A New Definition of A Priori Knowledge: In Search of a Modal Basis

In this paper I will offer a novel understanding of a priori knowledge. My claim is that the sharp distinction that is usually made between a priori and a posteriori knowledge is groundless. It will be argued that a plausible understanding of a priori and a posteriori knowledge has to acknowledge that they are in a constant bootstrapping relationship. It is also crucial that we distinguish between a priori propositions that hold in the actual world and merely possible, non-actual a priori propositions, as we will see when considering cases like Euclidean geometry. Furthermore, contrary to what Kripke seems to suggest, a priori knowledge is intimately connected with metaphysical modality, indeed, grounded in it. The task of a priori reasoning, according to this account, is to delimit the space of metaphysically possible worlds in order for us to be able to determine what is actual.

A Priori and the A Posteriori

The distinction between a priori and a posteriori knowledge has been the subject of an enormous amount of discussion, but the literature is biased against recognizing the intimate relationship between these forms of knowledge. For instance, it seems to be almost impossible to find a sample of pure a priori or a posteriori knowledge. In this paper it will be suggested that distinguishing between a priori and a posteriori is more problematic than is often suggested, and that a priori and a posteriori resources are in fact used in parallel. We will define this relationship between a priori and a posteriori knowledge as the bootstrapping relationship. As we will see, this relationship gives us reasons to seek for an altogether novel definition of a priori and a posteriori knowledge. Specifically, we will have to analyse the relationship between a priori knowledge and a priori reasoning, and it will be suggested that the latter serves as a more promising starting point for the analysis of aprioricity. We will also analyse a number of examples from the natural sciences and consider the role of a priori reasoning in these examples. The focus of this paper is the analysis of the concepts of a priori and a posteriori knowledge rather than the epistemic domain of a posteriori and a priori justification.

Agreement Patterns in English : Diachronic Corpus Studies on Common-Number Pronouns

This study reports a diachronic corpus investigation of common-number pronouns used to convey unknown or otherwise unspecified reference. The study charts agreement patterns in these pronouns in various diachronic and synchronic corpora. The objective is to provide base-line data on variant frequencies and distributions in the history of English, as there are no previous systematic corpus-based observations on this topic. This study seeks to answer the questions of how pronoun use is linked with the overall typological development in English and how their diachronic evolution is embedded in the linguistic and social structures in which they are used. The theoretical framework draws on corpus linguistics and historical sociolinguistics, grammaticalisation, diachronic typology, and multivariate analysis of modelling sociolinguistic variation. The method employs quantitative corpus analyses from two main electronic corpora, one from Modern English and the other from Present-day English. The Modern English material is the Corpus of Early English Correspondence, and the time frame covered is 1500-1800. The written component of the British National Corpus is used in the Present-day English investigations. In addition, the study draws supplementary data from other electronic corpora. The material is used to compare the frequencies and distributions of common-number pronouns between these two time periods. The study limits the common-number uses to two subsystems, one anaphoric to grammatically singular antecedents and one cataphoric, in which the pronoun is followed by a relative clause. Various statistical tools are used to process the data, ranging from cross-tabulations to multivariate VARBRUL analyses in which the effects of sociolinguistic and systemic parameters are assessed to model their impact on the dependent variable. This study shows how one pronoun type has extended its uses in both subsystems, an increase linked with grammaticalisation and the changes in other pronouns in English through the centuries. The variationist sociolinguistic analysis charts how grammaticalisation in the subsystems is embedded in the linguistic and social structures in which the pronouns are used. The study suggests a scale of two statistical generalisations of various sociolinguistic factors which contribute to grammaticalisation and its embedding at various stages of the process.

