Second Order Logic and Logical Form

This thesis explores several related issues surrounding second order logic. The central problem running throughout is whether second order logic should provide the underlying logic for formalizations of natural language. A prior problem is determining the significance of this choice. Such controversies over the adoption of a logic usually involve assessing the merits of challengers to first order logic. In some of these rival systems various first order logical truths do not hold. The failure of the Law of the Excluded Middle in intuitionistic systems is the most common example. The other alternatives to first order logic accept it as a part of the truth, but extend it by adding new logical constants. Some modal systems of logic are formed by adding to first order logic a symbol intended to be read as \u27it is logically necessary that.\u27 The first order semantics is extended to provide truth conditions for sentences containing this new symbol. In such cases the debate is whether we are justified in expanding the list of logical constants provided by first order logic. We accept the first order logical constants and are deciding whether, e.g., \u27it is logically necessary that\u27 should be added to the list

Frege's Basic Law V and Cantor's Theorem

The following essay reconsiders the ontological and logical issues around Frege’s Basic Law (V). If focuses less on Russell’s Paradox, as most treatments of Frege’s Grundgesetze der Arithmetik (GGA)1 do, but rather on the relation between Frege’s Basic Law (V) and Cantor’s Theorem (CT). So for the most part the inconsistency of Naïve Comprehension (in the context of standard Second Order Logic) will not concern us, but rather the ontological issues central to the conflict between (BLV) and (CT). These ontological issues are interesting in their own right. And if and only if in case ontological considerations make a strong case for something like (BLV) we have to trouble us with inconsistency and paraconsistency. These ontological issues also lead to a renewed methodological reflection what to assume or recognize as an axiom

From mathematics in logic to logic in mathematics : Boole and Frege

This project proceeds from the premise that the historical and logical value of Boole's logical calculus and its connection with Frege's logic remain to be recognised. It begins by discussing Gillies' application of Kuhn's concepts to the history oflogic and proposing the use of the concept of research programme as a methodological tool in the historiography oflogic. Then it analyses'the development of mathematical logic from Boole to Frege in terms of overlapping research programmes whilst discussing especially Boole's logical calculus. Two streams of development run through the project: 1. A discussion and appraisal of Boole's research programme in the context of logical debates and the emergence of symbolical algebra in Britain in the nineteenth century, including the improvements which Venn brings to logic as algebra, and the axiomatisation of 'Boolean algebras', which is due to Huntington and Sheffer. 2. An investigation of the particularity of the Fregean research programme, including an analysis ofthe extent to which certain elements of Begriffsschrift are new; and an account of Frege's discussion of Boole which focuses on the domain common to the two formal languages and shows the logical connection between Boole's logical calculus and Frege's. As a result, it is shown that the progress made in mathematical logic stemmed from two continuous and overlapping research programmes: Boole's introduction ofmathematics in logic and Frege's introduction oflogic in mathematics. In particular, Boole is regarded as the grandfather of metamathematics, and Lowenheim's theorem ofl915 is seen as a revival of his research programme

The Logical Problem of Identity

Abstract Keith A. Coleman Department of Philosophy, February 2008 University of Kansas A traditional problem concerning the meaning or logical content of statements of identity received its modern formulation in Gottlob Frege's "On Sense and Reference." Identity is taken either as a relation between objects or a relation between terms. If identity is interpreted as a relation between objects, then identity statements seem to be of little value since everything is clearly identical to itself. Assertions of identity are thought to convey significant information, but it is hard to see how they can on this interpretation. If identity is instead interpreted as a relation between terms, then identity statements still seem to be of little value since apparently they only convey a linguistic pronouncement to use certain terms interchangeably. Assertions of identity do not appear to be about the use of language, but, on this understanding of identity, they evidently are. I examine the nature of the problem (and what it would take to solve it) and the advantages and disadvantages of each one of the two approaches to interpreting the content of identity statements. I then investigate two approaches for solving the problem from the perspective of identity as a relation between objects. The first of these represents the account provided by Gottlob Frege, and the second represents the account provided by Saul Kripke. I conclude that neither one of these accounts finally solves the problem of identity in its entirety. I then examine Michael Lockwood's approach to resolving the problem of identity based on the idea of identity as a relation between terms. I discuss and critically evaluate Lockwood's account together with a modified version of that account. After arguing for the inadequacy of the views examined as ultimate solutions to the problem of identity, I end by suggesting a strategy prompted by treating identity as indiscernibility

The logical anti-psychologism of Frege and Husserl

Frege and Husserl are both recognized for their significant contributions to the overthrowing of logical psychologism, at least in its 19th century forms. Between Frege's profound impact on modern logic that extended the influence of his anti-psychologism and Husserl's extensive attempts at the refutation of logical psychologism in the Prolegomena to Logical Investigations, these arguments are generally understood as successful. This paper attempts to account for the development of these two anti-psychologistic conceptions of logical objects and for some of the basic differences between them. It identifies some problems that are common to strongly anti-psychologistic conceptions of logic and compares the extent to which Frege's and Husserl's views are open to these problems. Accordingly, this paper is divided into two parts. Part I develops a conception of the problems of logical psychologism as they are distinctively understood by each philosopher, out of the explicit arguments and criticisms made against the view in the texts. This conception is in each case informed by the overall historical trajectories of each philosopher's philosophical development. Part II examines the two views in light of common problems of anti-psychologism

Danielle Macbeth, "Realizing Reason: A Narrative of Truth and Knowing"

This substantial book is a highly original and thorough work of synthetic first philosophy. Although it has some recognizable roots in the Kantian/Sellarsian tradition of the Pittsburgh school, it adds a wealth of precise discussion of examples from science and mathematics, made possible by Macbeth's dual training in arts and sciences. It presents a developmental story of human reason bootstrapping itself towards greater power and clarity through the Western tradition (which is the sole purview of the discussion). This development is divided into three distinct stages, which might be summarized very roughly as knowledge of: i) Objects, ii) Concepts applied to Objects and iii) Concepts alone

Second-order logic is logic

"Second-order logic" is the name given to a formal system. Some claim that the formal system is a logical system. Others claim that it is a mathematical system. In the thesis, I examine these claims in the light of some philosophical criteria which first motivated Frege in his logicist project. The criteria are that a logic should be universal, it should reflect our intuitive notion of logical validity, and it should be analytic. The analysis is interesting in two respects. One is conceptual: it gives us a purchase on where and how to draw a distinction between logic and other sciences. The other interest is historical: showing that second-order logic is a logical system according to the philosophical criteria mentioned above goes some way towards vindicating Frege's logicist project in a contemporary context

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