Consistency, Models, and Soundness

This essay consists of two parts. In the first part, I focus my attention on the remarks that Frege makes on consistency when he sets about criticizing the method of creating new numbers through definition or abstraction. This gives me the opportunity to comment also a little on H. Hankel, J. Thomae—Frege’s main targets when he comes to criticize “formal theories of arithmetic” in Die Grundlagen der Arithmetik (1884) and the second volume of Grundgesetze der Arithmetik (1903)—G. Cantor, L. E. J. Brouwer and D. Hilbert (1899). Part 2 is mainly devoted to Hilbert’s proof theory of the 1920s (1922–1931). I begin with an account of his early attempt to prove directly, and thus not by reduction or by constructing a model, the consistency of (a fragment of) arithmetic. In subsequent sections, I give a kind of overview of Hilbert’s metamathematics of the 1920s and try to shed light on a number of difficulties to which it gives rise. One serious difficulty that I discuss is the fact, widely ignored in the pertinent literature on Hilbert’s programme, that his language of finitist metamathematics fails to supply the conceptual resources for formulating a consistency statement qua unbounded quantification. Along the way, I shall comment on W. W. Tait’s objection to an interpretation of Hilbert’s finitism by Niebergall and Schirn, on G. Gentzen’s allegedly finitist consistency proof for Peano Arithmetic as well as his ideas on the provability and unprovability of initial cases of transfinite induction in pure number theory. Another topic I deal with is what has come to be known as partial realizations of Hilbert’s programme, chiefly advocated by S. G. Simpson. Towards the end of this essay, I take a critical look at Wittgenstein’s views about (in)consistency and consistency proofs in the period 1929–1933. I argue that both his insouciant attitude towards the emergence of a contradiction in a calculus and his outright repudiation of metamathematical consistency proofs are unwarranted. In particular, I argue that Wittgenstein falls short of making a convincing case against Hilbert’s programme. I conclude with some philosophical remarks on consistency proofs and soundness and raise a question concerning the consistency of analysis.     

Emotional expression on a profile: Feature height, mouth angle, and tilt

Profiles may vary in expression with changes in the height of the profile’s features, the mouth’s angle at the profile contour, and head tilt. In three experiments, we varied these factors. Features low on the profile, with mouths at an obtuse angle to the profile line and head tilted 15° forehead-forward, appear especially sad. High features with acute mouth angles suit forward, upright, and backward head tilts. Feature height, mouth angle, and head tilt on a profile are physical variables that can specify postures such as downcast heads and turned-down mouths, and thereby psychological factors. We point out alternative accounts of the height and tilt effects.     

Kritische Anmerkungen zur Einweisung von Jugendlichen und Heranwachsenden in den Maßregelvollzug

In this paper the preconditions for placing adolescents and young adults in forensic psychiatric institutions are analyzed. According to the law such placements are only justifiable in exceptional cases. Even from a psychiatric point of view it is difficult to delineate a severe and most notably lasting mental disorder at a young age and distinguish it from a lack of maturity, although only the first would justify an indefinite placement in an institution. Nevertheless both broadly pathologizing certain behaviors and loosely interpreting preconditions for placements in institutions encourage commitment to institutions.     

Virtual social interactions in social anxiety--the impact of sex, gaze, and interpersonal distance

In social interactions, interpersonal distance between interaction partners plays an important role in determining the status of the relationship. Interpersonal distance is an important nonverbal behavior, and is used to regulate personal space in a complex interplay with other nonverbal behaviors such as eye gaze. In social anxiety, studies regarding the impact of interpersonal distance on within-situation avoidance behavior are so far rare. Thus the present study aimed to scrutinize the relationship between gaze direction, sex, interpersonal distance, and social anxiety in social interactions. Social interactions were modeled in a virtual-reality (VR) environment, where 20 low and 19 high socially anxious women were confronted with approaching male and female characters, who stopped in front of the participant, either some distance away or close to them, and displayed either a direct or an averted gaze. Gaze and head movements, as well as heart rate, were measured as indices of avoidance behavior and fear reactions. High socially anxious participants showed a complex pattern of avoidance behavior: when the avatar was standing farther away, high socially anxious women avoided gaze contact with male avatars showing a direct gaze. Furthermore, they showed avoidance behavior (backward head movements) in response to male avatars showing a direct gaze, regardless of the interpersonal distance. Overall, the current study proved that VR social interactions might be a very useful tool for investigating avoidance behavior of socially anxious individuals in highly controlled situations. This might also be the first step in using VR social interactions in clinical protocols for the therapy of social anxiety disorder.     

Gentzen Calculi for the Existence Predicate

We introduce Gentzen calculi for intuitionistic logic extended with an existence predicate. Such a logic was first introduced by Dana Scott, who provided a proof system for it in Hilbert style. We prove that the Gentzen calculus has cut elimination in so far that all cuts can be restricted to very simple ones. Applications of this logic to Skolemization, truth value logics and linear frames are also discussed.     

