The dissertation has two parts, each dealing with a problem, namely: 1) What is the most adequate account of fuzziness -the so-called phenomenon of vagueness?, and 2) what is the most plausible solution to the sorites, or heap paradox? I will try to show that fuzzy properties are those which are gradual, amenable to be possessed in a greater or smaller extent. Acknowledgement of degrees in the instantiation of a property allows for a gradual transition from one opposite (...) to the other, each intermediate stage constituting an overlap in certain proportion of both contraries. Hence, degrees in the possession of a property give rise to simple contradictions. The reason why I have chosen those two questions is that they provide the main philosophical motivation for a particular brand of an infinite valued and paraconsistent logic. I will claim that Classical logic (CL) is not adequate to handle fuzzy situations, and, being deficient, is in need of being expanded to make room for degrees of truth and weak contradictions. One can hardly deny the importance of the debate, since what is ultimately at stake is what the limits of truth, rationality, intelligibility and possibility are. The main disciplines within which the research moves are the philosophy of language, philosophy of logic, and ontology. (shrink)

The dissertation has two parts, each dealing with a problem, namely: 1) What is the most adequate account of fuzziness -the so-called phenomenon of vagueness?, and 2) what is the most plausible solution to the sorites, or heap paradox? I will try to show that fuzzy properties are those which are gradual, amenable to be possessed in a greater or smaller extent. Acknowledgement of degrees in the instantiation of a property allows for a gradual transition from one opposite (...) to the other, each intermediate stage constituting an overlap in certain proportion of both contraries. Hence, degrees in the possession of a property give rise to simple contradictions. The reason why I have chosen those two questions is that they provide the main philosophical motivation for a particular brand of an infinite valued and paraconsistent logic. I will claim that Classical logic (CL) is not adequate to handle fuzzy situations, and, being deficient, is in need of being expanded to make room for degrees of truth and weak contradictions. One can hardly deny the importance of the debate, since what is ultimately at stake is what the limits of truth, rationality, intelligibility and possibility are. The main disciplines within which the research moves are the philosophy of language, philosophy of logic, and ontology. (shrink)

This paper focuses on an explanation in a newspaper article: why new European Union citizens will come to the UK from Eastern Europe (e.g., because of available jobs). Using a corpus-based method of analysis, I show how regular target readers have been positioned to generate premises in dialogue with the explanation propositions, and thus into an understanding of the explanation as an argument, one which contains a biased conclusion not apparent in the text. Employing this method, and in particular ‘corpus (...) comparative statistical keywords’, I show how two issues can be freshly looked at: implicit premise recovery; the argument/explanation distinction. (shrink)

A logic is a doctrine of the logos, that is, of meaningful discourse; hence the first thing we expect from it is an account of what makes the logos meaningful — of what a meaning is. There is no single such doctrine or account: it is part of the immense richness of meaningful discourse that we can shift back and forth between several logics — several organized ways of reasoning, of providing reasons or grounds for our claims. Building on (...) previous work on Hegel's dialectical logic, the author here identifies three distinct logics simultaneously in play in our conversations. Analytic logic structures its organization of discourse around negation, contraries, and hence arguments forcing a conclusion to follow (under threat of inconsistency) from some premises. Dialectical logic's main tool is the construction of narratives, hence the attempted incorporation of interlocutors within one's own story. The third option, here labeled gradual logic, sees sorites (which are recalcitrant anomalies for the analytic approach) as ideal cases, since the bleeding of a predicate into an alleged contrary points the way to reaching an agreement among initially conflicting parties: to them eventually coming to regard themselves as stressing different aspects of one and the same thing. (shrink)

Recent work by Marchisella exposed a gap in the literature on causal non-monotonic reasoning: what is needed is an approach whose primary motivation is the formal representation of the way in which humans typically reason with cause and effect. We extend the work of Marchisella, and propose a new type of causal non-monotonic reasoning, _Plausible Causal Reasoning_, which fills the gap in the literature. We propose some new principles which help characterise Plausible Causal Reasoning, and suggest (...) a family of non-monotonic logics which provides a suitable starting point for its formalisation. (shrink)

