Some investigations into the algebraic constructive aspects of a decision procedure for various fragments of Relevant Logics are presented. Decidability of these fragments relies on S. Kripke's gentzenizations and on his combinatorial lemma known as Kripke's lemma that B. Meyer has shown equivalent to Dickson's lemma in number theory and to his own infinite divisor lemma, henceforth, Meyer's lemma or IDP. These investigations of the constructive aspects of the Kripke's-Meyer's decision procedure originate in the development of Paul Thistlewaite's “Kripke” theorem (...) prover that had been devised to tackle the decision problem of the Relevant Logic R. A. Urquhart's pen and paper solution that relies on a sophisticated algebraic and geometric treatment of the problem shows the usefulness of an algebraic approach in Logic. Here, the study of the constructive aspects of the Kripke-Meyer decision procedure relies on various algebraic constructive results in the theory of polynomials rings. (shrink)

This unique textbook states and proves all the major theorems of many-valued propositional logic and provides the reader with the most recent developments and trends, including applications to adaptive error-correcting binary search. The book is suitable for self-study, making the basic tools of many-valued logic accessible to students and scientists with a basic mathematical knowledge who are interested in the mathematical treatment of uncertain information. Stressing the interplay between algebra and logic, the book contains material never before published, such (...) as a simple proof of the completeness theorem and of the equivalence between Chang's MV algebras and Abelian lattice-ordered groups with unit - a necessary prerequisite for the incorporation of a genuine addition operation into fuzzy logic. Readers interested in fuzzy control are provided with a rich deductive system in which one can define fuzzy partitions, just as Boolean partitions can be defined and computed in classical logic. Detailed bibliographic remarks at the end of each chapter and an extensive bibliography lead the reader on to further specialised topics. (shrink)

The variety MBA of monadic bounded algebras consists of Boolean algebras with a distinguished element E, thought of as an existence predicate, and an operator ∃ reflecting the properties of the existential quantifier in free logic. This variety is generated by a certain class FMBA of algebras isomorphic to ones whose elements are propositional functions. We show that FMBA is characterised by the disjunction of the equations ∃E = 1 and ∃E = 0. We also define a weaker notion of (...) "relatively functional" algebra, and show that every member of MBA is isomorphic to a relatively functional one. (shrink)

The paper presents a formal model of the system of number representations as a multiplicity of mental number axes with a hierarchical structure. The hierarchy is determined by the mind as it acquires successive types of mental number axes generated by virtue of some algebraic mechanisms. Three types of algebraic structures, responsible for functioning these mechanisms, are distinguished: BASAN-structures, CASAN-structures and CAPPAN-structures. A foundational order holds between these structures. CAPPAN-structures are derivative from CASAN-structures which are extensions of BASAN-structures. The constructed (...) formal model unifies two competitive conceptions of cognitive arithmetic: namely, the conception of the mental number line and the conception of parallel individuation. The paper is the continuation of a paper entitled Representational structures of arithmetical thinking, in which rich empirical evidence supporting the model is presented. The main result achieved in the present paper may be philosophically interpreted as an attempt to formalize the Kantian conception of the pure idea of time, understood as the a priori form of human arithmetical thinking. In this way, our theory may be comprehended as a result of applying the hard method of logical reconstruction of fundamental epistemological categories. (shrink)

Jones' proposed application of Optimality Theory assumes the primary kinship data are genealogical definitions of kin terms. This, however, ignores the fact that these definitions can be predicted from the computational, algebralike structural logic of kinship terminologies, as has been discussed and demonstrated in numerous publications. The richness of human kinship systems derives from the cultural knowledge embedded in kinship terminologies as symbolic computation systems, not the post hoc constraints devised by Jones.

