This comprehensive text shows how various notions of logic can be viewed as notions of universal algebra providing more advanced concepts for those who have an introductory knowledge of algebraic logic, as well as those wishing to delve into more theoretical aspects.

This paper explores allowing truth value assignments to be undetermined or "partial" and overdetermined or "inconsistent", thus returning to an investigation of the four-valued semantics that I initiated in the sixties. I examine some natural consequence relations and show how they are related to existing logics, including ukasiewicz's three-valued logic, Kleene's three-valued logic, Anderson and Belnap's relevant entailments, Priest's "Logic of Paradox", and the first-degree fragment of the Dunn-McCall system "R-mingle". None of these systems have nested implications, and I investigate (...) twelve natural extensions containing nested implications, all of which can be viewed as coming from natural variations on Kripke's semantics for intuitionistic logic. Many of these logics exist antecedently in the literature, in particular Nelson 's "constructible falsity". (shrink)

We give a set of postulates for the minimal normal modal logicK + without negation or any kind of implication. The connectives are simply , , , . The postulates (and theorems) are all deducibility statements . The only postulates that might not be obvious are.

We study an application of gaggle theory to unary negative modal operators. First we treat negation as impossibility and get a minimal logic system Ki that has a perp semantics. Dunn 's kite of different negations can be dealt with in the extensions of this basic logic Ki. Next we treat negation as “unnecessity” and use a characteristic semantics for different negations in a kite which is dual to Dunn 's original one. Ku is the minimal logic that has a (...) characteristic semantics. We also show that Shramko's falsification logic FL can be incorporated into some extension of this basic logic Ku. Finally, we unite the two basic logics Ki and Ku together to get a negative modal logic K-, which is dual to the positive modal logic K+ in [7]. Shramko has suggested an extension of Dunn 's kite and also a dual version in [12]. He also suggested combining them into a “united” kite. We give a united semantics for this united kite of negations. (shrink)

Given classical (2 valued) structures and and a homomorphism h of onto , it is shown how to construct a (non-degenerate) 3-valued counterpart of . Classical sentences that are true in are non-false in . Applications to number theory and type theory (with axiom of infinity) produce finite 3-valued models in which all classically true sentences of these theories are non-false. Connections to relevant logic give absolute consistency proofs for versions of these theories formulated in relevant logic (the proof for (...) number theory was obtained earlier by R. K. Meyer and suggested the present abstract development). (shrink)

Results of a study investigating the relation between personality traits and ethical-orientation indicate sex is not an good predictor for differences in Machiavellian-, Type A personality- or ethical-orientation. Intelligence is found to be positively associated with Machiavellian- and Type A personality-orientation but negatively associated with ethical-orientation. Machiavellians tend to have Type A personalities, but tend to be less ethically-oriented than Nonmachiavellians. Type A personalities are more ethically-orientated than Type B personalities.

This paper explores the development of mathematics on a quantum logical base when mathematical postulates are taken as necessary truths. First it is shown that first-order Peano arithmetic formulated with quantum logic has the same theorems as classical first-order Peano arithmetic. Distribution for first-order arithmetical formulas is a theorem not of quantum logic but rather of arithmetic. Second, it is shown that distribution fails for second-order Peano arithmetic without extensionality. Third, it is shown that distribution holds for second-order Peano arithmetic (...) (second-order quantum logic) with extensionality. Some remarks about extensions to quantum set theory are made. (shrink)

Multiple concepts of nature are at play in environmental theory and practice. One that has gripped several theorists is the idea of nature as referring to that which is independent of humans and human activity. This concept has been subject to forceful criticism, notably in the recent work of Steven Vogel. After clarifying problematic and promising ways of charac­terizing independent nature, I engage Vogel’s critique. While the critique is compelling in certain respects, I argue that it fails to appreciate what (...) I take to be an important motivating concern of those drawn to the concept of independent nature, or something like it. I offer a characterization of that concern—a worry about problematic instrumentalization of the nonhuman world—and suggest why this concern, and the idea of independent nature which helps to make it intelligible, should continue to inform environmental theory and practice. In offering a qualified defense of the concept of independent nature, and of its value, I assume that such a concept is only one possible tool in a multi-pronged approach to environmental theorizing, deliberation, action, and policy. (shrink)

In this paper we introduce canonical extensions of partially ordered sets and monotone maps and a corresponding discrete duality. We then use these to give a uniform treatment of completeness of relational semantics for various substructural logics with implication as the residual(s) of fusion.

