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)

The approximate number system (ANS) enables organisms to represent the approximate number of items in an observed collection, quickly and independently of natural language. Recently, it has been proposed that the ANS goes beyond representing natural numbers by extracting and representing rational numbers (Clarke & Beck, 2021a). Prior work demonstrates that adults and children discriminate ratios in an approximate and ratio-dependent manner, consistent with the hallmarks of the ANS. Here, we use a well-known “connectedness illusion” to provide evidence that (...) these ratio-dependent ratio discriminations are (a) based on the perceived number of items in seen displays (and not just non-numerical confounds), (b) are not dependent on verbal working memory, or explicit counting routines, and (c) involve representations with a part-whole (or subset-superset) format, like a fraction, rather than a part-part format, like a ratio. These results vindicate key predictions of the hypothesis that the ANS represents rational numbers. (shrink)

To investigate mechanisms of rational representation, I consider construction of an ordered continuum of psychophysical scale of magnitude of sensation; counting mechanism leading to an approximate numerosity scale for integers; and conjoint measurement structure pitting the denominator against the numerator in tradeoff positions. Number sense of resulting rationals is neither intuitive nor expedient in their manipulation.

We agree with Clarke and Beck that the approximate number system represents rational numbers, and we demonstrate our support by highlighting the case of the empty set – the non-symbolic manifestation of zero. It is particularly interesting because of its perceptual and semantic uniqueness, and its exploration reveals fundamental new insights about how numerical information is represented.

We call an $\alpha \in \mathbb {R}$ regainingly approximable if there exists a computable nondecreasing sequence $(a_n)_n$ of rational numbers converging to $\alpha $ with $\alpha - a_n n}$ for infinitely many n. Similarly, there exist regainingly approximable sets whose initial segment complexity infinitely often reaches the maximum possible for c.e. sets. Finally, there is a uniform algorithm splitting regular real numbers into two regainingly approximable numbers that are still regular.

Approximate coherentism suggests that imperfectly rational agents should hold approximately coherent credences. This norm is intended as a generalization of ordinary coherence. I argue that it may be unable to play this role by considering its application under learning experiences. While it is unclear how imperfect agents should revise their beliefs, I suggest a plausible route is through Bayesian updating. However, Bayesian updating can take an incoherent agent from relatively more coherent credences to relatively less coherent credences, depending on (...) the data observed. Thus, comparative rationality judgments among incoherent agents are unduly sensitive to luck. (shrink)

In this paper, we investigate the problem of truth approximation via belief merging, i.e., we ask whether, and under what conditions, a group of inquirers merging together their beliefs makes progress toward the truth about the underlying domain. We answer this question by proving some formal results on how belief merging operators perform with respect to the task of truth approximation, construed as increasing verisimilitude or truthlikeness. Our results shed new light on the issue of how rational (dis)agreement affects (...) the inquirers’ quest for truth. In particular, they vindicate the intuition that scientific inquiry, and rational discussion in general, benefits from some heterogeneity in opinion and interaction among different viewpoints. The links between our approach and related analyses of truth tracking, judgment aggregation, and opinion dynamics, are also highlighted. (shrink)

A new approach for a uniform classification of the computably approximable real numbers is introduced. This is an important class of reals, consisting of the limits of computable sequences of rationals, and it coincides with the 0'-computable reals. Unlike some of the existing approaches, this applies uniformly to all reals in this class: to each computably approximable real x we assign a degree structure, the structure of all possible ways available to approximate x. So the main criterion for such classification (...) is the variety of the effective ways we have to approximate a real number. We exhibit extreme cases of such approximation structures and prove a number of related results. (shrink)

The so-called Preface Paradox seems to show that one can rationally believe two logically incompatible propositions. We address this puzzle, relying on the notions of truthlikeness and approximate truth as studied within the post-Popperian research programme on verisimilitude. In particular, we show that adequately combining probability, approximate truth, and truthlikeness leads to an explanation of how rational belief is possible in the face of the Preface Paradox. We argue that our account is superior to other solutions of the paradox, (...) including a recent one advanced by Hannes Leitgeb (Analysis 74.1). (shrink)

We study Δ2 reals x in terms of how they can be approximated symmetrically by a computable sequence of rationals. We deal with a natural notion of ‘approximation representation’ and study how these are related computationally for a fixed x. This is a continuation of earlier work; it aims at a classification of Δ2 reals based on approximation and it turns out to be quite different than the existing ones (based on information content etc.).

