Logic as the Universal Science : Bertrand Russell's Early Conception of Logic and Its Philosophical Context

Bertrand Russell (1872 1970) introduced the English-speaking philosophical world to modern, mathematical logic and foundational study of mathematics. The present study concerns the conception of logic that underlies his early logicist philosophy of mathematics, formulated in The Principles of Mathematics (1903). In 1967, Jean van Heijenoort published a paper, Logic as Language and Logic as Calculus, in which he argued that the early development of modern logic (roughly the period 1879 1930) can be understood, when considered in the light of a distinction between two essentially different perspectives on logic. According to the view of logic as language, logic constitutes the general framework for all rational discourse, or meaningful use of language, whereas the conception of logic as calculus regards logic more as a symbolism which is subject to reinterpretation. The calculus-view paves the way for systematic metatheory, where logic itself becomes a subject of mathematical study (model-theory). Several scholars have interpreted Russell s views on logic with the help of the interpretative tool introduced by van Heijenoort,. They have commonly argued that Russell s is a clear-cut case of the view of logic as language. In the present study a detailed reconstruction of the view and its implications is provided, and it is argued that the interpretation is seriously misleading as to what he really thought about logic. I argue that Russell s conception is best understood by setting it in its proper philosophical context. This is constituted by Immanuel Kant s theory of mathematics. Kant had argued that purely conceptual thought basically, the logical forms recognised in Aristotelian logic cannot capture the content of mathematical judgments and reasonings. Mathematical cognition is not grounded in logic but in space and time as the pure forms of intuition. As against this view, Russell argued that once logic is developed into a proper tool which can be applied to mathematical theories, Kant s views turn out to be completely wrong. In the present work the view is defended that Russell s logicist philosophy of mathematics, or the view that mathematics is really only logic, is based on what I term the Bolzanian account of logic . According to this conception, (i) the distinction between form and content is not explanatory in logic; (ii) the propositions of logic have genuine content; (iii) this content is conferred upon them by special entities, logical constants . The Bolzanian account, it is argued, is both historically important and throws genuine light on Russell s conception of logic.

Leibniz on Rational Decision-Making

In this study I discuss G. W. Leibniz's (1646-1716) views on rational decision-making from the standpoint of both God and man. The Divine decision takes place within creation, as God freely chooses the best from an infinite number of possible worlds. While God's choice is based on absolutely certain knowledge, human decisions on practical matters are mostly based on uncertain knowledge. However, in many respects they could be regarded as analogous in more complicated situations. In addition to giving an overview of the divine decision-making and discussing critically the criteria God favours in his choice, I provide an account of Leibniz's views on human deliberation, which includes some new ideas. One of these concerns is the importance of estimating probabilities in making decisions one estimates both the goodness of the act itself and its consequences as far as the desired good is concerned. Another idea is related to the plurality of goods in complicated decisions and the competition this may provoke. Thirdly, heuristic models are used to sketch situations under deliberation in order to help in making the decision. Combining the views of Marcelo Dascal, Jaakko Hintikka and Simo Knuuttila, I argue that Leibniz applied two kinds of models of rational decision-making to practical controversies, often without explicating the details. The more simple, traditional pair of scales model is best suited to cases in which one has to decide for or against some option, or to distribute goods among parties and strive for a compromise. What may be of more help in more complicated deliberations is the novel vectorial model, which is an instance of the general mathematical doctrine of the calculus of variations. To illustrate this distinction, I discuss some cases in which he apparently applied these models in different kinds of situation. These examples support the view that the models had a systematic value in his theory of practical rationality.

