In the literature on vagueness one finds two very different kinds of degree theory. The dominant kind of account of gradable adjectives in formal semantics and linguistics is built on an underlying framework involving bivalence and classical logic: its degrees are not degrees of truth. On the other hand, fuzzy logic based theories of vagueness—largely absent from the formal semantics literature but playing a significant role in both the philosophical literature on vagueness and in the contemporary logic literature—are logically nonclassical (...) and give a central role to the idea of degrees of truth. Each kind of degree theory has a strength: the classical kind allows for rich and subtle analyses of the comparative form of gradable adjectives and of various types of gradable precise adjectives, while the fuzzy kind yields a compelling solution to the sorites paradox. This paper argues that the fuzzy kind of theory can match the benefits of the classical kind and hence that the burden is on the latter to match the advantages of the former. In particular, we develop a new version of the fuzzy logic approach that—unlike existing fuzzy theories—yields a compelling analysis of the comparative as well as an adequate account of gradable precise predicates, while still retaining the advantage of genuinely solving the sorites paradox. (shrink)

Christoph Jäger has argued that Dretske’s information-based account of knowledge is committed to both knowledge and information closure under known entailment. However, in a reply to Jäger, Dretske defended his view on the basis of a discrepancy between the relation of information and the relation of logical implication. This paper shares Jäger’s criticism that Dretske’s externalist notion of information implies closure, but provides an analysis based on different grounds. By means of a distinction between two perspectives, the mathematical perspective and (...) the epistemological perspective, I present, in the former, a notion of logical implication that is compatible with the notion of information in the mathematical theory of information, and I show how, in the latter, Dretske’s logical reading of the closure principle is incompatible with his information-theoretic epistemological framework. (shrink)

One holistic thesis about symbols is that a symbol cannot exist singly, but only as apart of a symbol system. There is also the plausible view that symbol systems emerge gradually in an individual, in a group, and in a species. The problem is that symbol holism makes it hard to see how a symbol system can emerge gradually, at least if we are considering the emergence of a first symbol system. The only way it seems possible is if being (...) a symbol can be a matter of degree, which is initially problematic. This article explains how being a cognitive symbol can be a matter of degree after all. The contrary intuition arises from the way a process of interpretation forces an all-or-nothing character on symbols, leaving room for underlying material to realize symbols to different degrees in a way that Daniel Dennett’s work can help illuminate. Holism applies to symbols as interpreted, while gradualism applies to how the underlying material realizes symbols.Selon une thèse holistique sur les symboles, un symbole ne peut exister isolément mais doit faire partie d’un systéme symbolique. Une opinion, elle aussi plausible, veut que les systèmes symboliques émergent graduellement chez un individu, un groupe ou une espèce. Le problème c’est qu’on voit mal, si le holisme des systèmes symboliques tient, comment un système symbolique peut émerger graduellement, du moins pour la première fois. Ce n’est possible, semble-t-il, que si être un symbole est affaire de degré, thèse au départ problématique. Cet article explique comment être un symbole cognitif peut après tout être affaire de degré. L’intuition contraire vient de ce que le processus d’interprétation nous force au tout ou rien, ce qui laisse un jeu dans la façon dont le matériel sous-jacent réalise les symboles à des degrés divers. Les travaux de Daniel Dennett sont à cet égard éclairants. Le holisme vaut pour les symboles tels qu’ils sont interprétés, tandis que le gradualisme vaut pour la façon dont le matériel sous-jacent réalise les symboles. (shrink)

