We construct an algebra of generalized functions endowed with a canonical embedding of the space of Schwartz distributions.We offer a solution to the problem of multiplication of Schwartz distributions similar to but different from Colombeau’s solution.We show that the set of scalars of our algebra is an algebraically closed field unlike its counterpart in Colombeau theory, which is a ring with zero divisors. We prove a Hahn–Banach extension principle which does not hold in Colombeau theory. (...) We establish a connection between our theory with non-standard analysis and thus answer, although indirectly, a question raised by Colombeau. This article provides a bridge between Colombeau theory of generalized functions and non-standard analysis. (shrink)

One of the problems we face in many-valued logic is the difficulty of capturing the intuitive meaning of the connectives introduced through truth tables. At the same time, however, some logics have nice ways to capture the intended meaning of connectives easily, such as four-valued logic studied by Belnap and Dunn. Inspired by Dunn’s discovery, we first describe a mechanical procedure, in expansions of Belnap-Dunn logic, to obtain truth conditions in terms of the behavior of the Truth and the False, (...) which gives us intuitive readings of connectives, out of truth tables. Then, we revisit the notion of functional completeness, which is one of the key notions in many-valued logic, in view of Dunn’s idea. More concretely, we introduce a generalized notion of functional completeness which naturally arises in the spirit of Dunn’s idea, and prove some fundamental results corresponding to the classical results proved by Post and Słupecki. (shrink)

We offer a defense of one aspect of Paul Horwich’s response to the Liar paradox—more specifically, of his move to preserve classical logic. Horwich’s response requires that the full intersubstitutivity of ‘ ‘A’ is true’ and A be abandoned. It is thus open to the objection, due to Hartry Field, that it undermines the generalization function of truth. We defend Horwich’s move by isolating the grade of intersubstitutivity required by the generalization function and by providing a new reading of the (...) biconditionals of the form “ ‘A’ is true iff A.”. (shrink)

Scientific collaboration has increased over the past two centuries, a fact for which various explanations have been proposed. We offer a novel functional explanation of this increase in collaboration, grounded in a sequential model of scientific research where the priority rule applies. Robust patterns concerning the differential success of collaborative groups with respect to their competitors are derived, and it is argued that these patterns feed the development of collaboration. This general mechanism may trigger an ‘arms race’ and is compatible (...) with a certain amount of decreased productivity in collaborative groups as well as some over-collaboration. The proposed explanation succeeds in integrating various factors usually associated with the rise of collaboration. (shrink)

It has been known for many years that the matrix representation of the one-dimensional position-momentum commutator calculated with the position and momentum matrices in a finite basis is not proportional to the diagonal matrix, contrary to what one expects from the continuous-space commutator. This discrepancy has correctly been ascribed to the incompleteness of any finite basis, but without the details of exactly why this happens. Understanding why the discrepancy occurs requires calculating the position, momentum, and commutator matrix elements in the (...) continuous position basis, in which all are generalized functions. The reason for the discrepancy is revealed by replacing the generalized functions with sequences approaching them as their parameter approaches zero. Besides explaining the discrepancy in the discrete and continuous models, this investigation finds an unusual double-peaked sequence for the Dirac delta function. (shrink)

We can think of functional completeness in systems of propositional logic as a form of expressive completeness: while every logical constant in such system expresses a truth-function of finitely many arguments, functional completeness garantees that every truth-function of finitely many arguments can be expressed with the constants in the system. From this point of view, a functionnaly complete system of propositionnal logic can thus be seen as one where no logical constant is missing. Can a similar question be formulated for (...) quantified first-order logics ? How to make sense of the question whether, e.g., ordinary first-order logic is "functionaly" complete or have no logical constant missing ? In this note, we build on a suggestive proposal made by Bonnay(2006) and shows that it is equivalent to the criterion that a first-order logic L be functionaly complete if and only if every class of structures closed under L-elementary equivalence is L-elementary. Ordinary first-order logic is not complete in this sense. We raise the question whether any logic can be. -/- . (shrink)

Evolutionary psychology tends to be associated with a massively modular cognitive architecture. On this framework of human cognition, an assembly of specialized information processors called modules developed under selection pressures encountered throughout the phylogenic history of hominids. The coordinated activity of domain-specific modules carries out all the processes of belief fixation, abstract reasoning, and other facets of central cognition. Against the massive modularity thesis, I defend an account of systemic functional adaptedness, according to which non-modular systems emerged because of adaptive (...) problems imposed by the intrinsic physiology of the evolving human brain. The proposed reformulation of evolutionary theorizing draws from neural network models and Cummins’ (1975) account of systemic functions to identify selection pressures that gave rise to non-modular, domain-general mechanisms in cognitive architecture. (shrink)

