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 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)

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)

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)

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)

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)

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)

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)

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)

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)

This article offers a general definition of self-knowledge that embraces all forms and levels of self-knowledge in animals and humans. It is hypothesized that various levels of self-knowledge constitute an ordinal scale such that each species in a lineage displays the forms of self-knowledge found in related species as well as new forms it and its sister species may have evolved. Likewise, it is hypothesized that these various forms of levels of self-knowledge develop in the sequence in which they evolved. (...) Finally, a general hypothesis for the functional significance of self-knowledge is proposed along with subhypotheses regarding the adaptive significance of various levels of self-knowledge in mammals including human and nonhuman primates. The general hypothesis is that self-knowledge serves as a standard for assessing the qualities of conspecifics compared to those of the self. Such assessment is crucial to deciding among alternative reproductive and subsistence strategies. The qualities that are assessed, which vary across taxa, range from the size and strength of the self to its mathematical or musical abilities. This so-called assessment model of self-knowledge is based on evolutionary biological models for social selection and the role of assessment in animal communication. (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)

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)

Logics for ‘generally’ are intended to express some vague notions, such as ‘generally’, ‘several’, ‘many’, ‘most’, etc., by means of the new generalized quantifier ∇ and to reason about assertions with ‘generally’ . We introduce the idea of functional interpretation for ‘generally’ and show that representative functions enable elimination of ∇ and reduce consequence to classical theories. Thus, one can use proof procedures and theorem provers for classical first-order logic to reason about assertions involving ‘generally’.

Closure space has been proven to be a useful tool to restructure lattices and various order structures. This paper aims to provide an approach to characterizing domains by means of closure spaces. The notion of an interpolative generalized closure space is presented and shown to generate exactly domains, and the notion of an approximable mapping between interpolative generalized closure spaces is identified to represent Scott continuous functions between domains. These produce a category equivalent to that of domains (...) with Scott continuous functions. Meanwhile, some important subclasses of domains are discussed, such as algebraic domains, _L_-domains, bounded-complete domains, and continuous lattices. Conditions are presented which, when fulfilled by an interpolative generalized closure space, make the generated domain fulfill some restrictive conditions. (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.

We prove a Gleason-type theorem for the quantum probability rule using frame functions defined on positive-operator-valued measures, as opposed to the restricted class of orthogonal projection-valued measures used in the original theorem. The advantage of this method is that it works for two-dimensional quantum systems and even for vector spaces over rational fields—settings where the standard theorem fails. Furthermore, unlike the method necessary for proving the original result, the present one is rather elementary. In the case of a qubit, (...) we investigate similar results for frame functions defined upon various restricted classes of POVMs. For the so-called trine measurements, the standard quantum probability rule is again recovered. (shrink)

It has been proved that total generalized variation can better preserve edges while suppressing staircase effect. In this paper, we propose an effective hybrid regularization model based on second-order TGV and wavelet frame. The proposed model inherits the advantages of TGV regularization and wavelet frame regularization, can eliminate staircase effect while protecting the sharp edge, and simultaneously has good capability of sparsely estimating the piecewise smooth functions. The alternative direction method of multiplier is employed to solve the new (...) model. Numerical results show that our proposed model can preserve more details and get higher image visual quality than some current state-of-the-art methods. (shrink)

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

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)

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)

In his article Forme et Fonction Barge wrote in 1936 that living matter cannot be totally understood in terms of causality. In this paper we argue on the contrary that this is in principle possible.In order to develop our arguments, we discuss some basic and derived concepts used in morphology and functional anatomy. We also formulate comments on the so-called formal, functional and final elucidations.

Generalized anxiety disorder (GAD) is characterized by excessive and uncontrollable worry about everyday life. Prior neuroimaging studies have demonstrated that GAD is associated with disruptions in specific brain regions; however, little is known about the global functional connectivity maps in adolescents with GAD. Here, first-episode, medication-naive, adolescent GAD patients ( N = 36) and healthy controls ( N = 28) (HCs) underwent resting-state functional MRI (R-fMRI) and completed a package of questionnaires to assess clinical symptoms. Functional connectivity strength and (...) seed-based functional connectivity were employed to investigate the functional connectivity architecture. GAD patients showed reduced functional connectivity strength in right supramarginal gyrus (SMG) and right superior parietal gyrus (SPG) compared with HCs. Further seed-based functional connectivity analysis revealed that GAD patients displayed decreased functional connectivity between right SMG and left fusiform gyrus, inferior temporal gyrus, parahippocampal gyrus, bilateral precuneus and cuneus, and between right SPG and bilateral supplementary motor area and middle cingulate gyrus, as well as between the SMG-based network and the SPG-based network. Moreover, the disrupted intra-network connectivity (i.e., the SMG-based network and the SPG-based network) and inter-network connectivity between the SMG-based network and the SPG-based network accounted for 25.5% variance of the State and Trait Anxiety Inventory (STAI) and 39.5% variance of the trait subscale of STAI. Our findings highlight the abnormal functional architecture in the SMG-based network and the SPG-based network in GAD, providing novel insights into the pathological mechanisms of this disorder. (shrink)