I comment on two analyses of the Simonshaven case: one by Prakken (2019), based on arguments, and the other by van Koppen and Mackor (2019), based on scenarios (or stories, narratives). I argue that both analyses lack a clear account of proof beyond a reasonable doubt because they lack a clear account of the notion of plausibility. To illustrate this point, I focus on the defense argument during the appeal trial and show that both analyses face difficulties in modeling key (...) features of this argument. (shrink)

De Cock and Kerre, in considering Poincaré paradox, observed that the intuitive notion of "approximate similarity" cannot be adequately represented by the fuzzy equivalence relations. In this note we argue that the deduction apparatus of fuzzy logic gives adequate tools with which to face the question. Indeed, a first-order theory is proposed whose fuzzy models are plausible candidates for the notion of approximate similarity. A connection between these structures and the point-free metric spaces is also established.

Computational approaches to the law have frequently been characterized as being formalistic implementations of the syllogistic model of legal cognition: using insufficient or contradictory data, making analogies, learning through examples and experiences, applying vague and imprecise standards. We argue that, on the contrary, studies on neural networks and fuzzy reasoning show how AI & law research can go beyond syllogism, and, in doing that, can provide substantial contributions to the law.

How should we define the feasible set? Even when individuals agree on facts and values, as traditionally construed, different views on feasibility may suffice to produce very different policy conclusions. Focusing on the difficulties in the feasibility concept may help us resolve some policy disagreements, or at least identify the sources of those disagreements. Feasibility is most plausibly a matter of degree rather than of kind. Normative economic reasoning therefore faces a fuzzy social budget constraint. Iterative reasoning (...) about feasibility and desirability may help us overcome these problems. (shrink)

The term fuzzy logic is used in this paper to describe an imprecise logical system, FL, in which the truth-values are fuzzy subsets of the unit interval with linguistic labels such as true, false, not true, very true, quite true, not very true and not very false, etc. The truth-value set, , of FL is assumed to be generated by a context-free grammar, with a semantic rule providing a means of computing the meaning of each linguistic truth-value in (...) as a fuzzy subset of [0, 1].Since is not closed under the operations of negation, conjunction, disjunction and implication, the result of an operation on truth-values in requires, in general, a linguistic approximation by a truth-value in . As a consequence, the truth tables and the rules of inference in fuzzy logic are (i) inexact and (ii) dependent on the meaning associated with the primary truth-value true as well as the modifiers very, quite, more or less, etc. (shrink)

Metrics and their weaker forms are used to measure the difference between two data. There are many metrics that are available but not desired by a practitioner. This paper recommends in a plausible reasoning manner an easy-to-understand method to construct desired distance-like measures: to fuse easy-to-obtain pseudo-semi-metrics, pseudo-metrics, or metrics by making full use of well-known t-norms, t-conorms, aggregation operators, and similar operators. The simple reason to do this is that data for a real world problem are sometimes (...) from multiagents. A distance-like notion, called weak interval-valued pseudo-metrics, is defined by using known notions of pseudo-semi-metrics, pseudo-metrics, and metrics; this notion is topologically good and shows precision, flexibility, and compatibility than single pseudo-semi-metrics, pseudo-metrics, or metrics. Propositions and detailed examples are given to illustrate how to fabricate an expected or demanded WIVP-metric in practical problems, and WIVP-metric and its special cases are characterized by using axioms. Moreover, some WIVP-metrics pertinent to quantitative logic theory or interval-valued fuzzy graphs are constructed, and fixed point theorems and common fixed point theorems in weak interval-valued metric spaces are also presented. Topics and strategies for further study are also put forward concretely and clearly. (shrink)

In 1685, in The Art of Discovery, Leibniz set down an extraordinary idea: "The only way to rectify our reasonings is to make them as tangible as those of the Mathematicians, so that we can find our error at a glance, and when there are disputes among persons, we can simply say: Let us calculate [calculemus], without further ado, to see who is right." Calculemus.

