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 connection between similarity logic and the theory of closure operators is examined. Indeed one proves that the consequence relation defined in [14] can be obtained by composing two closure operators and that the resulting operator is still a closure operator. Also, we extend any similarity into a similarity which is compatible with the logical equivalence, and we prove that this gives the same consequence relation.

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)

Several methods of approximate reasoning are known, and many remain to be discovered across domains. Some are tied to reasoning with uncertainty. The developments over the last fifty years or so in the development of approximate reasoning methods, and their application across multiple STEM domains, suggest that it is necessary to introduce them early in school. While efforts towards building the infrastructure for the process have been limited, the bigger question is “What should be introduced?”. (...) If concepts are always approximated in real life, then how should they be done mathematically? If the real numbers are not always necessary to approximate concepts or even quantities, then how should they be approximated, and what should be taught in schools? Specifically, functional representations of approximate reasoning are not introduced early. This additionally impedes diversity, ethnomathematical explorations, learning from experience, and through models. The objective of this research is to explain the problem with a focus on the ethnomathematics of cooking, and associated haptic methods. Further, it is argued through the context that of the many approaches to approximate reasoning, intrusive methods are best avoided, and that general rough sets is better suited for modeling such knowledge. Intrusive methods, such as those based on fuzzy or probability theory, are those that indulge in an excess of unjustified numeric assumptions. This builds on earlier work of the present author on modeling approximate reasoning. Her axiomatic approach to granularity are additionally applicable to approximate recipes that possess desired properties. In the contexts of ethnomathematical discourse involving haptic reasoning (as in cooking), it is shown that such an approach can model the essence of context without oversimplifications or outright dismissal of the efforts involved. Further, it has the potential to help in inventing expressive structured languages for the ethnomathematics, and associated model eliciting activities. Distinct facets of three specific contexts are theoretically explored to illustrate aspects of the reasoning, and meta aspects. (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.

This paper describes an experimental platform for approximate knowledge databases called the Approximate Knowledge Database, based on a semantics inspired by rough sets. The implementation is based upon the use of a standard SQL database to store logical facts, augmented with several query interface layers implemented in JAVA through which extensional, intensional and local closed world nonmonotonic queries in the form of crisp or approximate logical formulas can be evaluated tractably. A graphical database design user interface is (...) also provided which simplifies the design of databases, the entering of data and the construction of queries. The theory and semantics for AKDBs is presented in addition to application examples and details concerning the database implementation. (shrink)

This short paper about fuzzy set-based approximate reasoning first emphasizes the three main semantics for fuzzy sets: similarity, preference and uncertainty. The difference between truth-functional many-valued logics of vague or gradual propositions and non fully compositional calculi such as possibilistic logic or similarity logics is stressed. Then, potentials of fuzzy set-based reasoning methods are briefly outlined for various kinds of approximate reasoning: deductive reasoning about flexible constraints, reasoning under uncertainty and inconsistency, hypothetical (...) class='Hi'>reasoning, exception-tolerant plausible reasoning using generic knowledge, interpolative reasoning, and abductive reasoning. Open problems are listed in the conclusion. (shrink)

It has long been recognized that the concept of inconsistency is a central part of commonsense reasoning. In this issue, a number of authors have explored the idea of reasoning with maximal consistent subsets of an inconsistent stratified knowledge base. This paradigm, often called “coherent-based reasoning", has resulted in some interesting proposals for para-consistent reasoning, non-monotonic reasoning, and argumentation systems. Unfortunately, coherent-based reasoning is computationally very expensive. This paper harnesses the approach of approximate (...) entailment by Schaerf and Cadoli [SCH 95] to develop the concept of “approximate coherent-based reasoning". To this end, we begin to present a multi-modal propositional logic that incorporates two dual families of modalities: □S and?S defined for each subset S of the set of atomic propositions. The resource parameter S indicates what atoms are taken into account when evaluating formulas. Next, we define resource-bounded consolidation operations that limit and control the generation of maximal consistent subsets of a stratified knowledge base. Then, we present counterparts to existential, universal, and argumentative inference that are prominent in coherence-based approaches. By virtue of modalities □S and?S, these inferences are approximated from below and from above, in an incremental fashion. Based on these features, we show that an anytime view of coherent-based reasoning is tenable. (shrink)

