We examine a general class of variable-value population principles. Our particular focus is on the extent to which such principles can avoid the repugnant and sadistic conclusions. We show that if a mild limit property is imposed, avoidance of the repugnant conclusion implies the sadistic conclusion. This result generalizes earlier observations by showing that they apply to a substantially larger class of principles. Our second theorem states that, under the limit property, the axiom of mere addition also conflicts (...) with avoidance of the repugnant conclusion. This result is a consequence of a similar observation that appears in the earlier literature. (shrink)

Variable-Value axiologies avoid Parfit’s Repugnant Conclusion while satisfying some weak instances of the Mere Addition principle. We apply calibration methods to two leading members of the family of Variable-Value views conditional upon: first, a very weak instance of Mere Addition and, second, some plausible empirical assumptions about the size and welfare of the intertemporal world population. We find that such facts calibrate these two Variable-Value views to be nearly totalist, and therefore imply conclusions that (...) should seem repugnant to anyone who opposes Total Utilitarianism only due to the Repugnant Conclusion. (shrink)

The notion of a proposition as a set of possible worlds or states occupies central stage in probability theory, semantics and epistemology, where it serves as the fundamental unit both of information and meaning. But this fact should not blind us to the existence of prospects with a different structure. In the paper I examine the use of random variables—in particular, proposition-valued random variables— in these fields and argue that we need a general account of rational attitude formation with respect (...) to them. (shrink)

If all differences in behavior are explainable in terms of universal values pursued under variable constraints, then much ethical theorizing is pointless. A strong presumption in favor of universal values can be established by showing that differences in behavior that were previously thought to be explainable only in terms of differences in values, can in fact be explained in terms of differences in constraints. Eleven such cases are briefly discussed, including cases of differences among racial, religious and other groups (...) in crime, culinary practices and the acceptance of innovation. (shrink)

We can use language to say what people did, often describing the same action in different complex ways. Davidson offered an illuminating analysis of action reports like “Miss Scarlet stabbed Colonel Mustard with a dagger in the library,” which involve adverbial modifiers. Part of the challenge here is to say how such modifiers are semantically related to the rest of the sentence. Building on the ancient observation that verbs are often used to describe what happened, Davidson argued that an action (...) report implies the occurrence of an eventthat meets the conditions specified by the report; where an event can be described in endlessly many ways. In this chapter, I review Davidson's account, his arguments in favor of it, and his responses to some initial objections. I also discuss some subsequent work that supports central aspects of Davidson's proposal while also presenting challenges for combining his analysis of action reports with his broader conception of linguistic meaning. (shrink)

What does it mean to ‘give’ the value of a variable in an algebraic context, and how does giving the value of a variable differ from merely describing it? I argue that to answer this question, we need to examine the role that giving the value of a variable plays in problem-solving practice. I argue that four different features are required for a statement to count as giving the value of a variable (...) in the context of solving an elementary algebra problem: the variable must be in the scope opened by the problem statement; the values given must be in the range of the variable, which is determined by the problem; the statement giving the values must represent a complete solution; and it must be in a canonical form. This account helps us better understand elementary algebra itself, as well as the use of algebraic tools to analyze phenomena in natural language. (shrink)

Prior research has identified age 9–11 as a critical period for the development of the control-of-variables strategy (CVS). We examine the stability of interindividual differences in children's CVS skills with regard to their precursor skills during this critical developmental period. To this end, we relate two precursor skills of CVS at age 9 to four skills constituting fully developed CVS more than 2 years later, controlling for children's more general cognitive development. Note thatN= 170 second- to fourth-graders worked on multiple (...) choice-assessments of their understanding of indeterminacy of evidence and of confounding. We find relations between these two precursor skills and children's CVS skills 2 years later at age 11 inplanning, identifying, andinterpretingcontrolled experiments, and in recognizing the inconclusiveness of confounded comparisons (understanding). In accordance with the perspective that both indeterminacy and confounding constitute critical, related yet distinct elements of CVS, both precursor skills contribute to the prediction of later CVS. Together, the two precursor skills can explain 39% of students' later CVS mastery. Overall, the understanding of indeterminacy is a stronger predictor of fully developed CVS than that of confounding. The understanding of confounding, however, is a better predictor of the more difficult CVS sub-skills of understanding the inconclusiveness of confounded comparisons, and of planning a correctly controlled experiment. Importantly, both precursor skills maintain interactive predictive strength when controlling for children's general cognitive abilities and reading comprehension, showing that the developmental dynamics of CVS and its precursor skills cannot be fully ascribed to general cognitive development. We discuss implications of these findings for theories about the development of CVS and broader scientific reasoning. (shrink)

It is shown that Bell's theorem fails for the Clifford algebra valued local realistic variables. This is made evident by exactly reproducing quantum mechanical expectation value for the EPR-Bohm type spin correlations observable by means of a local, deterministic, Clifford algebra valued variable, without necessitating either remote contextuality or backward causation. Since Clifford product of multivector variables is non-commutative in general, the spin correlations derived within our locally causal model violate the CHSH inequality just as strongly as their (...) quantum mechanical counterparts. (shrink)

