On abstraction in a Carnapian system

Rudolf Carnap (1891-1970) rejects two philosophical distinctions that have been made and admitted by Gottlob Frege (1848-1925), namely the object-concept and the sense-reference distinctions. In the analytic tradition and upon these distinctions, a family of analytic systems have been constructed and developed (which we call Fregean systems), within which a number of notions have been employed including the notion of abstraction. It has been claimed (by Neo- Fregeans) that the Fregean notion of abstraction has been captured by what is commonly known as the “principle of abstraction”. The goal of this dissertation is to present the notion of Carnapian abstraction, in particular, and the Carnapian system, in general, in distinction to the Fregean counterparts. We will argue that the admission and rejection of these distinctions will entail fundamentally different analytic systems. Hence, we will show how each system undertakes a different notion of abstraction. Abstraction in a Fregean system will be characterized as a mind-independent process subject to its own rules, whereas in a Carnapian system, abstraction will be characterized as a defined process of distancing from meaning in a linguistic framework. We will conclude that the Carnapian system has advantages over the Fregean one (among which is its simplicity), and that its technical aspect is yet to be developed.Rudolf Carnap (1891-1970) rejette deux distinctions philosophiques conçues par Gottlob Frege (1848-1925) : la distinction objet-concept et la distinction sens-référence. Dans la tradition analytique et parmi ces distinctions, une famille de systèmes analytiques a été construite et développée (appelée les « systèmes frégéen »), dans lesquels plusieurs notions ont été employées, incluant la notion d’abstraction. En fait, les néo- frégéen ont déclaré que la notion d’abstraction de Frege est capturée par ce qu’on appelle le « principe d’abstraction ». Le but de cette dissertation est de présenter la notion d’abstraction de Carnap en particulier et le système de Carnap en général, en comparaison aux notions de Frege. Nous allons argumenter que l’admission et le rejet de ces distinctions entraîneront des systèmes analytiques fondamentalement différents. Ainsi, nous allons démontrer comment chaque système utilise différentes notions d’abstraction. L’abstraction dans un système frégéen sera caractérisée comme un processus indépendant qui est confiné à ses propres règles, tandis que dans un système carnapien, l’abstraction sera caractérisée comme un processus défini d’éloignement du sens. Nous arriverons à la conclusion que le système carnapien a plus d’avantages que celui de Frege (comme la simplicité du système) et que son aspect technique a besoin d’être développé davantage

Metaontological Studies relating to the Problem of Universals

My dissertation deals with metaontology or metametaphysics. This is the subdiscipline of philosophy that is concerned with the investigation of metaphysical concepts, statements, theories and problems on the metalevel. It analyses the meaning of metaphysical statements and theories and discusses how they are to be justified. The name "metaontology" is recently coined, but the task of metaontology is the same as Immanuel Kant already dealt with in his Critique of Pure Reason. As methods I use both historical research and logical (or rather semantical) analysis. In order to understand clearly what metaphysical terms or theories mean or should mean we must both look at how they have been characterized in the course of the history of philosophy and then analyse the meanings that have historically been given to them with the methods of modern formal semantics. Metaontological research would be worthless if it could not in the end be applied to solving some substantive ontological questions. In the end of my dissertation, therefore, I give arguments for a solution to the substantively ontological problem of universals, a form of realism about universals called promiscuous realism. To prepare the way for that argument, I argue that the metaontological considerations most relevant to the problem of universals are considerations concerning ontological commitment, as the American philosophers Quine and van Inwagen have argued, not those concerning truthmakers as such philosophers as the Australian realist D. M. Armstrong have argued or those concerning verification conditions as such philosophers as Michael Dummett have argued. To justify this conclusion, I go first through well-known objections to verificationism, and show that they apply also to current verificationist theories such as Dummett's theory and Field's deflationist theory of truth. In the process I also respond to opponents of metaphysics who try to show with the aid of verificationism or structuralism that metaphysical questions would be meaningless or illegitimate in some other way. Having justified the central role of ontological commitment, I try to develop a detailed theory of it. The core of my work is a rigorous formal development of a theory of ontological commitment. I construct it by combining Alonzo Church's theory of ontological commitment with Tarski's theory of truth.Väitöskirjani käsittelee metaontologiaa eli metametafysiikkaa. Tämä on se metafilosofian osa-alue, joka tutkii metafyysisten väitteiden ja termien merkitystä ja sitä, miten metafyysiset väitteet ja teoriat voitaisiin oikeuttaa. Metafysiikka tai ontologia on taas tiede, joka tutkii olevaa yleensä tai kaikkeutta kokonaisuutena. Menetelminä käytän sekä historiallista tutkimusta että loogista (tai pikemminkin semanttista) analyysiä. On olemassa kolme pääasiallista teoriaa siitä, mikä on metaontologian keskeisin käsite. Sellaiset filosofit kuin australialainen Armstrong ovat väittäneet, että se on totuustekijöiden (truthmakers) käsite. Sellaiset anti-realistiset filosofit kuin englantilainen filosofi Michael Dummett ovat taas väittäneet että se on todennettavuusehtojen (verification conditions) käsite. Argumentoin näitä kahta käsitystä vastaan ja kolmannen puolesta, jonka mukaan keskeisin käsite on ontologisten sitoumusten käsite, kuten amerikkalainen filosofi Quine on väittänyt. Argumentoin, että Quinen ontologisten sitoumusten teoria voidaan erottaa hänen muista ontologisista näkemyksistään, kuten hänen semanttisesta holismistaan, ontologisesta relativismistaan tai strukturalismistaan, mitkä ovat mielestäni virheellisiä. Väitöskirjani ydin on täsmällinen teoria ontologisista sitoumuksista, jonka rakennan yhdistämällä Alonzo Churchin teoriaa ontologisista sitoumuksista Alfred Tarskin totuusteoriaan. Metaontologinen tutkimus olisi arvotonta, ellei sitä voisi lopulta käyttää substantiivisten ontologisten kysymysten ratkaisemiseen. Käsittelen siksi väitöskirjani loppupuolella yhtä perinteistä ontologian ongelmaa, universaalien ongelmaa. Jo Aristoteles määritteli teoksessaan Tulkinnasta universaalien olevan olioita, jotka (Lauri Carlsonin käännöksen mukaan) luonnostaan predikoidaan (sanotaan) monesta. Universaaliongelma koskee sitä, ovatko tällaiset universaalit vain kielellisiä ilmauksia, kuten yleisnimet, verbit ja adjektiivit, tai ihmismielestä riippuvia olioita, kuten yleiskäsitteet, vai voidaanko myös sanoa, että maailmassa itsessään olevia olioita voidaan predikoida jostakin. Realistin mukaan vastaus on myöntävä. Esitän väitöskirjan lopussa alustavan argumentin universaaleja koskevan realismin puolesta

