Assume that all algebras are atomless. (1) $Spind(A x B) = Spind(A) \cup Spind(B)$ . (2) $(\prod_{i\inI}^{w} = {\omega} \cup \bigcup_{i\inI}$ $Spind(A_{i})$ . Now suppose that $\kappa$ and $\lambda$ are infinite cardinals, with $kappa$ uncountable and regular and with $\kappa \textless \lambda$ . (3) There is an atomless Boolean algebra A such that $\mathfrak{u}(A) = \kappa$ and $i(A) = \lambda$ . (4) If $\lambda$ is also regular, then there is an atomless Boolean algebra A such that $t(A) = \mathfrak{s}(A) = (...) \kappa$ and $\mathfrak{a}(A) = \lambda$ . All results are in ZFC, and answer some problems posed in Monk [01] and Monk [ $\infty$ ]. (shrink)

The least element 0 of a finite meet semi-distributive lattice is a meet of meet-prime elements. We investigate conditions under which the least element of an algebraic, meet semi-distributive lattice is a (complete) meet of meet-prime elements. For example, this is true if the lattice has only countably many compact elements, or if |L| < 2ℵ0, or if L is in the variety generated by a finite meet semi-distributive lattice. We give an example of an algebraic, meet semi-distributive lattice that (...) has no meet-prime element or join-prime element. This lattice L has |L| = |LC| = 2ℵ0 where Lc is the set of compact elements of L. (shrink)

We exhibit a construction which produces for every Turing machine T with two halting states μ 0 and μ -1 , an algebra B(T) (finite and of finite type) with the property that the variety generated by B(T) is residually large if T halts in state μ -1 , while if T halts in state μ 0 then this variety is residually bounded by a finite cardinal.

This book presents the foundations of a general theory of algebras. Often called “universal algebra”, this theory provides a common framework for all algebraic systems, including groups, rings, modules, fields, and lattices. Each chapter is replete with useful illustrations and exercises that solidify the reader's understanding. The book begins by developing the main concepts and working tools of algebras and lattices, and continues with examples of classical algebraic systems like groups, semigroups, monoids, and categories. The essence of the book lies (...) in Chapter 4, which provides not only basic concepts and results of general algebra, but also the perspectives and intuitions shared by practitioners of the field. The book finishes with a study of possible uniqueness of factorizations of an algebra into a direct product of directly indecomposable algebras. There is enough material in this text for a two semester course sequence, but a one semester course could also focus primarily on Chapter 4, with additional topics selected from throughout the text. (shrink)

Bradd Hart, Matthew Valeriote, A Structure Theorem for Strongly Abelian Varieties with Few Models.Bradd Hart, Sergei Starchenko, Addendum to "A Structure Theorem for Strongly Abelian Varieties.".Bradd Hart, Sergei Starchenko, Matthew Valeriote, Vaught's Conjecture for Varieties.B. Hart, S. Starchenko, Superstable Quasi-Varieties.B. Hart, A. Pillay, S. Starchenko, Triviality, NDOP and Stable Varieties.

We determine precisely those locally finite varieties of unary algebras of finite type which, when augmented by a ternary discriminator, generate a variety with a decidable theory.

Let be a finite collection of finite algebras of finite signature such that SP( ) has meet semi-distributive congruence lattices. We prove that there exists a finite collection 1 of finite algebras of the same signature, , such that SP( 1) is finitely axiomatizable.We show also that if , then SP( 1) is finitely axiomatizable. We offer new proofs of two important finite basis theorems of D. Pigozzi and R. Willard. Our actual results are somewhat more general than this abstract (...) indicates. (shrink)

Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\mathcal{K}$$ \end{document} be a finite collection of finite algebras of finite signature such that SP(\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\mathcal{K}$$ \end{document}) has meet semi-distributive congruence lattices. We prove that there exists a finite collection \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\mathcal{K}$$ \end{document}1 of finite algebras of the same signature, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\mathcal{K}_1 \supseteq (...) \mathcal{K}$$ \end{document}, such that SP(\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\mathcal{K}$$ \end{document}1) is finitely axiomatizable.We show also that if \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$HS(\mathcal{K}) \subseteq SP(\mathcal{K})$$ \end{document}, then SP(\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\mathcal{K}$$ \end{document}1) is finitely axiomatizable. We offer new proofs of two important finite basis theorems of D. Pigozzi and R. Willard. Our actual results are somewhat more general than this abstract indicates. (shrink)

Alfred Tarski (1901--1983) is widely regarded as one of the two giants of twentieth-century logic and also as one of the four greatest logicians of all time (Aristotle, Frege and Gödel being the other three). Of the four, Tarski was the most prolific as a logician. The four volumes of his collected papers, which exclude most of his 19 monographs, span over 2500 pages. Aristotle's writings are comparable in volume, but most of the Aristotelian corpus is not about logic, whereas (...) virtually everything written by Tarski concerns logic more or less directly. There is no doubt that Tarski wrote more on logic than any other author; he started publishing on logic in 1921 at the age of 20 and continued until his death at the age of 82. Two of his works appeared posthumously [Hist. Philos. Logic 7 (1986), no. 2, 143--154; MR0868748 (88b:03010); Tarski and Givant, A formalization of set theory without variables, Amer. Math. Soc., Providence, RI, 1987; MR0920815 (89g:03012)]. Tarski's voluminous writings were widely scattered in numerous journals, some quite rare. It has been extremely difficult to study the development of Tarski's thought and to trace the interconnections and interdependence of his various papers. Thanks to the present collection all this has changed, and it is likely that the increased accessibility of Tarski's papers will have the effect of increasing Tarski's already enormous influence. (shrink)

Let us take an example that Bernard Williams (1981: 102) made famous. Suppose that you want a gin and tonic, and you believe that the stuff in front of you is gin. In fact, however, the stuff is not gin but petrol. So if you drink the stuff (even mixed with tonic), it will be decidedly unpleasant, to say the least. Should you choose to drink the stuff or not?

Here is a definition of knowledge: for you to know a proposition p is for you to have an outright belief in p that is correct precisely because it manifests the virtue of rationality. This definition resembles Ernest Sosa’s “virtue theory”, except that on this definition, the only virtue that must be manifested in all instances of knowledge is rationality, and no reductive account of rationality is attempted—rationality is assumed to be an irreducibly normative notion. This definition is compatible with (...) “internalism” about rationality, and with a form of “pragmatic encroachment” on the conditions of rational outright belief. An interpretation is given of this definition, and especially of the sense of ’because’ that it involves. On this interpretation, this definition entails that both safety and adherence are necessary conditions on knowledge; it supports a kind of contextualism about terms like ‘knowledge’; and it provides resources to defend safety, adherence, and contextualism, against some recent objections. (shrink)

In this paper, I apply the "conceptual role semantics" approach that I have proposed elsewhere (according to which the meaning of normative terms is given by their role in practical reasoning or deliberation) to the meaning of the term 'ought'. I argue that this approach can do three things: It can give an adequate explanation of the special connection that normative judgments have to practical reasoning and motivation for action. It can give an adequate account of why the central principles (...) of deontic logic are correct. It can give an explanation of the precise ways in which the term 'ought' is systematically context-sensitive, so that the term expresses different (but systematically related) concepts in different contexts. (shrink)

Often, when one faces a choice between alternative actions, there are reasons both for and against each alternative. On one way of understanding these words, what one “ought to do all things considered (ATC)” is determined by the totality of these reasons. So, these reasons can somehow be “combined” or “aggregated” to yield an ATC verdict on these alternatives. First, various assumptions about this sort of aggregation of reasons are articulated. Then it is shown that these assumptions allow for the (...) proof of a “Reasons Aggregation Theorem” – parallel to John Harsanyi’s 1955 “Social Aggregation Theorem”. All reasons for action are grounded in reason-providing values; and, for every such reason-providing value, there is in principle a way of measuring how strongly this value counts against each alternative. The theorem tells us that the appropriate measure of how much reason ATC there is against each alternative is simply a weighted sum of the strengths with which these values count against that alternative; the agent ought not ATC to perform an action iff there is ATC more reason against it than against some available alternative. (shrink)

