" There are 31 chapters in 5 parts and approximately 320 exercises marked by difficulty and whether or not they are necessary for further work in the book.

For any Boolean Algebra A, let cmm be the smallest size of an infinite partition of unity in A. The relationship of this function to the 21 common functions described in Monk [4] is described, for the class of all Boolean algebras, and also for its most important subclasses. This description involves three main results: the existence of a rigid tree algebra in which cmm exceeds any preassigned number, a rigid interval algebra with that property, and the construction of (...) an interval algebra in which every well-ordered chain has size less than cmm. (shrink)

For cardinals we let be the smallest size of a subset B of unbounded in the sense of ; that is, such that there is no function such that has size less than for all . Similarly for , the general dominating number, which is the smallest size of a subset B of such that for every there is an such that the above set has size less than . These cardinals are generalizations of the usual ones for . When (...) all three are the same regular cardinal, the relationships between them have been completely described by Cummings and Shelah. We also consider some variants of the functions, following van Douwen, in particular the version of in which B is required to consist of strictly increasing functions. Some of the main results of this paper are: (1) ; (2) for , always exists; (3) if , then ; (4) . For background see Section 1 of the paper. Several open problems are stated. (shrink)

The main notion dealt with in this article is where A is a Boolean algebra. A partition of 1 is a family ofnonzero pairwise disjoint elements with sum 1. One of the main reasons for interest in this notion is from investigations about maximal almost disjoint families of subsets of sets X, especially X=ω. We begin the paper with a few results about this set-theoretical notion.Some of the main results of the paper are:• (1) If there is a maximal family (...) of size λ≥κ of pairwise almost disjoint subsets of κ each of size κ, then there is a maximal family of size λ of pairwise almost disjoint subsets of κ+ each of size κ.• (2) A characterization of the class of all cardinalities of partitions of 1 in a product in terms of such classes for the factors; and a similar characterization for weak products.• (3) A cardinal number characterization of sets of cardinals with a largest element which are for some BA the set of all cardinalities of partitions of 1 of that BA.• (4) A computation of the set of cardinalities of partitions of 1 in a free product of finite-cofinite algebras. (shrink)

We give some results concerning various generalized continuum cardinals. The results answer some natural questions which have arisen in preparing a new edition of 5. To make the paper self-contained we define all of the cardinal functions that enter into the theorems here. There are many problems concerning these new functions, and we formulate some of the more important ones.

We consider eight special kinds of subalgebras of Boolean algebras. In Section 1 we describe the relationships between these subalgebra notions. In succeeding sections we consider how the subalgebra notions behave with respect to the most common cardinal functions on Boolean algebras.

Recall that a subset X of a Boolean algebra A is independent if for any two finite disjoint subsets F , G of X we have ∏ x∈F x ∏ y∈G −y≠0. The independence of a BA A , denoted by Ind, is the supremum of cardinalities of its independent subsets. We can also consider the maximal independent subsets. The smallest size of an infinite maximal independent subset is the cardinal invariant i , well known in the case A= P (...) / fin . In this article we consider the collection of all cardinalities of infinite maximal independent subsets of a BA A ; we call this set the spectrum of infinite maximal independent subsets , denoted by Spind. Note that infinite maximal independent subsets exist in any BA which is not superatomic. The main result is that any set of infinite cardinals can occur as Spind for some infinite BA A . Beyond this we give results concerning the way that Spind changes under various algebraic operations. However, the basic components of most algebras that we deal with are free algebras. (shrink)

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)

What Is Political Theory? provides students with a comprehensive overview of the current state of the discipline. Ten substantive chapters address the most pressing topics in political theory today, including: - what resources do the classic texts still provide for political theorists? - what areas will political theorists focus on in the future? - can western political theory alone continue to provide a framework for responding to the challenges of modern political life? The authors assess the intellectual challenges to conventional (...) political theory, such as post-structuralism and the scientific study of politics that have revitalized the field in the last 30 years. They also broaden the perspective to take in non-western ideas and to reconceptualize political theory in the light of specifically global challenges. Students and teachers of political theory and political philosophy will find this book invaluable in understanding the factors that have shaped current political theory and which will guide its future development. (shrink)

In developing a new theory of political and moral community, J. Donald Moon takes questions of cultural pluralism and difference more seriously than do many other liberal thinkers of our era: Moon is willing to confront the problem of how ...

This critical examination of the rights of private property contrasts two types of arguments about rights: those based on historical entitlement, and those based on the importance of property to freedom. The text explores the concept of ownership, and the relation between property and equality.