First order logic as a formal language : an investigation of categorial grammar.

Thesis. 1976. Ph.D.--Massachusetts Institute of Technology. Dept. of Philosophy.Microfiche copy available in Archives and Humanities.Bibliography: leaves 165-170.Ph.D

Synonymy and Identity of Proofs - A Philosophical Essay

The main objective of the dissertation is to investigate from a strictly philosophical perspective different approaches and results related to the problem of identity of proofs, which is a problem of general proof theory at the intersection of mathematics and philosophy. The author characterizes,compares and evaluates a range of formal criteria of proof-identity that have been proposed in the proof-theoretic literature. While these proposals come from mathematical logicians, the author’s background in both mathematical logic and philosophy allows him to present and discuss these proposals in a manner that is accessible to and fruitful for philosophers, especially those working in logic and philosophy of mathematics, as well as mathematical logicians. The dissertation is structured into a prologue and five sections. In the prologue, the author traces the development of the concept of a proof in ancient philosophy, culminating in the work of Aristotle. In Section I, the author turns to the roots of proof theory in modern philosophy, offering a detailed interpretation of Kant’s “Die falsche Spitzfindigkeit der vier syllogistischen Figuren”, which uncovers interesting links between Kant’s inferences of understanding and of reason and modern proof-theoretic semantics. In Section II, the author turns from historical to systematic considerations concerning different kinds of identity-criteria of proofs, ranging from overly liberal criteria that trivialize proof identity to overly strict, syntactical criteria. In Section III, the heart of the dissertation, the author offers a thorough philosophical discussion of the normalisation thesis. In Section IV, the author considers the difficulties encountered in his discussion of identity of proofs --- particularly of the normalisation thesis --- through the lens of a discussion of the notion of synonymy, and compares this thesis with other possible formal accounts of identity of proofs. In particular, by recourse to Carnap’s notion of synonymy, developed in “Meaning and Necessity”, the author proposes a notion of synonymy of proofs. In Section V, the final substantial section, the author compares the normalisation thesis to the Church-Turing thesis, thereby adducing another dimension of evaluation of the former

Relations between logic and mathematics in the work of Benjamin and Charles S. Peirce.

Charles Peirce (1839-1914) was one of the most important logicians of the nineteenth century. This thesis traces the development of his algebraic logic from his early papers, with especial attention paid to the mathematical aspects. There are three main sources to consider. 1) Benjamin Peirce (1809-1880), Charles's father and also a leading American mathematician of his day, was an inspiration. His memoir Linear Associative Algebra (1870) is summarised and for the first time the algebraic structures behind its 169 algebras are analysed in depth. 2) Peirce's early papers on algebraic logic from the late 1860s were largely an attempt to expand and adapt George Boole's calculus, using a part/whole theory of classes and algebraic analogies concerning symbols, operations and equations to produce a method of deducing consequences from premises. 3) One of Peirce's main achievements was his work on the theory of relations, following in the pioneering footsteps of Augustus De Morgan. By linking the theory of relations to his post-Boolean algebraic logic, he solved many of the limitations that beset Boole's calculus. Peirce's seminal paper `Description of a Notation for the Logic of Relatives' (1870) is analysed in detail, with a new interpretation suggested for his mysterious process of logical differentiation. Charles Peirce's later work up to the mid 1880s is then surveyed, both for its extended algebraic character and for its novel theory of quantification. The contributions of two of his students at the Johns Hopkins University, Oscar Mitchell and Christine Ladd-Franklin are traced, specifically with an analysis of their problem solving methods. The work of Peirce's successor Ernst Schröder is also reviewed, contrasting the differences and similarities between their logics. During the 1890s and later, Charles Peirce turned to a diagrammatic representation and extension of his algebraic logic. The basic concepts of this topological twist are introduced. Although Peirce's work in logic has been studied by previous scholars, this thesis stresses to a new extent the mathematical aspects of his logic - in particular the algebraic background and methods, not only of Peirce but also of several of his contemporaries

Proceedings of the Workshop on the lambda-Prolog Programming Language

The expressiveness of logic programs can be greatly increased over first-order Horn clauses through a stronger emphasis on logical connectives and by admitting various forms of higher-order quantification. The logic of hereditary Harrop formulas and the notion of uniform proof have been developed to provide a foundation for more expressive logic programming languages. The λ-Prolog language is actively being developed on top of these foundational considerations. The rich logical foundations of λ-Prolog provides it with declarative approaches to modular programming, hypothetical reasoning, higher-order programming, polymorphic typing, and meta-programming. These aspects of λ-Prolog have made it valuable as a higher-level language for the specification and implementation of programs in numerous areas, including natural language, automated reasoning, program transformation, and databases