Constructing Cantorian counterexamples

Cantors diagonal argument provides an indirect proof that there is no one-one function from the power set of a set A into A. This paper provides a somewhat more constructive proof of Cantors theorem, showing how, given a function f from the power set of A into A, one can explicitly define a counterexample to the thesis that f is one-one.     

Saving safety from counterexamples

Synthese - In this paper I will offer a comprehensive defense of the safety account of knowledge against counterexamples that have been recently put forward. In Sect. 2, I will discuss...     

Some counterexamples in multidimensional scaling

Given a set X with elements x, y,… which has a partial order < on the pairs of the Cartesian product X², one may seek a distance function ? on such pairs (x, y) which satisfies ?(x₁, y₁) < ?(x₂, y₂) precisely when (x₁, y₁) < (x₂, y₂), and even demand a metric space (X, ?) with some such compatible ? which has an isometric imbedding into a finite-dimensional Euclidean space or a separable Hilbert space. We exhibit here systems (X, <) which cannot meet the latter demand. The space of real m-tuples (ξ₁,…,ξ_m) with either the “city-block” norm Σ_i ∥ξ_i∥ or the “dominance” norm max_i, ∥ξ_i∥ cannot possibly become a subset of any finite-dimensional Euclidean space. The set of real sequences (ξ₁, ξ₂,…) with finitely many nonzero elements and the supremum norm sup_i, ∥ξ_i∥ cannot even become a subset of any separable Hilbert space.     

Structural equations and causation: six counterexamples

Hall [(2007), Philosophical Studies, 132, 109–136] offers a critique of structural equations accounts of actual causation, and then offers a new theory of his own. In this paper, I respond to Hall’s critique, and present some counterexamples to his new theory. These counterexamples are then diagnosed.

Modified Frankfurt-type counterexamples and flickers of freedom

A great deal of attention has been paid recently to the claim that traditional Frankfurt-type counterexamples to the Principle of Alternative Possibilities (PAP), which depend for their success on the presence of a perfectly reliable indicator (or prior sign) of what an agent will freely do if left to act on his own, are guilty of begging the question against incompatibilists, since such indicators seem to presuppose a deterministic relation between an agent’s free action and its causal antecedents. Objections of this sort have given rise to considerable efforts to construct alternative Frankfurt-type counterexamples that do not rely on prior signs of this kind and so do not presuppose determinism in a way that incompatibilists should find objectionable. One consequence of this shift in the way Frankfurt-type counterexamples are formulated is that it provides an opportunity for the forceful resurgence of certain versions of the so-called flicker defense of PAP. In this paper I develop two versions of the flicker defense, indicate their advantages over other versions of this strategy, and defend them against objections. Insofar as either of these is successful, it will show not only that PAP has yet to be falsified by any of the modified Frankfurt-type counterexamples currently on offer but that cases of this sort are in principle incapable of falsifying PAP.     

The difference between generating counterexamples and using them during reasoning

The aim of this article is to provide insight into the types of long-term knowledge that are used for solving causal conditional inferences. Two taxonomies were constructed to map the types of counterexample. The available counterexamples are traditionally probed via a counterexample generation task. We observed that there are some significant differences in the types of counterexample retrieved in the reasoning task versus the generation task. The generation task can be used for predicting answers that sprout from a reasoning process that takes counterexample into account, but some participants use a different reasoning process in which the available semantic information is not used as contrasting evidence. Nonetheless, we found that the results of the generation task validly predicted the proportion of inferences accepted as well as the number of counterexamples used during reasoning.     

Conditional reasoning,frequency of counterexamples,and the effect of response modality

Geiger and Oberauer (2007) found that when asked to reason with conditionals, people are very sensitive to information about the relative frequency of exceptions to conditional rules and quite insensitive to the relative number of disabling conditions. They asked participants to rate their degree of certainty in a conclusion. In the following studies, we investigated the possibility that this kind of response encourages a more probabilistic mode of processing compared with the usual dichotomous response. In Study 1, participants were given a variant of the problems used by Geiger and Oberauer with either the samescaled response format or a dichotomouscategorical response. The results with the scaled response were identical to those of Geiger and Oberauer. However, the results with the categorical response presented a very different profile. In Study 2, we presented similar problems using only frequency information, followed by a set of abstract conditional reasoning problems. The participants who performed better on the abstract problems showed a significantly different response profile than those who did worse on the abstract problems in the categorical response condition. No such difference was observed in the scaled response condition. These results show that response modality strongly affects the way in which information is processed in otherwise identical inferential problems and they are consistent with the idea that scaled responses promote a probabilistic mode of processing.     

On some putative graph-theoretic counterexamples to the Principle of the Identity of Indiscernibles

Recently, several authors have claimed to have found graph-theoretic counterexamples to the Principle of the Identity of Indiscernibles (PII). In this paper, I argue that their counterexamples presuppose a certain view of what unlabeled graphs are, and that this view is optional at best.     

The four free-operant freedoms

This article reviews early free-operant conditioning laboratory research and applications. The seldom-mentioned four free-operant freedoms are described for the first time in detail. Most current behavior analysts do not realize that the freedom to form responses and the freedom to speed responses were crucial steps in designing free-operant operanda in the 1950s. These four freedoms were known by the laboratory researchers of the 1950s to the point that, along with operanda design, Sidman (1960) did not feel the need to detail them in his classic, Tactics of Scientific Research. The dimensions of freedom in the operant were so well understood and accepted in the 1950s that most thought it redundant to use the term free operant. These issues are reviewed in some detail for younger behavior analysts who did not have the opportunity of learning them firsthand.     

Concurrent validity of four androgyny instruments

Four sex-role instruments were administered to a group of 281 graduate and undergraduate students at an urban university. Concurrent validity was examined through the use of correlational and factor-analytic techniques. Seven problems in interpretation of the categorizations made by the several instruments were identified: (1) item homogeneity of individual scales; (2) convergence of corresponding scales and divergence of scales from their complements; (3) multifactoriality; (4) factor complexity; (5) distribution of content across instruments; (6) interaction between factor complexity and content balance; and (7) correspondence of classifications of subjects across instruments.     

Sex-role preferences in four age levels

Brown's (1956a; 1956b) influential finding that girls of all ages exhibit masculine rather than feminine sex-role preferences has been challenged by the argument that the scale used in his studies—the It Scale for Children (ITSC)—is masculine biased. The present investigation reexamined this issue with an instrument designed to be free of the limitations inherent in the ITSC. Boys and girls from grades 2, 5, 8, and 11 examined drawings of activities scaled for sex-role stereotypy and selected those activities they would like to engage in. Consistent with Brown's earlier work, boys exhibited strong masculine preferences at all ages. Contrary to Brown, however, girls did not show similar masculine preferences. Instead, their choices reflected strong feminine preferences throughout the spectrum of ages tested.