The personhood of the physician is a crucial element in accomplishing the goals of medicine. We review claims made on behalf of the humanities in guiding professional identity formation. We explore the dichotomy that has evolved, since the Renaissance, between the humanities and the natural sciences. The result of this evolution is an historic misconstrual, preoccupying educators and diverting them from the moral development of physicians. We propose a curricular framework based on the recovery of Aristotelian concepts that bridge identity (...) and activity. The humanities and the natural sciences, jointly and severally, can fulfill developmental, characterological and instrumental purposes. (shrink)

Francis’ view of nature has been seen as positive in an ecological sense even by those who are for the most part critical of Christianity’s attitude to nature, such as Lynn White, Jr. I argue that one element of Francis’ uniqueness was that he saw the diversity of life as an expression of God’s creativity and benevolence and attempted to carry out that vision in ethical behavior. Much of what has been written about him has precedents in traditional hagiography, but (...) there remains an unmistakable impression of originality. It has been noted that Francis insisted on the goodness of creation, used terms of family relationship to refer to creatures other than human, and preached to them. However, another element has escaped notice: his emphasis on the presence of God in the diversity of created entities and his desire that humans should rejoice in this diversity and glorify God for it and with it. His devotion did not immediately dissolve multiplicity into oneness, but glorified God in each created being and delighted in their individuality. He advocated that praise be expressed by acting in ways consistent with respect for created diversity, not only by observing a strict rule of abstaining from harm to living beings, but also in positive treatment of all creatures. Nature took its meaning not from its serviceability to mankind, but from its expression of the multiple forms of God’s benevolent presence. (shrink)

This article investigates the characteristic attitudes of the Greeks toward nature, which formed the perceptual framework for their ecological thinking. Two major attitudes are discerned. One regarded nature as the theatre of the gods, whose interplay produced observed phenomena, but whose localization gave them particular, restricted roles. The other attitude viewed nature as the theatre of reason, and made the beginnings of ecological thought possible. The contributions of several Greek forerunners in the field of ecology are characterized. The most consistent, (...) balanced ecological writer in ancient Greece was Theophrastus, but his conception of an autonomous nature, interacting with man, was overshadowed in the history of ancient and medieval thought by the anthropocentric teleology of Aristotle. (shrink)

The image of the classical Mediterranean environment of the Greeks and Romans had a formative influence on the art, literature, and historical perception of modern Europe and America. How closely does is this image congruent with the ancient environment as it in reality existed? In particular, how forested was the ancient Mediterranean world, was there deforestation, and if so, what were its effects? The consensus of historians, geographers, and other scholars from the mid-nineteenth century through the first three quarters of (...) the twentieth century was that human activities had depleted the forests to a major extent and caused severe erosion. My research confirmed this general picture. Since then, revisionist historians have questioned these conclusions, maintaining instead that little environmental damage was done to forests and soils in ancient Greco-Roman times. In a reconsideration of the question, this paper looks at recent scientific work providing proxy evidence for the condition of forests at various times in ancient history. I look at three scientific methodologies, namely anthracology, palynology, and computer modeling. Each of these avenues of research offers support for the concept of forest change, both in abundance and species composition, and episodes of deforestation and erosion, and confirms my earlier work. (shrink)

Two conflicting tendencies may be discerned in Pythagorean ethics as applied to the environment: on the one hand, a sense of reverence for nature and kinship with all life that opposed killing and other forms of interference in the natural world, and on the other hand, a doctrine of the separability of soul and body which denigrates the body and the external world of which it is apart. The prescriptive content of Pythagorean ethics includes prohibitions against taking life, even in (...) sacrifices to the gods, and against eating anything that has been killed. Pollution of certain kinds is forbidden. These strictures were based on an organic, cyclical view of the world, emphasizing its harmony and balance. The Pythagoreans investigated some questions that would today be called ecological. Perhaps most importantly, they evinced a genuine respect for living things, deriving in part from the belief that animals and plants contain the reborn souls of human beings. These doctrines may have been derived from the attitudes and practices of ancestral hunters and gatherers in southeast Europe, with traditional Greek religion serving as the means of transmission from tribaI cultures to c1assical philosophy. The followers of Pythagoras split into two schools: a “scientific” school that neglected biology and therefore ecology, and a “religious” school that emphasized purity of soul and rejected any concern with physical nature. The more “environmentalist”teachings were gradually abandoned as the Pythagoreans accommodated themselves to the general attitudes of Greco-Roman culture. For instance, the objections to animal sacrifice, and to most plants as food, were dropped. The divorce of body and soul in later Pythagorean thought, wherever its influence was strong, brought with it indifference not only to the body, but to all the rest of the natural environment. (shrink)