Achieving completeness: from constructive set theory to large cardinals

This thesis is an exploration of several completeness phenomena, both in the constructive and the classical settings. After some introductory chapters in the first part of the thesis where we outline the background used later on, the constructive part contains a categorical formulation of several constructive completeness theorems available in the literature, but presented here in an unified framework. We develop them within a constructive reverse mathematical viewpoint, highlighting the metatheory used in each case and the strength of the corresponding completeness theorems. The classical part of the thesis focuses on infinitary intuitionistic propositional and predicate logic. We consider a propositional axiomatic system with a special distributivity rule that is enough to prove a completeness theorem, and we introduce weakly compact cardinals as the adequate metatheoretical assumption for this development. Finally, we return to the categorical formulation focusing this time on infinitary first-order intuitionistic logic. We propose a first-order system with a special rule, transfinite transitivity, that embodies both distributivity as well as a form of dependent choice, and study the extent to which completeness theorems can be established. We prove completeness using a weakly compact cardinal, and, like in the constructive part, we study disjunction-free fragments as well. The assumption of weak compactness is shown to be essential for the completeness theorems to hold.

Why is Werner Sombart not part of the core of classical sociologists? From fame to (near) oblivion:From Fame to (Near) Oblivion

The life and work of Werner Sombart poses an intellectual puzzle in the genealogy of modern social theorists. During his lifetime, Sombart was probably the most influential and prominent social scientist in Germany as well as in many other countries. Today he is among the least known social scientists. Why did he lose his status as one of the most brilliant and influential scholars and intellectuals of the 20th century? Why is his work almost forgotten today? While Weber's thesis about the influence of Protestantism on the development of capitalism is widely known, even beyond sociological circles, few sociologists today know that Sombart had an alternative explanation. An obvious explanation for Sombart's fall from grace is his embrace of Nazism. As Heidegger provides a counter-example, Sombart's fate requires a more complex explanation. In addition, we explore the different reception of his work in economic and sociological circles as compared to cultural theory and history. © 2001, SAGE Publications. All rights reserved.

Psychophysical evidence for two routes to suppression before binocular summation of signals in human vision

Visual mechanisms in primary visual cortex are suppressed by the superposition of gratings perpendicular to their preferred orientations. A clear picture of this process is needed to (i) inform functional architecture of image-processing models, (ii) identify the pathways available to support binocular rivalry, and (iii) generally advance our understanding of early vision. Here we use monoptic sine-wave gratings and cross-orientation masking (XOM) to reveal two cross-oriented suppressive pathways in humans, both of which occur before full binocular summation of signals. One is a within-eye (ipsiocular) pathway that is spatially broadband, immune to contrast adaptation and has a suppressive weight that tends to decrease with stimulus duration. The other pathway operates between the eyes (interocular), is spatially tuned, desensitizes with contrast adaptation and has a suppressive weight that increases with stimulus duration. When cross-oriented masks are presented to both eyes, masking is enhanced or diminished for conditions in which either ipsiocular or interocular pathways dominate masking, respectively. We propose that ipsiocular suppression precedes the influence of interocular suppression and tentatively associate the two effects with the lateral geniculate nucleus (or retina) and the visual cortex respectively. The interocular route is a good candidate for the initial pathway involved in binocular rivalry and predicts that interocular cross-orientation suppression should be found in cortical cells with predominantly ipsiocular drive. © 2007 IBRO.

Binocular summation of contrast remains intact in strabismic amblyopia

PURPOSE. Strabismic amblyopia is typically associated with several visual deficits, including loss of contrast sensitivity in the amblyopic eye and abnormal binocular vision. Binocular summation ratios (BSRs) are usually assessed by comparing contrast sensitivity for binocular stimuli (sens BIN) with that measured in the good eye alone (sensGOOD), giving BSR = sensBIN/sensGOOD. This calculation provides an operational index of clinical binocular function, but does not assess whether neuronal mechanisms for binocular summation of contrast remain intact. This study was conducted to investigate this question. METHODS. Horizontal sine-wave gratings were used as stimuli (3 or 9 cyc/deg; 200 ms), and the conventional method of assessment (above) was compared with one in which the contrast in the amblyopic eye was adjusted (normalized) to equate monocular sensitivities. RESULTS. In nine strabismic amblyopes (mean age, 32 years), the results confirmed that the BSR was close to unity when the conventional method was used (little or no binocular advantage), but increased to approximately √2 or higher when the normalization method was used. The results were similar to those for normal control subjects (n = 3; mean age, 38 years) and were consistent with the physiological summation of contrast between the eyes. When the normal observers performed the experiments with a neutral-density (ND) filter in front of one eye, their performance was similar to that of the amblyopes in both methods of assessment. CONCLUSIONS. The results indicate that strabismic amblyopes have mechanisms for binocular summation of contrast and that the amblyopic deficits of binocularity can be simulated with an ND filter. The implications of these results for best clinical practice are discussed. Copyright © Association for Research in Vision and Ophthalmology.