This essay offers an account of the truth conditions of sentences involving deontic modals like ‘ought’, designed to capture the difference between objective and subjective kinds of ‘ought’ This account resembles the classical semantics for deontic logic: according to this account, these truths conditions involve a function from the world of evaluation to a domain of worlds (equivalent to a so-called “modal base”), and an ordering of the worlds in such domains; this ordering of the worlds itself arises from two (...) further elements – a probability function and a value function – since this ordering ranks the worlds in accordance with the expected value of certain propositions that are true at those worlds. Thus, a proposition of the form ‘Ought (p)’ is true at a world of evaluation w if and only if p is true at all the top-ranked worlds in the domain assigned to w. This domain of worlds consists of metaphysically possible worlds, while the probability function is defined over a space of epistemically possible worlds (which may include metaphysically impossible worlds, such as worlds where Hesperus is not Phosphorus). Evidence is given that this account assigns the correct truth conditions to a wide range of sentences involving ‘ought’. Since these truth conditions involve both a domain of metaphysically possible worlds and a space of epistemically possible worlds, there are two corresponding kinds of conditional involving ‘ought’, depending on which space of worlds is restricted by the conditional. Finally, some objections that might be raised against this account are answered. (shrink)

Intuitively, it seems that some belief-forming practices have the following three properties: 1. They are rational practices, and the beliefs that we form by means of these practices are themselves rational or justified beliefs. 2. Even if in most cases these practices reliably lead to correct beliefs (i.e., beliefs in true propositions), they are not infallible: it is possible for beliefs that are formed by means of these practices to be incorrect (i.e., to be beliefs in false propositions). 3. The (...) rationality of these practices is basic or primitive. That is, the rationality of these practices is not due simply to the availability, by means of some process of reasoning that relies purely on other practices, of a rational or justified belief in the reliability of these practices. -/- How can there be such practices? This paper offers an answer to that question. (shrink)

Richard Price’s moral epistemology provides a distinctive account, not only of the sources of our moral knowledge, but also of its limits – that is, of the moral truths that we do not and even cannot know. According to this moral epistemology, the fundamental moral truths are necessary rather than contingent; if they are knowable at all, they are knowable a priori. In general, fundamental moral truths are akin to mathematical truths. Specifically, these necessary moral truths are grounded in the (...) essences of act-types, which are knowable through ‘intuition’ if knowable at all. However, Price firmly rejects utilitarianism, recognizing several different ‘branches of virtue’, which can conflict with each other. As he argues, we cannot know the truth about act-types that exemplify such conflicting branches of virtue if the competing considerations are too finely balanced. This view of moral epistemology is compared with those of several other philosophers, including W. D. Ross. It is argued that Price has a more unified view than Ross: for Price, all moral knowledge flows from a priori intuition, but this is compatible with our being ignorant of many of the fine details of the fundamental truths of morality. (shrink)

This chapter is concerned with the distinction that most contemporary epistemologists express by distinguishing between “propositional” and “doxastic” justification. The goal is to develop an account of this distinction that applies, not just to full or outright beliefs, but also to partial credences—and indeed, in principle, to attitudes of all kinds. The standard way of explaining this distinction, in terms of the “basing relation”, is criticized, and an alternative account—the “virtue manifestation” account—is proposed in its place. This account has a (...) certain kinship with the epistemological approach that is known as “reliabilism”, but it also has certain key differences—above all, it is compatible with an internalist view of rationality. (shrink)