Idealähtöistä koulualgebraa : IDEAA-opetusmallin kehittäminen algebran opetukseen peruskoulun 7. luokalla

This research is based on the problems in secondary school algebra I have noticed in my own work as a teacher of mathematics. Algebra does not touch the pupil, it remains knowledge that is not used or tested. Furthermore the performance level in algebra is quite low. This study presents a model for 7th grade algebra instruction in order to make algebra more natural and useful to students. I refer to the instruction model as the Idea-based Algebra (IDEAA). The basic ideas of this IDEAA model are 1) to combine children's own informal mathematics with scientific mathematics ("math math") and 2) to structure algebra content as a "map of big ideas", not as a traditional sequence of powers, polynomials, equations, and word problems. This research project is a kind of design process or design research. As such, this project has three, intertwined goals: research, design and pedagogical practice. I also assume three roles. As a researcher, I want to learn about learning and school algebra, its problems and possibilities. As a designer, I use research in the intervention to develop a shared artefact, the instruction model. In addition, I want to improve the practice through intervention and research. A design research like this is quite challenging. Its goals and means are intertwined and change in the research process. Theory emerges from the inquiry; it is not given a priori. The aim to improve instruction is normative, as one should take into account what "good" means in school algebra. An important part of my study is to work out these paradigmatic questions. The result of the study is threefold. The main result is the instruction model designed in the study. The second result is the theory that is developed of the teaching, learning and algebra. The third result is knowledge of the design process. The instruction model (IDEAA) is connected to four main features of good algebra education: 1) the situationality of learning, 2) learning as knowledge building, in which natural language and intuitive thinking work as "intermediaries", 3) the emergence and diversity of algebra, and 4) the development of high performance skills at any stage of instruction.

Boundedness of weakly singular integral operators on domains

The monograph dissertation deals with kernel integral operators and their mapping properties on Euclidean domains. The associated kernels are weakly singular and examples of such are given by Green functions of certain elliptic partial differential equations. It is well known that mapping properties of the corresponding Green operators can be used to deduce a priori estimates for the solutions of these equations. In the dissertation, natural size- and cancellation conditions are quantified for kernels defined in domains. These kernels induce integral operators which are then composed with any partial differential operator of prescribed order, depending on the size of the kernel. The main object of study in this dissertation being the boundedness properties of such compositions, the main result is the characterization of their Lp-boundedness on suitably regular domains. In case the aforementioned kernels are defined in the whole Euclidean space, their partial derivatives of prescribed order turn out to be so called standard kernels that arise in connection with singular integral operators. The Lp-boundedness of singular integrals is characterized by the T1 theorem, which is originally due to David and Journé and was published in 1984 (Ann. of Math. 120). The main result in the dissertation can be interpreted as a T1 theorem for weakly singular integral operators. The dissertation deals also with special convolution type weakly singular integral operators that are defined on Euclidean spaces.

Power and Authority. Birgitta of Sweden and her Revelations

In the High Middle Ages female saints were customarily noble virgins. Thus, as a wife and a mother of eight children, the Swedish noble lady Birgitta (1302/3 1373) was an atypical candidate for sanctity. However, in 1391 she was canonized only 18 years after her death and became a role model for many late medieval women, who were mothers and widows. The dissertation Power and Authority Birgitta of Sweden and Her Revelations investigates how Birgitta went about establishing her power and authority during the first ten years of her career as a living saint, in 1340 1349. It is written from the perspectives of gender, authority, and power. The sources consist of approximately seven hundred revelations, hagiographical texts and other medieval documents. This work concentrates on the interaction between Birgitta and her audience. During her lifetime Birgitta was already regarded as a holy woman, as a living saint. A living saint could be given no formal papal or other recognition, for one could never be certain about his or her future activities. Thus, the living saint needed an audience for whom to perform signs of sanctity. In this study particular attention is paid to situations within which the power relations between the living saint and her audience can be traced and are open to critical analysis. Situations of conflict that arose in Birgitta s life are especially fruitful for this purpose. During the Middle Ages, institutional power and authority were exclusively in the hands of secular male leaders and churchmen. In this work it is argued, however, that Birgitta used different kinds of power than men. It is evident that she exercized influence on lay people as well as on secular and clerical authorities. The second, third, and fourth chapter of this study examine the beginning of Birgitta s career as a visionary, what factors and influences lay behind it, and what kind of roles they played in establishing her religious authority. The fifth, sixth, and seventh chapter concentrate on Birgitta s exercising of power in specific situations during her time in Sweden until she left on a pilgrimage to Rome in 1349. The central question is how she exercised power with different people. As a result, this book will offer a narrative of Birgitta s social interactions in Sweden seen from the perspectives of power and authority. Along with the concept of power, authority is a key issue. By definition, one who has power also has authority but a person who does not have official power can, nevertheless, have authority. Authority in action is defined here as meaning that a person was listened to. Birgitta acted both in situations of open conflict and where no conflict was evident. Her strategies included, for example, inducement, encouragement and flattery. In order to make people do as she felt was right she also threatened them openly with divine wrath. Sometimes she even used both positive persuasion and threats. Birgitta s power seems very similar to that of priests and ascetics. Common to all of them was that their power demanded interaction with other people and audiences. Because Birgitta did not have power and authority ex officio she had to persuade people to believe in her powers. She did this because she was convinced of her mission and sought to make people change their lives. In so doing, she moved from the domestic field to the public fields of religion and politics.