There is good reason to suppose that our best physical theories, quantum mechanics and special relativity, are false if taken together and literally. If they are in fact false, then how should they count as providing knowledge of the physical world? One might imagine that, while strictly false, our best physical theories are nevertheless in some sense probably approximately true. This paper presents a notion of local probable approximate truth in terms of descriptive nesting relations between current and subsequent (...) theories. This notion helps explain how false physical theories might nevertheless provide physical knowledge of a variety that is particularly salient to diachronic empirical inquiry. (shrink)

Since Aristotle it is recognised that a valid syllogism cannot have two particular premises. However, that is not how a lay person sees it; at least as long as the premises read many, most etc, instead of a plain some. The lay people are right if one considers that these syllogisms do not have strict but approximate (Zadeh) validity. Typically there are only particular premises available in everyday life and one is dependent on such syllogisms. – Some rules on (...) the usage of particular premises are given below. (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.

The main question addressed in this paper is whether some false sentences can constitute evidence for the truth of other propositions. In this paper it is argued that there are good reasons to suspect that at least some false propositions can constitute evidence for the truth of certain other contingent propositions. The paper also introduces a novel condition concerning propositions that constitute evidence that explains a ubiquitous evidential practice and it contains a defense of a particular condition concerning the possession (...) of evidence. The core position adopted here then is that false propositions that are approximately true reports of measurements can constitute evidence for the truth of other propositions. So, it will be argued that evidence is only quasi-factive in this very specific sense. (shrink)

We investigate the logical and conceptual connections between abductive reasoning construed as a process of belief change, on the one hand, and truth approximation, construed as increasing (estimated) verisimilitude, on the other. We introduce the notion of ‘(verisimilitude-guided) abductive belief change’ and discuss under what conditions abductively changing our theories or beliefs does lead them closer to the truth, and hence tracks truth approximation conceived as the main aim of inquiry. The consequences of our analysis for some recent discussions (...) concerning belief revision aiming at truth approximation and inference to the best explanation are also highlighted. (shrink)

The standard theory for belief revision provides an elegant and powerful framework for reasoning about how a rational agent should change its beliefs when confronted with new information. However, the agents considered are extremely idealized. Some recent models attempt to tackle the problem of plausible belief revision by adding structure to the belief bases and using nonstandard inference operations. One of the key ideas is that not all of an agent's beliefs are relevant for an operation of belief change.In (...) this paper we incorporate the insights pertaining to local change and relevance sensitivity with the use of approximate inference relations. These approximate inference relations offer us partial solutions at any stage of the revision process. The quality of the approximations improves as we allow for more and more resources to be used. We are provided with upper and lower bounds to what would be obtained with the use of classical inference. (shrink)

What role does our knowledge about the ideal society play in guiding policymaking in the real world? One intuitive answer is to approximate. Namely, we have a duty to approximate the ideal within the relevant constraints of feasibility. However, political philosophers seem to have what might be called ‘approximatophobia'. Many philosophers, including idealists such as David Estlund, warn against approximation. Their criticism is chiefly motivated by ‘the problem of second best’, which points out that your second-best option may (...) not be closest to your best option. This paper aims to dispel ‘approximatophobia'. The difficulty posed by the problem of second best is often overstated. More positively, I present a novel defence of approximation, arguing that approximation of an ideal can be a reasonably reliable default strategy of action guidance in the real world. Difficulties that may afflict the project of approximation can be mitigated by sophistication of the project of approximation. After showing that critics of approximation overstep their mark in issuing the strong or moderate warning against approximation, I propose an account of sophisticated approximation. It seeks a series of reforms that make existing social institutions closer to the ideal based on careful selection of frame of analysis. Three virtuous correlations are identified as indicators for successful project of approximation, linking descriptive similarity to desirability, feasibility and knowledge of the ideal. I also explain a two-staged strategy of sophisticating the way you approximate the ideal, with an expected positive feedback effect. (shrink)

