoul’s Priestley structures. On the other hand, our generalized Priestley morphisms are similar to Celani’s meet-relations and are more general than Hansoul’s morphisms. As a result, our duality extends Hansoul’s duality and is an improvement of Celani’s duality.

In this paper we describe the Priestley space of a quasi-Stone algebra and use it to show that the class of finite quasi-Stone algebras has the amalgamation property. We also describe the Priestley space of the free quasi-Stone algebra over a finite set.

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.

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)

The Priestley duality for Wajsberg algebras is developed. The Wajsberg space is a De Morgan space endowed with a family of functions that are obtained in rather natural way.As a first application of this duality, a theorem about unicity of the structure is given.

In 1765, Joseph Priestley created what may be the world's first modern timeline, A Chart of Biography. This paper offers the first study of the philosophy underlying Priestley's timeline. It argues that Priestley was pushed towards representing times as lines by his views on abstract ideas and time, and there is no reason to believe that Newtonian absolutism grounds his uniform depiction of time. Further, the Chart confirms, and even advances, Priestley's views on human progress. Finally, (...) this study shows that Priestley's conception of time as a line comprises a landmark contribution to the history of space-time parallelism. (shrink)

A Priestley duality is developed for the variety j of all modal lattices. This is achieved by restricting to j a known Priestley duality for the variety of all bounded distributive lattices with a meet-homomorphism. The variety j was first studied by R. Beazer in 1986.The dual spaces of free modal lattices are constructed, paralleling P.R. Halmos'' construction of the dual spaces of free monadic Boolean algebras and its generalization, by R. Cignoli, to distributive lattices with a quantifier.

This paper compares and contrasts the main metaphysical and religious ideas of the eighteenth-century political theorist and chemist Joseph Priestley (1733–1804) with those of his correspondent Ruder Boscovic (1711–1787), astronomer, poet, mathematician, diplomat and Jesuit priest. It points out the theological differences between the two thinkers resulting from the divergent ontological and metaphysical implications of the theory of point atomism that they shared. This theory they placed in a wider context, considering both its limits and its value as a (...) contribution to ongoing speculations concerning the nature of space and time, theological conundrums such as free will and the mind-body problem. Where they differed, however, was that Priestley, a chemist rather than a mathematician, used point atomism mainly to support his campaign to further materialism and discredit Christianity. Boscovic, on the other hand, carefully distinguished the truths of faith and reason and was therefore indignant at Priestley's misuse of his speculations. Boscovic's protest was probably motivated in part by the materialism of atheists such as the Baron d’Holbach (1723–1789). Priestley was no atheist, however. Interestingly, François Arouet de Voltaire (1698–1778), unaware of the works of either Boscovic or Priestley, devised a theistic materialism in his last years that in many respects resembled Priestley's. I begin with brief biographies of the two thinkers, and outline their intermittent relationship. (shrink)

In this note we introduce and study algebras of type such that is a bounded distributive lattice and ⌝ is an operator that satisfies the condition ⌝ = a ⌝ b and ⌝ 0 = 1. We develop the topological duality between these algebras and Priestley spaces with a relation. In addition, we characterize the congruences and the subalgebras of such an algebra. As an application, we will determine the Priestley spaces of quasi-Stone algebras.

In this paper we obtain characterizations of subalgebras of Heyting algebras and De Morgan Heyting algebras. In both cases we obtain these characterizations by defining certain equivalence relations on the Priestley-type topological representations of the corresponding algebras. As a particular case we derive the characterization of maximal subalgebras of Heyting algebras given by M. Adams for the finite case.