This article evaluates whether Rescher's rules for plausible reasoning or other rules used in artificial intelligence for "confidence factors" can be extended to deal with arguments where the linked-convergent distinction is important.

Standard models of adolescent risk taking posit that the cognitive abilities of adolescents and adults are equivalent, and that increases in risk taking that occur during adolescence are the result of socio emotional differences in impulsivity, sensation seeking, and lack of self-control. Fuzzy-trace theory incorporates these socio emotional differences. However, it predicts that there are also cognitive differences between adolescents and adults, specifically that there are developmental increases in gist-based intuition that reflects understanding. Gist understanding, as opposed to verbatim-based (...) analysis, generally has been hypothesized to have a protective effect on risk taking in adolescence. Gist understanding is also an essential element of informed consent regarding risks in medical decision- making. Evidence thus supports the argument that adolescents’ status as mature minors should be treated as an exception rather than a presumption, because accuracy in verbatim analysis is not mature gist understanding. Use of the exception should be accompanied by medical experts’ input on the bottom-line gist of risks involved in treatment. (shrink)

As an introduction to our work, we emphasize the parallel interpretation of abstract tools and the concepts of undetermined and vague information. Imprecision, uncertainty and their relationships are inspected. Suitable interpretations of the fuzzy sets theory are applied to legal phenomena in an attempt to clearly circumscribe the possible applications of the theory. The fundamental notion of reference sets is examined in detail, hence highlighting their importance. A systematic and combinatorial classification of the relevant subsets of the legal field (...) is supplied for practical application. Although the use of the fuzzy sets theory is sometimes suggested as a palliative measure (no competition exists), it can also be complementary (serve as a building block to improve modelisation). An Appendix gives a brief recall of the key-concepts of the axiomatic theory of fuzziness and its developments: fuzzy sets, fuzzy logic, fuzzy control and theory of possibility. (shrink)

Analogy making from examples is a central task in intelligent system behavior. A lot of real world problems involve analogy making and generalization. Research investigates these questions by building computer models of human thinking concepts. These concepts can be divided into high level approaches as used in cognitive science and low level models as used in neural networks. Applications range over the spectrum of recognition, categorization and analogy reasoning. A major part of legal reasoning could be formally interpreted (...) as an analogy making process. Because it is not the same as reasoning in mathematics or the physical sciences, it is necessary to use a method, which incorporates first the ability to specify likelihood and second the opportunity of including known court decisions. We use for modelling the analogy making process in legal reasoning neural networks and fuzzy systems. In the first part of the paper a neural network is described to identify precedents of immaterial damages. The second application presents a fuzzy system for determining the required waiting period after traffic accidents. Both examples demonstrate how to model reasoning in legal applications analogous to recent decisions: first, by learning a system with court decisions, and second, by analyzing, modelling and testing the decision making with a fuzzy system. (shrink)

Medical knowledge as well as clinical practice are characterized by inescapable uncertainty. There are many reasons this is the case, but foremost among them is that almost everything in medicine is inevitably vague, be it something linguistic such as the term “illness”, or something extra-linguistic such as the condition referred to as illness. If we ask ourselves, then, what the term “illness” means exactly, on the one hand; and how we may precisely delimit the condition illness, on the other; we (...) shall realize that to answer these and similar questions requires specific methods that enable us to adequately cope with vagueness. As we shall see below, fuzzy logic provides us with just such methods. (shrink)