This unique anthology of new, contributed essays offers a range of perspectives on various aspects of ontic vagueness. It seeks to answer core questions pertaining to onticism, the view that vagueness exists in the world itself. The questions to be addressed include whether vague objects must have vague identity, and whether ontic vagueness has a distinctive logic, one that is not shared by semantic or epistemic vagueness. The essays in this volume explain the motivations behind onticism, such as the plausibility (...) of mereological vagueness and indeterminacy in quantum mechanics and they offer various arguments both for and against ontic vagueness; onticism is also compared with other, competing theories of vagueness such as semanticism, the view that vagueness exists only in our linguistic representation of the world. Gareth Evans’s influential paper of 1978, “Can There Be Vague Objects?” gave a simple but cogent argument against the coherence of ontic vagueness. Onticism was subsequently dismissed by many. However, in recent years, researchers have become aware of the logical gaps in Evans’s argument and this has triggered a new wave of interest in onticism. Onticism is now widely regarded as at least a coherent view. Reflecting this growing consensus, the present anthology for the first time puts together essays that are focused on onticism and its various facets and it fills in the lacuna in the literature on vagueness, a much-discussed subject in contemporary philosophy. (shrink)

This paper presents a new theory of vagueness, which is designed to retain the virtues of the fuzzy theory, while avoiding the problem of higher-order vagueness. The theory presented here accommodates the idea that for any statement S₁ to the effect that 'Bob is bald' is x true, for x in [0, 1], there should be a further statement S₂ which tells us how true S₁ is, and so on - that is, it accommodates higher-order vagueness without resorting to the (...) claim that the metalanguage in which the semantics of vagueness is presented is itself vague, and without requiring us to abandon the idea that the logic - as opposed to the semantics - of vague discourse is classical. I model the extension of a vague predicate P as a blurry set, this being a function which assigns a degree of membership or degree function to each object o, where a degree function in turn assigns an element of [0, 1] to each finite sequence of elements of [0, 1]. The idea is that the assignment to the sequence (0.3, 0.2), for example, represents the degree to which it is true to say that it is 0.2 true that o is P to degree 0.3. The philosophical merits of my theory are discussed in detail, and the theory is compared with other extensions and generalisations of fuzzy logic in the literature. (shrink)

In a previous paper the concept of disease was fuzzy-logically analyzed and a sketch was given of a prototype resemblance theory of disease (Sadegh-Zadeh (2000). J. Med. Philos., 25:605–38). This theory is outlined in the present paper. It demonstrates what it means to say that the concept of disease is a nonclassical one and, therefore, not amenable to traditional methods of inquiry. The theory undertakes a reconstruction of disease as a category that in contradistinction to traditional views is not based (...) on a set of common features of its members, that is individual diseases, but on a few best examples of the category, called its prototypes, and a similarity relationship such that a human condition is considered a disease if it resembles a prototype. It enables new approaches to resolving many of the stubborn problems associated with the concept of disease. (shrink)

This paper is specialized with fuzzy random multi-objective unbalanced transportation data problems by using of fuzzy programming technique when the sources and destination parameters are fuzzy random variables in inequality type of constraints. In this paper, we focus on the solution procedure of the specified transportation data problems where the objective functions are minimization type and supplies and demands are replaced by the fuzzy random variables. We first convert the fuzzy random multi-objective unbalanced transportation problem into deterministic problem by using (...) fuzzy random chance constraints approach. By introducing the concept of linear membership function of fuzzy programming, multi-objective deterministic transportation problem is converted into single objective deterministic problem and then we solve it and we obtain the optimal compromise solution. Lastly a numerical example is provided for illustration the methodology. (shrink)

Inspired by the operation of human social organisation, this paper presents a new architecture—a pyramid-committee—for developing society-oriented intelligence, whose structure imitates the organisation of human society in its decision making. The system takes a pyramid-like hierarchical structure with links in the pyramid forming a semi-lattice, which relate not only to nodes in the same layer, but also to others in different layers. The output of the system is a result of the negotiation and balancing of different interests. For such a (...) system to function, the main difficulties concern the complicated relationships between different factors or agents. Focussing on the airport environment audit, we discuss the development of a model framework and the role of neural networks. (shrink)