It was shown in [3] (see also [5]) that there is a duality between the category of bounded distributive lattices endowed with a join-homomorphism and the category of Priestley spaces endowed with a Priestley relation. In this paper, bounded distributive lattices endowed with a join-homomorphism, are considered as algebras and we characterize the congruences of these algebras in terms of the mentioned duality and certain closed subsets of Priestley spaces. This enable us to characterize the simple and (...) subdirectly irreducible algebras. In particular, Priestley relations enable us to characterize the congruence lattice of the Q-distributive lattices considered in [4]. Moreover, these results give us an effective method to characterize the simple and subdirectly irreducible monadic De Morgan algebras [7].The duality considered in [4], was obtained in terms of the range of the quantifiers, and such a duality was enough to obtain the simple and subdirectly irreducible algebras, but not to characterize the congruences. (shrink)

In this paper we introduce the class of weak Heyting–Brouwer algebras (WHB-algebras, for short). We extend the well known duality between distributive lattices and Priestley spaces, in order to exhibit a relational Priestley-like duality for WHB-algebras. Finally, as an application of the duality, we build the tense extension of a WHB-algebra and we employ it as a tool for proving structural properties of the variety such as the finite model property, the amalgamation property, the congruence extension property and (...) the Maehara interpolation property. (shrink)

In this paper we study the variety \(\textsf{WL}\) of bounded distributive lattices endowed with an implication, called weak Lewis distributive lattices. This variety corresponds to the algebraic semantics of the \(\{\vee,\wedge,\Rightarrow,\bot,\top \}\) -fragment of the arithmetical base preservativity logic \(\mathsf {iP^{-}}\). The variety \(\textsf{WL}\) properly contains the variety of bounded distributive lattices with strict implication, also known as weak Heyting algebras. We introduce the notion of WL-frame and we prove a representation theorem for WL-lattices by means of WL-frames. We extended (...) this representation to a topological duality by means of Priestley spaces endowed with a special neighbourhood relation between points and closed upsets of the space. These results are applied in order to give a representation and a topological duality for the variety of weak Heyting–Lewis algebras, i.e., for the algebraic semantics of the arithmetical base preservativity logic \(\textsf{iP}^{-}\). (shrink)

A topological duality is presented for a wide class of lattice-ordered structures including lattice-ordered groups. In this new approach, which simplifies considerably previous results of the author, the dual space is obtained by endowing the Priestley space of the underlying lattice with two binary functions, linked by set-theoretical complement and acting as symmetrical partners. In the particular case of l-groups, one of these functions is the usual product of sets and the axiomatization of the dual space (...) is given by very simple first-order sentences, saying essentially that both functions are associative and that the space is a residuated semigroup with respect to each of them. (shrink)

We give topological proofs of Görnemann’s adaptation to Heyting algebras of the Rasiowa-Sikorski Lemma for Boolean algebras; and of the Rauszer-Sabalski generalisation of it to distributive lattices. The arguments use the Priestley topology on the set of prime filters, and the Baire category theorem. This is preceded by a discussion of criteria for compactness of various spaces of subsets of a lattice, including spaces of filters, prime filters etc.

Eighteenth-Century english scientists, Poets, And philosophers extended the meaning of 'distance' beyond a concept of space and time to include psychological and aesthetic meanings. Berkeley (1709), Priestley (1772), And thomas wedgwood (1818) showed that it was not a self-Evident idea but a complex intellectual construction. The poets denham (1655), Pope (1711), Dyer (1726), Collins (1747), Gray (1747), Campbell (1799) and wordsworth (1805-1827) used distance to represent a mental perspective, An aesthetic attitude, Nostalgia, Hope, Fancy, And imagination. Hume (1739), (...) Hartley (1749), Adam smith (1761), Burke (1757), And blair (1783) discussed early versions of aesthetic distance. The expanding meaning of 'distance' corresponds with the changing understanding of space and time in the philosophies of locke, Hume, And kant. Distance provided the eighteenth century with a means of exploring the relationship between subjective and objective realms of experience. (shrink)