Oral Health among Young Adults and the Middle-aged in Iran

The aim of the present study was to assess oral health and treatment needs among adult Iranians according to socio-demographic status, smoking, and oral hygiene, and to investigate the relationships between these determinants and oral health. Data for 4448 young adult (aged 18) and 8301 middle-aged (aged 35 to 44) Iranians were collected in 2002 as part of a national survey using the World Health Organization (WHO) criteria for sampling and clinical diagnoses, across 28 provinces by 33 calibrated examiners. Gender, age, place of residence, and level of education served as socio-demographic information, smoking as behavioural and modified plaque index (PI) as the biological risk indicator for oral hygiene. Number of teeth, decayed teeth (DT), filled teeth (FT), decayed, missing, filled teeth (DMFT), community periodontal index (CPI), and prosthodontic rehabilitation served as outcome variables of oral health. Mean number of DMFT was 4.3 (Standard deviation (SD) = 3.7) in young adults and 11.0 (SD = 6.4) among middle-aged individuals. Among young adults the D-component (DT = 70%), and among middle-aged individuals the M-component (60%) dominated in the DMFT index. Among young adults, visible plaque was found in nearly all subjects. Maximum (max) PI was associated with higher mean number of DT, and higher periodontal treatment needs. A healthy periodontium was a rare condition, with 8% of young adults and 1% of middle-aged individuals having a max CPI = 0. The majority of the CPI findings among young adults consisted of calculus (48%) and deepened periodontal pockets (21%). Respective values for middle-aged individuals were 40% and 53%. Having a deep pocket (max CPI = 4) was more likely among young adults with a low level of education (Odds ratio (OR) = 2.7, 95% Confidence interval (CI) = 1.9–4.0) than it was among well-educated individuals. Among middle-aged individuals, having calculus or a periodontal pocket was more likely in men (OR = 1.8, 95% CI = 1.6–2.0) and in illiterate subjects (OR = 6.3, 95% CI = 5.1–7.8) than it was for their counterparts. Among young adults, having 28 teeth was more (p < 0.05) prevalent among men (72% vs. 68% for women), urban residents (71% vs. 67% for rural residents), and those with a high level of education (73% vs. 60% for those with a low level). Among middle-aged individuals, having a functional dentition was associated with younger age (OR = 2.0, 95% CI = 1.7−2.5) and higher level of education (OR = 1.8, 95% CI = 1.6−2.1). Of middle-aged individuals, 2% of 35- to 39-year-olds and 5% of those aged 40 to 44 were edentulous. Among the dentate subjects (n = 7,925), prosthodontic rehabilitation was more prevalent (p < 0.001) among women, urban residents, and those with a high level of education than it was among their counterparts. Among those having 1 to 19 teeth, a removable denture was the most common type of prosthodontic rehabilitation. Middle-aged individuals lacking a functional dentition were more likely (OR = 6.0, 95% CI = 4.8−7.6) to have prosthodontic rehabilitation than were those having a functional dentition. In total, 81% of all reported being non-smokers, and 32% of men and 5% of women were current smokers. Heavy smokers were the most likely to have deepened periodontal pockets (max CPI ≥ 3, OR = 2.9, 95% CI = 1.8−4.7) and to have less than 20 teeth (OR = 2.3, 95% CI = 1.5−3.6). The findings indicate impaired oral health status in adult Iranians, particularly those of low socio-economic status and educational level. The high prevalence of dental plaque and calculus and considerable unmet treatment needs call for a preventive population strategy with special emphasis on the improvement of oral self-care and smoking cessation to tackle the underlying risk factors for oral diseases in the Iranian adult population.

Oral health behaviour, conditions and care among dentate elderly patients in Lithuania: preventive aspects