The plausible reasoning class (called the JSM-reasoning in honour of John Stuart Mill) is described. It implements interaction of three forms of non-deductive procedures  induction, analogy and abduction. Empirical induction in the JSM-reasoning is the basis for generation of hypotheses on causal relations (determinants of social behaviour). Inference by analogy means that predictions about previously unknown properties of objects (individual’s behaviour) are inferred from causal relations. Abductive inference is performed to check on the explanatory adequacy (...) of generated hypotheses. To recognize rationality of respondents’ opinion deductive inference is used. Plausible reasoning, semantics of argumentation logic and deductive recognition of opinion rationality represent logical tool for cognitive sociology problems. (shrink)

A common reason for giving research participants post-trial access to the trial intervention appeals to reciprocity, the principle, stated most generally, that if one person benefits a second, the second should reciprocate: benefit the first in return. Many authors consider it obvious that reciprocity supports PTA. Yet their reciprocity principles differ, with many authors apparently unaware of alternative versions. This article is the first to gather the range of reciprocity principles. It finds that: most are false. The most plausible (...) principle, which is also problematic, applies only when participants experience significant net risks or burdens. Seldom does reciprocity support PTA for participants or give researchers stronger reason to benefit participants than equally needy non-participants. Reciprocity fails to explain the common view that it is bad when participants in a successful trial have benefited from the trial intervention but lack PTA to it. (shrink)

Plausible (eikotic) reasoning known from ancient Greek (late Academic) skeptical philosophy is shown to be a clear notion that can be analyzed by argumentation methods, and that is important for argumentation studies. It is shown how there is a continuous thread running from the Sophists to the skeptical philosopher Carneades, through remarks of Locke and Bentham on the subject, to recent research in artificial intelligence. Eleven characteristics of plausible reasoning are specified by analyzing key examples of (...) it recognized as important in ancient Greek skeptical philosophy using an artificial intelligence model called the Carneades Argumentation System (CAS). By applying CAS to ancient examples it is shown how plausible reasoning is especially useful for gaining a better understanding of evidential reasoning in law, and argued that it can also be applied to everyday argumentation. Our analysis of the snake and rope example of Carneades is also used to point out some ways CAS needs to be extended if it is to more fully model the views of this ancient philosopher on argumentation. (shrink)

The book can also be used as an excellent text for graduate-level courses in AI, operations research, or applied probability.

This paper has two goals. First, we develop frameworks for logical systems which are able to reflect not only non-monotonic patterns of reasoning, but also paraconsistent reasoning. Our second goal is to have a better understanding of the conditions that a useful relation for nonmonotonic reasoning should satisfy. For this we consider a sequence of generalizations of the pioneering works of Gabbay, Kraus, Lehmann, Magidor and Makinson. These generalizations allow the use of monotonic nonclassical logics as the (...) underlying logic upon which nonmonotonic reasoning may be based. Our sequence of frameworks culminates in what we call plausible, nonmonotonic, multiple-conclusion consequences relations . Our study yields intuitive justifications for conditions that have been proposed in previous frameworks and also clarifies the connections among some of these systems. In addition, we present a general method for constructing plausible nonmonotonic relations. This method is based on a multiple-valued semantics, and on Shoham's idea of preferential models. (shrink)

A new criterion is introduced for judging the suitability of various fuzzy logics for practical uncertain reasoning in a probabilistic world and the relationship of this criterion to several established criteria, and its consequences for truth functional belief, are investigated.

Despite overwhelming evidence suggesting that quantifier scope is a phenomenon that must be treated at the pragmatic level, most computational treatments of scope ambiguities have thus far been a collection of syntactically motivated preference rules. This might be in part due to the prevailing wisdom that a commonsense inferencing strategy would require the storage of and reasoning with a vast amount of background knowledge. In this paper we hope to demonstrate that the challenge in developing a commonsense inferencing strategy (...) is in the discovery of the relevant commonsense data and in a proper formulation of the inferencing strategy itself, and that a massive amount of background knowledge is not always required. In particular, we present a very effective procedure for resolving quantifier scope ambiguities at the pragmatic level using simple quantitative data that is readily available in most database environments. (shrink)