In mathematics, when we want to introduce classes which fulfill certain conditions, we usually prove beforehand that classes fulfilling such conditions do exist, and that such classes are uniquely determined by the conditions. The statements which state such unicity and existence of classes are in mathematical logic consequences of the principles of extensionality and class existence. In order to illustrate how these principles enable us to introduce classes into systems of mathematical logic, let us consider the manner in which Gödel (...) introduces classes in his book on set theory.For instance, before introducing the definition of the non-ordered pair of two classesGödel puts down as its justification the following two axioms:By A4, for every two classesyandzthere exists at least one non-ordered pairwof them; and by A3,wis uniquely determined in A4. (shrink)

The p value is the probability under the null hypothesis of obtaining an experimental result that is at least as extreme as the one that we have actually obtained. That probability plays a crucial role in frequentist statistical inferences. But if we take the word ‘extreme’ to mean ‘improbable’, then we can show that this type of inference can be very problematic. In this paper, I argue that it is a mistake to make such an interpretation. Under minimal assumptions (...) about the alternative hypothesis, I explain why ‘extreme’ means ‘outside the most precise predicted range of experimental outcomes for a given upper bound probability of error’. Doing so, I rebut recent formulations of recurrent criticisms against the frequentist approach in statistics and underscore the importance of random variables. (shrink)

For a graph G, its variable sum exdeg index is defined as SEI a G = ∑ x y ∈ E G a d x + a d y, where a is a real number other than 1 and d x is the degree of a vertex x. In this paper, we characterize all trees on n vertices with first three maximum and first three minimum values of the SEI a index. Also, we determine all the trees of order (...) n with given diameter d and having first three largest values of the SEI a index. (shrink)

This paper presents the interpersonal variability of harm challenge to Jerome Wakefield’s harmful-dysfunction account (HDA) of disorder. This challenge stems from the seeming fact that what promotes well-being or is harmful to someone varies much more across individuals than what is intuitively healthy or disordered. This makes it at least prima facie difficult to see how judgments about health and disorder could, as harm-requiring accounts of disorder like the HDA maintain, be based on, or closely linked to, judgments about well-being (...) and harm. This interpersonal variability of harm challenge is made salient by the difficulty faced by harm-requiring accounts of disorder in dealing satisfactorily with cases of intuitively disordered conditions that seem harmless because they do not deprive the individuals that they affect of anything that they value (e.g., desired infertility). I argue that this challenge is made more serious for the HDA by some clarifications Wakefield has recently made on harm. In recent publications, Wakefield dissociates himself from the sheer cultural-relativist view of harm attributed to him by some critics based on his linkage of harm to social values, and adopts a more qualified social-values-based view of harm that leaves room for criticism of the values endorsed by members of a cultural group at a given time. I show how Wakefield’s qualified view makes it more difficult for the HDA to deal with the interpersonal variability of harm challenge, at least when applied to a Western cultural context where a high value is placed on autonomy and individual choice. (shrink)

Fixed-rate versions of rule-consequentialism and rule-utilitarianism evaluate rules in terms of the expected net value of one particular level of social acceptance, but one far enough below 100% social acceptance to make salient the complexities created by partial compliance. Variable-rate versions of rule-consequentialism and rule-utilitarianism instead evaluate rules in terms of their expected net value at all different levels of social acceptance. Brad Hooker has advocated a fixed-rate version. Michael Ridge has argued that the variable-rate version (...) is better. The debate continues here. Of particular interest is the difference between the implications of Hooker's and Ridge's rules about doing good for others. (shrink)

The phenomenon of quantification into attitude ascriptions has haunted broadly Fregean views, according to which co-referential proper names are not always substitutable salva veritate in attitude ascriptions. Opponents of Fregeanism argue that a belief ascription containing a proper name such as ‘Michael believes that Lindsay is charitable’ is equivalent to a quantified sentence such as ‘there is someone such that Michael believes that she is charitable, and that person is Lindsay’. They conclude that the semantic contribution of a name such (...) as ‘Lindsay’ is the same as the semantic contribution of a variable under an assignment, which these opponents suggest is merely the object assigned to that variable. However, renewed interest in variables suggests that they make a more complicated contribution to the semantic processing of sentences that contain them. In particular, a variable contributes both an assignment-unsaturated and an assignment-saturated semantic value. I use this dual role of the semantics of variables to develop a response to the argument from quantifying in. I take as my point of departure Cumming's () view that an attitude ascription relates the subject of an attitude to the assignment-unsaturated semantic value of an open sentence. I argue that this approach fails. I propose an alternative, according to which the truth of a belief ascription depends on both the assignment-saturated and the assignment-unsaturated semantic value of the open sentence in its that-clause. This approach reverses standard assumptions concerning the relation between quantification and substitution. (shrink)

The development of the dynamic semantics of natural languagehas put issues of variable control on the agenda of formal semantics. Inthis paper we regard variables as names for stacks of values and makeexplicit several control actions as push and pop actions on stacks. Weapply this idea both to static and dynamic languages and compare theirfinite variable hierarchies, i.e., the relation between the number ofvariable stacks that is available and the expressivity of the language.This can be compared in natural (...) languages with relating the number ofpronouns available to the expressivity of the language.The results are obtained using techniques from static and dynamic modeltheory: model theoretic games, transition systems and bisimulation. (shrink)

In this paper it will be shown that when a conditional statement is understood or known to be true, a number of implicitly specified variables are given more or less concrete values. Each of the variables will be defined and examples will be employed to demonstrate their use in conditional evaluation. From time to time this analysis in terms of variables will be contrasted with a 'possible worlds' analysis of conditionals. The purpose of this paper is not to argue against (...) the possible worlds analysis but rather to provide an alternative to that analysis. (shrink)