Reconstructed Empiricism

According to Bas van Fraassen, scientific realists and anti-realists disagree about whether accepting a scientific theory involves believing that the theory is true. On van Fraassen’s own anti-realist empiricist position, accepting a theory involves believing only that the theory is correct in its claims about observable aspects of the world. However, a number of philosophers have argued that acceptance and belief cannot be distinguished and thus that the debate is either confused or trivially settled in favor of the realist. In addition, another set of philosophers have argued that van Fraassen’s empiricist position appeals to an unmotivated distinction between observable and unobservable aspects of the world. This paper aims to reconstruct a van Fraassen-style empiricism about scientific acceptance that avoids these two objections – reconstructed empiricism.     

Epistemic characterizations of iterated deletion of inferior strategy profiles in preference-based type spaces

Bonanno (Logics and the foundations of game and decision theory, Amsterdam University Press, Amsterdam, 2008) provides an epistemic characterization for the solution concept of iterated deletion of inferior strategy profiles (IDIP) by embedding strategic-form games with ordinal payoffs in non-probabilistic epistemic models which are built on Kripke frames. In this paper, we will follow the event-based approach to epistemic game theory and supplement strategic games with type space models, where each type is associated with a preference relation on the state space. In such a framework, IDIP can be characterized by the conditions that at least one player has correct beliefs about the state of the world and that there is common belief that every player is rational, has correct beliefs about the state of the world and has strictly monotone preferences. Moreover, we shall compare the epistemic motivations for IDIP and its mixed strategy variant known as strong rationalizability (SR). Presuppose the above conditions. Whenever there is also common belief that players’ preferences are representable by some expected utility function IDIP still applies. But if there is common belief that players’ preferences are representable by some expected payoff function, then SR results.     

Towards an Account of Epistemic Luck for Necessary Truths

Modal epistemologists parse modal conditions on knowledge in terms of metaphysical possibilities or ways the world might have been. This is problematic. Understanding modal conditions on knowledge this way has made modal epistemology, as currently worked out, unable to account for epistemic luck in the case of necessary truths, and unable to characterise widely discussed issues such as the problem of religious diversity and the perceived epistemological problem with knowledge of abstract objects. Moreover, there is reason to think that this is a congenital defect of orthodox modal epistemology. This way of characterising modal epistemology is however optional. It is shown that one can non-circularly characterise modal conditions on knowledge in terms of epistemic possibilities, or ways the world might be for the target agent. Characterising the anti-luck condition in terms of epistemic possibilities removes the impediment to understanding epistemic luck in the case of necessary truths and opens the door to using these conditions to shed new light on some longstanding epistemological problems.     

Budget-constrained optimal insurance with belief heterogeneity

We re-examine the problem of budget-constrained demand for insurance indemnification when the insured and the insurer disagree about the likelihoods associated with the realizations of the insurable loss. For ease of comparison with the classical literature, we adopt the original setting of Arrow (1971), but allow for divergence in beliefs between the insurer and the insured; and in particular for singularity between these beliefs, that is, disagreement about zero-probability events. We do not impose the no sabotage condition on admissible indemnities. Instead, we impose a state-verification cost that the insurer can incur in order to verify the loss severity, which rules out ex post moral hazard issues that could otherwise arise from possible misreporting of the loss by the insured. Under a mild consistency requirement between these beliefs that is weaker than the Monotone Likelihood Ratio (MLR) condition, we characterize the optimal indemnity and show that it has a simple two-part structure: full insurance on an event to which the insurer assigns zero probability, and a variable deductible on the complement of this event, which depends on the state of the world through a likelihood ratio. The latter is obtained from a Lebesgue decomposition of the insured’s belief with respect to the insurer’s belief.     

Moore's Paradox in Belief and Desire

Is there a Moore’s paradox in desire? I give a normative explanation of the epistemic irrationality, and hence absurdity, of Moorean belief that builds on Green and Williams’ normative account of absurdity. This explains why Moorean beliefs are normally irrational and thus absurd, while some Moorean beliefs are absurd without being irrational. Then I defend constructing a Moorean desire as the syntactic counterpart of a Moorean belief and distinguish it from a ‘Frankfurt’ conjunction of desires. Next I discuss putative examples of rational and irrational desires, suggesting that there are norms of rational desire. Then I examine David Wall’s groundbreaking argument that Moorean desires are always unreasonable. Next I show against this that there are rational as well as irrational Moorean desires. Those that are irrational are also absurd, although there seem to be absurd desires that are not irrational. I conclude that certain norms of rational desire should be rejected.     