Research into expertise is relatively common in cognitive science concerning expertise existing across many domains. However, much less research has examined how experts within the same domain assess the performance of their peer experts. We report the results of a modified think-aloud study conducted with 18 pilots . Pairs of same-ranked pilots were asked to rate the performance of a captain flying in a critical pre-recorded simulator scenario. Findings reveal considerable variance within performance categories, differences in the process used as (...) evidence in support of a performance rating, different numbers and types of facts identified, and differences in how specific performance events affect choice of performance category and gravity of performance assessment. Such variance is consistent with low inter-rater reliability. Because raters exhibited good, albeit imprecise, reasons and facts, a fuzzy mathematical model of performance rating was developed. The model provides good agreement with observed variations. (shrink)

Propositional dynamic logic (PDL) provides a natural setting for semantics of means-end relations involving non-determinism, but such models do not include probabilistic features common to much practical reasoning involving means and ends. We alter the semantics for PDL by adding probabilities to the transition systems and interpreting dynamic formulas 〈α〉 ϕ as fuzzy predicates about the reliability of α as a means to ϕ. This gives our semantics a measure of efficacy for means-end relations.

Many quantitative scales are constructed using cutoffs on a continuum with scores assigned to the cutoffs. This paper develops a framework for using or constructing such scales from a decision-making standpoint. It addresses questions such as: How many distinct thresholds or cutoffs on a scale (i.e., what levels of granularity) are useful for a rational agent? Where should these thresholds be placed given a rational agent’s preferences and risk-orientation? Do scale score assignments have any bearing on decision-making and if so, (...) how should scores be assigned? Given two possible states of nature , an ordered collection of alternatives from which one is to be selected depending on the probability that A is the case, a simple expected utility condition stipulates when adjacent alternatives are distinguishable and determines the threshold odds separating them. Threshold odds and utilities are mapped onto scale scores via a simple distance model. The placement of the thresholds reflects relative concern over decisional consequences given A versus consequences given ∼ A. Likewise, it is shown that scale scores reflect risk-aversion or risk-seeking not only with respect to A versus ∼ A but also with respect to the rank of the R j . Connections are drawn between this framework and rank-dependent expected utility (RDEU) theory. Implications are adumbrated for both machine and human decision-making. (shrink)

In this paper, notions of fuzzy closure system and fuzzy closure L—system on L—ordered sets are introduced from the fuzzy point of view. We first explore the fundamental properties of fuzzy closure systems. Then the correspondence between fuzzy closure systems and fuzzy closure operators is established. Finally, we study the connections between fuzzy closure systems and fuzzy Galois connections. © 2011 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim.

This paper offers the Boolean many-valued solution to the Sorites Paradox. According to the precisification-based Boolean many-valued theory, from which this solution arises, sentences have not only two truth values, truth (or 1) and falsity (or 0), but many Boolean values between 0 and 1. The Boolean value of a sentence is identified with the set of precisifications in which the sentence is true. Unlike degrees fuzzy logic assigns to sentences, Boolean many values are not linearly but only partially ordered; (...) so there are values that are incomparable. Despite this fact, the sentences in a sorites series can be taken as having values in a linear order, and losing (or gaining) value as we move from sentence to sentence in the series. So there is no sharp borderline between 0 and 1. Assigning Boolean many values instead of the two truth values to sentences for their vagueness values is analogous to assigning propositions instead of the truth values to sentences for their semantic values, as propositions are, from our viewpoint, also Boolean many values. (shrink)

This thesis will utilise the literature on the distinction between doing harm and allowing harm to develop a novel system of moral sequencing that can be applied to general moral problems to decide if, when, and how an agent should intervene to prevent harm from occurring to another agent. Off the back of this discussion, this thesis will offer a way of determining the responsibility of certain agents for their actions within a moral sequence. These motivations will be at the (...) centre of the discussions in this thesis and will be accomplished by ultimately updating the system of moral sequencing so that it makes sense of practical cases of inner-agent change, which itself provides a justification for intervening earlier, and potentially diminishing the responsibility for agents who have experienced an inner-agent change, in a moral sequence. (shrink)