Functional imaging of peripheral vision and dorsal stream function in the human cerebral cortex

Visual information processing in brain proceeds in both serial and parallel fashion throughout various functionally distinct hierarchically organised cortical areas. Feedforward signals from retina and hierarchically lower cortical levels are the major activators of visual neurons, but top-down and feedback signals from higher level cortical areas have a modulating effect on neural processing. My work concentrates on visual encoding in hierarchically low level cortical visual areas in human brain and examines neural processing especially in cortical representation of visual field periphery. I use magnetoencephalography and functional magnetic resonance imaging to measure neuromagnetic and hemodynamic responses during visual stimulation and oculomotor and cognitive tasks from healthy volunteers. My thesis comprises six publications. Visual cortex forms a great challenge for modeling of neuromagnetic sources. My work shows that a priori information of source locations are needed for modeling of neuromagnetic sources in visual cortex. In addition, my work examines other potential confounding factors in vision studies such as light scatter inside the eye which may result in erroneous responses in cortex outside the representation of stimulated region, and eye movements and attention. I mapped cortical representations of peripheral visual field and identified a putative human homologue of functional area V6 of the macaque in the posterior bank of parieto-occipital sulcus. My work shows that human V6 activates during eye-movements and that it responds to visual motion at short latencies. These findings suggest that human V6, like its monkey homologue, is related to fast processing of visual stimuli and visually guided movements. I demonstrate that peripheral vision is functionally related to eye-movements and connected to rapid stream of functional areas that process visual motion. In addition, my work shows two different forms of top-down modulation of neural processing in the hierachically lowest cortical levels; one that is related to dorsal stream activation and may reflect motor processing or resetting signals that prepare visual cortex for change in the environment and another local signal enhancement at the attended region that reflects local feed-back signal and may perceptionally increase the stimulus saliency.

Lightcurve inversion for asteroid spins and shapes

Knowledge of the physical properties of asteroids is crucial in many branches of solar-system research. Knowledge of the spin states and shapes is needed, e.g., for accurate orbit determination and to study the history and evolution of the asteroids. In my thesis, I present new methods for using photometric lightcurves of asteroids in the determination of their spin states and shapes. The convex inversion method makes use of a general polyhedron shape model and provides us at best with an unambiguous spin solution and a convex shape solution that reproduces the main features of the original shape. Deriving information about the non-convex shape features is, in principle, also possible, but usually requires a priori information about the object. Alternatively, a distribution of non-convex solutions, describing the scale of the non-convexities, is also possible to be obtained. Due to insufficient number of absolute observations and inaccurately defined asteroid phase curves, the $c/b$-ratio, i.e., the flatness of the shape model is often somewhat ill-defined. However, especially in the case of elongated objects, the flatness seems to be quite well constrained, even in the case when only relative lightcurves are available. The results prove that it is, contrary to the earlier misbelief, possible to derive shape information from the lightcurve data if a sufficiently wide range of observing geometries is covered by the observations. Along with the more accurate shape models, also the rotational states, i.e., spin vectors and rotation periods, are defined with improved accuracy. The shape solutions obtained so far reveal a population of irregular objects whose most descriptive shape characteristics, however, can be expressed with only a few parameters. Preliminary statistical analyses for the shapes suggests that there are correlations between shape and other physical properties, such as the size, rotation period and taxonomic type of the asteroids. More shape data of, especially, the smallest and largest asteroids, as well as the fast and slow rotators is called for in order to be able to study the statistics more thoroughly.