Rough sets are used in numerous knowledge representation contexts and are then empowered with varied ontologies. These may be intrinsically associated with ideas of rationality under certain conditions. In recent papers, specific granular generalisations of graded and variable precision rough sets are investigated by the present author from the perspective of rationality of approximations (and the associated semantics of rationality in approximate reasoning). The studies are extended to ideal-based approximations (sometimes referred to as subsethood-based approximations). It is additionally (...) shown that co-granular or point-wise approximations defined by σ-ideals/filters (for an arbitrary relation σ) fit easily into the entire scheme. Concepts of the rationality of objects (vague or crisp) and their types are introduced and are shown to be applicable to most general rough sets by the present author. Surprising results on these are proved on these by her in this part of the research paper. The present paper is the first of a three part study on the topic. (shrink)

Journal of Mathematical Logic, Volume 22, Issue 03, December 2022. We study a new class of NP search problems, those which can be proved total using standard combinatorial reasoning based on approximate counting. Our model for this kind of reasoning is the bounded arithmetic theory [math] of [E. Jeřábek, Approximate counting by hashing in bounded arithmetic, J. Symb. Log. 74(3) (2009) 829–860]. In particular, the Ramsey and weak pigeonhole search problems lie in the new class. We (...) give a purely computational characterization of this class and show that, relative to an oracle, it does not contain the problem CPLS, a strengthening of PLS. As CPLS is provably total in the theory [math], this shows that [math] does not prove every [math] sentence which is provable in bounded arithmetic. This answers the question posed in [S. Buss, L. A. Kołodziejczyk and N. Thapen, Fragments of approximate counting, J. Symb. Log. 79(2) (2014) 496–525] and represents some progress in the program of separating the levels of the bounded arithmetic hierarchy by low-complexity sentences. Our main technical tool is an extension of the “fixing lemma” from [P. Pudlák and N. Thapen, Random resolution refutations, Comput. Complexity, 28(2) (2019) 185–239], a form of switching lemma, which we use to show that a random partial oracle from a certain distribution will, with high probability, determine an entire computation of a [math] oracle machine. The introduction to the paper is intended to make the statements and context of the results accessible to someone unfamiliar with NP search problems or with bounded arithmetic. (shrink)

What role does our knowledge about the ideal society play in guiding policymaking in the real world? One intuitive answer is to approximate. Namely, we have a duty to approximate the ideal within the relevant constraints of feasibility. However, political philosophers seem to have what might be called ‘approximatophobia'. Many philosophers, including idealists such as David Estlund, warn against approximation. Their criticism is chiefly motivated by ‘the problem of second best’, which points out that your second-best option may (...) not be closest to your best option. This paper aims to dispel ‘approximatophobia'. The difficulty posed by the problem of second best is often overstated. More positively, I present a novel defence of approximation, arguing that approximation of an ideal can be a reasonably reliable default strategy of action guidance in the real world. Difficulties that may afflict the project of approximation can be mitigated by sophistication of the project of approximation. After showing that critics of approximation overstep their mark in issuing the strong or moderate warning against approximation, I propose an account of sophisticated approximation. It seeks a series of reforms that make existing social institutions closer to the ideal based on careful selection of frame of analysis. Three virtuous correlations are identified as indicators for successful project of approximation, linking descriptive similarity to desirability, feasibility and knowledge of the ideal. I also explain a two-staged strategy of sophisticating the way you approximate the ideal, with an expected positive feedback effect. (shrink)

Movements, in the last fifty years, have been bringing knowledge closer to the new knowledge coming from the neurosciences. In this brief essay, we intend to analyze neuropsychoanalysis and neuroaesthetics, as promising areas in these terms. Such an analysis will be based on critical theory, conceptualized mainly by Adorno and Horkheimer, especially considering that “On the path to modern science, men renounced meaning and replaced the concept with the formula, the cause with the rule and probability”. Therefore, in relation to (...) neuropsychoanalysis, its historical and scientific foundations, the international movement that gave rise to the area, its purposes and objections attributed to it will be analyzed. Regarding neuroaesthetics, its emergence, its relations with conceptual art and the neurohistory of art will be presented, as well as possible intrinsic parallels to sensation and experience. Finally, following the objective of the essay, the epistemological critique of these areas will be done through the concepts of Gestell (Heidegger) and Ecosofia (Guattari), as well as supported by Valéry and Teresa D’Ávila. (shrink)