A many-valued sentential logic with truth values in an injective MV-algebra is introduced and the axiomatizability of this logic is proved. The paper develops some ideas of Goguen and generalizes the results of Pavelka on the unit interval. The proof for completeness is purely algebraic. A corollary of the Completeness Theorem is that fuzzy logic on the unit interval is semantically complete if and only if the algebra of the truth values is a complete MV-algebra. In the well-defined (...) class='Hi'>fuzzy sentential logic holds the Compactness Theorem, while the Deduction Theorem and the Finiteness Theorem in general do not hold. Because of its generality and good-behaviour, MV-valued logic can be regarded as a mathematical basis of fuzzy reasoning. (shrink)

In a legal expert system based on CBR (Case-Based Reasoning), legal statute rules are interpreted on the basis of precedents. This interpretation, because of its vagueness and uncertainty of the interpretation cannot be handled with the means used for crisp cases. In our legal expert system, on the basis of the facts of precedents, the statute rule is interpreted as a form of case rule, the application of which involves the concepts of membership and vagueness. The case rule is (...) stored in a data base by means of fuzzy frames. The inference based on a case rule is made by fuzzy YES and fuzzy NO, and the degree of similarity of cases. The system proposed here will be used for legal education; its main area of application is contract, especially in relation to the United Nations Convention on Contracts for the International Sale of Goods (CISG). (shrink)

Although Fuzzy logic and Fuzzy Mathematics is a widespread subject and there is a vast literature about it, yet the use of Fuzzy issues like Fuzzy sets and Fuzzy numbers was relatively rare in time concept. This could be seen in the Fuzzy time series. In addition, some attempts are done in fuzzing Turing Machines but seemingly there is no need to fuzzy time. Throughout this article, we try to change this picture and (...) show why it is helpful to consider the instants of time as Fuzzy numbers. In physics, though there are revolutionary ideas on the time concept like B theories in contrast to A theory also about central concepts like space, momentum… it is a long time that these concepts are changed, but time is considered classically in all well-known and established physics theories. Seemingly, we stick to the classical time concept in all fields of science and we have a vast inertia to change it. Our goal in this article is to provide some bases why it is rational and reasonable to change and modify this picture. Here, the central point is the modified version of “Unexpected Hanging” paradox as it is described in "Is classical Mathematics appropriate for theory of Computation".This modified version leads us to a contradiction and based on that it is presented there why some problems in Theory of Computation are not solved yet. To resolve the difficulties arising there, we have two choices. Either “choosing” a new type of Logic like “Para-consistent Logic” to tolerate contradiction or changing and improving the time concept and consequently to modify the “Turing Computational Model”. Throughout this paper, we select the second way for benefiting from saving some aspects of Classical Logic. In chapter 2, by applying quantum Mechanics and Schrodinger equation we compute the associated fuzzy number to time. (shrink)

In their unifying theory to model uncertainty, Friedman and Halpern (1995–2003) applied plausibility measures to default reasoning satisfying certain sets of axioms. They proposed a distinctive condition for plausibility measures that characterizes “qualitative” reasoning (as contrasted with probabilistic reasoning). A similar and similarly fundamental, but more general and thus stronger condition was independently suggested in the context of “basic” entrenchment-based belief revision by Rott (1996–2003). The present paper analyzes the relation between the two approaches to formalizing basic (...) notions of plausibility as used in qualitative default reasoning. While neither approach is a special case of the other, translations can be found that elucidate their relationship. I argue that Rott’s notion of plausibility allows for a more modular set-up and has a better philosophical motivation than that of Friedman and Halpern. (shrink)

