Priestley Duality for Bilattices

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 for Quasi-Stone Algebras

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. This revised version was published online in June 2006 with corrections to the Cover Date.     

Free Modal Lattices via Priestley Duality

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.     

The Priestley duality for Wajsberg algebras

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.     

Priestley duality for some subalgebra lattices

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).     

Self-concern From Priestley To Hazlitt

This article argues that the Fourth and Fifth of John Toland's Letters to Serena are best understood as a creative confrontation of Spinoza and Leibniz – one in which crucial aspects of Leibniz's thought are extracted from their original context and made to serve a purpose that is ultimately Spinozistic. Accordingly, it suggests that the critique of Spinoza that takes up so much of the fourth Letter, in particular, should be read as a means of `perfecting' Spinoza (via Leibniz), rather than as the outright dismissal it might appear to be. In order to make its case, the article outlines: the supposed problems that Toland finds in Spinoza; what Toland takes from Leibniz, and what he discards, in order to solve these `problems'; and the imprint of Spinoza's naturalism on the eventual `solution' that Toland offers. The article concludes that, whatever the success of this `solution', Toland's speculative labours should still be treated as creative, perspicuous and intrinsically significant.     

Duality and Canonical Extensions of Bounded Distributive Lattices with Operators,and Applications to the Semantics of Non-Classical Logics II

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.     

Duality and Canonical Extensions of Bounded Distributive Lattices with Operators,and Applications to the Semantics of Non-Classical Logics I

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. This revised version was published online in June 2006 with corrections to the Cover Date.     

Distributive Luck

Abstact

This article explores the Rawlsian goal of ensuring that distributions are not influenced by the morally arbitrary. It does so by bringing discussions of distributive justice into contact with the debate over moral luck initiated by Williams and Nagel. Rawls’ own justice as fairness appears to be incompatible with the arbitrariness commitment, as it creates some equalities arbitrarily. A major rival, Dworkin’s version of brute luck egalitarianism, aims to be continuous with ordinary ethics, and so is (a) sensitive to non-philosophical beliefs about free will and responsibility, and (b) allows inequalities to arise on the basis of option luck. But Dworkin does not present convincing reasons in support of continuity, and there are compelling moral reasons for justice to be sensitive to the best philosophical account of free will and responsibility, as is proposed by the revised brute luck egalitarianism of Arneson and Cohen. While Dworkinian brute luck egalitarianism admits three sorts of morally arbitrary disadvantaging which correspond to three forms of moral luck (constitutive, circumstantial, and option luck), revised brute luck egalitarian-ism does not disadvantage on the basis of constitutive or circumstantial luck. But it is not as sensitive to responsibility as it needs to be to fully extinguish the influence of the morally arbitrary, for persons under it may exercise their responsibility equivalently yet end up with different outcomes on account of option luck. It is concluded that egalitarians should deny the existence of distributive luck, which is luck in the levels of advantage that individuals are due.     

A Duality for Involutive Bisemilattices

Studia Logica - We establish a duality between the category of involutive bisemilattices and the category of semilattice inverse systems of Stone spaces, using Stone duality from one side and the...     

Duality for algebras of relevant logics

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 author gratefully acknowledges the support of the National Sciences and Engineering Research Council of Canada.     

A Theory of Providence for Distributive Justice

Distributive justice assumes a morally critical judgment of nature, which typically contradicts providential conceptions. Hence, simple conceptions of divine Providence cannot support distributive justice. This essay analyzes and develops a complex strand of theorizing about Providence within Jewish philosophy that is compatible with distributive justice. According to this conception, the actions of divine Providence express different and mutually exclusive considerations of justice. Therefore, the moral value of outcomes is intransitive between the situations of different people. And while each providential action is justified from an ethical perspective, the total outcome is distinct from God's ultimate desire. Human ethics responds to this disparity by redistribution. This conception of Providence also contributes to the additional issue of intergenerational justice through the concomitant idea of life missions. The classical rendering of missions creates problems, however, for distributive justice. I conclude by formulating a conception of life missions that is compatible with both distributive and intergenerational justice.     

Reid and Priestley on Method and the Mind

Reid said little in his published writings about his contemporary Joseph Priestley, but his unpublished work is largely devoted to the latter. Much of Priestley's philosophical thought — his materialism, his determinism, his Lockean scientific realism - was as antithetical to Reid's as was Hume's philosophy in a very different way. Neither Reid nor Priestley formulated a full response to the other. Priestley's response to Reid came very early in his career, and is marked by haste and immaturity. In his last decade Reid worried much about Priestley's materialism, but that concern never reached publication. I document Reid's unpublished response to Priestley, and also view Reid's response from Priestley's perspective, as deduced from his published works. Both thinkers attempted to base their arguments on Newtonian method. Reid's position is the more puzzling of the two, since he nowhere makes clear how Newtonian method favours mind-body dualism over materialism, which is the central debate between them.     

Global Distributive Justice

The paper discusses the problem of global distributive justice. It proposes to distinguish between principles for the domestic and for the global or intersocietal distribution of wealth. It is argued that there may be a plurality of partly diverging domestic conceptions of distributive justice, not all of which need to be liberal egalitarian conceptions. It is maintained, however, that principles regulating the intersocietal distribution of wealth have to be egalitarian principles. This claim is defended against Rawls's argument in The Law of Peoples that egalitarian principles of distributive justice should not be applied globally. Moreover, it is explained in detail, why Rawls's "duty of assistance to burdened societies" cannot be an appropriate substitute for a global principle of distributive justice.     

Intertemporal Distributive Judgement

This paper considers the simple two-person two-period case of distributive judgement, and argues (a) that sensible intertemporal distributive principle should consider both the distribution of people's life time well-being and the distribution of people's well-being at each period and (b) that, if (a) is correct, Egalitarianism is more acceptable than Prioritarianism since the latter must choose either one.