A real number is recursively approximable if there is a computable sequence of rational numbers converging to it. If some extra condition to the convergence is added, then the limit real number might have more effectivity. In this note we summarize some recent attempts to classify the recursively approximable real numbers by the convergence rates of the corresponding computable sequences ofr ational numbers.

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)

In many areas of scientific inquiry, the phenomena under investigation are viewed as functions on the real numbers. Since observational precision is limited, it makes sense to view these phenomena as bounded functions on the rationals. One may translate the basic notions of recursion theory into this framework by first interpreting a partial recursive function as a function on Q. The standard notions of inductive inference carry over as well, with no change in the theory. When considering the class of (...) computable functions on Q, there are a number of natural ways in which to define the distance between two functions. We utilize standard metrics to explore notions of approximate inference — our inference machines will attempt to guess values which converge to the correct answer in these metrics. We show that the new inference notions, NV∞, EX∞, and BC∞, infer more classes of functions than their standard counterparts, NV, EX, and BC. Furthermore, we give precise inclusions between the new inference notions and those in the standard inference hierarchy. We also explore weaker notions of approximate inference, leading to inference hierarchies analogous to the EXn and BCn hierarchies. Oracle inductive inference is also considered, and we give sufficient conditions under which approximate inference from a generic oracle G is equivalent to approximate inference with only finitely many queries to G. (shrink)

We study the first-order theories of some natural and important classes of coloured trees, including the four classes of trees whose paths have the order type respectively of the natural numbers, the integers, the rationals, and the reals. We develop a technique for approximating a tree as a suitably coloured linear order. We then present the first-order theories of certain classes of coloured linear orders and use them, along with the approximating technique, to establish complete axiomatisations of the four classes (...) of trees mentioned above. (shrink)

Clarke and Beck argue that the approximate number system represents rational numbers, like 1/3 or 3.5. I think this claim is not supported by the evidence. Rather, I argue, ANS should be interpreted as representing natural numbers and ratios among them; and we should view the contents of these representations are genuinely approximate.

This monograph presents new ideas in nomic truth approximation. It features original and revised papers from a philosopher of science who has studied the concept for more than 35 years. Over the course of time, the author's initial ideas evolved. He discovered a way to generalize his first theory of nomic truth approximation, viz. by dropping an unnecessarily strong assumption. In particular, he first believed to have to assume that theories were maximally specific in the sense that they did not (...) only exclude certain conceptual possibilities, but also that all non-excluded possibilities were in fact claimed to be nomically possible. Now, he argues that the exclusion claim alone, or for that matter the inclusion claim alone, is sufficient to motivate the formal definition of being closer to the nomic truth. The papers collected here detail this generalized view of nomic truthlikeness or verisimilitude. Besides this, the book presents, in adapted form, the relation with several other topics, such as, domain revision, aesthetic progress, abduction, inference to the best explanation, pragmatic aspects, probabilistic methods, belief revision and epistemological positions, notably constructive realism. Overall, the volume presents profound insight into nomic truth approximation. This idea seeks to determine how one theory can be closer to, or more similar to, the truth about what is nomically, e.g. physically, chemically, biologically, possible than another theory. As a result, it represents the ultimate goal of theory oriented empirical science. Theo Kuipers is the author of Studies in Inductive Probability and Rational Expectation, From Instrumentalism to Constructive Realism and Structures in Science. He is the volume-editor of the Handbook on General Philosophy of Science. In 2005 there appeared two volumes of Essays in Debate with Theo Kuipers, entitled Confirmation, Empirical Progress, and Truth Approximation and Cognitive Structures in Scientific Inquiry. (shrink)