Hájek (Erkenntnis 70(2):211–235, 2009) argues that probabilities cannot be the limits of relative frequencies in counterfactual infinite sequences. I argue for a different understanding of these limits, drawing on Norton’s (Philos Sci 79(2):207–232, 2012) distinction between approximations (inexact descriptions of a target) and idealizations (separate models that bear analogies to the target). Then, I adapt Hájek’s arguments to this new context. These arguments provide excellent reasons not to use hypothetical frequencies as idealizations, but no reason not to use them as (...) approximations. (shrink)

Frames for interpreting situations are necessary in the face of time constraints for action and indeterminacy of the “right or optimal thing to do” given multiple objectives but not all frames are equally useful. We need a way of modeling representational frames according to the informational gain of using them and the computational cost of synthesizing a decisive reason for acting from them.

Some theories of rational belief assume that beliefs should be closed under conjunction. I motivate the rejection of the conjunction closure, and point out that the consequences of this rejection are not as severe as it is usually thought. An often raised objection is that without the conjunction closure people are unable to reason. I outline an approach in which we can – in usual cases – reason using conjunctions without accepting the closure in its whole generality. This solution is (...) based on the notion of confidence levels, which can be defined using probabilities. Moreover, on this approach, reasoning has a scalable computational complexity adaptable to cognitive abilities of both rationally bounded and perfectly rational agents. I perform a simulation to assess its error rate, and compare it to reasoning with conjunction closure. (shrink)

Three related intuitions are explicated in this paper. The first is the idea that there must be some kind of probabilistic version of the HD-method, a ‘Hypothetico-Probabilistic (HP-) method’, in terms of something like probabilistic consequences, instead of deductive consequences. According to the second intuition, the comparative application of this method should also be functional for some probabilistic kind of empirical progress, and according to the third intuition this should be functional for something like probabilistic truth approximation. In all three (...) cases, the guiding idea is to explicate these intuitions by explicating the crucial notions as appropriate ‘concretizations’ of their deductive analogs, being ‘idealizations’. It turns out that the comparative version of the proposed HP-method amounts to the likelihood comparison (LC-) method applied to the cumulated evidence. This method turns out to be not only functional for probabilistic empirical progress but also for probabilistic truth approximation. The latter is based on a probabilistic threshold theorem constituting for this reason the analog of the deductive success theorem. (shrink)

Three related intuitions are explicated in this paper. The first is the idea that there must be some kind of probabilistic version of the HD-method, a 'Hypothetico-Probabilistic method', in terms of something like probabilistic consequences, instead of deductive consequences. According to the second intuition, the comparative application of this method should also be functional for some probabilistic kind of empirical progress, and according to the third intuition this should be functional for something like probabilistic truth approximation. In all three cases, (...) the guiding idea is to explicate these intuitions by explicating the crucial notions as appropriate 'concretizations' of their deductive analogs, being 'idealizations'. It turns out that the comparative version of the proposed HP-method amounts to the likelihood comparison method applied to the cumulated evidence. This method turns out to be not only functional for probabilistic empirical progress but also for probabilistic truth approximation. The latter is based on a probabilistic threshold theorem constituting for this reason the analog of the deductive success theorem. (shrink)