Utilities of health states are often estimated to support public decisions in health care. People’s preferences may be imprecise, for lack of actual trade-off experience. We show how to elicit the utilities accounting for imprecision, discover the main drivers of imprecision, and compare several approaches to modelling health state utility data in the fuzzy setting. We extended the time trade-off questionnaire, to elicit utilities of states defined in the EQ-5D-3L descriptive system in184 respondents. Our study demonstrates that respondents are capable (...) of assessing their own imprecision and rigorous mathematical modelling is possible. The imprecision is larger than as inferred from the standard TTO method and is larger than estimation error, even in our smallish sample. Non-trading in TTO often results from imprecision, rather than lexicographic preferences for longevity over quality. People are especially imprecise in assessing the impact of usual activities on utility; also, the internal inconsistency of a health state increases the imprecision. Fuzzy least squares method seems best suited to assign disutilities to individual dimensions, while separately modelling the location of utility and amount of imprecision seems best to produce value sets. If crisp parameters are estimated, accounting for imprecision changes the results little. (shrink)

Can identity itself be vague? Can there be vague objects? Does a positive answer to either question entail a positive answer to the other? In this paper we answer these questions as follows: No, No, and Yes. First, we discuss Evans’s famous 1978 argument and argue that the main lesson that it imparts is that identity itself cannot be vague. We defend the argument from objections and endorse this conclusion. We acknowledge, however, that the argument does not by itself establish (...) either that there cannot be vague objects or that there cannot be identity statements that are indeterminate for ontic reasons. And we further acknowledge that it does not by itself establish that there cannot be identity statements that are indeterminate in virtue of the existence of vague objects. We then go on to argue that, despite this, one who believes in vague objects cannot endorse Evans’s argument. To establish this we offer supplementary arguments that show that if vague objects exist then identity is vague, and that if identity is vague then vague objects exist. Finally we draw attention to an argument parallel to that of Evans’s, but safer, which can be employed against the putative ontic indeterminacy in identity of vague objects which can be differentiated by identity-free properties. (shrink)

This work proposes an understanding of deductive, default and abductive reasoning as different instances of the same phenomenon: epistemic dynamics. It discusses the main intuitions behind each one of these reasoning processes, and suggest how they can be understood as different epistemic actions that modify an agent’s knowledge and/or beliefs in a different way, making formal the discussion with the use of the dynamic epistemic logic framework. The ideas in this paper put the studied processes under the same umbrella, thus (...) highlighting their relationship and allowing a better understanding of how they interact together. (shrink)

In many applications we have a set of objects together with their properties. Since the available information is usually incomplete and/or imprecise, the true knowledge about subsets of objects can be determined approximately only. In this paper, we discuss a fuzzy generalisation of two pairs of relation-based operators suitable for fuzzy set approximations, which have been recently investigated by Düntsch and Gediga. Double residuated lattices, introduced by Orlowska and Radzikowska, are taken as basic algebraic structures. Main properties of these operators (...) are presented. (shrink)

Logics of binary relations corresponding, among others, to the class RRA of representable relation algebras and the class FRA of full relation algebras are presented together with the proof systems in the style of dual tableaux. Next, the logics are extended with relational constants interpreted as point relations. Applications of these logics to reasoning in non-classical logics are recalled. An example is given of a dual tableau proof of an equation which is RRA-valid, while not RA-valid.

This paper aims to briefly introduce the main idea behind the fuzzy approach and to identify the areas and problems encountered in the humanities that might profit from using this approach. Based on a short overview of selected applications of fuzzy in psychology we identify key areas in which the fuzzy approach has already been applied, and propose a list of general types of problems that the fuzzy approach may provide solutions for in psychology and the humanities in general. These (...) types of problems are illustrated using practical examples. The benefits and possible shortcomings of using the fuzzy approach compared to classical approaches in use today are discussed. The goal of this paper is to indicate areas in research and practice in the humanities, where modern mathematical tools—in this case linguistic fuzzy modeling—have already been used or might prove promising. (shrink)

