The Logic of Principia Mathematica

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Linguistics and Philosophy, 1995.Includes bibliographical references (p. 181-184).by Darryl Jung.Ph.D

Traditional logic and the early history of sets, 1854–1908

Ministerio de Educación y Ciencia (España

Frege's paradox.

Thesis. 1977. Ph.D.--Massachusetts Institute of Technology. Dept. of Linguistics and Philosophy.MICROFICHE COPY AVAILABLE IN ARCHIVES AND HUMANITIES.Vita.Bibliography : leaves 231-236.Ph.D

Omnis Determinatio est Negatio: A Genealogy and Defense of the Hegelian Conception of Negation

This dissertation explores a line of philosophical thought that accords negation a fundamental role in the determination of concepts and the kinds, universals and particulars to which concepts provide access. Though it takes the largely historical route of philosophical genealogy, focusing on Plato, Boethius and Hegel, the dissertation is also intended as a defense of the philosophical significance of negation itself, and of a principle Hegel famously attributes to Spinoza, omnis determinatio est negatio. I argue that in explaining negation as difference (to heteron) Plato reveals its originary function as the characteristic operation of cognition, and that this discursive role is the basis of the logical and grammatical roles it is more conventionally assigned. So understood, negation also underpins the traditional, scholastic treatment of differentiae to which it is historically and logically linked. It is also tied to a view of concepts (and kinds) as hierarchically determined, and to a method for mapping out this hierarchy, i.e., the so-called method of division. The method of division is introduced by Plato, developed by Aristotle, institutionalized by Boethius and, as I argue, reconceived by Hegel, in part in terms of his notion of sublation (Aufhebung). I argue, as well, that, given the role of negation, division and the inherence model of the proposition it supports show that concepts are grounded as much in negation (and non-being) as in the bedrock of being. This groundedness of the concept in negation, which I call the problem of negation, is a defining concern of the Hegelian lineage

Propositions : an ontological inquiry

It is more or less agreed that propositions are the meanings of sentences, the fundamental truth-bearers, and the objects of propositional attitudes. Associated with these roles, there are the following three questions: the Composition Question, the Representation Question, and the Attitude Question. Roughly, the first concerns the metaphysical relation between propositions and propositional constituents, the second concerns the ability of representing things as being such-and-so, and the third concerns how propositions can be the objects of propositional attitudes. I examine three mainstream theories of propositions: the Russellian theories, the possible-world accounts, and the Neo-Russellian theories, and argue that each fails to answer at least one of the questions and thus is incapable of providing an account of these propositional roles. Therefore, if a theory of propositions is able to answer these questions in a uniform manner, it would be a better theory of propositions. For what can be explained by other theories can also be explained by this theory, and it can also answer more questions than any other theory. In this dissertation, I defend a broadly Fregean theory of propositions, according to which propositions are sui generis, multi-analysable, and necessary beings, and argue that with respect to these propositional roles, it can provide a better account than other theories of propositions

BASIC LOGICAL KNOWLEDGE AND ITS JUSTIFICATION

This thesis contributes to the area within the philosophy of logic that concerns epistemological questions about fundamental logical laws. The state of the literature has been dominated by meaning-theoretic approaches that attempt to answer such questions in terms of conditions for understanding words or for possessing concepts, and empiricist approaches that attempt to characterize logical knowledge as a posteriori. The theoretical contribution made by my work amounts to the presentation of an alternative theory that does not appeal to the notion of rational insight, and which is not psychologistic, conventionalist, or inductivist. The theory that I propose involves a philosophical analysis of logical concepts which facilitates a transcendental argument in support of basic logical knowledge. I arrive at this theory by radically readapting methods and insights from the theories of geometry of Kant and Helmholtz, and from Frege’s logicist explanation of the basis of our knowledge of arithmetic. I argue that the meaning-theoretic approaches of Christopher Peacocke and Paul Boghossian overestimate the epistemological power of the conditions for understanding logical constants. I argue that the naturalistic approach of Penelope Maddy is psychologistic. To lay the groundwork for my own proposal 1 investigate the role of Kant’s transcendental method in his theory of geometry, and the role of the theory of analytic definitions in a reconstruction of Frege’s logicism. My own proposal posits that logical laws are analytic of logical concepts, and that logical concepts are conditions of the possibility of possessing a minimally reasonable conceptual scheme