Arnon Avron has written: “Dunn-McCall logic RM is by far the best understood and the most well-behaved in the family of logics developed by the school of Anderson and Belnap.” I agree. There is the famous saying: “Do not let the perfect become the enemy of the good.” I might say: “good enough.” In this spirit, I will examine the logic R-Mingle, exploring how it is only a “semi-relevant logic” but still a paraconsistent logic. I shall discuss the history of (...) RM, and compare RM to Anderson and Belnap’s system R of relevant implication and to classical two-valued logic. There is a “consumer’s guide,” evaluating these logics as “tools,” in the light of my recent work on “Humans as Rational Toolmaking Animals.”. (shrink)

Sexual size dimorphism is generally associated with sexual selection via agonistic male competition in nonhuman primates. These primate models play an important role in understanding the origins and evolution of human behavior. Human size dimorphism is often hypothesized to be associated with high rates of male violence and polygyny. This raises the question of whether human dimorphism and patterns of male violence are inherited from a common ancestor with chimpanzees or are uniquely derived. Here I review patterns of, and causal (...) models for, dimorphism in humans and other primates. While dimorphism in primates is associated with agonistic male mate competition, a variety of factors can affect male and female size, and thereby dimorphism. The causes of human sexual size dimorphism are uncertain, and could involve several non-mutually-exclusive mechanisms, such as mate competition, resource competition, intergroup violence, and female choice. A phylogenetic reconstruction of the evolution of dimorphism, including fossil hominins, indicates that the modern human condition is derived. This suggests that at least some behavioral similarities with Pan associated with dimorphism may have arisen independently, and not directly from a common ancestor. (shrink)

This volume includes three previously untranslated transcripts of Kant's logic lectures.

John O’Neill, Alan Holland, and Andrew Light usefully distinguish two ways of thinking about environmental values, namely, end-state and historical views. To value nature in an end-state way is to value it because it instantiates certain properties, such as complexity or diversity. In contrast, a historical view says that nature’s value is (partly) determined by its particular history. Three contemporary defenses of a historical view are explored in order to clarify: (1) the normatively relevant history; (2) how historical considerations are (...) supposed to instruct environmental decision making; and (3) the relative importance of historical and end-state considerations. There are multiple reasons for including historical considerations in an account of environmental values. For example, knowledge of a natural object’s history can add depth and texture to our appreciation of that object. Further, if we were blind to the relevant history, we could not adequately understand and defend environmental policy goals such as preserving the potentials of natural systems or maintaining ecological health, for these goals appear to have irreducibly historical aspects. While historical considerations are important, such considerations are insufficient to guide our normative thinking about nature and how it should be dealt with practically. But they succeed in broadening and deepening our understanding of the nature and sources of environmental value. (shrink)

Prominent philosophers have argued that contradictions contain either too much or too little information to be useful. We dispute this with what we call the “Paradox of the Two Firefighters.” Suppose you are awakened in your hotel room by a fire alarm. You open the door. You see three possible ways out: left, right, straight ahead. You see two firefighters. One says there is exactly one safe route and it is to your left. The other says there is exactly one (...) safe route and it is to your right. While the two firemen are giving you contradictory information, they are also both giving you the perhaps useful information that there is a safe way out and it is not straight ahead. We give two analyses. The first uses the “Opinion Tetrahedron,” introduced by Dunn as a generalization of Audun Jøsang’s “Opinion Triangle.” The Opinion Tetrahedron in effect embeds the values of the “Belnap-Dunn 4-valued Logic” into a context of subjective probability generalized to allow for degrees of belief, disbelief, and two kinds of uncertainty—that in which the reasoner has too little information and that in which the reasoner has too much information. Jøsang had only a single value for uncertainty. We also present an alternative solution, again based on subjective probability but of a more standard type. This solution builds upon “linear opinion pooling.” Kiefer had already developed apparatus for assessing risk using expert opinion, and this influences the second solution. Finally, we discuss how these solutions might apply to “Big Data” and the World Wide Web. (shrink)

In this paper two different formulations of Robinson's arithmetic based on relevant logic are examined. The formulation based on the natural numbers (including zero) is shown to collapse into classical Robinson's arithmetic, whereas the one based on the positive integers (excluding zero) is shown not to similarly collapse. Relations of these two formulations to R. K. Meyer's system R# of relevant Peano arithmetic are examined, and some remarks are made about the role of constant functions (e.g., multiplication by zero) in (...) relevant arithmetic. (shrink)

The validity of an entailment has nothing to do with whether or not the components are true, false, necessary, or impossible; it has to do solely with whether or not there is a necessary connection between antecedent and consequent. Hence it is a mistake (we feel) to try to build a sieve which will “strain out” entailments from the set of material or strict “implications” present in some system of truth-functions, or of truth-functions with modality. Anderson and Belnap (1962, p. (...) 47). (shrink)

An environmental ethic needs to have an answer to two basic questions: what nature should we care about, and why? A number of proposals have been made about how to answer these questions. In this paper, I consider in detail one such proposal, namely, biological or ecological integrity. Different characterizations of integrity can be found in the literature, but I will treat the following one as paradigmatic. Integrity refers to a property of landscapes that are relatively unmodified by human activity (...) and that have their native biota largely intact.1 By “native biota,” I mean the native plant and animal life in a particular place, and... (shrink)