Information Sharing in Banking: A Collusive Device?

We show that information sharing among banks may serve as a collusive device. An informational sharing agreement is an a-priori commitment to reduce informational asymmetries between banks in future lending. Hence, information sharing tends to increase the intensity of competition in future periods and, thus, reduces the value of informational rents in current competition. We contribute to the existing literature by emphasizing that a reduction in informational rents will also reduce the intensity of competition in the current period, thereby reducing competitive pressure in current credit markets. We provide a large class of economic environments, where a ban on information sharing would be strictly welfare-enhancing.

On the Modal Content of A Posteriori Necessities

This paper challenges the Kripkean interpretation of a posteriori necessities. It will be demonstrated, by an analysis of classic examples, that the modal content of supposed a posteriori necessities is more complicated than the Kripkean line suggests. We will see that further research is needed concerning the a priori principles underlying all a posteriori necessities. In the course of this analysis it will emerge that the modal content of a posteriori necessities can be best described in terms of a Finean conception of modality – by giving essences priority over modality. The upshot of this is that we might be able to establish the necessity of certain supposed a posteriori necessities by a priori means.

Paradigms in Statistical Inference for Finite Populations; Up to the 1950s

Modern sample surveys started to spread after statistician at the U.S. Bureau of the Census in the 1940s had developed a sampling design for the Current Population Survey (CPS). A significant factor was also that digital computers became available for statisticians. In the beginning of 1950s, the theory was documented in textbooks on survey sampling. This thesis is about the development of the statistical inference for sample surveys. For the first time the idea of statistical inference was enunciated by a French scientist, P. S. Laplace. In 1781, he published a plan for a partial investigation in which he determined the sample size needed to reach the desired accuracy in estimation. The plan was based on Laplace s Principle of Inverse Probability and on his derivation of the Central Limit Theorem. They were published in a memoir in 1774 which is one of the origins of statistical inference. Laplace s inference model was based on Bernoulli trials and binominal probabilities. He assumed that populations were changing constantly. It was depicted by assuming a priori distributions for parameters. Laplace s inference model dominated statistical thinking for a century. Sample selection in Laplace s investigations was purposive. In 1894 in the International Statistical Institute meeting, Norwegian Anders Kiaer presented the idea of the Representative Method to draw samples. Its idea was that the sample would be a miniature of the population. It is still prevailing. The virtues of random sampling were known but practical problems of sample selection and data collection hindered its use. Arhtur Bowley realized the potentials of Kiaer s method and in the beginning of the 20th century carried out several surveys in the UK. He also developed the theory of statistical inference for finite populations. It was based on Laplace s inference model. R. A. Fisher contributions in the 1920 s constitute a watershed in the statistical science He revolutionized the theory of statistics. In addition, he introduced a new statistical inference model which is still the prevailing paradigm. The essential idea is to draw repeatedly samples from the same population and the assumption that population parameters are constants. Fisher s theory did not include a priori probabilities. Jerzy Neyman adopted Fisher s inference model and applied it to finite populations with the difference that Neyman s inference model does not include any assumptions of the distributions of the study variables. Applying Fisher s fiducial argument he developed the theory for confidence intervals. Neyman s last contribution to survey sampling presented a theory for double sampling. This gave the central idea for statisticians at the U.S. Census Bureau to develop the complex survey design for the CPS. Important criterion was to have a method in which the costs of data collection were acceptable, and which provided approximately equal interviewer workloads, besides sufficient accuracy in estimation.