The evolution of "classical mythology" within marketing measure development

Purpose - This paper provides a deeper examination of the fundamentals of commonly-used techniques - such as coefficient alpha and factor analysis - in order to more strongly link the techniques used by marketing and social researchers to their underlying psychometric and statistical rationale. Design/methodology approach - A wide-ranging review and synthesis of psychometric and other measurement literature both within and outside the marketing field is used to illuminate and reconsider a number of misconceptions which seem to have evolved in marketing research. Findings - The research finds that marketing scholars have generally concentrated on reporting what are essentially arbitrary figures such as coefficient alpha, without fully understanding what these figures imply. It is argued that, if the link between theory and technique is not clearly understood, use of psychometric measure development tools actually runs the risk of detracting from the validity of the measures rather than enhancing it. Research limitations/implications - The focus on one stage of a particular form of measure development could be seen as rather specialised. The paper also runs the risk of increasing the amount of dogma surrounding measurement, which runs contrary to the spirit of this paper. Practical implications - This paper shows that researchers may need to spend more time interpreting measurement results. Rather than simply referring to precedence, one needs to understand the link between measurement theory and actual technique. Originality/value - This paper presents psychometric measurement and item analysis theory in easily understandable format, and offers an important set of conceptual tools for researchers in many fields. © Emerald Group Publishing Limited.

Binocular combination at threshold: temporal filtering and summation of signals in separate ON and OFF channels

How does the brain combine spatio-temporal signals from the two eyes? We quantified binocular summation as the improvement in 2AFC contrast sensitivity for flickering gratings seen by two eyes compared with one. Binocular gratings in-phase showed sensitivity up to 1.8 times higher, suggesting nearly linear summation of contrasts. The binocular advantage decreased to 1.4 at lower spatial and higher temporal frequencies (0.25 cycle deg-1, 30 Hz). Dichoptic, antiphase gratings showed only a small binocular advantage, by a factor of 1.1 to 1.2, but no evidence of cancellation. We present a signal-processing model to account for the contrast-sensitivity functions and the pattern of binocular summation. It has linear sustained and transient temporal filters, nonlinear transduction, and half-wave rectification that creates ON and OFF channels. Binocular summation occurs separately within ON and OFF channels, thus explaining the phase-specific binocular advantage. The model also accounts for earlier findings on detection of brief antiphase flashes and the surprising finding that dichoptic antiphase flicker is seen as frequency-doubled (Cavonius et al, 1992 Ophthalmic and Physiological Optics 12 153 - 156). [Supported by EPSRC project GR/S74515/01].

Interocular masking and summation indicate two stages of divisive contrast gain control

Our understanding of early spatial vision owes much to contrast masking and summation paradigms. In particular, the deep region of facilitation at low mask contrasts is thought to indicate a rapidly accelerating contrast transducer (eg a square-law or greater). In experiment 1, we tapped an early stage of this process by measuring monocular and binocular thresholds for patches of 1 cycle deg-1 sine-wave grating. Threshold ratios were around 1.7, implying a nearly linear transducer with an exponent around 1.3. With this form of transducer, two previous models (Legge, 1984 Vision Research 24 385 - 394; Meese et al, 2004 Perception 33 Supplement, 41) failed to fit the monocular, binocular, and dichoptic masking functions measured in experiment 2. However, a new model with two-stages of divisive gain control fits the data very well. Stage 1 incorporates nearly linear monocular transducers (to account for the high level of binocular summation and slight dichoptic facilitation), and monocular and interocular suppression (to fit the profound 42 Oral presentations: Spatial vision Thursday dichoptic masking). Stage 2 incorporates steeply accelerating transduction (to fit the deep regions of monocular and binocular facilitation), and binocular summation and suppression (to fit the monocular and binocular masking). With all model parameters fixed from the discrimination thresholds, we examined the slopes of the psychometric functions. The monocular and binocular slopes were steep (Weibull ߘ3-4) at very low mask contrasts and shallow (ߘ1.2) at all higher contrasts, as predicted by all three models. The dichoptic slopes were steep (ߘ3-4) at very low contrasts, and very steep (ß>5.5) at high contrasts (confirming Meese et al, loco cit.). A crucial new result was that intermediate dichoptic mask contrasts produced shallow slopes (ߘ2). Only the two-stage model predicted the observed pattern of slope variation, so providing good empirical support for a two-stage process of binocular contrast transduction. [Supported by EPSRC GR/S74515/01]

Contrast masking in strabismic amblyopia: attenuation, noise, interocular suppression and binocular summation

