The variety \(\mathbb{DHMSH}\) of dually hemimorphic semi-Heyting algebras was introduced in 2011 by the second author as an expansion of semi-Heyting algebras by a dual hemimorphism. In this paper, we focus on the variety \(\mathbb{DHMSH}\) from a logical point of view. The paper presents an extensive investigation of the logic corresponding to the variety of dually hemimorphic semi-Heyting algebras and of its axiomatic extensions, along with an equally extensive universal algebraic study of their corresponding algebraic semantics. Firstly, we present a (...) Hilbert-style axiomatization of a new logic called "Dually hemimorphic semi-Heyting logic" (\(\mathcal{DHMSH}\), for short), as an expansion of semi-intuitionistic logic \(\mathcal{SI}\) (also called \(\mathcal{SH}\)) introduced by the first author by adding a weak negation (to be interpreted as a dual hemimorphism). We then prove that it is implicative in the sense of Rasiowa and that it is complete with respect to the variety \(\mathbb{DHMSH}\). It is deduced that the logic \(\mathcal{DHMSH}\) is algebraizable in the sense of Blok and Pigozzi, with the variety \(\mathbb{DHMSH}\) as its equivalent algebraic semantics and that the lattice of axiomatic extensions of \(\mathcal{DHMSH}\) is dually isomorphic to the lattice of subvarieties of \(\mathbb{DHMSH}\). A new axiomatization for Moisil's logic is also obtained. Secondly, we characterize the axiomatic extensions of \(\mathcal{DHMSH}\) in which the "Deduction Theorem" holds. Thirdly, we present several new logics, extending the logic \(\mathcal{DHMSH}\), corresponding to several important subvarieties of the variety \(\mathbb{DHMSH}\). These include logics corresponding to the varieties generated by two-element, three-element and some four-element dually quasi-De Morgan semi-Heyting algebras, as well as a new axiomatization for the 3-valued Łukasiewicz logic. Surprisingly, many of these logics turn out to be connexive logics, only a few of which are presented in this paper. Fourthly, we present axiomatizations for two infinite sequences of logics namely, De Morgan Gödel logics and dually pseudocomplemented Gödel logics. Fifthly, axiomatizations are also provided for logics corresponding to many subvarieties of regular dually quasi-De Morgan Stone semi-Heyting algebras, of regular De Morgan semi-Heyting algebras of level 1, and of JI-distributive semi-Heyting algebras of level 1. We conclude the paper with some open problems. Most of the logics considered in this paper are discriminator logics in the sense that they correspond to discriminator varieties. Some of them, just like the classical logic, are even primal in the sense that their corresponding varieties are generated by primal algebras. (shrink)

An algebra A = ⟨A, ∨, ∧, →, 0, 1⟩ is a semi-Heyting algebra if ⟨A, ∨, ∧, 0, 1⟩ is a bounded lattice, and it satisfies the identities: x ∧ ≈ x ∧ y, x ∧ ≈ x ∧ [ → ], and x → x ≈ 1.

The purpose of this paper is to define and investigate the new class of quasi-Stone algebras . Among other things we characterize the class of simple QSA's and the class of subdirectly irreducible QSA's. It follows from this characterization that the subdirectly irreducible QSA's form an elementary class and that the variety of QSA's is locally finite. Furthermore we prove that the lattice of subvarieties of QSA's is an -chain. MSC: 03G25, 06D16, 06E15.

The variety \ of implication zroupoids and a constant 0) was defined and investigated by Sankappanavar :21–50, 2012), as a generalization of De Morgan algebras. Also, in Sankappanavar :21–50, 2012), several subvarieties of \ were introduced, including the subvariety \, defined by the identity: \, which plays a crucial role in this paper. Some more new subvarieties of \ are studied in Cornejo and Sankappanavar that includes the subvariety \ of semilattices with a least element 0. An (...) explicit description of semisimple subvarieties of \ is given in Cornejo and Sankappanavar. It is a well known fact that there is a partial order ) induced by the operation ∧, both in the variety \ of semilattices with a least element and in the variety \ of De Morgan algebras. As both \ and \ are subvarieties of \ and the definition of partial order can be expressed in terms of the implication and the constant, it is but natural to ask whether the relation \ on \ is actually a partial order in some subvariety of \ that includes both \ and \. The purpose of the present paper is two-fold: Firstly, a complete answer is given to the above mentioned problem. Indeed, our first main theorem shows that the variety \ is a maximal subvariety of \ with respect to the property that the relation \ is a partial order on its members. In view of this result, one is then naturally led to consider the problem of determining the number of non-isomorphic algebras in \ that can be defined on an n-element chain -chains), n being a natural number. Secondly, we answer this problem in our second main theorem which says that, for each \, there are exactly n nonisomorphic \-chains of size n. (shrink)

