Existence Assumptions and Logical Principles: Choice Operators in Intuitionistic Logic

Hilbert’s choice operators τ and ε, when added to intuitionistic logic, strengthen it. In the presence of certain extensionality axioms they produce classical logic, while in the presence of weaker decidability conditions for terms they produce various superintuitionistic intermediate logics. In this thesis, I argue that there are important philosophical lessons to be learned from these results. To make the case, I begin with a historical discussion situating the development of Hilbert’s operators in relation to his evolving program in the foundations of mathematics and in relation to philosophical motivations leading to the development of intuitionistic logic. This sets the stage for a brief description of the relevant part of Dummett’s program to recast debates in metaphysics, and in particular disputes about realism and anti-realism, as closely intertwined with issues in philosophical logic, with the acceptance of classical logic for a domain reflecting a commitment to realism for that domain. Then I review extant results about what is provable and what is not when one adds epsilon to intuitionistic logic, largely due to Bell and DeVidi, and I give several new proofs of intermediate logics from intuitionistic logic+ε without identity. With all this in hand, I turn to a discussion of the philosophical significance of choice operators. Among the conclusions I defend are that these results provide a finer-grained basis for Dummett’s contention that commitment to classically valid but intuitionistically invalid principles reflect metaphysical commitments by showing those principles to be derivable from certain existence assumptions; that Dummett’s framework is improved by these results as they show that questions of realism and anti-realism are not an “all or nothing” matter, but that there are plausibly metaphysical stances between the poles of anti-realism (corresponding to acceptance just of intutionistic logic) and realism (corresponding to acceptance of classical logic), because different sorts of ontological assumptions yield intermediate rather than classical logic; and that these intermediate positions between classical and intuitionistic logic link up in interesting ways with our intuitions about issues of objectivity and reality, and do so usefully by linking to questions around intriguing everyday concepts such as “is smart,” which I suggest involve a number of distinct dimensions which might themselves be objective, but because of their multivalent structure are themselves intermediate between being objective and not. Finally, I discuss the implications of these results for ongoing debates about the status of arbitrary and ideal objects in the foundations of logic, showing among other things that much of the discussion is flawed because it does not recognize the degree to which the claims being made depend on the presumption that one is working with a very strong (i.e., classical) logic

Variations and Application Conditions Of the Data Type »Image« - The Foundation of Computational Visualistics

Few years ago, the department of computer science of the University Magdeburg invented a completely new diploma programme called 'computational visualistics', a curriculum dealing with all aspects of computational pictures. Only isolated aspects had been studied so far in computer science, particularly in the independent domains of computer graphics, image processing, information visualization, and computer vision. So is there indeed a coherent domain of research behind such a curriculum? The answer to that question depends crucially on a data structure that acts as a mediator between general visualistics and computer science: the data structure "image". The present text investigates that data structure, its components, and its application conditions, and thus elaborates the very foundations of computational visualistics as a unique and homogenous field of research. Before concentrating on that data structure, the theory of pictures in general and the definition of pictures as perceptoid signs in particular are closely examined. This includes an act-theoretic consideration about resemblance as the crucial link between image and object, the communicative function of context building as the central concept for comparing pictures and language, and several modes of reflection underlying the relation between image and image user. In the main chapter, the data structure "image" is extendedly analyzed under the perspectives of syntax, semantics, and pragmatics. While syntactic aspects mostly concern image processing, semantic questions form the core of computer graphics and computer vision. Pragmatic considerations are particularly involved with interactive pictures but also extend to the field of information visualization and even to computer art. Four case studies provide practical applications of various aspects of the analysis

Carnap ja tieteiden ykseys