To investigate amblyopic contrast vision at threshold and above we performed pedestal-masking (contrastdiscrimination) experiments with a group of eight strabismic amblyopes using horizontal sinusoidal gratings (mainly 3 c/deg) in monocular, binocular and dichoptic configurations balanced across eye (i.e. five conditions). With some exceptions in some observers, the four main results were as follows. (1) For the monocular and dichoptic conditions, sensitivity was less in the amblyopic eye than in the good eye at all mask contrasts. (2) Binocular and monocular dipper functions superimposed in the good eye. (3) Monocular masking functions had a normal dipper shape in the good eye, but facilitation was diminished in the amblyopic eye. (4) A less consistent result was normal facilitation in dichoptic masking when testing the good eye, but a loss of this when testing the amblyopic eye. This pattern of amblyopic results was replicated in a normal observer by placing a neutral density filter in front of one eye. The two-stage model of binocular contrast gain control [Meese, T.S., Georgeson, M.A. & Baker, D.H. (2006). Binocular contrast vision at and above threshold. Journal of Vision 6, 1224--1243.] was `lesioned' in several ways to assess the form of the amblyopic deficit. The most successful model involves attenuation of signal and an increase in noise in the amblyopic eye, and intact stages of interocular suppression and binocular summation. This implies a behavioural influence from monocular noise in the amblyopic visual system as well as in normal observers with an ND filter over one eye.

Area summation in human vision at and above detection threshold

The initial image-processing stages of visual cortex are well suited to a local (patchwise) analysis of the viewed scene. But the world's structures extend over space as textures and surfaces, suggesting the need for spatial integration. Most models of contrast vision fall shy of this process because (i) the weak area summation at detection threshold is attributed to probability summation (PS) and (ii) there is little or no advantage of area well above threshold. Both of these views are challenged here. First, it is shown that results at threshold are consistent with linear summation of contrast following retinal inhomogeneity, spatial filtering, nonlinear contrast transduction and multiple sources of additive Gaussian noise. We suggest that the suprathreshold loss of the area advantage in previous studies is due to a concomitant increase in suppression from the pedestal. To overcome this confound, a novel stimulus class is designed where: (i) the observer operates on a constant retinal area, (ii) the target area is controlled within this summation field, and (iii) the pedestal is fixed in size. Using this arrangement, substantial summation is found along the entire masking function, including the region of facilitation. Our analysis shows that PS and uncertainty cannot account for the results, and that suprathreshold summation of contrast extends over at least seven target cycles of grating. © 2007 The Royal Society.

The classical theory of field evaporation

The field evaporation literature has been carefully analysed and is shown to contain various confusions. After redefining consistent terminology, this thesis investigates the mechanisms of field evaporation, in particular, the relevance of the theoretical mechanisms by analysing the available experimental data. A new formalism `extended image-hump formalism' is developed and is used to devise several tests of whether the image-hump mechanism is operating. The general conclusion is that in most cases the Mueller mechanism is not operating and escape takes place via Gomer-type mechanisms.

Neuronal convergence in early contrast vision: binocular summation is followed by response nonlinearity and area summation

We assessed summation of contrast across eyes and area at detection threshold ( C t). Stimuli were sine-wave gratings (2.5 c/deg) spatially modulated by cosine- and anticosine-phase raised plaids (0.5 c/deg components oriented at ±45°). When presented dichoptically the signal regions were interdigitated across eyes but produced a smooth continuous grating following their linear binocular sum. The average summation ratio ( C t1/([ C t1+2]) for this stimulus pair was 1.64 (4.3 dB). This was only slightly less than the binocular summation found for the same patch type presented to both eyes, and the area summation found for the two different patch types presented to the same eye. We considered 192 model architectures containing each of the following four elements in all possible orders: (i) linear summation or a MAX operator across eyes, (ii) linear summation or a MAX operator across area, (iii) linear or accelerating contrast transduction, and (iv) additive Gaussian, stochastic noise. Formal equivalences reduced this to 62 different models. The most successful four-element model was: linear summation across eyes followed by nonlinear contrast transduction, linear summation across area, and late noise. Model performance was enhanced when additional nonlinearities were placed before binocular summation and after area summation. The implications for models of probability summation and uncertainty are discussed.

Anisotropy for spatial summation of elongated patches of grating:A tale of two tails

Studies of spatial summation often use sinusoidal gratings with blurred edges. When the envelope is elongated (i) along the grating stripes and (ii) at right angles to the grating stripes, we refer to the stimuli as skunk-tails and tiger-tails respectively. Previous work [Polat & Tyler, 1999; Vision Research, 39, 887-895.] has found that sensitivity to skunk-tails is greater than for tiger-tails, but there have been several failures to replicate this result within a subset of the conditions. To address this we measured detection thresholds for skunk-tails, tiger-tails and squares of grating with sides matched to the lengths of the tails. For foveal viewing, we found a contrast sensitivity advantage in the order of 2 dB for skunk-tails over tiger-tails, but only for horizontal gratings. For vertical gratings, sensitivity was very similar for both tail-types. When the stimuli were presented parafoveally (upper right visual field), a small advantage was found for skunk-tails over tiger-tails at both orientations, and spatial summation slopes were close to that of the ideal observer. We did not replicate the findings of Polat & Tyler, but our results are consistent with (i) those of Foley et al. [Foley, J. M., Varadharajan, S., Koh, C. C., & Farias, C. Q. (2007) Vision Research, 47, 85-107.] who used only vertical gratings and (ii) those from modelfest, where only horizontal gratings were used. The small effect of tail-type here suggests an anisotropy in the underlying physiology. © 2007 Elsevier Ltd. All rights reserved.