We determine the number of non‐isomorphic semi‐Heyting algebras on an n‐element chain, where n is a positive integer, using a recursive method. We then prove that the numbers obtained agree with those determined in [1]. We apply the formula to calculate the number of non‐isomorphic semi‐Heyting chains of a given size in some important subvarieties of the variety of semi‐Heyting algebras that were introduced in [5]. We further exploit this recursive method to calculate the numbers of non‐isomorphic semi‐Heyting chains with (...) n elements such that removing the mth element () we are left with a subalgebra. We also solve a related problem posed in [1] of determining the number of ways a semi‐Heyting chain with elements can be extended to a n element semi‐Heyting chain by adding a new element in the mth place. Finally we combine these results by finding a second way to calculate the numbers that provides some extra information. (shrink)

This paper is a contribution toward developing a theory of expansions of semi-Heyting algebras. It grew out of an attempt to settle a conjecture we had made in 1987. Firstly, we unify and extend strikingly similar results of [ 48 ] and [ 50 ] to the (new) equational class DHMSH of dually hemimorphic semi-Heyting algebras, or to its subvariety BDQDSH of blended dual quasi-De Morgan semi-Heyting algebras, thus settling the conjecture. Secondly, we give a criterion for a unary expansion (...) of semi-Heyting algebras to be a discriminator variety and give an algorithm to produce discriminator varieties. We then apply the criterion to exhibit an increasing sequence of discriminator subvarieties of BDQDSH . We also use it to prove that the variety DQSSH of dually quasi-Stone semi- Heyting algebras is a discriminator variety. Thirdly, we investigate a binary expansion of semi-Heyting algebras, namely the variety DblSH of double semi-Heyting algebras by characterizing its simples, and use the characterization to present an increasing sequence of discriminator subvarieties of DblSH . Finally, we apply these results to give bases for “small” subvarieties of BDQDSH , DQSSH , and DblSH. (shrink)

This paper augments Hailperin's substantial efforts to place Boole's algebra of logic on a solid footing. Namely Horn sentences are used to give a modern formulation of the principle that Boole adopted in 1854 as the foundation for his algebra of logic—we call this principle The Rule of 0 and 1.

In ”Skepticism,” Peter Klein distinguishes between the “Academic Skeptic” who proposes that we cannot have knowledge of a certain set of propositions and the “Pyrrhonian Skeptic” who refrains from opining about whether we can have knowledge. Klein argues that Academic Skepticism is plausibly supported by a “Closure Principle‐style” argument based on the claim that if x entails y and S has justification for x, then S has justification for y. He turns to contextualism to see if it can contribute to (...) the discussion between one who claims that we can have knowledge about some epistemically interesting class of propositions and the Academic Skeptic. He outlines the background of Pyrrhonian Skepticism, pointing out that the Pyrrhonist withholds assent concerning our knowledge‐bearing status because reason cannot provide an adequate basis for assent. He assesses three possible patterns of reasoning, and concludes that the Pyrrhonist view, that reason cannot resolve matters concerning the nonevident, is vindicated. (shrink)

Bioethics needs an expanded moral vision. It is now time for bioethics to take full account of the problems of health disparities and structural injustice that are made newly urgent by the COVID-19 pandemic and the effects of climate change. Nancy M. P. King, Gail E. Henderson, and Larry R. Churchill make the case for a more social understanding and application of justice, a deeper humility in assessing expertise in bioethics consulting, a broader and more relevant research agenda, and greater (...) appreciation of the profound health implications of global warming. (shrink)

First published in 1982, this reissue contains a critical exposition of the views of Frege, Dedekind and Peano on the foundations of arithmetic. The last quarter of the 19th century witnessed a remarkable growth of interest in the foundations of arithmetic. This work analyses both the reasons for this growth of interest within both mathematics and philosophy and the ways in which this study of the foundations of arithmetic led to new insights in philosophy and striking advances in logic. This (...) historical-critical study provides an excellent introduction to the problems of the philosophy of mathematics - problems which have wide implications for philosophy as a whole. This reissue will appeal to students of both mathematics and philosophy who wish to improve their knowledge of logic. (shrink)

This book views Green's philosophical opus through his public life and political commitments. It demonstrates how his main ethical and political conceptions -- his idea of 'self realisation' and his theory of individuality within community -- were informed by evangelical theology, popular Protestantism and an idea of the English national consciousness as formed by religious conflict. While the significance of Kant and Hegel is acknowledged, it is argued that 'indigenous' qualities of Green's teachings resonated with Victorian Liberal values.

First published in 1987. Routledge is an imprint of Taylor & Francis, an informa company.