Articulating an account of food justice in isolation from broader questions about sustainability would leave many important normative issues unaddressed. This chapter explores the reasons for thinking that questions of food justice need to be framed within the context of the broader set of social and environmental goals that comprise sustainability. An initial difficulty faced by this proposal is that many philosophers (among others) have viewed the concept and norm of sustainability with suspicion. Reasons for this range from concern about (...) sustainability being hopelessly vague and hence useless for policy, to concern that interest in sustainability is just the latest cover for business as usual and thus a kind of betrayal of the environmental cause. While I believe such concerns are unconvincing, there is no question that sustainability is a contested concept—one that needs careful specification and defense if it is to do any work helping to frame discussions of food justice. This chapter maps three types of sustainability view, ranging from a minimalist to a very demanding view. Depending on the view of sustainability one adopts, there can be significantly different implications for how we should think about, and try to realize in practice, food justice. Some of these implications are explored with respect to each type of sustainability view sketched. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:714 BOOK REVIEWS learning; the Jesuits lean to the voluntarist. Possessed of a unitary academic model, he is really arguing for Aristotle's analogy of attribution. Apparently, no one of his 200 plus Jesuit contacts told him that Nastri prefer St. Thomas and his analogy of proper proportionality. The historian Daniel Boorstin spent 25 yeari!l writing his trilogy on The Americans. What emerges from this analysis? Boorstin points out that (...) Americans like ambiguity, that they function best amid flux; ambivalence becomes their proper medium. For this reason, I have accepted Nathaniel Hawthorne as our most central writer. He situates American questions within a landscape of chiaroscuro. The Jesuit experience in the United States belongs essentially to the same landscape. But today, as formerly, it remains a landscape of questions. McDonough, on the other hand, has too many certitudes, too many answers, too much high definition-what Henry Adams regards as "mere facts." Instead of illuminating the landscape from within, his study inevitably recedes into the larger ambiguity reserved for what Wallace Stevens has called, " Prologues to what is possible." Saint Paul University Ottawa, Ontario JOHN P. McINTYRE, S.J. The Dilemma of Freedom and Foreknowledge. By LINDA TRINKAUS ZAGZEBSKI. New York: Oxford University Press, 1991. viii + 215 pp. $29.95. As a Christian and a philosopher, Linda Trinkaus Zagzebski wants to understand why it is reasonable to affirm that we remain free despite God's infallible knowledge of our future acts. Moreover, she is not content simply to demonstrate that this affirmation involves no logical con· tradiction, for "reasonable people expect more" (p. 33) than a merely negative result. Hence the author sets herself the more ambitious task of working toward a positive solution that would explain why the compatibility of divine foreknowledge and human freedom is plausible and not a bare logical possibility. The hook sorts out recent opinions on the subject and sketches an original position that she hopes will advance the discussion. Zagzehski begins by constructing the most formidable version of the dilemma. Assuming that God is infallible and essentially omniscient, that the past has a kind of necessity that the future lacks ("accidental necessity"), and that this necessity can he transferred (in some sense) from the antecedent to the consequent of a conditional proposition, the dilemma may he stated as follows: BOOK REVIEWS (1) Necessarily, God believes at time t 1 that I will do S at t 3 if and only if I do S at t 3. (2) It is accidentally necessary at t 2 that God believes at t 1 that I will do S at t 3. (3) If (1) and (2) are true, then it is accidentally necessary at t 2 that I do Sat t 3. (4) It follows that it is accidentally necessary at t 2 that I do S at t 3. (5) But if it is accidentally necessary at t 2 that I do S at t 3, I do not do S at t 3 freely. (6) Therefore, I do not do S at t 3 freely. 715 Much of the first chapter is devoted to establishing that this form of the dilemma is more resistant to solution than any other, especially the so-called problem of future truth. Zagzebski goes on to evaluate three broad approaches to solving the dilemma. In Chapter 2 she takes up the " timelessness " solution. Its proponents, among whom the author numbers Boethius, Thomas Aquinas, Eleonore Stump, and Norman Kretzmann, negate premise (2) of the foreknowledge dilemma by contending that God is not in time. Zagzebski considers this an acceptable negative solution to the dilemma and sees in it the potential to ground a more comprehensive, positive solution. Nonetheless, she refrains from subscribing wholeheartedly to the timelessness solution because it confronts one with a dilemma at least as perplexing as the one it solves. She puts the problem this way: like the past, eternity is fixed, for it is constituted by a state of affairs about which we cannot do anything; therefore my future acts are determined no less by God's timeless knowledge than by God's foreknowing them in a strictly temporal sense. In Chapter 3, Zagzebski considers... (shrink)

This chapter begins with an examination of Kierkegaard's historical political contributions, before it describes and evaluates the ways in which some recent scholarship has understood Kierkegaard's positive contributions to social and political thought. It limits the examination to three distinct understandings—postconventional identity as expressed by Habermas, a politics of difference as expressed by Caputo, and religious teleology as expressed by Westphal—as three possible ways to understand Kierkegaard's contribution to social and political thought. The primary aim is to identify common features (...) of these political interpretations of Kierkegaard so as to provide an account of what it would take to describe Kierkegaard's thought as a political theory. (shrink)