Evidence Theory is a branch of mathematics that concerns combination of empirical evidence in an individual’s mind in order to construct a coherent picture of reality. Designed to deal with unexpected empirical evidence suggesting new possibilities, evidence theory is compatible with Shackle’s idea of decision-making as a creative act. This essay investigates this connection in detail, pointing to the usefulness of evidence theory to formalise and extend Shackle’s decision theory. In order to ease a proper framing of the issues involved, (...) evidence theory is compared with sub-additive probability theory and Ewens’s infinite alleles model. Furthermore, the original version of evidence theory is presented along with its most recent developments. (shrink)

Learning the basics of the logic of certain and the uncertain is presented as the result of a work of an interdisciplinary team. Our experimentation involves essentially two aspects: language comprehension of a statement and analysis of the information. The first aspect is to see how children interpret a sentence with subject and predicate, that is, if they believe that the truth values that can be assigned are those of bivalent logic, i.e. true or false, or truth-values of a multivalent (...) logic i.e. if there is the possibility/need to consider truth values intermediate between true and false such as: more true than false, more false than true, halfway between true and false. It is also required children to identify phrases not complete and therefore that are not linguistic statements. Regarding the second aspect it comes to analyzing the information on the concept expressed by a proposition, i.e. to see if it is possible to immediately assign to it a truth value, or if it is necessary to acquire a further information. Moreover we highlight the distinction between uncertainty due to incomplete information, which leads to probability assessments, and semantic uncertainty that leads to the theory of fuzzy sets. (shrink)

We sketch the development of fuzzy sets and fuzzy logic in Austria during the last 50 years which started, after some early traces, in 1976 in Linz with my own work and in Vienna with Klaus-Peter Adlassnig’s work. Therefore we first discuss the history of our research group at the Johannes Kepler University Linz and at the JKU-Softwarepark Hagenberg. Next we have a closer look at the developments at the Vienna University Medical School, at the logic group at the Vienna (...) University of Technology, and at the work of Reinhard Viertl in statis- tics. Finally we present some achievements made at the University of Innsbruck. (shrink)

Philosophers, linguists and others interested in problems concerning natural language frequently employ tools from logic and model theory. The question arises as to the proper interpretation of the formal methods employed—of the relationship between, on the one hand, the formal languages and their set-theoretic models and, on the other hand, the objects of ultimate interest: natural language and the meanings and truth conditions of its constituent words, phrases and sentences. Two familiar answers to this question are descriptivism and instrumentalism. More (...) recently, a third answer has been proposed: the logic as modelling view. This paper seeks to clarify and assess this view of logic. The conclusion is that we can successfully adopt the modelling perspective on a given piece of logical machinery only if we have to hand some other machinery to which we take the descriptive attitude. Thus, logic as modelling is not a full-fledged alternative to the descriptive view— for it cannot stand alone: it can at best be an addition to the descriptive perspective. The paper first presents the argument in a general, abstract form, before working through a detailed case study. The case examined is the one with respect to which the logic as modelling view has been developed in the greatest detail in the literature: the case of fuzzy model theory as an account of vagueness in natural language. (shrink)

On the occasion of the 120th anniversary of Mathesis, this work wants to be a memory, a tribute to two great presidents of Mathesis: Bruno de Finetti and Angelo Fadini. Both have pursued the idea of interdisciplinary teaching and research. Bruno de Finetti, with his books on The invention of truth,, and on Logic and Intuitive Mathematics,, and his very famous "Theory of probability",, shows a rejection of formal education, comfortable, monodisciplinary, made of certainties, and chooses the impervious way of (...) addressing the problems that are to the base of science. Angelo Fadini, with his papers and books on Theory of Fuzzy Sets, shows first in Italy several logical questions which puts as the basis for practical applications in Architecture. This paper is an attempt to experiment, in an interdisciplinary framework, the basic ideas of Bruno de Finetti and Angelo Fadini in primary school, in the belief that in the Primary School are formed ideas and intuitions, while in the secondary school the attention is focused mainly on specific issues of Mathematics. We shows some results of a still ongoing experimentation. (shrink)