Drawing on early work on ethical leadership, we argue that when leaders engage in leader moral hypocrisy (i.e., ethical promotion without ethical demonstration), followers can experience psychological reactance—a negative response to a perceived restriction of freedom—which can have negative downstream consequences. In a survey of employee–manager dyads (study 1), we demonstrate that leader moral hypocrisy is positively associated with follower psychological reactance, which increases follower deviance. In two subsequent laboratory experiments, we find similar patterns of results (study 2) and explore (...) potential alternative mechanisms (study 3). We demonstrate in a final experiment with working adults that the relationship between leader moral hypocrisy and psychological reactance is partly explained by increased perceptions of a leader’s use of power (study 4). We discuss the implications of our findings and advocate for further understanding of the risks associated with psychological reactance in response to leaders and other workplace situations. (shrink)

v. 1. Inheritance and vocation.--v. 2. Philosophy and psychology.

What reasons for action do we have? What explains why we have these reasons? This paper articulates some of the basic structural features of a theory that would provide answers to these questions. According to this theory, reasons for action are all grounded in intrinsic values, but in a way that makes room for a thoroughly non-consequentialist view of the way in which intrinsic values generate reasons for aaction.

Many philosophers have claimed that the intentional is normative. (This claim is the analogue, within the philosophy of mind, of the claim that is often made within the philosophy of language, that meaning is normative.) But what exactly does this claim mean? And what reason is there for believing it? In this paper, I shall first try to clarify the content of the claim that the intentional is normative. Then I shall examine a number of the arguments that philosophers have (...) advanced for this claim (and for the parallel claim that meaning is normative). As we shall see, many of these arguments are unsuccessful. However, I shall close by giving a sketch of what may be a successful argument for this claim. (shrink)

Unfortunately, business is often associated with unethical behavior. While research has offered a number of explanations for why business might encourage unethical behavior, we argue that how a person frames a situation may provide important insight. Drawing on the decision frame literature, the goal of the current research is to identify the differences in cognitive processing associated with two decision frames dominant in the business ethics literature—business and ethical—and, with that knowledge, examine ways to mitigate the detrimental influence of frame (...) on unethical behavior. We first demonstrate the causal link between frame and misrepresentation, and then identify several differences in cognitive processing—cost–benefit analyses, concern for others, and construal level—that distinguish business and ethical frames, and investigate their effects on misrepresentation. In our final set of studies, we demonstrate that the influence of these frames on misrepresentation can be altered by manipulating these cognitive processes, both mitigating and exacerbating a decision maker’s engagement in misrepresentation. We conclude by summarizing our findings and their potential impact on unethical behavior more broadly. (shrink)

Hierocles, a Stoic of the second century CE, famously deployed an image of the ‘concentric circles’ that surround each of us. The image should not be read as advocating absolute impartiality (in the style of classical utilitarianism) or as illustrating the Stoic theory of oikeiōsis. Instead, it is designed to illustrate how it is ‘appropriate to act’ in certain cases. Like other Stoics, Hierocles bases his investigation of appropriate acts on what is ‘in accordance with nature’. According to his view, (...) each of us has a duty of mutual aid towards every other human being—but a weightier duty towards those who are socially closer than those who are more distant. The fundamental principle is that each of us has the duty of being an ‘ally’ of every other human being in serving the natural purposes of all the circles to which we and that other human being both belong. (shrink)

A variant of the process dissociation procedure was coupled with a manipulation of response signal lag to assess whether manipulations of context affect one or both of the familiarity and search processes described by the dual process model of recognition. Participants studied a list of word pairs followed by a recognition test with target words presented in the same or different context, and in the same or different form as study . Participants were asked to recognize any target word regardless (...) of changes to form , or to only recognise words that were presented in the same form . The standard context reinstatement effect was evident even at the short response lags. Analyses of the estimates of the contributions of familiarity and search processes suggest that the context effect demonstrated here can be attributed in part to the influence of familiarity on recognition, whereas the effect on recollection was less clear. (shrink)