This dissertation concentrates on a particular exemplification of the ideal of the unity of science in the history of twentieth-century philosophy. Taking Rudolf Carnap (1891--1970) as an exemplar of a scholar whose work in philosophy of science was at bottom motivated by the ideal of a unified conception of science, it attempts to distillate the essential characteristics and methodological significance of such a conception by a combination of historical and systematic analysis. Given the conspicuously holoscopic character of Carnap's philosophical orientation, there arises an interesting question about the relation of his work to that of other prominent ``seekers of the wider view'' in the history of philosophy (and history of science). On a more general level, we ask what kind of intellectual and moral characteristics are associated with a scholar who is motivated by the unification of science. Making it explicit: if a coherent conception of a unified conception of science is conceivable, what kind of normative criteria can then be applied to a scholar and his actions? In other words, what are the external and internal qualifications of scholar's vocation under the unified conception of science? In the first part of the dissertation we provide a general account of the problem's background in the intersection of intellectual history and systematics. In the first chapter main emphasis will be put to the dialectic between agent-based and structural explanations in historiography. The survey of a few exemplars of models of historical explanation is intended to provide a background framework for discussing the relation between descriptive analysis and analysis of values. In as much as our modern scientific world conception and the general, essentially human, consciousness of the domain of validity seem to be in a fundamental conflict, a philosophical clarification of the issues that depend on this fundamental distinction is contingent on having proper tools at its disposal. Indeed, it is necessary to acknowledge -- with respect to both scientific knowledge and moral positions -- that the issues of genesis and validity have little in common. Both the image of nature, built upon the masses of scientific and technological knowledge gathered, and the modern conceptions of the moral have developed in the course of history. The lesson that historicism can teach us is the possibility to adopt a symmetrical attitude with respect to the status of the questions of genesis and validity within these (very different) domains. This symmetric attitude enables us to see that the validity of a theory or position (in science or in moral philosophy) cannot depend on the diachronic aspects of its genesis. Rather, it is precisely the case that the late appearance of certain scientific theories and certain moral positions is an index that they are complex and presuppose a great deal genetically, and this is seen to be a common feature of all good theories. Thus, in order to approach the evolution of these ideas from a general perspective, we have to acknowledge their fundamental ontological difference and adopt a variety of tools to study these domains. I present four different approaches to the study of historical phenomena that appertain to the themes of this dissertation. In the second chapter we provide a synopsis of the important thematic about the relationship between morals and science. After a brief examination of the concepts of the moral and the scientific, we proceed to give an account of the concept of scientific self which acts as a kind of normative meta-concept co-ordinating the interaction between the epistemic and the ethical requirements appertaining to the education and professional formation of a scientist. From a historical perspective it is easy to to see that the intension of the concept of scientific self varies according to the contingent factors such as the external conditions of education and the requirements set by new experimental techniques, but the essential, axiologically relevant, internal determinates of the concept are seen to accumulate over time in a conservative manner. Especially interesting here are the determinates that can be traced back to the complementary intellectual traditions of Enlightenment and Romanticism. One of the most important exemplifications of an articulated conception of scientific self can be found in J. G. Fichte's ``Vorlesungen über die Bestimmung des Gelehrten'' of 1794. In these lectures Fichte develops a beautiful -- and still highly relevant -- conception of the true goals of a scholar as well as the qualifications he must fulfill to attain those goals. From Fichte we turn to study the history of one particular intellectual virtue that has direct relevance for the questions tackled in the second part of the dissertation, viz. tolerance. In the third chapter we focus on the importance of a priori knowledge for both ethics and science. These themes are developed only in their barest outlines in order to provide some theoretical support to the fundamental philosophical thesis of the dissertation concerning the distinction between Is and Ought, and its relevance for the question of the unity of science. We will briefly touch upon the question about the relationship of a priori and empirical knowledge in ethics, and provide a brief synopsis of the relevance of the distinction analytic/synthetic in this domain. Finally we address cursorily the ontologically crucial problem about the moral element in man and present -- with a view to the Enlightenment virtues -- a synopsis of the