On a now orthodox view, humans and many other animals possess a “number sense,” or approximate number system, that represents number. Recently, this orthodox view has been subject to numerous critiques that question whether the ANS genuinely represents number. We distinguish three lines of critique – the arguments from congruency, confounds, and imprecision – and show that none succeed. We then provide positive reasons to think that the ANS genuinely represents numbers, and not just non-numerical confounds or exotic substitutes for (...) number, such as “numerosities” or “quanticals,” as critics propose. In so doing, we raise a neglected question: numbers of what kind? Proponents of the orthodox view have been remarkably coy on this issue. But this is unsatisfactory since the predictions of the orthodox view, including the situations in which the ANS is expected to succeed or fail, turn on the kind of number being represented. In response, we propose that the ANS represents not only natural numbers, but also non-natural rational numbers. It does not represent irrational numbers, however, and thereby fails to represent the real numbers more generally. This distances our proposal from existing conjectures, refines our understanding of the ANS, and paves the way for future research. (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)

It is well known that the computational complexity of propositional knowledge base revision is at the second level of polynomial hierarchy. A way to solve this kind of problems is to introduce approximate algorithms. In this paper, an approximate approach is introduced for belief change. Operators, which satisfy the AGM rational postulates, are defined to change belief sets or belief bases. Furthermore, approximate algorithms to implement the revision of finite belief bases are presented. The time complexities of the approximate (...) algorithms shown in this paper are at lower level than the time complexities of the existed approaches in literatures, although they may not generate the optimal solution, and this is meaningful from the theoretical point of view. (shrink)

Some epistemologists think that the Bayesian ideals matter because we can approximate them. That is, our attitudes can be more or less close to the ones of our ideal Bayesian counterpart. In this paper, I raise a worry for this justification of epistemic ideals. The worry is this: In order to correctly compare agents to their ideal counterparts, we need to imagine idealized agents who have the same relevant information, knowledge, or evidence. However, there are cases in which one’s ideal (...) counterpart cannot have one’s information, knowledge, or evidence. In these situations, agents cannot compare themselves to their ideal counterpart. (shrink)

Whereas classic work in judgment and decision making has focused on the deviation of intuition from rationality, more recent research has focused on the performance of intuition in real-world environments. Borrowing from both approaches, we investigate to which extent competing models of intuitive probabilistic decision making overlap with choices according to the axioms of probability theory and how accurate those models can be expected to perform in real-world environments. Specifically, we assessed to which extent heuristics, models implementing weighted additive information (...) integration (WADD), and the parallel constraint satisfaction (PCS) network model approximate the Bayesian solution and how often they lead to correct decisions in a probabilistic decision task. PCS and WADD outperform simple heuristics on both criteria with an approximation of 88.8% and a performance of 73.7%. Results are discussed in the light of selection of intuitive processes by reinforcement learning. (shrink)

This paper addresses a common challenge with computational cognitive models: identifying parameter values that are both theoretically plausible and generate predictions that match well with empirical data. While computational models can offer deep explanations of cognition, they are computationally complex and often out of reach of traditional parameter fitting methods. Weak methodology may lead to premature rejection of valid models or to acceptance of models that might otherwise be falsified. Mathematically robust fitting methods are, therefore, essential to the progress of (...) computational modeling in cognitive science. In this article, we investigate the capability and role of modern fitting methods—including Bayesian optimization and approximate Bayesian computation—and contrast them to some more commonly used methods: grid search and Nelder–Mead optimization. Our investigation consists of a reanalysis of the fitting of two previous computational models: an Adaptive Control of Thought—Rational model of skill acquisition and a computational rationality model of visual search. The results contrast the efficiency and informativeness of the methods. A key advantage of the Bayesian methods is the ability to estimate the uncertainty of fitted parameter values. We conclude that approximate Bayesian computation is (a) efficient, (b) informative, and (c) offers a path to reproducible results. (shrink)

Numbers are symbols manipulated in accord with the axioms of arithmetic. They sometimes represent discrete and continuous quantities, but they are often simply names. Brains, including insect brains, represent the rational numbers with a fixed-point data type, consisting of a significand and an exponent, thereby conveying both magnitude and precision.