The paper studies Heyting algebras within the framework of computable structure theory. We prove that the class _K_ containing all Heyting algebras with distinguished atoms and coatoms is complete in the sense of the work of Hirschfeldt et al. (Ann Pure Appl Logic 115(1-3):71-113, 2002). This shows that the class _K_ is rich from the computability-theoretic point of view: for example, every possible degree spectrum can be realized by a countable structure from _K_. In addition, there is no simple (...) syntactic characterization of computably categorical members of _K_ (i.e., structures from _K_ possessing a unique computable copy, up to computable isomorphisms). (shrink)

In this paper, we investigate the Lindenbaum algebra ℒ of the theory T fin = Th of the class M fin of all finite models of a finite rich signature. We prove that this algebra is an atomic Boolean algebra while its Gödel numeration γ is a [Formula: see text]-numeration. Moreover, the quotient algebra /ℱ, γ/ℱ) modulo the Fréchet ideal ℱ is a [Formula: see text]-algebra, which is universal over the class of all [Formula: (...) see text] Boolean algebras. These conditions characterize uniquely the algebra ℒ; moreover, these conditions characterize up to recursive isomorphism the numerated Boolean quotient algebra /ℱ, γ/ℱ). These results extend the work of Trakhtenbrot [17] and Vaught [18] on the first order theory of the class of all finite models of a finite rich signature. (shrink)

Henkin and Tarski proved that an atomic cylindric algebra in which every atom is a rectangle must be representable . This theorem and its analogues for quasi-polyadic algebras with and without equality are formulated in Henkin, Monk and Tarski [13]. We introduce a natural and more general notion of rectangular density that can be applied to arbitrary cylindric and quasi-polyadic algebras, not just atomic ones. We then show that every rectangularly dense cylindric algebra is representable, and we extend (...) this result to other classes of algebras of logic, for example quasi-polyadic algebras and substitution-cylindrification algebras with and without equality, relation algebras, and special Boolean monoids. The results of op. cit. mentioned above are special cases of our general theorems. We point out an error in the proof of the Henkin-Monk-Tarski representation theorem for atomic, equality-free, quasi-polyadic algebras with rectangular atoms. The error consists in the implicit assumption of a property that does not, in general, hold. We then give a correct proof of their theorem. Henkin and Tarski also introduced the notion of a rich cylindric algebra and proved in op. cit. that every rich cylindric algebra of finite dimension satisfying certain special identities is representable. We introduce a modification of the notion of a rich algebra that, in our opinion, renders it more natural. In particular, under this modification richness becomes a density notion. Moreover, our notion of richness applies not only to algebras with equality, such as cylindric algebras, but also to algebras without equality. We show that a finite dimensional algebra is rich iff it is rectangularly dense and quasi-atomic; moreover, each of these conditions is also equivalent to a very natural condition of point density . As a consequence, every finite dimensional rich algebra of logic is representable. We do not have to assume the validity of any special identities to establish this representability. Not only does this give an improvement of the Henkin-Tarski representation theorem for rich cylindric algebras, it solves positively an open problem in op. cit. concerning the representability of finite dimensional rich quasi-polyadic algebras without equality. (shrink)

It is well known that there is a correspondence between sets and complete, atomic Boolean algebras (\(\textit{CABA}\)s) taking a set to its power-set and, conversely, a complete, atomic Boolean algebra to its set of atomic elements. Of course, such a correspondence induces an equivalence between the opposite category of \(\textbf{Set}\) and the category of \(\textit{CABA}\)s. We modify this result by taking multialgebras over a signature \(\Sigma\), specifically those whose non-deterministic operations cannot return the empty-set, to \(\textit{CABA}\)s with their zero (...) element removed (which we call a \(\textit{bottomless Boolean algebra}\)) equipped with a structure of \(\Sigma\)-algebra compatible with its order (that we call \(\textit{ord-algebras}\)). Conversely, an ord-algebra over \(\Sigma\) is taken to its set of atomic elements equipped with a structure of multialgebra over \(\Sigma\). This leads to an equivalence between the category of \(\Sigma\)-multialgebras and the category of ord-algebras over \(\Sigma\). The intuition, here, is that if one wishes to do so, non-determinism may be replaced by a sufficiently rich ordering of the underlying structures. (shrink)