Arguments for the existence of God in Anselm's Proslogion chapter II and III

Anselm's argument for the existence of God in Proslogion Chap.II starts from the contention that `lq when a Fool hears `something-than-which-nothing-greater-can-be-thought', he understands what he hears, and what he understands is in his mind. This is a special feature of the Pros.II argument which distinguishes the argument from other ontological arguments set up by, for example, Descartes and Leibniz. This is also the context which makes semantics necessary for evaluation of the argument. It is quite natural to ask `lq What is understood by the Fool, and what is in his mind? It is essential for a proper consideration of the argument to identify the object which is understood by the Fool, and so, is in his mind. A semantics gives answers to the questions of `lq What the Fool understands? and `lq What is in the Fool's mind? If we choose a semantics as a meta-theory to interpret the Pros.II argument, it makes an effective guide to identify the object. It is a necessary condition for a proper evaluation of the Pros.II argument to fix our universe of discourse, especially since, in the argument, we are involved in such talk about existing objects as Anselm's contention that `when a Fool hears `something-than-which-nothing-greater-can-be-thought', he understands what he hears, and what he understands is in his mind. The ontology to which a semantic theory commits us will be accepted as our scope of objects when we introduce our semantic theory to interpret the Pros.II argument, and this ontological boundary constrains us to identify the object in a certain way. Consistent application of an ontology, most of all, is needed for the evaluation of the logical validity of an argument. If we take Frege's three-level semantics, we are ontologically committed to intensional entities, like meaning, as well as extensional entities. Sluga contends that Frege's anti-psychologism for meanings should not be interpreted as vindicating reification of intensional entities in relation to Frege's contextualism, that Frege's anti-psychologism with his contextualism is nothing but a linguistic version of Kantian philosophy for the transcendental unity of a judgement. There is, however, another possible interpretation of Frege's contextualism. According to Dummett, the significance of Frege's contextualism must be understood as a way of explanation for a word's having meaning. If Dummett's view is cogent, we could say that Frege's contextualism does not prevent our interpreting his semantics as being committed to intensional entities. We need not worry that Frege's over all semantics, especially with his contextualism, would internally deny the ontological interpretation of his theory. We see Anselm's argument for the existence of God in Pros.II is an invalid argument if we introduce Frege's three-level semantics, i.e. if we acknowledge meanings of words as entities in our universe of discourse. We can also employ extensional semantics for the interpretation of the Pros.II argument. According to extensionalists, like Quine and Kripke, we need not assume intensional entities, like meaning, to be part of our ontological domain. They argue that we can employ our language well enough without assuming intensional entities. If we choose extensional semantics as a meta-theory to interpret the Pros.II argument, it commits us only to extensional entities as objects in the Universe of our interpretation. In Sections 1.4 and 1.5, I show that extensional semantics makes the Pros.II argument a valid argument for the existence of God. `lq Necessary existence is the central concept of Anselm's argument for the existence of God in Proslogion Chap.III. It has been said that, even if the argument is formally valid, it cannot stand as a valid argument for the existence of God, since `lq necessary existence is an absurd concept like `lq round square. And further that even if there is a meaningful combination of concepts for `lq necessary existence, it cannot quality as a subject of an a priori argument. As objections to the interpretations which make the Pros.III argument valid, it has been argued that even if there is a concept of `lq necessary existence which is meaningful and there is another concept of `lq necessary existence which is suitable as a subject of an a priori argument, there is no concept of `lq necessary existence which is meaningful and at the same time suitable as a subject of an a priori argument. In Chap.2 and Chap.3, I try to show that there can be concepts of `lq necessary existence which are proof against these objections. Anselm's arguments for the existence of God in Proslogian Chap.II and Chap.III are logically valid arguments on some logical principles. Some fideists, K. Barth, for example, argue that Anselm's arguments for the existence of God in Proslogion are not proofs for the existence of God even if they are logically valid arguments. I raise the question how this attitude could be possible, in Chap.4 and Chap.5. Barth's fideistic interpretation of Anselm's Proslogion arguments does not find any flaw in the validity of the arguments, and it accepts the meaningfulness and truth of the premises even to the fool in Proslogion. If this is the case, i.e. if Barth's interpretation accepts the validity of the arguments and the truth of the premises, I raise the question, how can the arguments not be interpreted as proofs for the existence of God? How is it possible that the function of the arguments is not that of proving the existence of God? According to Wittgensteinian fideism, premises in the arguments should not be intelligible to those who do not believe in God's existence already, and so the real function of the arguments is the elucidation, the understanding of believer's belief, rather than proving articles of belief to unbelievers. Barth's fideistic interpretation of the arguments, however, fully recognizes the meaningfulness and truth of the premises in the arguments as well as the validity of the arguments. I argue that there could be a justification for the Barthian fideism. As Malcolm notices, there are still atheists who understand Anselm's arguments as valid, but the only possibility for the people who recognize the validity of Anselm's arguments still to remain atheists has been thought to be to challenge the truth of premises employed in the arguments. Now, of the atheistic possibility, we can change the direction of our attention, that is, to the question about the function of a logically valid argument itself. What has not been thought of in relation to Anselm's arguments is the significance of logical truth or the logical validity of an argument. We have not asked such questions as `lq What does a logical truth say? and `lq What does a logically valid argument guarantee with true premises? Let us assume that even the premises are accepted by atheists. Do they all convert to theism? If that were so, the disagreement between atheist and believer over the ontological arguments should turn only on the truth of premises. If that is not so, there is some point in raising this other question. If there are people who, recognizing the premises and validity of an argument, are still reluctant to accept the conclusion, we have reason to question the function of a valid argument. I argue that there is a way of being consistently reasonable while accepting the premises and the validity of the ontological arguments and yet remaining an atheist or an agnostic

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

Finitism--an essay on Hilbert's programme

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Linguistics and Philosophy, 1991.Includes bibliographical references (p. 213-219).by David Watson Galloway.Ph.D