process of the dissociation of the concept of the moral from the concept of the scientific. We describe the characteristics of ethical impulse in modern times and the quite idiosyncratic view on morals and especially on moral justification advocated by the members of Vienna Circle. We will see how the dissociation of the moral from the domain of the rational discourse inevitably results in the philosophically poverished stance of moral non-cognitivism which Carnap maintained throughout his career. In the second part of the dissertation we can finally address the adduced problem in its particular ramifications in the philosophy of Rudolf Carnap. Given this general problematic, we attempt to vindicate the underlying overall motivation of Carnap's philosophy and to reconstruct the architectonic of Carnap's systematic thought in the light of recent research. One of the main tasks is to evaluate the coherence of interpretations provided in the research literature which place Carnap in the continuum of thinkers that are, in some sense, committed to the ideals and values of Enlightenment. The most explicit rendering of this line of thought is the recent monograph, Carnap and Twentieth-Century Thought, by A.W. Carus which puts Carnap's method of explication on center stage. I critically examine this line of interpretation indicated by Carus and explore more deeply its historical dimensions. Over and above the interpretation of Carus, we assess to what extent Carnap's philosophical program fulfills the criteria that are imposed upon it by the requirement of an Enlightenment conception of unified science. The central significance of logic and mathematics in Carnap's philosophical program is seen to derive from the fundamental conception of Carnap that within the total system of knowledge logic and mathematics are performing the essential role of supplying the forms of concepts, statements, and inferences, forms which are then applicable everywhere, hence also to non-logical knowledge. Therefore, the demarcation between logical and non-logical expressions, along with the Principle of Tolerance and logical pluralism, constitutes one of the central strands of Carnap's thought. Indeed, the Principle of Tolerance and the logicality criterion are seen to be two inextricably entwined aspects of a solution to a fundamental problem that Carnap searches a solution to and which characterizes his aspirations throughout the period under consideration here, i.e. the problem of the rationality of scientific discourse under the variability of linguistic systems of knowledge representation. I depict the overall development of Carnap's philosophy with this central idea continually in focus. As a supplement to the interpretation of Carnap's program as a concerted attempt to look for the fundamental invariants of thought and experience, I provide the view that a necessary condition for implementing his ideal of explication is a coherent formulation of what might be called the task of providing genealogies of important scientific concepts and ideas. This complies with the attractive account represented by Howard Stein about the two basic functions of philosophy, i.e., a distinction between ``the enterprise of knowledge'' and the ``enterprise of understanding''. It is argued here that an essential ingredient of Carnap's method of explication is a variety of philosophical history of science which provides the necessary insight into the problem complex one is tackling with under the purview of explication. Therefore, a significant role is bestowed upon historical knowledge and historiography. I attempt to accommodate this aspect of the ``enterprise of understanding'' within the more explicitly confined ``enterprise of knowledge'' that Carnap was overtly concerned with. However, it is argued that the ``enterprise of understanding'' constituted an equally important aspect of Carnap's philosophical program, although it remained covert in his publications.Väitöskirjan aihepiirinä on tieteen ykseyden ideaalin merkitys Rudolf Carnapin (1891 - 1970) ajattelussa ja tuotannossa, sekä tämän ideaalin ja sen julkilausumattomien ennakkoedellytysten analyysi tieteenfilosofian ja -historian modernissa traditiossa. Keskittymällä Rudolf Carnapin ajatteluun työssä pyritään tuomaan esille tieteen ykseyden ideaalin olennaiset piirteet ja metodologinen merkitys historiallisen ja systemaattisen analyysin valossa. Yleisellä tasolla työssä kysytään, minkälaisia intellektuaalisia ja moraalisia piirteitä voidaan liittää tutkijaan ja hänen työhönsä, erityisesti sellaisen tutkijaan, joka ammentaa vaikutteensa tieteen ykseyden ideaalista. Työssä pyritään eksplikoiman ne normatiiviset ehdot, joita voidaan soveltaa tutkijaan ja hänen työhönsä carnapilaisen tieteenkäsityksen puitteissa; toisin sanoen esitetään ne ulkoiset sekä sisäiset epistemologiset ja moraalifilosofiset kriteerit, jotka tutkijan työtä määrittävät tieteen ykseyden ideaalin mukaisesti. Työn ensimmäisessä osassa tarkastellaan tieteiden ykseyden ja tieteentekijän moraalisen kehityksen välistä suhdetta historiallisesta ja systemaattisesta perspektiivistä. Keskeisenä tieteenhistoriallisena vertailukohtana on objektiivisuuden käsitteen kehitys, jota mm. Lorraine Daston ja Peter Galison ovat ansiokkaasti kuvanneet teoksessaan Objectivity (2007). Sen lisäksi että Daston ja Galison esittävät tieteen metodologiaan liittyviä keskeisiä huomioita objektiivisuuden käsitteen merkityksestä ja soveltamisesta eri aikoina, he asettavat keskeiselle sijalle minän, persoonan tai itsen käsitteen tieteellisessä tutkimusprosessissa. Tarkastelemalla kuinka erilaiset käsitykset itsestä -- ja etenkin nk. "tieteellisestä itsestä" ("scientific self") -- ovat kehittyneet objektiivisuuden käsitteen rinnalla, he päätyvät tulokseen, että ns. episteemiset hyveet ovat kiinteässä vuorovaikutuksessa eettisiin arvoihin sekä yksilön moraaliseen kehitykseen, ja muodostavat välttämättömät ehdot tieteenharjoittamiselle. Episteemisten hyveiden keskeisenä tehtävänä on ohjata tieteellistä tutkimusta siten, että ne vahvistavat tiedonhankinnan pragmaattista tehokkuutta ja edistävät "totuuteen pyrkivää tutkimusta" vetoamalla eettisiin arvoihin -- arvoihin, jotka ohjaavat yksilön intellektuaalista ja moraalista kehitystä ja luovat pohjan erilaisille tavoille ("techniques of the self") harjoittaa kykyjä, taitoja ja taipumuksia, jotka ovat tieteenharjoittamisessa keskeisiä. Historian valossa on selvää, että käsitykset "tieteellisestä itsestä" ovat muuttuneet aikojen kuluessa. Syynä tähän ovat olleet mm. kehittyneet käsitykset objektiivisuudesta sekä tieteen tavoitteista ja menetelmistä. Myös havaintolaitteiden kehityksellä on ollut tässä ratkaiseva merkitys (esimerkiksi kameran (daguerrotypia) ja fonografin keksiminen 1800-luvulla). Erityisen kiinnostavia ovat ne käsitykset itsestä, jotka liittyvät valistuksen ja romantiikan rikkaisiin traditiohin. Yhden merkittävän näkemyksen tieteellisestä itsestä ja tutkijan ammatin merkityksestä esitti J. G. Fichte luennoissaan ``Vorlesungen über die Bestimmung des Gelehrten'' vuodelta 1794. Näissä luennoissa Fichte esittelee kauniin -- ja edelleen ajankohtaisen -- näkemyksen tieteellisestä tutkimuksesta ja niistä ehdoista, jotka tutkijan tulisi täyttää saavuttaakseen tavoitteensa. Fichten näkemysten esittelyn jälkeen siirrytään tutkimaan yhden Carnapin filosofianäkemyksen kannalta keskeisen käsitteen, suvaitsevaisuusperiaatteen (Principle of Tolerance), historiaa. Lisäksi eritellään niitä apriorisen tiedon ja empiirisen tiedon lajeja, joilla on merkitystä moraalifilosofian kannalta ja toisaalta analyysin käsitteen historiaa siltä osin, kuin se muodostaa olennaisen taustan Carnapin filosofian ja filosofisen metodin ymmärtämiselle. Työn toisessa osassa siirrytään tarkastelemaan eksplisiittisesti Rudolf Carnapin filosofiaa ja sen kehitystä 1900-luvun ensimmäisiltä vuosikymmeniltä 1950-luvulle. Työn ensimmäisessä osassa esitellyn problematiikan valossa yritetään kuvata ja perustella Carnapin filosofianäkemyksen taustalla vaikuttava keskeinen motivaatio sekä esittää hänen ajattelunsa kokonaisrakenne, sen "arkkitektoniikka" viimeaikaisen tutkimuksen pohjalta. Yksi tutkimuksen keskeisiä tavoitteita on arvioida tutkimuskirjallisuudessa esiintyneitä tulkintoja, jotka asettavat Carnapin niiden ajattelijoiden jatkumoon, jotka ovat jossakin mielessä sidoksissa valistuksen traditioon, sen arvoihin ja ideaaleihin. Yksi tässä suhteessa tärkeimmistä tulkinnoista esitetään A. W. Caruksen monografiassa Carnap and Twentieth-Century Thought (2007), joka asettaa Carnapin eksplikaation metodin tutkimukselliseen keskiöön. Arvioin kriittisesti Caruksen tulkintaa ja tutkin syvemmin sen historiallisia ennakkoehtoja ja aatetaustaa. Pyrin arvioimaan sitä, missä määrin Carnapin filosofinen ohjelma täyttää ne ehdot, jotka liittyvät valistukselle ominaiseen tieteiden ykseyden käsitykseen. Logiikan ja matematiikan keskeinen merkitys Carnapin filosofiassa seuraa siitä Carnapin perustavasta näkemyksestä, että tiedon kokonaisesityksessä logiikan ja matematiikan tehtävänä on tarjota käsitteiden, lausumien ja päättelyjen formaaliset mallit (muodot), mallit jotka ovat siten sovellettavissa kaikkialla, myös ei--loogisen tiedon piirissä. Siten jako loogisten ja ei-loogisten ilmaisujen välillä sekä toleranssiperiaate ja tähän liittyvä looginen pluralismi muodostavat Carnapin ajattelun yhden keskeisen säikeen. Kuvaan Carnapin ajattelun kehitystä tämä keskeinen ajatus silmämääränäni. Täydentävänä näkökulmana käsitykselle, että Carnapin ohjelma tarkoitti keskitettyä yritystä etsiä systemaattisesti ajattelun ja kokemuksen perustavia invariansseja, tuon esille näkemyksen, jonka mukaan eksplikaation ideaalin välttämättömänä ehtona on koherentti muotoilu ja määritelmä sille, mitä voitaisiin kutsua tärkeiden tieteellisten käsitteiden ja ideoiden syntyhistorioiden esitykseksi. Tämä on yhdenmukaista sen Howard Steinin esittämän näkemyksen kanssa, jonka mukaan Carnapin filosofiaa luonnehtii kaksi toisiaan täydentävää tutkimuslinjaa. Stein erottelee yhtäältä "tiedon hankkeen" ("enterprise of knowledge") ja "ymmärtämisen hankkeen" ("enterprise of understanding"). Esitän, että olennainen osa Carnapin eksplikaation metodia on filosofisen tieteenhistorian tutkimusote, joka tarjoaa välttämättömät ennakkoedellytykset -- käsitteelliset mahdollisuudet eksplikaatioavaruudessa -- annetun eksplikaatiotoiminnan piirissä. Tästä syystä historiallisella tiedolla ja "ymmärtämisen hankkeella" on keskeinen merkitys Carnapin metodin kannalta, vaikkakin tämä ulottuvuus säilyi pitkälti julkilausumattomana hänen julkaistuissa teoksissaan