The present paper contains some technical results on a many-valued logic with truth values from the interval of real numbers [0; 1]. This logic, discussed originally in [1], latter in [2] and [3], was called the logic of fuzzy concepts. Our aim is to give an algebraic axiomatics for fuzzy propositional logic. For this purpose the variety of L-algebras with signature en- riched with a unary operation { involution is stud- ied. A one-to-one correspondence between congruences on an LI-algebra (...) and lters of a special kind is used to prove the representation theorem for LI-algebras. By this theorem every LI-algebra is isomorphic to a subdirect product of chains. The full characteristic of the subdirectly irreducible LI-algebras is given . It turns out that the variety of all L-algebras, as well as any of its subvarieties, is generated by its nite algebras. (shrink)

We consider countable so‐called rich subsemigroups of ; each such semigroup T gives a variety CPEAT that is axiomatizable by a finite schema of equations taken in a countable subsignature of that of ω‐dimensional cylindric‐polyadic algebras with equality where substitutions are restricted to maps in T. It is shown that for any such T, if and only if is representable as a concrete set algebra of ω‐ary relations. The operations in the signature are set‐theoretically interpreted like in polyadic (...) equality set algebras, but such operations are relativized to a union of cartesian spaces that are not necessarily disjoint. This is a form of guarding semantics. We show that CPEAT is canonical and atom‐canonical. Imposing an extra condition on T, we prove that atomic algebras in CPEAT are completely representable and that CPEAT has the super amalgamation property. If T is rich and finitely represented, it is shown that CPEAT is term definitionally equivalent to a finitely axiomatizable Sahlqvist variety. Such semigroups exist. This can be regarded as a solution to the central finitizability problem in algebraic logic for first order logic with equality if we do not insist on full fledged commutativity of quantifiers. The finite dimensional case is approached from the view point of guarded and clique guarded (relativized) semantics of fragments of first order logic using finitely many variables. Both positive and negative results are presented. (shrink)

Using representations of nonflat Scott domains to model type systems, it is natural to wish that they be “linear”, in which case the complexity of the fundamental test for entailment of information drops from exponential to linear, the corresponding mathematical theory becomes much simpler, and moreover has ties to models of computation arising in the study of sequentiality, concurrency, and linear logic. Earlier attempts to develop a fully nonflat semantics based on linear domain representations for a rich enough type (...) system allowing inductive types, were designed in a way that felt rather artificial, as it featured certain awkward and counter‐intuitive properties; eventually, the focus turned on general, nonlinear representations. Here we try to turn this situation around, by showing that we can work linearly in a systematic way within the nonlinear model, and that we may even restrict to a fully linear model whose objects are in a bijective correspondence with the ones of the nonlinear and are easily seen to form a prime algebraic domain. To obtain our results we study mappings of finite approximations of objects that can be used to turn approximations into normal and linear forms. (shrink)

Modern philosophy is, for what appear to be good reasons, uniformly hostile to sui generis final causes. And motivated to develop philosophically and scientifically plausible interpretations, scholars have increasingly offered reductivist and eliminitivist accounts of Aristotle's teleological commitment. This trend in contemporary scholarship is misguided. We have strong grounds to believe Aristotle accepted unreduced sui generis teleology, and reductivist and eliminitivist accounts face insurmountable textual and philosophical difficulties. We offer Aristotelians cold comfort by replacing his apparent view with failed accounts. (...) And so we ought to admit Aristotle’s prima facie commitments and deal with — if not accept — the consequences. (shrink)

Among the most significant findings of SUPPORT was that 50% of ICU patients suffered from moderate to severe pain during the last days of life. At the time of its publication late in 1995, SUPPORT was merely the latest in a long series of articles in the medical literature documenting the widespread and significant undertreatment of pain, beginning with a 1973 study of hospital inpatients. Much has been written about the phenomenon of undertreated pain and inadequate care of patients at (...) the end of life, and many positive suggestions for reform of clinical education and clinical practice have been iterated and reiterated in the two decades separating the studies. Proposals for modifying clinician behaviors in this aspect of patient care have tended to focus on particular barriers to effective pain management and palliative care. (shrink)