If you were to list the perennial issues in philosophy, the nature of time would no doubt be on that list. The essays in the present volume all touch upon the problem of time. The volume includes four contributions from different perspectives within the history of philosophy of time.Jani Hakkarainen and Todd Ryan delve into David Hume's account of time. Hume thinks there can be no time without succession. Consequently, unchanging, steadfast objects do not have a duration. They are stationary, (...) not subject to any change. Although they lack a duration, they coexist with a temporal succession. This introduces a problem: What is the relation between unchanging objects and time? Hakkarainen and Ryan draw on what they call Hume's Principle of Quantitative Comparison. Each indivisible temporal extent is equal to any other temporal extent; differences in duration are due to the different number of temporal parts. Steadfast objects may be thought to undergo a certain number of calculable changes when they are compared to ordinary successions we experience. This experience is provided, for example, by the ticking of a clock. We have the capacity to think that steadfast objects have precise durations. We humans nevertheless cannot give any deep explanation as how we come to this conclusion.Emily Thomas focuses on the graphic representation of time. Whereas the visual representation of time has typically been done with matrices or grids, Joseph Priestley represented time in a line in 1765. Priestley's line divides time “by an equal scale” into centuries. Many have interpreted this as being evidence of a Newtonian influence, as if there is an absolute time flowing equably. Thomas situates Priestley's position in the modern debates on the nature of our ideas. She argues that the absolutist reading does not have merit. Thomas thinks rather that Priestley sides with David Hartley, who explained how we acquire the ideas of duration and time from sensation. Thomas's essay advances our understanding of the history of space-time parallelism. With timelines, the analogy between time and space becomes a possibility. In my essay I assess Ernst Mach's criticism of Isaac Newton's substantivalist account of time. Newton's argument for absolute time relates to his mathematical-empirical physics and theology. In his Science of Mechanics, Mach largely neglects the laws of motion, calculus and the omnipresence theology that is at the background of Newton's concept of time. Mach did however criticize Newton's absolutist argument on the basis that the self-existing flow cannot be observed, measured, or experimented with, and he provided an alternative relational account of time. I point out that this was not only extreme positivism or phenomenalism, but an epistemology and ontology suited for the subsequent physical theories of relativity. Finally, I argue that Mach's denial of the independent atomic structure of matter should not be entirely assimilated to his denial of substantivalism about time. The two phenomena are different, so we should evaluate Mach's reasoning concerning them accordingly. Matyáš Moravec and Peter West go through Susan Stebbing's criticism of Arthur Eddington's popular account of the passage of time. Stebbing's target, Moravec and West argue, is Eddington's largely Bergsonian view on time. She objects to the view that our minds have “a private door” to the physical reality which no measurements can disclose. Stebbing believes that our experience of the passage of time belongs to our ordinary, every-day experience, and she thinks Eddington mystifies the whole idea by leaning on a peculiar form of introspection. Moravec and West build a case for Bergson's influence on Eddington, arguing that Bergson probably did contribute to Eddington's idea of “the intuition of becoming.” Both seem to think that we may access the dynamic temporal reality in ways not captured by scientific investigation, by means of an introspective internal experience of duration. As a final remark, I think I should note, with regret, that out of the six authors in this volume only one is a woman. Let the reader know that as an editor I tried to strike a better balance. I still trust that the quality of the articles is high, and most importantly, worthwhile for readers interested in the history of philosophy of time. (shrink)

The main aim of the present paper is to explain a nature of relationships exist between Nelson and Heyting algebras. In the realization, a topological duality theory of Heyting and Nelson algebras based on the topological duality theory of Priestley for bounded distributive lattices are applied. The general method of construction of spaces dual to Nelson algebras from a given dual space to Heyting algebra is described. The algebraic counterpart of this construction being a generalization of the Fidel-Vakarelov (...) construction is also given. These results are applied to compare the equational category N of Nelson algebras and some its subcategories with the equational category H of Heyting algebras. It is proved that the category N is topological over the category H. (shrink)

We give a coalgebraic view of the restricted Priestley duality between Heyting algebras and Heyting spaces. More precisely, we show that the category of Heyting spaces is isomorphic to a full subcategory of the category of all -coalgebras, based on Boolean spaces, where is the functor which maps a Boolean space to its hyperspace of nonempty closed subsets. As an appendix, we include a proof of the characterization of Heyting spaces and the morphisms between them.

