A Quasi-Fregean Solution to ‘The Concept Horse’ Paradox

In this paper I offer a conceptually tighter, quasi-Fregean solution to the concept horse paradox based on the idea that the unterfallen relation is asymmetrical. The solution is conceptually tighter in the sense that it retains the Fregean principle of separating sharply between concepts and objects, it retains Frege’s conclusion that the sentence ‘the concept horse is not a concept’ is true, but does not violate our intuitions on the matter. The solution is only ‘quasi’- Fregean in the sense that it rejects Frege’s claims about the ontological import of natural language and his analysis thereof

Psychologism And Its History Revalued

A hundred years ago Frege had published most of his arguments against psychologism and Husserl was busy writing his Logical Investigations, which was to appear at the turn of the century and open with a long onslaught on psychologism. The arguments of these two logicians against the psychologistic view - of Mill, Erdmann and many others - that the discipline of logic, its sentences, or its "laws", deal with psychological phenomena met with widespread approval from those best qualified to judge (for example Lukasiewicz). They set the agenda for most twentieth century work in exact, "scientific", or analytic philosophy. As the century draws to its close, many of the arguments of Frege and Husserl have been found wanting by analytic philosophers and cognitive scientists who are prepared to argue that the laws of logic are just laws of human thought

Willard Van Orman Quine's Philosophical Development in the 1930s and 1940s

As analytic philosophy is becoming increasingly aware of and interested in its own history, the study of that field is broadening to include, not just its earliest beginnings, but also the mid-twentieth century. One of the towering figures of this epoch is W.V. Quine (1908-2000), champion of naturalism in philosophy of science, pioneer of mathematical logic, trying to unite an austerely physicalist theory of the world with the truths of mathematics, psychology, and linguistics. Quine's posthumous papers, notes, and drafts revealing the development of his views in the forties have recently begun to be published, as well as careful philosophical studies of, for instance, the evolution of his key doctrine that mathematical and logical truth are continuous with, not divorced from, the truths of natural science. But one central text has remained unexplored: Quine's Portuguese-language book on logic, his 'farewell for now' to the discipline as he embarked on an assignment in the Navy in WWII. Anglophone philosophers have neglected this book because they could not read it. Jointly with colleagues, I have completed the first full English translation of this book. In this accompanying paper I draw out the main philosophical contributions Quine made in the book, placing them in their historical context and relating them to Quine's overall philosophical development during the period. Besides significant developments in the evolution of Quine's views on meaning and analyticity, I argue, this book is also driven by Quine's indebtedness to Russell and Whitehead, Tarski, and Frege, and contains crucial developments in his thinking on philosophy of logic and ontology. This includes early versions of some arguments from 'On What There Is', four-dimensionalism, and virtual set theory

Hesperus and Phosphorus: Sense, Pretense, and Reference

In “On Sense and Reference,” surrounding his discussion of how we describe what people say and think, identity is Frege’s first stop and his last. We will follow Frege’s plan here, but we will stop also in the land of make-believe

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

Logic and Proof for Mathematicians: A Twentieth Century Perspective

If there is one aspect of mathematics education that frustrates both students and teachers alike, it has got to be learning how to do valid proofs. Students often feel they really know the mathematics they\u27re studying but that their teachers place some unreasonably stringent demands upon their arguments. Teachers, on the other hand, can\u27t understand where their students get some of the wacky arguments they come up with. They argue in circles, they end up proving a different result from what they claim, they make false statements, they draw invalid inferences - it can be quite exasperating at times! Unfortunately, constructing a genuine demonstration is not a side issue in mathematics; deduction forms the backbone and hygiene of mathematical intercourse and simply must be learned by mathematics majors. I\u27d like to focus attention on this topic in this paper first by sharing with you how we attack the problem at Dordt. I will then attempt to give the issue a bit of historical and philosophical depth by tracing some developments in mathematics and logic which seem to have had a strong influence upon contemporary approaches to teaching proof. I hope this will contribute to further discussion of the problem

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

A critique of Tractarian semantics

This is a critique of the principal claims made within Ludwig Wittgenstein\u27s Tractatus Logico-Philosophicus. It traces the development of his thought from the time he dictated the pre-Tractarian Notes on Logic to Russell up until about 1932 when he began work on the Philosophical Grammar. The influence exercised upon him by Frege, Russell and Moore are considered at length. Chapter one examines Moore\u27s relational theory of judgment which Wittgenstein apparently accepted upon his arrival at Cambridge in 1911. From Moore Wittgenstein would inherit one of the fundamental metaphysical theses of the Tractatus, namely, that the world consists of facts rather than things. Wittgenstein\u27s attempt to overcome the relational theory\u27s inability to account for falsehood, negation, and the possibility of truly ascribing false beliefs to others would herald some of the principal theses of Tractarian semantics: that propositional signs must exhibit bipolarity, that a distinction must be drawn between Sinn and Bedeutung, and that a distinction holds between what can be said and what can only be shown. Chapter Two examines how these theses are sharpened by considering the influence of Frege and the manner in which Wittgenstein disposes of Russell\u27s Paradox. considerable attention is given to the issue of whether Frege is to be interpreted as a semantic Platonist. It is argued that he is not, and that Tractarian semantics shores up the problematic features of Frege\u27s philosophy which make it susceptible to the paradox. From Frege Wittgenstein derives the idea that all representation requires a structured medium. The chapter concludes by considering how this entails the falsehood of semantic Platonism. Chapter Three studies Wittgenstein\u27s argument for logical atomism and gives it a favorable assessment. The influence of Russell\u27s conception of logical analysis is considered. The chapter concludes by showing the way Wittgenstein\u27s thesis that there must be simple subsistent objects depends upon the truth of his Grundgedanke, i.e., the claim that the logical constants are not referring terms. Chapter Four examines the argument for the Grundgedanke, and defends it against criticism based upon phenomenological considerations for objectifying negativity. It is demonstrated that Wittgenstein\u27s view entails that a distinction must be drawn between propositions possessing sense and those that are senseless but no less a part of our language. Chapter Five examines Wittgenstein\u27s claim that the essence of a proposition consists in a propositional sign\u27s projective relation to the world, and it considers the Tractarian analysis of propositional attitude ascriptions. It is argued that the analysis of these sorts of sentences forms the principal problem with the Tractatus. The chapter includes a discussion of why the Color Exclusion Problem need not be considered problematic for the author of the Tractatus, and it defends the realistic interpretation given of the Tractatus throughout the dissertation against criticisms arising from a consideration of Wittgenstein\u27s remarks on solipsism

Why did Frege reject the theory of types?

I investigate why Frege rejected the theory of types, as Russell presented it to him in their correspondence. Frege claims that it commits one to violations of the law of excluded middle, but this complaint seems to rest on a dogmatic refusal to take Russell’s proposal seriously on its own terms. What is at stake is not so much the truth of a law of logic, but the structure of the hierarchy of the logical categories, something Frege seems to neglect. To come to a better understanding of Frege’s response, I proceed to investigate his conception of the nature of the logical categories, and how it differs from Russell’s. I argue that, for Frege, our grasp of the logical categories cannot be severed from our grasp of the Begriffsschrift notation itself. Russell, on the other hand, attaches no such importance to notation. From Frege’s point of view, Russell has not succeeded in presenting an alternative conception of the logical hierarchy, since such a conception must go in tandem with the development of a notation. Moreover, Frege has good reasons to think that Russell’s proposal does not admit of a suitable notation

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