To a first approximation, cognitive science agrees with everyday notions about reasoning: According to both views, reasoning is a special sort of relation between beliefs – a relation that holds when accepting (or rejecting) one or more beliefs causes others to be accepted (rejected). If you learn, for example, that everyone dislikes iguana pudding, that should increase the likelihood of your believing that Calvin, in particular, dislikes iguana pudding. Reasoning could produce an entirely new belief about Calvin's (...) attitude toward the pudding, or it could modify an old one. In either case, accepting the second idea on the basis of the first exemplifies reasoning of the simplest sort. More complex reasoning results from chains of such changes. (Café Maudit serves everything Calvin dislikes. So, since everybody dislikes iguana pudding, Calvin does; since Calvin does, Café Maudit serves it.). (shrink)

Causal reasoning is an aspect of learning, reasoning, and decision-making that involves the cognitive ability to discover relationships between causal relata, learn and understand these causal relationships, and make use of this causal knowledge in prediction, explanation, decision-making, and reasoning in terms of counterfactuals. Can we fully automate causal reasoning? One might feel inclined, on the basis of certain groundbreaking advances in causal epistemology, to reply in the affirmative. The aim of this paper is to demonstrate (...) that one still has good skeptical grounds for resisting any conclusions in favour of the automation of causal reasoning. If by causal reasoning is meant the entirety of the process through which we discover causal relationships and make use of this knowledge in prediction, explanation, decision-making, and reasoning in terms of counterfactuals, then one relies besides on tacit knowledge, as might be constituted by or derived from the epistemic faculty virtues and abilities of the causal reasoner, the value systems and character traits of the causal reasoner, the implicit knowledge base available to the causal reasoner, and the habits that sustain our causal reasoning practices. While certain aspects of causal reasoning may be axiomatized and formalized and algorithms may be implemented to approximate causal reasoning, one has to remain skeptical about whether causal reasoning may be fully automated. This demonstration will involve an engagement with Meno’s Paradox. (shrink)

It is shown here that the microcanonical ensemble for a system of noninteracting bosons and fermions contains a subensemble of state vectors for which all particles of the system are distinguishable. This “IQC” (inner quantum-classical) subensemble is therefore fully classical, except for a rather extreme quantization of particle momentum and position, which appears as the natural price that must be paid for distinguishability. The contribution of the IQC subensemble to the entropy is readily calculated, and the criterion for this to (...) be a good approximation to the exact entropy is a logarithmically strengthened form of the usual criterion for the validity of classical statistics in terms of the thermal de Broglie wavelength and the average volume per particle. Thus, it becomes possible to derive the Maxwell-Boltzmann distribution directly from the ensemble in the classical limit, using fully classical reasoning about the distinguishability of particles. The entropy is additive—theN! factor of the Boltzmann count cancels out in the course of the calculation, and the “N! paradox” is thereby resolved. The method of “correct Boltzmann counting” and the lowest term of the Wigner-Kirkwood series for the partition function are seen to be partly based on the IQC subensemble, and their partly nonclassical nature is clarified. The clear separation in the full ensemble of classical and nonclassical components makes it possible to derive the classical statistics of indistinguishable particles from their quantum statistics in a controlled, explicit way. This is particularly important for nonequilibrium theory. The treatment of molecular collisions along too-literally classical lines turns out to require exorbitantly high temperatures, although there are suggestions of indirect ways in which classical nonequilibrium theory might be justified at ordinary temperatures. The applicability of exact classical ergodic and mixing theory to systems at ordinary temperatures is called into question, although the general idea of coarse-graining is confirmed. The concepts on which the IQC idea is based are shown to give rise to a series development of thermostatistical quantities, starting with the distinguishable-particle approximation. (shrink)

There is good reason to suppose that our best physical theories are false: In addition to its own internal problems, the standard formulation of quantum mechanics is logically incompatible with special relativity. I will also argue that we have no concrete idea what it means to claim that these theories are approximately true.