General extenders are phrases like 'or something', 'and everything', 'and things ', 'and stuff ', and 'and so on'. Although they are an everyday feature of spoken language, are crucial in successful interpersonal communication, and have multiple functions in discourse, they have so far gone virtually unnoticed in linguistics. This pioneering work provides a comprehensive description of this new linguistic category. It offers new insights into ongoing changes in contemporary English, the effect of grammaticalization, novel uses as associative plural (...) markers and indicators of intertextuality, and the metapragmatic role of extenders in interaction. The forms and functions of general extenders are presented clearly and accessibly, enabling students to understand a number of different frameworks of analysis in discourse-pragmatic studies. From an applied perspective, the book presents a description of translation equivalents, an analysis of second language variation, and practical exercises for teaching second language learners of English. (shrink)

I present and defend the generalized selected effects theory (GSE) of function. According to GSE, the function of a trait consists in the activity that contributed to its bearer’s differential reproduction, or differential retention, within a population. Unlike the traditional selected effects (SE) theory, it does not require that the functional trait helped its bearer reproduce; differential retention is enough. Although the core theory has been presented previously, I go significantly beyond those presentations by providing a new argument for (...) GSE and defending it from a recent objection. I also sketch its implications for teleosemantics and philosophy of medicine. (shrink)

Part I of Frege’s Grundgesetze is devoted to the “exposition [Darlegung]” of his formal system.

Both unipolar and bipolar depression have been linked with impairments in executive functioning. In particular, mood symptom severity is associated with differences in common EF, a latent measure of general EF abilities. The relationship between mood disorders and EF is particularly salient in adolescence and young adulthood when the ongoing development of EF intersects with a higher risk of mood disorder onset. However, it remains unclear if common EF impairments have associations with specific symptom dimensions of mood pathology such as (...) blunted positive affect, mood instability, or physiological arousal, or if differences in common EF more broadly relate to what is shared across various symptom domains, such as general negative affect or distress. To address this question, bifactor models can be applied to simultaneously examine the shared and unique contributions of particular mood symptom dimensions. However, no studies to our knowledge have examined bifactor models of mood symptoms in relation to measures of common EF. This study examined associations between common EF and general vs. specific symptom dimensions using structural equation modeling in adolescents and young adults with varying severity of mood symptoms. A General Depression factor capturing shared variance across symptoms statistically predicted lower Common EF. Additionally, a factor specific to physiological arousal was associated with lower Common EF. Anhedonia-specific and Mania-specific factors were not significantly related to Common EF. Altogether, these results indicate that deficits in common EF are driven by, or reflect, general features of mood pathology that are shared across symptom dimensions but are also specifically associated with physiological arousal. (shrink)

1. IntroductionIn What Biological Functions Are and Why They Matter, Justin Garson offers a novel theory of biological functions, the generalized selected effects (GSE) theory.1 He presents the theory in a clear and comprehensive way, defends it against various objections and applies it to different areas of philosophy, including the philosophy of psychiatry, the debate about mechanisms and the debate about teleosemantic theories of mental content.2Like other proponents of the aetiological approach to functions, Garson maintains that (...) a trait’s biological functions are grounded in its selectional history. However, his GSE theory differs from standard versions of the aetiological approach – so-called selected effects (SE) theories – in a crucial respect: while SE theories assume that all biological functions must ultimately be grounded in phylogenetic processes of natural selection or other (closely analogous) processes of differential reproduction (Millikan 1984: 15–82, Neander 1991, Godfrey-Smith 1994), the GSE theory allows that functions may also be grounded in processes of differential retention (see §2 for details). Garson forcefully argues that this broader account is theoretically well-motivated and has significant advantages over standard SE accounts when it comes to accommodating certain important classes of ontogenetically determined functions. (shrink)

Motivated by aspects of reasoning in theories of physics, Robin Giles defined a characterization of infinite valued Łukasiewicz logic in terms of a game that combines Lorenzen-style dialogue rules for logical connectives with a scheme for betting on results of dispersive experiments for evaluating atomic propositions. We analyze this game and provide conditions on payoff functions that allow us to extract many-valued truth functions from dialogue rules of a quite general form. Besides finite and infinite valued Łukasiewicz logics, (...) also Meyer and Slaney’s Abelian logic and Cancellative Hoop Logic turn out to be characterizable in this manner. (shrink)

Looking for an adequate explication of the concept of a biological function, several authors have proposed to link function to design. Unfortunately, known explications of biological design in turn refer to functions. The concept of general design I will introduce here breaks up this circle. I specify design with respect to its ontogenetic role. This allows function to be based on design without making reference to the history of the design, or to the phylogeny of an organism, while retaining (...) the normative aspect of function ascriptions. The concept is applicable to the function and design of technical artifacts as well. Several problems well known with other definitions can be overcome by this approach. (shrink)

