A counterexample to the contrastive account of knowledge

Many epistemologists treat knowledge as a binary relation that holds between a subject and a proposition. The contrastive account of knowledge developed by Jonathan Schaffer maintains that knowledge is a ternary, contrastive relation that holds between a subject, a proposition, and a set of contextually salient alternative propositions the subject’s evidence must eliminate. For the contrastivist, it is never simply the case that S knows that p; in every case of knowledge S knows that p rather than q. This paper offers a counterexample to the contrastive account of knowledge. Part 1 summarizes the contrastive theory developed by Schaffer in a series of recent papers. Part 2 presents an example from a class of cases characterized by compatibility between the proposition p and each of the alternative propositions that occupy q. In such cases the alternative propositions that partially constitute the ternary contrastive relation play no role in the acquisition of knowledge. Part 3 considers and rejects potential responses to the counterexample. The paper concludes that the contrastive theory is not a general account of knowledge.     

A counterexample to Fishburn's conjecture on finite linear qualitative probability

Kraft, Pratt and Seidenberg (Ann. Math. Statist. 30 (1959) 408) provided an infinite set of axioms which, when taken together with de Finetti's axiom, gives a necessary and sufficient set of “cancellation” conditions for representability of an ordering relation on subsets of a set by an order-preserving probability measure. Fishburn (1996) defined f(n) to be the smallest positive integer k such that every comparative probability ordering on an n-element set which satisfies the cancellation conditions C₄,…,C_k is representable. By the work of Kraft, Pratt, and Seidenberg (1959) and Fishburn (J. Math. Psychol. 40 (1996) 64; J. Combin. Design 5 (1997) 353), it is known that n-1?f(n)?n+1 for all n?5. Also Fishburn proved that f(5)=4, and conjectured that f(n)=n-1 for all n?5. In this paper we confirm that f(6)=5, but give counter-examples to Fishburn's conjecture for n=7, showing that f(7)?7. We summarise, correct and extend many of the known results on this topic, including the notion of “almost representability”, and offer an amended version of Fishburn's conjecture.     

Vann McGee's counterexample to modus ponens

Conclusion After rejecting some unjustified criticism of McGee's claim about modus ponens, I argued for two conditionals: If the Adams-Appiah theory of conditionals is correct, then modus ponens is saved from McGee's counterexample. If the Lewis-Jackson theory is correct, the same holds. Furthermore, I raised some doubts whether the antecedents of these conditionals are true. But, if one of them were, then, by modus ponens, modus ponens would be generally valid.     

On the investigation of local identifiability: A counterexample

An assertion that the parameters of a covariance structure are locally identified at a certain point only if the rank of the Jacobian matrix at that point equals the number of parameters, is shown to be false by means of a counterexample.     

A counterexample to the rigidity assumption in the visual perception of structure from motion

It has been proposed that the human visual system prefers perceptions of objects that are rigid or undergo minimum form change. A counterexample is presented in which a rigid two-dimensional figure rotating in the frontal plane is perceived as a distorting three-dimensional shape. It is argued that this perception results from the stimulation of automatic processes for perceiving size change, and that these processes are not subject to a general rigidity assumption.     

Working memory and counterexample retrieval for causal conditionals

The present study is part of recent attempts to specify the characteristics of the counterexample retrieval process during causal conditional reasoning. The study tried to pinpoint whether the retrieval of stored counterexamples (alternative causes and disabling conditions) for a causal conditional is completely automatic in nature or whether the search process also demands executive working memory (WM) resources. In Experiment 1, participants were presented with a counterexample generation task and a measure of WM capacity. We found a positive relation between search efficiency, as measured by the number of generated counterexamples in limited time, and WM capacity. Experiment 2 examined the effects of a secondary WM load on the retrieval performance. As predicted, burdening WM with an attention-demanding secondary task decreased the retrieval efficiency. Both low and high spans were affected by the WM load but load effects were less pronounced for the most strongly associated counterexamples. Findings established that in addition to an automatic search component, the counterexample retrieval draws on WM resources.     

The brain-sex theory of occupational choice: a counterexample

The brain-sex theory of occupational choice suggests that males and females in male-typical careers show a male pattern of cognitive ability in terms of better spatial than verbal performance on cognitive tests with the reverse pattern for females and males in female-typical careers. These differences are thought to result from patterns of cerebral functional lateralisation. This study sought such occupationally related effects using synonym generation (verbal ability) and mental rotation (spatial ability) tasks used previously. It also used entrants to these careers as participants to examine whether patterns of cognitive abilities might predate explicit training and practice. Using a population of entrants to sex-differentiated university courses, a moderate occupational effect on the synonym generation task was found, along with a weak (p < .10) sex effect on the mental rotation task. Highest performance on the mental rotation task was by female students in fashion design, a female-dominated occupation which makes substantial visuospatial demands and attracts many students with literacy problems such as dyslexia. This group then appears to be a counterexample to the brain-sex theory. However, methodological issues surrounding previous studies are highlighted: the simple synonym task appears to show limited discrimination of the sexes, leading to questions concerning the legitimacy of inferences about lateralisation based on scores from that test. Moreover, the human figure-based mental rotation task appears to tap the wrong aspect of visuospatial skill, likely to be needed for male-typical courses such as engineering. Since the fashion-design career is also one that attracts disproportionately many male students whose sexual orientation is homosexual, data were examined for evidence of female-typical patterns of cognitive performance among that subgroup. This was not found. This study therefore provides no evidence for the claim that female-pattern cerebral functional lateralisation is likely in gay males.     