It has been argued recently that knowledge is the norm of practical reasoning. This norm can be formulated as a bi-conditional: it is appropriate to treat p as a reason for acting if and only if you know that p. Other proposals replace knowledge with warranted or justified belief. This paper gives counter-examples of both directions of any such bi-conditional. To the left-to-right direction: scientists can appropriately treat as reasons for action propositions of a theory they believe to be false (...) but good approximations to the truth for present purposes. Cases based on a variant of Pascal’s Wager and actions performed by a skeptic also illustrate the point. To the right-to-left direction: in certain circumstances, it can be unreasonable for a scientist to reason from propositions of a theory she knows to be true. (shrink)

An increasingly prevalent approach to studying human cognition is to construe the mind as optimally allocating limited cognitive resources among cognitive processes. Under this bounded rationality approach (Icard in Philos Sci 85(1):79–101, 2018; Simon in Utility and probability, Palgrave Macmillan, 1980), it is common to assume that resource-bounded cognitive agents approximate normative solutions to statistical inference problems, and that much of the bias and variability in human performance can be explained in terms of the approximation strategies we employ. In this (...) paper, we argue that this approach restricts itself to an unnecessarily narrow scope of cognitive models, which limits its ability to explain how humans flexibly adapt their representations to novel environments. We argue that more attention should be paid to how we form our cognitive representations in the first place, and advocate for pluralistic framework which jointly optimizes over both representations and algorithms for manipulating them. We identify several fundamental trade-offs that manifest in this joint optimization, and draw on recent work to motivate a unified formal framework for this analysis. We illustrate a simplified version of this analysis with a case study in social cognition, and outline several new directions for research that this approach suggests. (shrink)

The proposal by Clarke and Beck offers a new explanation for the association between the approximate number system and math. Previous explanations have largely relied on developmental arguments, an underspecified notion of the ANS as an “error detection mechanism,” or affective factors. The proposal that the ANS represents rational numbers suggests that it may directly support a broader range of math skills.

Research on “improper” linear models has shown that predetermined weighting schemes for the linear model, such as equally weighting all predictors, can be surprisingly accurate on cross-validation. We review recent advances that can characterize the optimal choice of an improper linear model. We extend this research to the understanding of fast and frugal heuristics, particularly to the ecologically rational goal of understanding in which task environments given heuristics are optimal. We demonstrate how to test this model using the Recognition (...) Heuristic and Take the Best heuristic, show how the model reconciles with the ecological rationality program, and discuss how our prescriptive, computational approach could be approximated by simpler mental rules that might be more descriptive. Echoing the arguments of van Rooij et al., we stress the virtue of having a computationally tractable model of strategy selection, even if one proposes that cognizers use a simpler heuristic process to approximate it. (shrink)

This paper considers contractarianism as a method of justification. The analysis accepts the key tenets of contractarianism: expected utility maximization, unanimity as the criteria of acceptance, and social-scientific uncertainty of modelled agents. In addition to these three features, however, the analysis introduces a fourth feature: a criteria of rational belief formation, viz. Bayesian belief updating. Using a formal model, this paper identifies a decisive objection to contractarian justification. Insofar as contractarian projects approximate the Agreement Model, therefore, they fail to (...) justify their conclusions. Insofar as they fail to approximate the Agreement Model, they must explain which modelling assumption they reject. (shrink)

Clarke and Beck propose that the approximate number system (ANS) represents rational numbers. The evidence cited supports only the view that it represents ratios (and positive integers). Rational numbers are extensive magnitudes (i.e., sizes), whereas ratios are intensities. It is also argued that WHAT a system represents and HOW it does so are not as independent of one another as the authors assume.

We define rational Hermite interpolants to vector-valued functions and show that, in the context of Clifford algebras, the numerator and denominator polynomials belong to a complex extension of the Lipschitz group. We also discuss the problem of constructing an algebraic representation for the generalized inverse of a vector, which is at the heart of the usual development of vector rational approximation.