In this paper we shall introduce the variety FWHA of frontal weak Heyting algebras as a generalization of the frontal Heyting algebras introduced by Leo Esakia in [ 10 ]. A frontal operator in a weak Heyting algebra A is an expansive operator τ preserving finite meets which also satisfies the equation $${\tau(a) \leq b \vee (b \rightarrow a)}$$, for all $${a, b \in A}$$. These operators were studied from an algebraic, logical and topological point of view by Leo Esakia (...) in [ 10 ]. We will study frontal operators in weak Heyting algebras and we will consider two examples of them. We will give a Priestley duality for the category of frontal weak Heyting algebras in terms of relational spaces $${\langle X, \leq, T, R \rangle}$$ where $${\langle X, \leq, T \rangle}$$ is a WH -space [ 6 ], and R is an additional binary relation used to interpret the modal operator. We will also study the WH -algebras with successor and the WH -algebras with gamma. For these varieties we will give two topological dualities. The first one is based on the representation given for the frontal weak Heyting algebras. The second one is based on certain particular classes of WH -spaces. (shrink)

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

The paper discusses regularisation of dualities. A given duality between (concrete) categories, e.g. a variety of algebras and a category of representation spaces, is lifted to a duality between the respective categories of semilattice representations in the category of algebras and the category of spaces. In particular, this gives duality for the regularisation of an irregular variety that has a duality. If the type of the variety includes constants, then the regularisation depends critically on the location or absence of constants (...) within the defining identities. The role of schizophrenic objects is discussed, and a number of applications are given. Among these applications are different forms of regularisation of Priestley, Stone and Pontryagin dualities. (shrink)

We are all used to drawing straight lines to represent time, and above them, we plot historical events or physical or economic data. What to us is a self-evident convention, is however of an astonishingly recent date: it emerged only in the second half of the eighteenth century. To us, this late date seems paradoxical and cries out for an explanation. How else did earlier periods measure change, if not as a function of time? it will be argued that since (...) Antiquity, time was taken to measure change, and change to occur in space. ‘Our’ idea of representing time as an independent dimension would have seemed aberrant. But then, a second issue arises. Did not medieval natural philosophers employ timelines, Oresme’s diagram of the mean speed theorem being the most famous case? However, as will be shown, our interpretation of his diagram is probably wrong. This insight, in turn, takes care of a third paradox, namely Galileo’s initial inability to represent the law of free fall correctly. This article will document that the timeline first emerged in the late sixteenth century in works on chronology, made its first appearance in physics in Galileo’s diagrams, and had its general breakthrough in the eighteenth century. (shrink)

If ethical expressivism is true, then moral judgements are motivational desire-like states and do not robustly represent reality. This gives rise to the problem of how to understand moral error. How can we be mistaken if there is no moral reality to be mistaken about? The standard expressivist explanation of moral doubt is couched in terms of our fear that our judgements may not survive improvements to our epistemic situation. There is a debate between Egan :205–219, 2007), Blackburn :201–213, 2009), (...) Köhler :161–165, 2015) and Ridge :1–21, 2015) on the adequacy of this explanation when it comes to the phenomenon of fundamental moral error. This paper is my contribution to the debate. I argue, contrary to Blackburn and Ridge, that expressivism is committed to some first-order anti-realist sounding claims, and thus quasi-realism fails. If expressivism is true, none of us can coherently believe we might be fundamentally mistaken. However, contrary to Egan and Köhler, I do not think this is bad. Expressivists can still do what motivated the project of quasi-realism in the first place: they can interpret, make sense of, and vindicate ordinary moral discourse. I end by showing it yields some positive results for moral philosophers too – it effectively amounts to a transcendental argument against unhealthy moral scepticism. (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)