This paper focuses on the cognitive/computational and evolutionary levels. It describes three proposals to make cognition computationally tractable, namely: Resource Rationality, the Adaptive Toolbox and Massive Modularity. While each of these proposals appeals to evolutionary considerations to dissolve the intractability of cognition, Rich, Blokpoel, de Haan, and van Rooij argue that, in each case, the intractability challenge is not resolved, but just relocated to the level of evolution.

Many interpretations of Hume's Dialogues Concerning Natural Religion have labored under the assumption that one of the characters represents Hume's view on the Design Argument, and Philo is often selected for this role. I reject this opinion by showing that Philo is inconsistent. He offers a decisive refutation of the Design Argument, yet later endorses this very argument. I then dismiss two prominent ways of handling Philo's reversal: first, I show that Philo is not ironic either in his skepticism or (...) in his theistic reversal. Second, I reject the suggestion that the Design Argument is a natural belief, since it differs significantly from causal and external world beliefs. Finally, I argue that the control the Design Argument exerts is the product of a youthful indoctrination that prevents Philo from consistently maintaining his skeptical position. (shrink)

Many interpretations of Hume's Dialogues Concerning Natural Religion have labored under the assumption that one of the characters represents Hume's view on the Design Argument, and Philo is often selected for this role. I reject this opinion by showing that Philo is inconsistent. He offers a decisive refutation of the Design Argument, yet later endorses this very argument. I then dismiss two prominent ways of handling Philo's reversal: first, I show that Philo is not ironic either in his skepticism or (...) in his theistic reversal. Second, I reject the suggestion that the Design Argument is a natural belief, since it differs significantly from causal and external world beliefs. Finally, I argue that the control the Design Argument exerts is the product of a youthful indoctrination that prevents Philo from consistently maintaining his skeptical position. (shrink)

In the mid to late 1980s a debate arose over the moral and legal authority of advance medical directives. At the center of this debate were two point-counterpoint law journal articles by Rebecca Dresser and Nancy Rhoden. What appeared to have the makings of an ongoing critical dialogue ended with the untimely death of Nancy Rhoden. Rebecca Dresser, however, has continued her challenge of advance directives in numerous publications, most recently in a critique of Ronald Dworkin's Life's Dominion. Like Rhoden, (...) Dworkin has been a staunch advocate of advance directives as an exercise of what has come to be referred to as prospective or precedent autonomy. In this paper I will consider a number of the issues that Dresser has repeatedly raised about the infirmities of advance directives, and suggest that it is from an understanding of and appreciation for the narrative dimension of the life of a person that advanced directives draw one of their most powerful justifications. (shrink)

Of the many opportunities and affordances that mobile technologies bring to our day-to-day lives, the ability to cheat physical separation and remain accessible to each other—in an instant—also brings pressure to bear on well-established social conventions as to how we should act when we are engaged with others in shared spaces. In this paper we explore some ethical dimensions of mobile communication by considering the manner in which individuals in everyday contexts balance interpretations of emergent social conventions with personal desires (...) to connect in the moment. As we later discuss, the decisions made in response to a ringing mobile phone or flashing text message emerge from consequential versus deontological ethical frames used to determine what to do versus what we ought to do. This is particularly true in western and North American cultural contexts from which our data are collected. Using Goffman's dramaturgy, we suggest that these conflicts occurring on an individual level provide evidence of social structure, and are simultaneously entwined with our less obvious ruminations on the maintenance of social order. (shrink)

This paper focuses on the cognitive/computational and evolutionary levels. It describes three proposals to make cognition computationally tractable, namely: Resource Rationality, the Adaptive Toolbox and Massive Modularity. While each of these proposals appeals to evolutionary considerations to dissolve the intractability of cognition, Rich, Blokpoel, de Haan, and van Rooij argue that, in each case, the intractability challenge is not resolved, but just relocated to the level of evolution.