Varieties of Pragmatic Encroachment

According to a recent view, known as the 'pragmatic encroachment' thesis, an agent’s non-truth-related factors are relevant to the epistemic status of her beliefs. In particular, in addition to truth-related factors, practical factors are said to be relevant to the question whether or not true belief amounts to knowledge. Despite the intuitive appeal of the thesis, however, it is puzzling how practical factors can impact the truth-related factors that ground the epistemic status of one's beliefs. In this paper, I will distinguish between a strong and a weak sense of the way in which practical factors are said to be thus relevant. Their differences are explicated in terms of the nature and the extent to which practical factors are said to impact the epistemic status of one's beliefs. I begin by considering a strong version of the thesis that suggests principles according to which the practical rationality of one's actions is a necessary condition on knowledge and justification. Having noted an inadequacy in the formulation of such principles, the arguments in their support are subsequently stated and criticized. Finally, I identify two modest versions of the thesis of pragmatic encroachment and argue that they, too, fail to explain how practical factors can bear on the epistemic status of one's beliefs.     

What Is Entitlement?

In his seminal paper, “Content Preservation,” Tyler Burge defends an original account of testimonial knowledge. The originality of the account is due, in part, to the fact that it is cast within a novel epistemic framework. The central feature of that framework is the introduction of the concept of entitlement, which is alleged to be a distinctive type of positive epistemic support or warrant. Entitlement and justification, according to Burge, are sub-species of warrant. Justification is the internalist form of warrant, but entitlement is epistemically externalist. My focus in this paper is Burge’s conception of entitlement, and there are three primary issues that I wish to address. What is the relationship between entitlement and the more traditional concept of justification? In what sense is entitlement epistemically externalist? Has Burge introduced a new epistemic concept or merely coined a new term for a familiar epistemic concept?     

Double-bosonization and Majid’s conjecture (IV): Type-crossings from A to BCD

Both in Majid's double-bosonization theory and in Rosso's quantum shuffle theory, the rankinductive and type-crossing construction for U_q(g)'s is still a remaining open question. In this paper, working in Majid's framework, based on the generalized double-bosonization theorem we proved before, we further describe explicitly the type-crossing construction of U_q(g)'s for(BCD)_n series directly from type An-1via adding a pair of dual braided groups determined by a pair of(R, R′)-matrices of type A derived from the respective suitably chosen representations. Combining with our results of the first three papers of this series, this solves Majid's conjecture, i.e., any quantum group U_q(g) associated to a simple Lie algebra g can be grown out of U_q(sl_2)recursively by a series of suitably chosen double-bosonization procedures.     

Characterization of the alternating groups by their order and one conjugacy class length

Let G be a finite group, and let N(G) be the set of conjugacy class sizes of G. By Thompson’s conjecture, if L is a finite non-abelian simple group, G is a finite group with a trivial center, and N(G) = N(L), then L and G are isomorphic. Recently, Chen et al. contributed interestingly to Thompson’s conjecture under a weak condition. They only used the group order and one or two special conjugacy class sizes of simple groups and characterized successfully sporadic simple groups (see Li’s PhD dissertation). In this article, we investigate validity of Thompson’s conjecture under a weak condition for the alternating groups of degrees p+1 and p+2, where p is a prime number. This work implies that Thompson’s conjecture holds for the alternating groups of degree p + 1 and p + 2.     

On extensions of some block designs

There are some results concerning t-designs in which the number of points in the intersection of two blocks takes less than t values. For example, if t = 2, then the design is symmetric (in such a design, v = b or, equivalently, k = r). In 1974, B. Gross described t-(v, k, l) designs that, for some integer s, 0 < s < t, do not contain two blocks intersecting at exactly s points. Below, it is proved that potentially infinite series of designs from the claim of Gross’ theorem are finite. Gross’ theorem is substantially sharpened.     

The Single-Period Inventory Problem with Triangular Demand Distribution

A simple method is presented for determining ‘closed-form’ solutions for an optimum (s, S) ordering policy for the single-period inventory problem with a set-up cost of ordering and the uncertain total demand over the period represented by a triangular probability density function. The distribution reflecting the decision maker's degree of belief that all values of total demand outside of two (possibly ‘soft’) limits are barely credible and all values within the limits have uniform increasing probability density towards a (possibly ‘soft’) modal value. The importance of the closed form solutions obtained is that they remove the need for enumeration over alternative values of s in determining the optimum (s, S) ordering policy, also they can be encoded in an algorithm simple to implement and they allow easy sensitivity analysis of the results to perturbations in the estimates of the problem parameters. A numerical example is presented to illustrate the essential features of the method.     

A Behavioral Interpretation of Belief Functions

Shafer’s belief functions were introduced in the seventies of the previous century as a mathematical tool in order to model epistemic probability. One of the reasons that they were not picked up by mainstream probability was the lack of a behavioral interpretation. In this paper, we provide such a behavioral interpretation and re-derive Shafer’s belief functions via a betting interpretation reminiscent of the classical Dutch Book Theorem for probability distributions. We relate our betting interpretation of belief functions to the existing literature.     