In his important recent book Schroeder proposes a Humean theory of reasons that he calls hypotheticalism. His rigourous account of the weight of reasons is crucial to his theory, both as an element of the theory and constituting his defence to powerful standard objections to Humean theories of reasons. In this paper I examine that rigourous account and show it to face problems of vacuity and consonance. There are technical resources that may be brought to bear on the problem of (...) vacuity but implementation is not simple and philosophical motivation a further difficulty. Even supposing vacuity is fixed, the problems of consonance bring to light a different obstruction lying in Schroeder’s path. There is a difference between the general weighing of reasons and the context specificity of the correct placing of weight on them in deliberation and this difference cannot be fixed by the resources in the account. For these reasons we are still waiting for a plausible Humean theory of reasons. (shrink)

Although Fuzzy logic and Fuzzy Mathematics is a widespread subject and there is a vast literature about it, yet the use of Fuzzy issues like Fuzzy sets and Fuzzy numbers was relatively rare in time concept. This could be seen in the Fuzzy time series. In addition, some attempts are done in fuzzing Turing Machines but seemingly there is no need to fuzzy time. Throughout this article, we try to change this picture and (...) show why it is helpful to consider the instants of time as Fuzzy numbers. In physics, though there are revolutionary ideas on the time concept like B theories in contrast to A theory also about central concepts like space, momentum… it is a long time that these concepts are changed, but time is considered classically in all well-known and established physics theories. Seemingly, we stick to the classical time concept in all fields of science and we have a vast inertia to change it. Our goal in this article is to provide some bases why it is rational and reasonable to change and modify this picture. Here, the central point is the modified version of “Unexpected Hanging” paradox as it is described in "Is classical Mathematics appropriate for theory of Computation".This modified version leads us to a contradiction and based on that it is presented there why some problems in Theory of Computation are not solved yet. To resolve the difficulties arising there, we have two choices. Either “choosing” a new type of Logic like “Para-consistent Logic” to tolerate contradiction or changing and improving the time concept and consequently to modify the “Turing Computational Model”. Throughout this paper, we select the second way for benefiting from saving some aspects of Classical Logic. In chapter 2, by applying quantum Mechanics and Schrodinger equation we compute the associated fuzzy number to time. These, provides a new interpretation of Quantum Mechanics.More exactly what we see here is "Particle-Fuzzy time" interpretation of quantum Mechanics, in contrast to some other interpretations of Quantum Mechanics like " Wave-Particle" interpretation. At the end, we propound a question about the possible solution of a paradox in Physics, the contradiction between General Relativity and Quantum Mechanics. (shrink)

Computability theorists have introduced multiple hierarchies to measure the complexity of sets of natural numbers. The Kleene Hierarchy classifies sets according to the first-order complexity of their defining formulas. The Ershov Hierarchy classifies Δ20\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varDelta ^0_2$$\end{document} sets with respect to the number of mistakes that are needed to approximate them. Biacino and Gerla extended the Kleene Hierarchy to the realm of fuzzy sets, whose membership functions range in a complete lattice L. (...) In this paper, we combine the Ershov Hierarchy and fuzzy set theory, by introducing and investigating the Fuzzy Ershov Hierarchy. In particular, we focus on the fuzzy n-c.e. sets which form the finite levels of this hierarchy. Intuitively, a fuzzy set is n-c.e. if its membership function can be approximated by changing monotonicity at most n-1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n-1$$\end{document} times. We prove that the Fuzzy Ershov Hierarchy does not collapse; that, in analogy with the classical case, each fuzzy n-c.e. set can be represented as a Boolean combination of fuzzy c.e. sets; but that, contrary to the classical case, the Fuzzy Ershov Hierarchy does not exhaust the class of all Δ20\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varDelta ^0_2$$\end{document} fuzzy sets. (shrink)

The method of approximate reasoning using a fuzzy logic introduced by Baldwin (1978 a,b,c), is used to model human reasoning in the resolution of two well known paradoxes. It is shown how classical propositional logic fails to resolve the paradoxes, how multiple valued logic partially succeeds and that a satisfactory resolution is obtained with fuzzy logic. The problem of precise representation of vague concepts is considered in the light of the results obtained.