Jay Asher’s novel Thirteen Reasons Why and its Netflix adaptation have enjoyed widespread popularity. While they draw needed attention to issues like bullying and teen estrangement, they may have an unintended effect: they mislead about the etiology of suicide and even glamorize it to a degree. The medical literature has shown that suicide is almost always the result of psychiatric disorder, not provocative stress, in much the same way an asthmatic crisis is primarily the result of an underlying medical condition, (...) not an allergic stimulus. Both the novel and Netflix series ignore this premise and even belittle the idea. Thus, while the story has artistic merits, it also has the potential to be destructive if accessed by young readers and viewers seeking guidance. Approximately ten percent of teens suffer from depression, and suicide recently surpassed homicide as the second-leading cause of death among persons ages ten to twenty-four in the United States. A more balanced view addressing these misconceptions is provided for teens, parents, and mental health professionals. (shrink)

The qualitative theory of nomic truth approximation, presented in Kuipers in his, in which ‘the truth’ concerns the distinction between nomic, e.g. physical, possibilities and impossibilities, rests on a very restrictive assumption, viz. that theories always claim to characterize the boundary between nomic possibilities and impossibilities. Fully recognizing two different functions of theories, viz. excluding and representing, this paper drops this assumption by conceiving theories in development as tuples of postulates and models, where the postulates claim to exclude nomic impossibilities (...) and the models claim to represent nomic possibilities. Revising theories becomes then a matter of adding or revising models and/or postulates in the light of increasing evidence, captured by a special kind of theories, viz. ‘data-theories’. Under the assumption that the data-theory is true, achieving empirical progress in this way provides good reasons for the abductive conclusion that truth approximation has been achieved as well. Here, the notions of truth approximation and empirical progress are formally direct generalizations of the earlier ones. However, truth approximation is now explicitly defined in terms of increasing truth-content and decreasing falsity-content of theories, whereas empirical progress is defined in terms of lasting increased accepted and decreased rejected content in the light of increasing evidence. These definitions are strongly inspired by a paper of Gustavo Cevolani, Vincenzo Crupi and Roberto Festa, viz., “Verisimilitude and belief change for conjunctive theories” :183–222, 2011). (shrink)

This is the first part of a two-part paper in which I conclude the process, initiated elsewhere, of tracking objective conditions of vagueness of representation from language to pictures, from philosophy to imaging science, from vagueness to approximation, from representation to reasoning, with a focus on the application of fuzzy set theory and its challenges.

We are concerned with formal models of reasoning under uncertainty. Many approaches to this problem are known in the literature e.g. Dempster-Shafer theory [29], [42], bayesian-based reasoning [21], [29], belief networks [29], many-valued logics and fuzzy logics [6], non-monotonic logics [29], neural network logics [14]. We propose rough mereology developed by the last two authors [22-25] as a foundation for approximate reasoning about complex objects. Our notion of a complex object includes, among others, proofs understood as (...) schemes constructed in order to support within our knowledge assertions/hypotheses about reality described by our knowledge incompletely. (shrink)

We are concerned with formal models of reasoning under uncertainty. Many approaches to this problem are known in the literature e.g. Dempster-Shafer theory [29], [42], bayesian-based reasoning [21], [29], belief networks [29], many-valued logics and fuzzy logics [6], non-monotonic logics [29], neural network logics [14]. We propose rough mereology developed by the last two authors [22-25] as a foundation for approximate reasoning about complex objects. Our notion of a complex object includes, among others, proofs understood as (...) schemes constructed in order to support within our knowledge assertions/hypotheses about reality described by our knowledge incompletely. (shrink)

The study of deductive reasoning has been a major paradigm in psychology for approximately the past 40 years. Research has shown that people make many logical errors on such tasks and are strongly influenced by problem content and context. It is argued that this paradigm was developed in a context of logicist thinking that is now outmoded. Few reasoning researchers still believe that logic is an appropriate normative system for most human reasoning, let alone a model for (...) describing the process of human reasoning, and many use the paradigm principally to study pragmatic and probabilistic processes. It is suggested that the methods used for studying reasoning be reviewed, especially the instructional context, which necessarily defines pragmatic influences as biases. (shrink)

Many philosophers appeal to the “fallacy of approximation”, or “problem of second best”. However, despite the pervasiveness of such appeals, there has been only a single attempt to provide a systematic account of what the fallacy is. We identify the shortcomings of this account and propose a better one in its place. Our account not only captures all the contexts in which approximation-based reasoning occurs but also systematically explains the several different ways in which it can be in error.