Calculation of Bezout’s coefficients for the <Emphasis Type="Italic">k</Emphasis>-ary algorithm of finding GCD

Bezout’s equation is a representation of the greatest common divisor d of integers A and B as a linear combination Ax + By = d, where x and y are integers called Bezout’s coefficients. The task of finding Bezout’s coefficients has numerous applications in the number theory and cryptography, for example, for calculation of multiplicative inverse elements in modular arithmetic. Usually Bezout’s coefficients are caclulated using the extended version of the classical Euclidian algorithm.We elaborate a new algorithm for calculating Bezout’s coefficients based on the k-ary GCD algorithm.     

A simple logic for reasoning about incomplete knowledge

The semantics of modal logics for reasoning about belief or knowledge is often described in terms of accessibility relations, which is too expressive to account for mere epistemic states of an agent. This paper proposes a simple logic whose atoms express epistemic attitudes about formulae expressed in another basic propositional language, and that allows for conjunctions, disjunctions and negations of belief or knowledge statements. It allows an agent to reason about what is known about the beliefs held by another agent. This simple epistemic logic borrows its syntax and axioms from the modal logic KD. It uses only a fragment of the S5 language, which makes it a two-tiered propositional logic rather than as an extension thereof. Its semantics is given in terms of epistemic states understood as subsets of mutually exclusive propositional interpretations. Our approach offers a logical grounding to uncertainty theories like possibility theory and belief functions. In fact, we define the most basic logic for possibility theory as shown by a completeness proof that does not rely on accessibility relations.     

Reliabilist Justification: Basic,Easy, and Brute

Process reliabilists hold that in order for a belief to be justified, it must result from a reliable cognitive process. They also hold that a belief can be basically justified: justified in this manner without having any justification to believe that belief is reliably produced. Fumerton (1995), Vogel (2000), and Cohen (2002) have objected that such basic justification leads to implausible easy justification by means of either epistemic closure principles or so-called track record arguments. I argue that once we carefully distinguish closure principles from transmission principles, and epistemic consequences from epistemic preconditions, neither version of this objection succeeds.

An analysis of the a priori and a posteriori

I present and defend a unified, non-reductive analysis of the a priori and a posteriori. It is a mistake to remove all epistemic conditions from the analysis of the a priori (as, for example, Alvin Goldman has recently suggested doing). We can keep epistemic conditions (like unrevisability) in the analysis as long as we insist that a priori and a posteriori justification admit of degrees. I recommend making the degree to which a belief’s justification is a priori or a posteriori solely dependent on the revisability relations that obtain among the faculties that deliver the belief and all other faculties.     

Approximation of almost Sahoo–Riedel’s points by Sahoo–Riedel’s points

In this paper, we present an extention of Hyers–Ulam stability of Sahoo–Riedel’s points for real-valued differentiable functions on [a, b] and then we obtain stability results of Flett’s points for functions in the class of continuously differentiable functions on [a, b] with f′(a) = f′(b).     

Modified Kantorovich theorem and asymptotic approximations of solutions to singularly perturbed systems of ordinary differential equations

The functional equation f(x,ε) = 0 containing a small parameter ε and admitting regular and singular degeneracy as ε → 0 is considered. By the methods of small parameter, a function x _n ⁰(ε) satisfying this equation within a residual error of O(ε ⁿ⁺¹) is found. A modified Newton’s sequence starting from the element x _n ⁰(ε) is constructed. The existence of the limit of Newton’s sequence is based on the NK theorem proven in this work (a new variant of the proof of the Kantorovich theorem substantiating the convergence of Newton’s iterative sequence). The deviation of the limit of Newton’s sequence from the initial approximation x _n ⁰(ε) has the order of O(ε ⁿ⁺¹), which proves the asymptotic character of the approximation x _n ⁰(ε). The method proposed is implemented in constructing an asymptotic approximation of a system of ordinary differential equations on a finite or infinite time interval with a small parameter multiplying the derivatives, but it can be applied to a wider class of functional equations with a small parameters.     

Bivariate Binomial Moments and Bonferroni-Type Inequalities

We obtain bivariate forms of Gumbel’s, Fréchet’s and Chung’s linear inequalities for P(S ≥ u, T ≥ v) in terms of the bivariate binomial moments {S _{i, j} }, 1 ≤ i ≤ k,1 ≤ j ≤ l of the joint distribution of (S, T). At u = v = 1, the Gumbel and Fréchet bounds improve monotonically with non-decreasing (k, l). The method of proof uses combinatorial identities, and reveals a multiplicative structure before taking expectation over sample points.