This new edition of Introduction to Lattices and Order presents a radical reorganization and updating, though its primary aim is unchanged. The explosive development of theoretical computer science in recent years has, in particular, influenced the book's evolution: a fresh treatment of fixpoints testifies to this and Galois connections now feature prominently. An early presentation of concept analysis gives both a concrete foundation for the subsequent theory of complete lattices and a glimpse of a methodology for data analysis that is (...) of commercial value in social science. Classroom experience has led to numerous pedagogical improvements and many new exercises have been added. As before, exposure to elementary abstract algebra and the notation of set theory are the only prerequisites, making the book suitable for advanced undergraduates and beginning graduate students. It will also be a valuable resource for anyone who meets ordered structures. (shrink)

_This tribute to Nathan Sivin (1931-2022), a pathbreaking scholar of Chinese science, medicine, and philosophy, highlights his distinctive contributions to historical understanding and his memorable wit. I offer a brief biographical sketch followed by an analysis of the main themes in Sivin’s voluminous publications and a personal note of appreciation. It is no exaggeration to say that Sivin transformed his field of study. His contrarian spirit, relentless curiosity, and humanistic insight helped make the history of East Asian science, technology, and (...) medicine the vibrant area of study that it is today._. (shrink)

Recipes from the Garden of Contentment. By Yuan Mei, translated and annotated by Sean J. S. Chen. Great Barrington, MA: Berkshire, 2019. Pp. xxxiii + 428. $125.

Mind and Body in Early China: Beyond Orientalism and the Myth of Holism. By Edward Slingerland. New York: Oxford University Press, 2019. Pp. xiv + 385. $35.

The techniques of natural duality theory are applied to certain finitely generated varieties of Heyting algebras to obtain optimal dualities for these varieties, and thereby to address algebraic questions about them. In particular, a complete characterisation is given of the endodualisable finite subdirectly irreducible Heyting algebras. The procedures involved rely heavily on Priestley duality for Heyting algebras.

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)

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)

Effective documentation is considered a core competency for clinical ethics consultation. Ethics consultants within the Cleveland Clinic in Cleveland, Ohio, observed variation in the formatting of ethics chart notes across consultants and realized that this formatting was based on their own views of effectiveness. To minimize variation and optimize the readability and understandability of ethics chart notes for end users, a team undertook a quality improvement project to assess the formatting preferences of healthcare professionals who rely on ethics consultation notes. (...) The team developed three sample templates and conducted interviews with stakeholders to understand their preferences. A single standardized template was developed based on the preferences that emerged, which all consultants on the ethics consultation service then utilized. In the first five months of implementation, the percentage of end user respondents marking the highest Likert scale option on a post-consultation survey regarding whether the ethics consultation service provided helpful documentation increased from 60 percent to 72 percent compared to the same five-month period in the year prior. (shrink)

This paper focuses on natural dualities for varieties of bilattice-based algebras. Such varieties have been widely studied as semantic models in situations where information is incomplete or inconsistent. The most popular tool for studying bilattices-based algebras is product representation. The authors recently set up a widely applicable algebraic framework which enabled product representations over a base variety to be derived in a uniform and categorical manner. By combining this methodology with that of natural duality theory, we demonstrate how to build (...) a natural duality for any bilattice-based variety which has a suitable product representation over a dualisable base variety. This procedure allows us systematically to present economical natural dualities for many bilattice-based varieties, for most of which no dual representation has previously been given. Among our results we highlight that for bilattices with a generalised conflation operation. Here both the associated product representation and the duality are new. Finally we outline analogous procedures for pre-bilattice-based algebras. (shrink)

This paper is a study of duality in the absence of canonicity. Specifically it concerns double quasioperator algebras, a class of distributive lattice expansions in which, coordinatewise, each operation either preserves both join and meet or reverses them. A variety of DQAs need not be canonical, but as has been shown in a companion paper, it is canonical in a generalized sense and an algebraic correspondence theorem is available. For very many varieties, canonicity (as traditionally defined) and correspondence lead on (...) to topological dualities in which the topological and correspondence components are quite separate. It is shown that, for DQAs, generalized canonicity is sufficient to yield, in a uniform way, topological dualities in the same style as those for canonical varieties. However topology and correspondence are no longer separable in the same way. (shrink)

First published in 1978, this reissue presents a seminal philosophical work by professor Putnam, in which he puts forward a conception of knowledge which makes ethics, practical knowledge and non-mathematic parts of the social sciences just as much parts of 'knowledge' as the sciences themselves. He also rejects the idea that knowledge can be demarcated from non-knowledge by the fact that the former alone adheres to 'the scientific method'. The first part of the book consists of Professor Putnam's John Locke (...) lectures, delivered at the University of Oxford in 1976, offering a detailed examination of a 'physicalist' theory of reference against a background of the works of Tarski, Carnap, Popper, Hempel and Kant. The analysis then extends to notions of truth, the character of linguistic enquiry and social scientific enquiry in general, interconnecting with the great metaphysical problem of realism, the nature of language and reference, and the character of ourselves. (shrink)

Both syntactic and semantic solutions are given for the entailment problem of duality theory. The test algebra theorem provides both a syntactic solution to the entailment problem in terms of primitive positive formulae and a new derivation of the corresponding result in clone theory, viz. the syntactic description of $\operatorname{Inv(Pol}(R))$ for a given set R of finitary relations on a finite set. The semantic solution to the entailment problem follows from the syntactic one, or can be given in the form (...) of an algorithm. It shows, in the special case of a purely relational type, that duality-theoretic entailment is describable in terms of five constructs, namely trivial relations, intersection, repetition removal, product, and retractive projection. All except the last are concrete, in the sense that they are described by a quantifier-free formula. It is proved that if the finite algebra $\underline{M}$ generates a congruence-distributive variety and all subalgebras of $\underline{M}$ are subdirectly irreducible, then concrete constructs suffice to describe entailment. The concept of entailment appropriate to strong dualities is also introduced, and described in terms of coordinate projections, restriction of domains, and composition of partial functions. (shrink)

The twentieth century has witnessed an unprecedented 'crisis in the foundations of mathematics', featuring a world-famous paradox (Russell's Paradox), a challenge to 'classical' mathematics from a world-famous mathematician (the 'mathematical intuitionism' of Brouwer), a new foundational school (Hilbert's Formalism), and the profound incompleteness results of Kurt Gödel. In the same period, the cross-fertilization of mathematics and philosophy resulted in a new sort of 'mathematical philosophy', associated most notably (but in different ways) with Bertrand Russell, W. V. Quine, and Gödel himself, (...) and which remains at the focus of Anglo-Saxon philosophical discussion. The present collection brings together in a convenient form the seminal articles in the philosophy of mathematics by these and other major thinkers. It is a substantially revised version of the edition first published in 1964 and includes a revised bibliography. The volume will be welcomed as a major work of reference at this level in the field. (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)

Naturalism in philosophy has a long and distinguished heritage. This is no less true in epistemology than it is in other areas of philosophy. At the same time, epistemology in the English speaking world in the first half of die twentieth century was dominated by an approach quite hostile to naturalism. Now, at the close of the twentieth century, naturalism is resurgent.