The present cross-sectional study aimed to assess oral health behaviour, dental and periodontal conditions, dental care, and their relationships among elderly dentate patients in Lithuania. The target population in the study were dentate patients aged 60 and older attending public dental services in Kedainiai, Lithuania. The data collection took place between the autumn of 1999 and the winter of 2001. Data were collected by means of a self-administered questionnaire for all (n=174) and a clinical examination targeting about half of the subjects (n=100). The questionnaire inquired about oral health behaviour, the life-first and also the most recent dental treatments, sources on and self-assessed knowledge of oral self-care, a self-reported number of teeth, and socio-demographic information. The clinical examination included basic dental and periodontal conditions. A total of 82 women and 92 men completed the questionnaire; their mean age was 69.2 and their average number of teeth was 16.2 (CI 95% 15.4-17.1). In all, 25% had 21 or more teeth and 32% indicated wearing removable dentures. The oral health behaviour, the participants reported, was poor: 30% reported twice daily toothbrushing, 57% responded that they always use fluoride toothpaste, 19% indicated daily interdental cleaning, nearly all said they take sugar in their coffee and tea, and 30% indicated going for check-ups. As the main source of information on oral self-care, the subjects indicated health professionals (82%), followed by social contacts (72%), broadcasted media (58%), and printed media (42%). A total of 34% assessed their knowledge of oral self-care as good, and their self-assessed knowledge correlated (r=0.52) with professional guidance they had received about oral self-care. In their most recent treatment, conservative (39%) and non-conservative (34%) treatments dominated, and preventive ones were the least reported (7%). Regarding guidance in oral self-care, 54% reported having received such about toothbrushing, 32% about interdental cleaning, and 33% had been given visual information. Clinical examinations revealed the presence of plaque, calculus, bleeding on probing and deepened pockets in all of the subjects; 70% of the subjects were diagnosed with pockets of 6mm and deeper, 94% with caries, and 73% with overhangs of restorations. Those subjects assessing their knowledge of oral self-care as good and reporting a higher intensity of guidance in oral self-care as received, indicated practicing the recommended oral self-care more frequently. Twice daily toothbrushing was associated with good self-assessed knowledge of oral self-care (OR 4.1, p<0.001) and a university education (OR 5.6, p<0.001). Those subjects with better oral health behaviour had a greater number of teeth. Having 21 or more teeth was associated with good self-assessed knowledge of oral self-care (OR 4.1, p=0.03). Better periodontal conditions were associated with a higher frequency of toothbrushing. The presence of periodontal pockets of 6mm and deeper was associated with the level of self-assessed knowledge of oral self-care being below good (OR=3.0, p=0.04) and the level of dental cleanliness being poor (OR=2.7, p=0.02). To conclude, oral health behaviour and conditions call for improvement in elderly subjects in Lithuania. To improve the oral health of their elderly dentate patients, dentists should apply all the available tools of chair-side prevention and active guidance. The latter would be an effective means of updating the knowledge of oral self-care and supporting recommended oral health behaviour. A preventive approach should be strongly emphasized in countries with limited resources for oral health care, such as Lithuania. Author’s address: Sonata Vyšniauskaite, Department of Oral Public Health, Institute of Dentistry, University of Helsinki, P.O.Box 41, FI-00014 Helsinki, Finland. E-mail: sonata.vysniauskaite@helsinki.fi

Finitary Abstract Elementary Classes

The research in model theory has extended from the study of elementary classes to non-elementary classes, i.e. to classes which are not completely axiomatizable in elementary logic. The main theme has been the attempt to generalize tools from elementary stability theory to cover more applications arising in other branches of mathematics. In this doctoral thesis we introduce finitary abstract elementary classes, a non-elementary framework of model theory. These classes are a special case of abstract elementary classes (AEC), introduced by Saharon Shelah in the 1980's. We have collected a set of properties for classes of structures, which enable us to develop a 'geometric' approach to stability theory, including an independence calculus, in a very general framework. The thesis studies AEC's with amalgamation, joint embedding, arbitrarily large models, countable Löwenheim-Skolem number and finite character. The novel idea is the property of finite character, which enables the use of a notion of a weak type instead of the usual Galois type. Notions of simplicity, superstability, Lascar strong type, primary model and U-rank are inroduced for finitary classes. A categoricity transfer result is proved for simple, tame finitary classes: categoricity in any uncountable cardinal transfers upwards and to all cardinals above the Hanf number. Unlike the previous categoricity transfer results of equal generality the theorem does not assume the categoricity cardinal being a successor. The thesis consists of three independent papers. All three papers are joint work with Tapani Hyttinen.

A Possible and Necessary Consistency Proof

After Gödel's incompleteness theorems and the collapse of Hilbert's programme Gerhard Gentzen continued the quest for consistency proofs of Peano arithmetic. He considered a finitistic or constructive proof still possible and necessary for the foundations of mathematics. For a proof to be meaningful, the principles relied on should be considered more reliable than the doubtful elements of the theory concerned. He worked out a total of four proofs between 1934 and 1939. This thesis examines the consistency proofs for arithmetic by Gentzen from different angles. The consistency of Heyting arithmetic is shown both in a sequent calculus notation and in natural deduction. The former proof includes a cut elimination theorem for the calculus and a syntactical study of the purely arithmetical part of the system. The latter consistency proof in standard natural deduction has been an open problem since the publication of Gentzen's proofs. The solution to this problem for an intuitionistic calculus is based on a normalization proof by Howard. The proof is performed in the manner of Gentzen, by giving a reduction procedure for derivations of falsity. In contrast to Gentzen's proof, the procedure contains a vector assignment. The reduction reduces the first component of the vector and this component can be interpreted as an ordinal less than epsilon_0, thus ordering the derivations by complexity and proving termination of the process.