Given a finitely generated group Γ, we study the space Isom(Γ, ℚ������) of all actions of Γ by isometries of the rational Urysohn metric space ℚ������, where Isom(Γ, ℚ������) is equipped with the topology it inherits seen as a closed subset of Isom(ℚ������) Γ . When Γ is the free group ������ n on n generators this space is just Isom(ℚ������) n , but is in general significantly more complicated. We prove that when Γ is finitely generated Abelian there (...) is a generic point in Isom(Γ, ℚ������), i.e., there is a comeagre set of mutually conjugate isometric actions of Γ on ℚ������. (shrink)

Accuracy-first epistemologists argue that rational agents have probabilistically coherent credences. But why should we care, given that we can’t help being incoherent? A common answer: probabilistic coherence is an ideal to be approximated as best one can. De Bona (in: Philos Sci 84(2), 189–213, 2017) and Staffel (in: Unsettled thoughts: a theory of degrees of rationality, Oxford University Press, 2019) show how accuracy-firsters can spell out this answer by adopting an appropriate notion of approximate coherence. In this essay, I (...) argue that De Bona and Staffel’s proposal is not entirely satisfactory, because it does not show whether, or why, it’s generally better to be more rather than less approximately coherent from an accuracy-first perspective. To address this point, I give a characterisation of their notion of approximate coherence in terms of accuracy. This shows that accuracy-firsters should maintain that more approximately coherent credences are better than less approximately coherent ones _if and only if_ they accept that it’s better to miss out on less rather than more guaranteed accuracy. (shrink)

Could age be a valid criterion for rationing? In Just health, Norman Daniels argues that under certain circumstances age rationing is prudent, and therefore a morally permissible strategy to tackle the problem of resource scarcity. Crucial to his argument is the distinction between two problem-settings of intergenerational equity: equity among age groups and equity among birth cohorts. While fairness between age groups can involve unequal benefit treatment in different life stages, fairness between birth cohorts implies enjoying approximate equality in benefit (...) ratios. Although both questions of fairness are distinct, the resolution of the one depends on resolution of the other. In this paper, I investigate whether Daniels’ account of age rationing could be defended as a fair way of setting limits to healthcare entitlements. I will focus on two main points. First, I will consider whether the age group problem could be resolved without appealing to a conception of the good. Second, I will demonstrate that the connection between the age group problem and the birth cohort problem runs deeper than Daniels initially thought—and that it ultimately suggests a method for prioritisation in problem solving strategies. (shrink)

L.A. Paul has recently argued that, on the standard model of rationality, individuals cannot make rational decisions about whether to have a child or not. In this paper, I show that Paul’s arguments do not plausibly demonstrate that the standard model of rationality precludes rational decisions to have a child. I argue that there are phenomenal and non-phenomenal values that can be used to determine the value that having a child will have for us and, in turn, that (...) can be used to make rational decisions about whether to have a child or not. I also argue that we can have an approximate idea of what it is like for us to have a child, even before we have a child and that, on the standard model, this is sufficient to make rational decisions to have a child. (shrink)

Computational models have shown how polarisation can rise among deliberating agents as they approximate epistemic rationality. This paper provides further support for the thesis that polarisation can rise under condition of epistemic rationality, but it does not depend on limitations that extant models rely on, such as memory restrictions or biased evaluation of other agents’ testimony. Instead, deliberation is modelled through agents’ purposeful introduction of arguments and their rational reactions to introductions of others. This process induces polarisation dynamics on (...) its own. A second result is that the effect size of polarisation dynamics correlates with particular types of argumentative behaviour. Polarisation effects can be soothed when agents take into account the opinions of others as premises, and they are amplified as agents fortify their own beliefs. These results underpin the relevance of argumentation as a factor in social-epistemic processes and indicate that rising issue polarisation is not a reliable indicator of epistemic shortcomings. (shrink)

In the spirit of recent analytic investigations, Bennett seeks to analyze the concept of rationality. He approaches this topic by first considering the behavior of honey-bees, which he claims is non-rational. Using this as a model he examines variations that more closely approximate the linguistic manifestation of rationality. Bennett's most interesting thesis is that while language is necessary for rationality, the possession of language is not sufficient for rationality. A good deal of familiar ground is covered here and while (...) Bennett has many novel insights, there seems to be little genuine advance over the work of the later Wittgenstein and his followers.—R. J. B. (shrink)