As a preliminary step to beginning to assess the usefulness of clinical vignettes to measure ethical sensitivity in undergraduate medical students, five clinical vignettes with seven to nine ethical issues each were created. The ethical issues in the vignettes were discussed and outlined by an expert panel. One randomly selected vignette was presented to first, second and third year students at the University of Toronto as part of another examination. The students were asked to list the issues presented by the (...) patient problem. Responses from 281 students were obtained. These students identified an average of 2.72 ethical issues per vignette. Each response was classified under the domains of autonomy, beneficence and justice. Comparisons were made between classes and between vignettes. There was considerable variation between classes and the responses to different vignettes seem to indicate that different vignettes measure the various domains in different ways. It does appear that the use of vignettes is one way to measure aspects of ethical sensitivity in medical students but more study is required to clarify exactly what is being measured. (shrink)

An instrument to assess 'ethical sensitivity' has been developed. The instrument presents four clinical vignettes and the respondent is asked to list the ethical issues related to each vignette. The responses are classified, post hoc, into the domains of autonomy, beneficence and justice. This instrument was used in 1990 to assess the ethical sensitivity of students in all four medical classes at the University of Toronto. Ethical sensitivity, as measured by this instrument, is not related to age or grade-point average. (...) Sensitivity increases between the 1st and 2nd year and then decreases throughout the rest of undergraduate medical training, such that the 4th-year students identify fewer issues than those entering medical school. Students expressing a career choice of family medicine identify more issues than their peers. Several problems with the use of the instrument and the interpretation of the data were found. Nonetheless, these findings, if reproducible, are important and their meaning needs further discussion. (shrink)

Engineering is 'the people-serving profession'. The work of engineers involves interaction with clients, other engineers, and the public at large. More than any other profession, their work also directly involves and affects the environment. This book makes the case that engineers have special professional obligations to protect and enhance the environment, and the authors - one, an engineer and the other, a philosopher - seek to provide an ethical basis for these obligations. In exploring these ethical issues, the authors aim (...) to show that engineers make a difference. The text opens with a series of case studies in which engineers face complex and challenging decisions about the environment. Succeeding chapters examine different ideas about environmental ethics for engineers, including professional codes and both modern and historical discussions of environmental responsibility. The book concludes with a collection of readings that complement the text. Students, as well as practising engineers, will find much of interest in this well-argued and thought-provoking book. (shrink)

This is the first book to take a comprehensive look at white collar criminal offenses from the perspective of moral and legal theory. Focussing on the way in which key white collar crimes such as fraud, perjury, false statements, obstruction of justice, bribery, extortion, blackmail, insider trading, tax evasion, and regulatory and intellectual property offenses are shaped and informed by a range of familiar, but nevertheless powerful, moral norms.

First published in 1987. Routledge is an imprint of Taylor & Francis, an informa company.

Gordon Baker and I had been colleagues at St John’s for almost ten years when we resolved, in 1976, to undertake the task of writing a commentary on Wittgenstein’s Philosophical Investigations. We had been talking about Wittgenstein since 1969, and when we cooperated in writing a long critical notice on the Philosophical Grammar in 1975, we found that working together was mutually instructive, intellectually stimulating and great fun. We thought that we still had much to say about Wittgenstein’s philosophy, and (...) it seemed to us that misinterpretations of passages in the Investigations were so extensive that it would be worth trying to write a detailed analytical commentary. It is difficult to recapture the excitement of those early days in being among the first to work on the microfilms and, subsequently, on the photocopies of Wittgenstein’s Nachlass. We spent many hundreds of hours poring over the typescripts and the often only semi-legible manuscripts, fascinated and privileged to be able to try to follow the development of the thoughts of a great philosophical genius. We talked endlessly about what we had found in Wittgenstein’s manuscripts and typescripts, and debated how it should be understood. The first fruit of our labours was Wittgenstein – Understanding and Meaning. Its guiding idea was to draw attention to the manner in which Wittgenstein linked the concepts of meaning, understanding and explanation, and so to bypass the connections between meaning, truth and truth-conditions that so fascinated philosophers of the 1970s, and to abandon the red-herring of assertion-conditions and anti-realism. After a hiatus of four years, during which time we wrote a controversial book entitled Frege – Logical Excavations and a polemical book on contemporary philosophy of language – Language, Sense. (shrink)

Differential scanning calorimetry (DSC) has been employed to investigate the glass transition activation energy E g, thermal stability and glass-forming ability (GFA) of Se90In10− x Sb x (x = 0, 2, 4, 6, 8, 10) chalcogenide glasses. DSC runs were performed at six different heating rates. Well-defined endothermic and exothermic peaks were obtained at glass transition and crystallization temperature. The dependence of glass transition temperature T g on heating rate (α), as well as composition of Sb, has been studied. From (...) the dependence of glass transition temperature on heating rate, the E g has been calculated on the basis of the Kissinger [Anal. Chem. 29 (1957) p.1702] and Moynihan [J. Phys. Chem. 78 (1974) p.267] models. Thermal stability has been monitored through the calculation of temperature differences T c–T g, the stability parameter S, and the enthalpy released during crystallization H c. The GFA has been investigated on the basis of the Hruby parameter H r, which is strong indicator of GFA. Results for GFA are in good agreement with fragility index F i calculations, indicating that Se90In6Sb4 is an excellent glass-former. (shrink)