“And the earth was without form, and void; and darkness was upon the face of the deep” Genesis This article examines the problem of belief as it relates to radical negativity and as such engages with two positions in regard to the Real of the void. The first, drawing from speculative realism seeks to conceptualise this void in positive terms, as something that can be reached and in a sense overcome. The second, Hegelian account, by contrast, situates the void as (...) something inherent to the very coordinates of reality and being itself. As such the following discussion specifically examines the status of objective reality within the continental philosophical paradigm. (shrink)

You meet someone new; you like them; you send them to your Facebook page. But how accurate is this representation of you? We all want to look our best, which is why we are drawn to the ability to fudge things a bit online. How does this projection of who we are distort us into who we want to be? Facebook allows us to hide our flaws that are all too visible in real life. We can embellish or correct what (...) we said earlier, edit out what we don’t like about ourselves, and only show photos where our chin is down and the lighting is blown out just enough to hide the bump on our nose. Our prospective employers can even turn to Facebook to assess our desirability. So might the constructed aesthetic of Facebook affect our approach to ethics – to interacting with other people? Does the way we design ourselves on Facebook directly impact the other people in our lives? Is each of us really just a rough draft that needs immediate editing? What about Facebook makes it possible to achieve genuine beauty? (shrink)

In the current American iconoclash, certain monuments are subject to vandalism and municipal removal from their pedestals. Phrases such as “the erasure of history” and “damnatio memoriae” point to concerns that iconoclasm is an attempt to censor history or even remove certain individuals from public memory altogether. Because these phrases beckon the past, this wave of iconoclasm calls for a close examination of previous image-breaking to establish motives. Drawing first from art history, we analyze Byzantine iconoclasm and anxieties over the (...) nature of icons’ power, before contextualizing these findings within image destruction from the Paleolithic to the present day. Each comparison is suggestive of an enduring aesthetic principle: that what appears inanimate is not always inert. Next, drawing from cultural anthropology, we argue that principles of sympathetic magic are at the heart of contemporary iconoclasms, but not in the way media outlets often suggest. Instead, the fear of history’s erasure betrays a deeply rooted equivalence between the representation and the represented. In perceiving their fates as shared, sympathetic magic is seen to persist in the way humans create, interpret, and desecrate images. We conclude with the speculative realist proposition that iconoclasm can produce new, original artworks, which carries implications for the autonomy of art and its distribution between artist and artwork. (shrink)

A cursory reading of Thomas Aquinas’s work can give the impression he condones a despotic or exploitative relationship between humans and the environment. Many philosophers and theologians have sought to dispel this impression and draw out a more robust Thomistic environmental ethic. In this paper, I support this endeavor by describing how, in Thomas’s work, the environment is God’s artistic property and how this notion qualifies our use of the environment. Next, I consider two concepts related to artistic property: vandalism (...) and showcasing. I explore these concepts as they relate to the environment and find they give us reasons not to deface or destroy creatures and to look to creation for guidance in problem-solving. (shrink)

There are two prominent viewpoints regarding the nature of rationality and how it should be evaluated in situations of interest: the traditional axiomatic approach and the newer ecological rationality. An obstacle to comparing and evaluating these seemingly opposite approaches is that they employ different language and formalisms, ask different questions, and are at different stages of development. I adapt a formal framework known as SCOP to address this problem by providing a comprehensive common framework in which both approaches may be (...) defined and compared. The main result is that the axiomatic and ecological approaches are in far greater agreement on fundamental issues than has been appreciated; this is supported by a pair of theorems to the effect that they will make accordant rationality judgements when forced to evaluate the same situation. I conclude that ecological rationality has some subtle advantages, but that we should move past the issues currently dominating the discussion of rationality. (shrink)