Many economic theorists hold that social institutions can lead otherwise irrational agents to approximate the predictions of traditional rational choice theory. But there is little consensus on how institutions do so. I defend an economic internalist account of the institution-actor relationship by explaining economic rationality as a feature of individuals whose decision-making is aided by institutional structures. This approach, known as the subjective transaction costs theory, represents apparently irrational behavior as a rational response to high subjective transaction costs (...) of thinking and deciding. The theory has two attractive features. First, it reconciles rational choice theory with the increasing body of evidence cataloguing putative errors in economic decision-making. Second, it vindicates the explanatory power of individual choice against externalist challenges; the subjective transaction costs theory keeps economic rationality in the head. (shrink)

While game theory has been transformative for decision making, the assumptions made can be overly restrictive in certain instances. In this work, we investigate some of the underlying assumptions of rationality, such as mutual consistency and best response, and consider ways to relax these assumptions using concepts from level-k reasoning and quantal response equilibrium (QRE) respectively. Specifically, we propose an information-theoretic two-parameter model called the quantal hierarchy model, which can relax both mutual consistency and best response while still approximating level-k, (...) QRE, or typical Nash equilibrium behavior in the limiting cases. The model is based on a recursive form of the variational free energy principle, representing higher-order reasoning as (pseudo) sequential decision-making in extensive-form game tree. This representation enables us to treat simultaneous games in a similar manner to sequential games, where reasoning resources deplete throughout the game-tree. Bounds in player processing abilities are captured as information costs, where future branches of reasoning are discounted, implying a hierarchy of players where lower-level players have fewer processing resources. We demonstrate the effectiveness of the quantal hierarchy model in several canonical economic games, both simultaneous and sequential, using out-of-sample modelling. (shrink)

The logical form of an inductive step figures as a deductive fallacy: concluding the antecedent from affirming a conditional and its consequent. In the sphere of practical rationality, however, where concerned with the presuppositions of action and the interactions between beliefs and desires, certain schemata have been proposed that express rational demands on agents who desire things to happen in the world. In this context, if agent A desires to φ and believes that ψ brings about φ, then, A (...) is rationally expected to desire to ψ. But it is equally rational to expect that A’s desire to φ presupposes her belief that a certain state of affairs obtains where φ is a possibility. It then follows that assuming a desire to φ and a belief that ψ brings about φ, we can derive an logical form that approximates that of the above inductive step: the beliefs that φ is possible and that ψ brings about φ entail the belief that ψ is possible. It remains an open question whether this conclusion adequately supports the rationality of the exact inductive schema. (shrink)

Contents: Preface. Introduction. J. ECHEVERRIA, A. IBARRA and T. MORMANN: The Long and Winding Road to the Philosophy of Science in Spain. REPRESENTATION AND MEASUREMENT. A. IBARRA and T. MORMANN: Theories as Representations. J. GARRIDO GARRIDO: The Justification of Measurement. O. FERNÁNDEZ PRAT and D. QUESADA: Spatial Representations and Their Physical Content. J.A. DIEZ CALZADA: The Theory-Net of Interval Measurement Theory. TRUTH, RATIONALITY, AND METHOD. J.C. GARCÍA-BERMEJO OCHOA: Realism and Truth Approximation in Economic Theory. W.J. GONZALEZ: Rationality in Economics and (...) Scientific Predictions: A Critical Reconstruction of Bounded Rationality and Its Role in Economic Predictions. J.P. ZAMORA BONILLA: An Invitation to Methodonomics. LOGICS, SEMANTICS, AND THEORETICAL STRUCTURES. J.L. FALGUERA: A Basis for a Formal Semantics of Linguistic Formulations of Science. A. SOBRINO and E. TRILLAS: Can Fuzzy Logic Help to Pose Some Problems in the Philosophy of Science? J. de LORENZO: Demonstrative Ways in Mathematical Doing. M. CASANUEVA: Genetics and Fertilization: A Good Marriage. C.U. MOULINES: The Concept of Universe from a Metatheoretical Point of View. List of Contributors. Index of Names. (shrink)

In contrast to Clarke and Beck's claim that that the approximate number system represents rational numbers, we argue for a more modest alternative: The ANS represents natural numbers, and there are separate, non-numeric processes that can be used to represent ratios across a wide range of domains, including natural numbers.