This book addresses three major questions about law and legal systems: (1) What are the defining and organizing forms of legal institutions, legal rules, interpretive methodologies, and other legal phenomena? (2) How does frontal and systematic focus on these forms advance understanding of such phenomena? (3) What credit should the functions of forms have when such phenomena serve policy and related purposes, rule of law values, and fundamental political values such as democracy, liberty, and justice? This is the first (...) book that seeks to offer general answers to these questions and thus gives form in the law its due. The answers not only provide articulate conversancy with the subject but also reveal insights into the nature of law itself, the oldest and foremost problem in legal theory and allied subjects. (shrink)

The classic logical positivist account of historical explanation, putting forward what is variously called the "regularity interpretation" (#Gardiner, The Nature of Historical Explanation), the "covering law model" (#Dray, Laws and Explanation in History), or the "deductive model" (Michael #Scriven, "Truisms as Grounds for Historical Explanations"). See also #Danto, Narration and Knowledge, for further criticisms of the model. Hempel formalizes historical explanation as involving (a) statements of determining (initial and boundary) conditions for the event to be explained, and (b) statements of (...) general laws that imply that whenever events of the kind described in (a) occur, an event of the kind to be explained will take place. He admits that in practice historians rarely explain historical events in this way, but instead give only fragmentary "explanation sketches." Such sketches can, however, be "scientifically acceptable," providing it points to where "more specific statements" are to be found (351). An interesting aspect of the article is its "late" positivist, proto‑Kuhnian assertion that "the separation of 'pure description' and 'hypothetical generalization and theory construction' in empirical science is unwarranted" (356). But Hempel does not work out the wider implications of this important conclusion. It is also worth noting that he himself was not much interested in the philosophy of history; other authors took up his model and discussed it within that context. Usefully discussed by #Ricoeur, Time and Narrative, 1:112‑17. (shrink)

Programming practice suggests a notion of general iteration corresponding to the while-do construct. This leads to new characterizations of general computable unary functions usable in computer science.

Psychology has benefited from an enormous wealth of knowledge about processes of cognition in relation to how the brain organizes information. Within the categorization literature, this behavior is often explained through theories of memory construction called exemplar theory and prototype theory which are typically based on similarity or rule functions as explanations of how categories emerge. Although these theories work well at modeling highly controlled stimuli in laboratory settings, they often perform less well outside of these settings, such as (...) explaining the emergence of background knowledge processes. In order to explain background knowledge, we present a non-similarity-based post-Skinnerian theory of human language called Relational Frame Theory (RFT) which is rooted in a philosophical world view called functional contextualism (FC). This theory offers a very different interpretation of how categories emerge through the functions of behavior and through contextual cues, which may be of some benefit to existing categorization theories. Specifically, RFT may be able to offer a novel explanation of how background knowledge arises, and we provide some mathematical considerations in order to identify a formal model. Finally, we discuss much of this work within the broader context of general semantic knowledge and artificial intelligence research. (shrink)

Function refers to a broad family of concepts of varying abstractness and range of application, from a many-one mathematical relation of great generality to, for example, highly specialized roles of designed elements in complex machines such as degaussing in a television set, or contributory processes to control mechanisms in complex metabolic pathways, such as the inhibitory function of the appropriate part of the lac-operon on the production of lactase through its action on the genome in the absence of lactose. We (...) would like a language broad enough, neutral enough, but yet powerful enough to cover all such cases, and at the same time to give a framework form explanation both of the family resemblances and differences. General logic and mathematics are too abstract, but more importantly, too broad, whereas other discourses of function, such as the biological and teleological contexts, are too narrow. Information is especially suited since it is mathematically grounded, but also has a wellknown physical interpretation through the Schr dinger/Brillouin Negentropy Principle of Information, and an engineering or design interpretation through Shannon's communication theory. My main focus will be on the functions of autonomous anticipatory systems, but I will try to demonstrate both the connections between this notion of function and the others, especially to dynamical systems with a physical interpretation on the one side and intentional systems on the other. The former are based in concepts like force, energy and work, while the latter involve notions like representation, control and purpose, traditionally, at least in Modern times, on opposite sides of the Cartesian divide. In principle, information can be reduced to energy, but it has the advantage of being more flexible, and easier to apply to higher level phenomena. (shrink)

Blair equates the constructs of working memory (WM), executive function, and general fluid intelligence (gF). We argue that there is good reason not to equate these constructs. We view WM and gF as separable but highly related, and suggest that the mechanism behind the relationship is controlled attention – an ability that is dependent on normal functioning of the prefrontal cortex. (Published Online April 5 2006).

A conceptualization of capitalism as a consequence of functional differentiation is proposed. The general theoretical model of a functionally differentiated capitalist society is outlined in four steps based on these keywords: functional differentiation; capitalist economy; capitalist society; welfare state. This model grasps the essential characteristics of the analytical prototype of a capitalist society. What are its basic components that, working together, generate this kind of society’s structural dynamics?