The English polymath Joseph Priestley wrote on a wide range of scientific, theological and pedagogical subjects. After the appearance of his influential Rudiments of English Grammar and A Course of Lectures on the Theory of Language and Universal Grammar, both of which are reissued in this series, Priestley produced in 1765 his Essay on a Course of Liberal Education, which is included and expanded on in this 1778 publication. Here he explains the reasons behind his decision to guide (...) the curriculum at Warrington Academy towards a greater focus on subjects with a more direct application to 'civil and active life'. He offers more general instruction on the cultivation of young men in various spheres, notably through foreign travel and developing 'knowledge of the world'. Priestley ends by outlining a suggested course of lectures on the history and laws of England. (shrink)

Dialetheists believe some sentences are both true and false. Objectors have argued that this makes it unclear how people can disagree with each other because, given the dialetheist’s commitments, if I make a claim and you tell me my claim is false, we might both be correct. Graham Priest (2006a) thinks that people disagree by rejecting or denying what is said rather than ascribing falsehood to it. We build on the work of Julien Murzi and Massimiliano Carrara (2015) and show (...) that Priest’s approach cannot succeed: given the same dialetheist’s commitments you may be correct to reject a claim that I correctly believe. We argue further that any attempt to solve the problem by identifying a new attitude of disagreement will also fail. The culprit, we claim, is the attempt to find a pair of attitudes that satisfy ‘exclusivity’—that is, attitudes such that both cannot be simultaneously correct. Instead of identifying disagreement by the kinds of attitudes involved, we propose dialetheists focus on the normative landscape and identify it in part by whether parties have reasons to change their attitudes. We offer our own normative theory of disagreement to help dialetheists with this challenge. (shrink)

Many people believe that if you identify as a particular gender, then you are that gender. This paper is my attempt at making sense of this claim. I propose to conceive of genders as names: determined by the individual and not requiring any specific biological or psychological qualities, yet still important to the bearer.

Joseph Priestley (1733-1804) was arguably the most important English theorist to focus on the issue of political liberty during the English Enlightenment. His concept of freedom is of crucial importance to two of the major issues of his day: the right of dissenters to religious toleration, and the right of the American colonists to self-government. Priestley's writings lack a modern edition and this new collection will be the first to render accessible his Essay on First Principles, The Present (...) State of Liberty and the Letter of Advice. An introduction and notes, and guides to further reading and key figures in the text provide the student with all the material necessary for approaching Priestley. (shrink)

The Free Discussion between Richard Price and Joseph Priestley (1778) originated as a correspondence between the two after the publication of Priestley's Disquisitions on Matter and Spirit, his most important philosophical work (1777). At the time it was thought remarkable that a controversey such as this could be conducted so amicably, but then the two were close friends. Nevertheless their philosophical, as opposed to their oft mentioned political views, were at opposite ends of a spectrum.

This work has been selected by scholars as being culturally important, and is part of the knowledge base of civilization as we know it. This work was reproduced from the original artifact, and remains as true to the original work as possible. Therefore, you will see the original copyright references, library stamps (as most of these works have been housed in our most important libraries around the world), and other notations in the work.This work is in the public domain in (...) the United States of America, and possibly other nations. Within the United States, you may freely copy and distribute this work, as no entity (individual or corporate) has a copyright on the body of the work.As a reproduction of a historical artifact, this work may contain missing or blurred pages, poor pictures, errant marks, etc. Scholars believe, and we concur, that this work is important enough to be preserved, reproduced, and made generally available to the public. We appreciate your support of the preservation process, and thank you for being an important part of keeping this knowledge alive and relevant. (shrink)

This Elibron Classics title is a reprint of the original edition published by J. Johnson in London, 1777.

This paper describes several emotion expressions in Koromu, a language of Papua New Guinea. As in other languages, emotions can be expressed by reference to body events and processes. Bodily images are used for common and pertinent emotion expressions in Koromu and the alternative grammatical constructions in which some of these expressions occur enable speakers to express varying emotions while still indicating that there are shared semantic components between the expressions. In addition, as the emotion expressions are examined and their (...) meanings explicated, a number of universal concepts and components of meaning can be observed. A study of these language specific expressions therefore contributes to a cross-linguistic understanding of the relationship between emotion and the body. (shrink)