The Potential of Anamnestic Comparative Self-Assessment (ACSA) to Reduce Bias in the Measurement of Subjective Well-Being

Background: The conventional question (CQ) on subjective well-being (SWB) is e.g. “How is life?”, with ratings between e.g. ‘Best’ and ‘Worst possible’. Disadvantages may be casualness of responses and biases of proximate, peer or cultural relativity. Alternatively, with Anamnestic Comparative Self-Assessment (ACSA), the scale anchors are the respondents’ self-defined memories of their best and worst periods in life. Thus ACSA uses life review and experiential scale anchors. Objective: To compare the validity, sensitivity and responsiveness of the CQ and ACSA. Method: ACSA and the CQ were administered in parallel to 2584 university-hospital patients suffering from a wide range of psychiatric and somatic diseases. Results: ACSA and CQ did not measure the same construct (r = 0.50). CQ ratings were almost normally distributed, whereas ACSA ratings were overall lower, and clearly positively skewed, suggesting greater sensitivity to the respondents’ diseased state. Contrary to CQ, ACSA ratings of critically ill patients with end-stage liver disease were very low. After life-saving liver transplantation, ACSA ratings increased significantly more than CQ ratings, suggesting better responsiveness of ACSA to objective change. Trait-like socio-demographic variables such as sex, age, and marital status influenced CQ, but not ACSA ratings. Conclusion: In between-subject studies, depending on one’s study objectives, ACSA should be considered as a complement or an alternative to conventional SWB instruments. The CQ is probably preferable when socio-demographic variables are study endpoints. In longitudinal or intervention studies and for intercultural comparisons, ACSA, which reduces the need for correction of several biases or confounders, seems more useful.     

Combining mixture distribution and multidimensional IRTree models for the measurement of extreme response styles

Personality constructs, attitudes and other non-cognitive variables are often measured using rating or Likert-type scales, which does not come without problems. Especially in low-stakes assessments, respondents may produce biased responses due to response styles (RS) that reduce the validity and comparability of the measurement. Detecting and correcting RS is not always straightforward because not all respondents show RS and the ones who do may not do so to the same extent or in the same direction. The present study proposes the combination of a multidimensional IRTree model with a mixture distribution item response theory model and illustrates the application of the approach using data from the Programme for the International Assessment of Adult Competencies (PIAAC). This joint approach allows for the differentiation between different latent classes of respondents who show different RS behaviours and respondents who show RS versus respondents who give (largely) unbiased responses. We illustrate the application of the approach by examining extreme RS and show how the resulting latent classes can be further examined using external variables and process data from computer-based assessments to develop a better understanding of response behaviour and RS.     

Scoring Depression on a Common Metric: A Comparison of EAP Estimation,Plausible Value Imputation,and Full Bayesian IRT Modeling

There are a growing number of item response theory (IRT) studies that calibrate different patient-reported outcome (PRO) measures, such as anxiety, depression, physical function, and pain, on common, instrument-independent metrics. In the case of depression, it has been reported that there are considerable mean score differences when scoring on a common metric from different, previously linked instruments. Ideally, those estimates should be the same. We investigated to what extent those differences are influenced by different scoring methods that take into account several levels of uncertainty, such as measurement error (through plausible value imputation) and item parameter uncertainty (through full Bayesian IRT modeling). Depression estimates from different instruments were more similar, and their corresponding confidence/credible intervals were larger when plausible value imputation or Bayesian modeling was used, compared to the direct use of expected a posteriori (EAP) estimates. Furthermore, we explored the use of Bayesian IRT models to update item parameters based on newly collected data.     

Reliability and validity of the Single‐Target IAT (ST‐IAT): assessing automatic affect towards multiple attitude objects

In contrast to the original Implicit Association Test (IAT), the Single‐Target Implicit Association Test (ST‐IAT) measures the evaluation of a target object without the need to simultaneously evaluate a counter‐category. The present research investigates (a) whether position within a series of several ST‐IATs affects reliability and validity, and (b) whether the ST‐IAT exhibits adequate construct validity if the target objects are closely interrelated. We address these questions by taking five interrelated yet distinct political parties in Germany as an exemplary domain. The ST‐IAT reliably and validly assessed attitudes towards political parties (Study 1). Serial position effects did not affect the results. The ST‐IATs mostly captured a specific party evaluation and exhibited discriminant validity. At the same time, discriminant validity was limited among parties within one wing of the political left–right spectrum that underlies implicit and explicit party evaluations (Study 2). If used with caution, the ST‐IAT can be a valuable supplement to implicit measures in the case of multiple single‐target assessments. Copyright © 2007 John Wiley & Sons, Ltd.