A two-dimensional computer-controlled visual stimulator

A computer-controlled instrument that creates complex two-dimensional patterns on a CRT monitor is described. These patterns are used to elicit visual evoked responses. Patterns are produced on a raster that is rotatable about its center. It is possible to assign to arbitrary regions in the raster any of four independent one-dimensional spatial-temporal functions. For each spatial-temporal function, the experimenter can select an arbitrary spatial profile, the spatial frequency of the profile, the starting phase of the profile, the temporal function, and the depth of modulation.     

Introduction to two-dimensional Fourier analysis

The application of two-dimensional Fourier analysis provides new avenues for research in visual perception. This tutorial serves as an introduction to some of the methods used in two-dimensional Fourier analysis and an introduction to two-dimensional image processing in general.     

A two-dimensional model of social representations

This paper reports a series of four experiments conducted to gain insight into students' social representations of the firm. The results suggested a possible revision of the central core theory. They showed that the central elements of the representation were treated differently by the subjects. More specifically, the notion of Profit and Hierarchy, both central, did not play the same role in the representation. The subjects were found to grant evaluative power to the notion of Profit but not to the notion of Hierarchy. This differentiation by evaluative potential was also found for two peripheral elements. These differences suggest that independently of their central or peripheral nature, the various elements of a representation can be placed along an evaluative continuum. By combining the evaluative dimension and the central or peripheral dimension of the elements, we obtain a two-dimensional model of representations which divides discourse and cognition into four fields.     

A logic for epistemic two-dimensional semantics

Epistemic two-dimensional semantics is a theory in the philosophy of language that provides an account of meaning which is sensitive to the distinction between necessity and apriority. While this theory is usually presented in an informal manner, I take some steps in formalizing it in this paper. To do so, I define a semantics for a propositional modal logic with operators for the modalities of necessity, actuality, and apriority that captures the relevant ideas of epistemic two-dimensional semantics. I also describe some properties of the logic that are interesting from a philosophical perspective, and apply it to the so-called nesting problem.     

A two-dimensional model that employs explicit and implicit attitudes to characterize prejudice

In the authors' 2-dimensional model of prejudice, explicit and implicit attitudes are used to create 4 profiles: truly low prejudiced (TLP: double lows), aversive racists (AR: low explicit modern racism/high implicit prejudice), principled conservatives (PC: high explicit modern racism/low implicit prejudice), and modern racists (MR: double highs). Students completed an Asian Modern Racism Scale and an Asian/White Implicit Association Test. The authors compared the 4 groups' prejudice-related ideologies (i.e., egalitarianism/humanism and social conservatism) and economic/political conservatism (Study 1, N=132). The authors also tested whether MR but not PC (Study 2, N=65) and AR but not TLP (Study 3, N=143) are more likely to negatively evaluate an Asian target when attributional ambiguity is high (vs. low). In support of the model, TLP did not hold prejudice-related ideologies and did not discriminate; AR were low in conservatism and demonstrated the attributional-ambiguity effect; PC did not strongly endorse prejudice-related ideologies and did not discriminate; MR strongly endorsed prejudice-related ideologies, were conservative, and demonstrated the attributional-ambiguity effect. The authors discuss implications for operationalizing and understanding the nature of prejudice.     

Inference suppression and semantic memory retrieval: Every counterexample counts

Reasoning with conditionals involving causal content is known to be affected by retrieval of counterexamples from semantic memory. In this study we examined the characteristics of this search process in everyday conditional reasoning. In Experiment 1 we manipulated the number (zero to four) of explicitly presented counterexamples (alternative causes or disabling conditions) for causal conditionals. In Experiment 2, using a generation pretest, we measured the number of counterexamples participants could retrieve for a set of causal conditionals. One month after the pretest, participants were presented a reasoning task with the same conditionals. The experiments indicated that acceptance of modus ponens linearly decreased with every additionally retrieved disabler, whereas affirmation of the consequent acceptance linearly decreased as a function of the number of retrieved alternatives. Results for denial of the antecedent and modus tollens were less clear. The findings show that the search process does not necessarily stop after retrieval of a single counterexample and that every additional counterexample has an impact on the inference acceptance.     

A two-dimensional logic for diagonalization and the a priori

Synthese - Two-dimensional semantics, which can represent the distinction between a priority and (one kind of) necessity, has wielded considerable influence in the philosophy of language. In this...