Abstract In the work, the author derives an integral representation of the Gauss hypergeometric functions F 1 2 ⁢ ( a - 1 2 , a ; a + 1 2 ; z ) {{}_{2}F_{1}(a-\frac{1}{2},a;a+\frac{1}{2};z)} by three approaches, applies the integral representation to give integral representations of several functions involving the inverse tangent function and including the Wilf function, and find out several combinatorial identities.

Abstract How should controversial issues be taught in schools? In the philosophy of education, this question is often answered along the following lines: if a topic is controversial, it should be taught non-directively. A common justification for this idea is that other ways of teaching controversial issues would simply amount to indoctrination. In this paper, I would like to critically discuss this idea in more detail. More specifically, I will argue that a general categorization of directive approaches to teaching controversial issues as indoctrinatory is not as convincing as one might initially think. Against this background, I will then develop an alternative explanation for the pedagogical illegitimacy of directive approaches to teaching controversial issues. The basic idea behind this explanation is that such approaches undermine an important educational goal of teaching controversial issues by depriving students of the opportunity to arrive at well-formed judgments on these issues.

Abstract Since the work of George Boolos in the 1980s, philosophers have become more comfortable making use of plural reference and plural quantification in their theorizing. Plausibly, part of the reason is that it sometimes has a philosophical payoff. Mereological nihilists, for example, can do justice to the apparent truth of everyday claims involving singular reference or singular quantification over composite objects by paraphrasing them as claims involving plural reference or plural quantification over only mereological simples. But the analogous question of whether predicates can plurally express multiple relations has not received attention. I show that answering affirmatively can have philosophical payoffs of its own, illustrating this by explaining how it allows the directionalist about relations to articulate their position in such a way that it avoids Timothy Williamson's (1985) challenge. The result is an expanded range of formal tools which the philosopher can make use of in their theorizing.

Abstract It is alleged that imprecise probabilism can render one unable to update one’s credence in light of new evidence. While such belief inertia by itself is already quite worrying, I argue that it has other worrying epistemic implications that were previously unnoticed. It opens the possibility that one’s doxastic state depends not only on the evidence one has but also on the temporal order between awareness growth and evidence acquisition. I find this implication difficult to accept.

Abstract Mathematical measures are often used to express facts about what is typical — about what the vast majority of cases, in other words, are like. Many different measures can be used to express typicality facts, however. So exactly what justifies choosing one measure over another? In this paper, I propose an empiricist answer to this question. Roughly put, the answer only appeals to observations of frequencies, along with the standard scientific assumption that the results of experiments are not quasi-miraculous, atypical flukes.

Abstract Kyle Blumberg has recently argued that (i) the ideal worlds account of desire — according to which for S to want p is for all of S's top-ranked worlds to be p-worlds — has difficulties accounting for certain cases involving the ascribee's ignorance. He takes these cases to be (ii) a reason to disprefer the Kratzerian account of `want' to its rivals, and (iii) to doubt that desire ascriptions are upward entailing. I challenge all three claims. Along the way, I motivate and develop a Kratzer-style account, according to which what a subject wants depends not just on their information and preferences, but also on the decision rule they embrace and the salient decision problem.