In diesem Buch werden Aspekte der Aussagenlogik und der Prädikaten­ logik der ersten Stufe behandelt. Eine Mathematisierung und Kalkü­ lisierung der Logik kann natürlich ganz verschieden ausfallen, je nach dem, von welchen Motiven man sich primär leiten läßt. Wir stellen drei Gesichtspunkte, die uns auch als die wesentlichsten erscheinen, in den Vordergrund: Die Formalisierung des Wahrheits­ begriffes, die Formalisierung des Beweisbarkeitsbegriffes und das Problem des Suchens nach Beweisen. Diese drei Aspekte führen zu drei verschiedenen Arten von Kalkülen. Die Betonung des (...) Wahrheitsbegriffes führte auf die untersuchung der Hilberttypkalküle von einem Standpunkt, wie er etwa auch im Buch von Rasiowa-Sikorski [Ra-Si] eingenommen wird. Hierbei wurde besonderen Wert auf die algebraischen Techniken gelegt, denn die Natur der Vollständigkeitsbeweise in diesen Kalkülen läßt sich u. E. eigentlich nur algebraisch verstehen. Etwas überspitzt könnte man formulieren, daß die Vollständigkeitsbeweise in Hilberttypkal­ külen Korollare zu Betrachtungen über Kongruenzrelationen in ge­ wissen Boole'schen Algebren sind. Bei den modelltheoretischen Be­ trachtungen haben wir uns kurz ge faßt und nur einige grundlegende Begriffe vorgestellt. (shrink)

Provability, Computability and Reflection.

Continuous recordings of brain electrical activity were obtained from a group of 176 patients throughout surgical procedures using general anesthesia. Artifact-free data from the 19 electrodes of the International 10/20 System were subjected to quantitative analysis of the electroencephalogram (QEEG). Induction was variously accomplished with etomidate, propofol or thiopental. Anesthesia was maintained throughout the procedures by isoflurane, desflurane or sevoflurane (N = 68), total intravenous anesthesia using propofol (N = 49), or nitrous oxide plus narcotics (N = 59). A set (...) of QEEG measures were found which reversibly displayed high heterogeneity of variance between four states as follows: (1) during induction; (2) just after loss of consciousness (LOC); (3) just before return of consciousness (ROC); (4) just after ROC. Homogeneity of variance across all agents within states was found. Topographic statistical probability images were compared between states. At LOC, power increased in all frequency bands in the power spectrum with the exception of a decrease in gamma activity, and there was a marked anteriorization of power. Additionally, a significant change occurred in hemispheric relationships, with prefrontal and frontal regions of each hemisphere becoming more closely coupled, and anterior and posterior regions on each hemisphere, as well as homologous regions between the two hemispheres, uncoupling. All of these changes reversed upon ROC. Variable resolution electromagnetic tomography (VARETA) was performed to localize salient features of power anteriorization in three dimensions. A common set of neuroanatomical regions appeared to be the locus of the most probable generators of the observed EEG changes. (shrink)

Let T 1 , T 2 be countable first-order theories, M i ⊨ T i , and ������ any regular ultrafilter on λ ≥ $\aleph_{0}$ . A longstanding open problem of Keisler asks when T 2 is more complex than T 1 , as measured by the fact that for any such λ, ������, if the ultrapower (M 2 ) λ /������ realizes all types over sets of size ≤ λ, then so must the ultrapower (M 1 ) λ /������. (...) In this paper, building on the author's prior work [12] [13] [14], we show that the relative complexity of first-order theories in Keisler's sense is reflected in the relative graph-theoretic complexity of sequences of hypergraphs associated to formulas of the theory. After reviewing prior work on Keisler's order, we present the new construction in the context of ultrapowers, give various applications to the open question of the unstable classification, and investigate the interaction between theories and regularizing sets. We show that there is a minimum unstable theory, a minimum TP 2 theory, and that maximality is implied by the density of certain graph edges (between components arising from Szemerédi-regular decompositions) remaining bounded away from 0, 1. We also introduce and discuss flexible ultrafilters, a relevant class of regular ultrafilters which reflect the sensitivity of certain unstable (non low) theories to the sizes of regularizing sets, and prove that any ultrafilter which saturates the minimal TP 2 theory is flexible. (shrink)

Some theorists, worried about liberalism’s potential as a foundation for public health ethics, suggest that republicanism provides a better background of justification for public health policies, interventions, etc. In this article, this suggestion is put to the test, and it is argued that (i) contemporary (civic) republicanism and liberalism are not nearly as opposed as it is sometimes suggested, and that (ii) the kind of republicanism which one leading scholar in the field, Bruce Jennings, as an alternative to liberalism, does (...) not reflect the contemporary understanding of republicanism as held by, e.g. Phillip Pettit et al. (shrink)

The characteristic sequence of hypergraphs Pn:n<ω associated to a formula φ, introduced in Malliaris [5], is defined by Pn=i≤nφ. We continue the study of characteristic sequences, showing that graph-theoretic techniques, notably Szemerédi’s celebrated regularity lemma, can be naturally applied to the study of model-theoretic complexity via the characteristic sequence. Specifically, we relate classification-theoretic properties of φ and of the Pn to density between components in Szemerédi-regular decompositions of graphs in the characteristic sequence. In addition, we use Szemerédi regularity to calibrate (...) model-theoretic notions of independence by describing the depth of independence of a constellation of sets and showing that certain failures of depth imply Shelah’s strong order property SOP3; this sheds light on the interplay of independence and order in unstable theories. (shrink)

BACKGROUND: Serotonin transporter promoter genotype appears to increase risk for depression in the context of stressful life events. However, the effects of this genotype on measures of stress sensitivity are poorly understood. Therefore, this study examined whether 5-HTTLPR genotype was associated with negative information processing biases in early childhood. METHOD: Thirty-nine unselected seven-year-old children completed a negative mood induction procedure and a Self-Referent Encoding Task designed to measure positive and negative schematic processing. Children were also genotyped for the 5-HTTLPR gene. (...) RESULTS: Children who were homozygous for the short allele of the 5-HTTLPR gene showed greater negative schematic processing following a negative mood prime than those with other genotypes. 5-HTTLPR genotype was not significantly associated with positive schematic processing. LIMITATIONS: The sample size for this study was small. We did not analyze more recently reported variants of the 5-HTTLPR long alleles. CONCLUSIONS: 5-HTTLPR genotype is associated with negative information processing styles following a negative mood prime in a non-clinical sample of young children. Such cognitive styles are thought to be activated in response to stressful life events, leading to depressive symptoms; thus, cognitive styles may index the "stress-sensitivity" conferred by this genotype. (shrink)