Popper's short essay about the rationality principle has been the target of many criticisms which have raised serious doubts about its consistency. How could the well-known promoter of falsificationism suggest that we not reject a principle that he himself describes as false? Nonetheless, the essay can be read in a way that makes it appear much more consistent. Better sense can be made of Popper's own examples (the flustered driver, the pedestrian, etc.), by taking seriously his view that the rationality (...) principle might be "approximately true" and falsified only in very rare cases, while also giving proper attention to his four rather elliptical arguments. (shrink)

In this commentary, we offer an additional function of rationalization. Namely, in certain social contexts, the proximal and ultimate function of beliefs and desires is social inclusion. In such contexts, rationalization often facilitates distortion of rather than approximation to truth. Understanding the role of social identity is not only timely and important, but also critical to fully understand the function of rationalization.

The distinction between non-symbolic and symbolic number is poorly addressed by the authors despite being relevant in numerical cognition, and even more important in light of the proposal that the approximate number system represents rational numbers. Although evidence on non-symbolic number and ratios fits with ANS representations, the case for symbolic number and rational numbers is still open.

In this article, 41 models of covariation detection from 2 × 2 contingency tables were evaluated against past data in the literature and against data from new experiments. A new model was also included based on a limiting case of the normative phi‐coefficient under an extreme rarity assumption, which has been shown to be an important factor in covariation detection (McKenzie & Mikkelsen, 2007) and data selection (Hattori, 2002; Oaksford & Chater, 1994, 2003). The results were supportive of the new (...) model. To investigate its explanatory adequacy, a rational analysis using two computer simulations was conducted. These simulations revealed the environmental conditions and the memory restrictions under which the new model best approximates the normative model of covariation detection in these tasks. They thus demonstrated the adaptive rationality of the new model. (shrink)

Probabilistic coherence is not an absolute requirement of rationality; nevertheless, it is an ideal of rationality with substantive normative import. An idealized rational agent who avoided making implicit logical errors in forming his preferences would be coherent. In response to the challenge, recently made by epistemologists such as Foley and Plantinga, that appeals to ideal rationality render probabilism either irrelevant or implausible, I argue that idealized requirements can be normatively relevant even when the ideals are unattainable, so long as (...) they define a structure that links imperfect and perfect rationality in a way that enables us to make sense of the notion of better approximations to the ideal. I then analyze the notion of approximation to the ideal of coherence by developing a generalized theory of belief functions that allows for incoherence, and showing how such belief functions can be ordered with regard to greater or lesser coherence. (shrink)

Since its inauguration in 1932, the Whitney Biennial has fostered contemporary artistic innovation and diversity, becoming a highly anticipated event in the art world. The 2010 Biennial is curated by Francesco Bonami and Gary Carrion-Murayari and features works by approximately 55 artists working in a variety of media and practices. Uniquely, this catalogue serves as both a handsome accompaniment to the 2010 exhibition and an insightful exploration of the significance of this acclaimed and often controversial event throughout its history. In (...) addition to presenting full-colour reproductions of the selected artists' recent work, the curators have prepared a joint essay on the 2010 exhibition, and a group of writers contributed brief entries on the represented artists' techniques, influences and recent work. Exhibition: Whitney Museum of American Art, 25 February - 30 May 2010. (shrink)

This is an excerpt from the contentIn the introduction to part I of the symposium we stated that a rational agent could be thought of as an agent who has good reasons for its actions. In formal analyses of economic, medical, political, military and forensic decisions rationality, that is the “goodness” of those reasons, is inextricably intertwined with probability. Typically, those analyses concern decisions in a particular class of uncertain situations, namely “risky” situations, where all the relevant available alternative (...) actions are known, each or most actions are non-deterministic , and the probability of each outcome is known to some degree of approximation. As yet, formal analyses have only begun to scratch the surface of the subtleties implicated in uncertain, but not risky, situations, where the available courses of actions, and/or their possible outcomes, and/or the probabilities of those outcomes, are not all. (shrink)