Charles Dickens began one of his many great works of literature with this seemingly paradoxical, self-contradictory statement. Reflecting on a jury verdict in Northern California in June of 2001, in the context of what has transpired during the decade of the 1990s with regard to the care of dying patients, observations in the genre of Dickens come readily to mind. In 1991, two of the most compelling books on the subject of pain, medicine, and society were published: Eric Cassell's The (...) Nature of Suffering and the Goals of Medicine and David Morris's The Culture of Pain. Both works unflinchingly recognized and explored the implications of the vast clinical literature documenting the failure of modern medicine to respond compassionately and effectively to what had by then become an epidemic of undertreated pain. (shrink)

When one considers the protracted and continuing struggle of the citizens of Oregon to include physician-assisted suicide among the panoply of measures available to dying patients and the physicians who care for them, the depth and breadth of the issue becomes inescapable. The potential intractability of the dispute is illustrated by the very fact, noted in the preceding parenthetical phrase, that consensus eludes us on even the most basic of semantic points—how we are to most aptly characterize the conduct in (...) question? The case featured in this issue's The Caduceus in Court must be examined as the latest battle in a long war of attrition waged by opponents of physician-assisted suicide against the Oregon Death with Dignity Act. After briefly reviewing that history, I will then consider the details of Oregon v. Ashcroft and the implications of the litigation not only for the care of dying patients but also for the regulation of medical practice in the United States and, more broadly still, the search for resolution of serious medical disputes in a democratic society. (shrink)

Sarah Moss’ thesis that we have probabilistic knowledge is from some perspectives unsurprising and from other perspectives hard to make sense of. The thesis is potentially transformative, but not yet elaborated in sufficient detail for epistemologists. This paper interprets Mossean probabilistic knowledge in a suitably-modified Kripke framework, thus filling in key details. It argues that probabilistic knowledge looks natural and plausible when so interpreted, and shows how the most pressing challenges to the thesis can be overcome. Most importantly, probabilistic knowledge (...) can satisfy factivity in the framework, though we are not forced to accept a specific account of probabilistic “facts”. The framework also reflects Moss’ claim that old-fashioned propositional knowledge is just a limiting case of probabilistic knowledge, and all knowledge is fundamentally probabilistic. Finally, Moss endorses a failure of contraposition: for example, p implies probably p, but not probably p does not imply not p. The framework makes clear the sense in which the valid inferences regarding probably p are as Moss claims. (shrink)

This paper addresses the problems that must be considered if computers are going to treat their users as individuals with distinct personalities, goals, and so forth. It first outlines the issues, and then proposes stereotypes as a useful mechanism for building models of individual users on the basis of a small amount of information about them. In order to build user models quickly, a large amount of uncertain knowledge must be incorporated into the models. The issue of how to resolve (...) the conflicts; hot will arise along such inferences is discussed. A system, Grundy, is described that builds models of its users, with the aid of stereotypes, and then exploits those models to guide it in its task, suggesting novels that people may find interesting. If stereotypes are to be useful to Grundy, they must accurately characterize the users of the system. Some techniques to modify stereotypes on the basis of experience are discussed. An analysis of Grundy's performance shows that its user models are effective in guiding its performance. (shrink)

Knowledge-first epistemology includes a knowledge norm of action: roughly, act only on what you know. This norm has been criticized, especially from the perspective of so-called standard decision theory. Mueller and Ross provide example decision problems which seem to show that acting properly cannot require knowledge. I argue that this conclusion depends on applying a particular decision theory which is ill-motivated in this context. Agents’ knowledge is often most plausibly formalized as an ambiguous epistemic state, and the theory of decision (...) under ambiguity is then the appropriate modeling tool. I show how to model agents as acting rationally on the basis of their knowledge according to such a theory. I conclude that the tension between the knowledge norm of action and formal decision theory is illusory; the knowledge-first paradigm should be used to actively select the decision-theoretical tools that can best capture the knowledge-based decisions in any given situation. (shrink)