In their correspondence in 1902 and 1903, after discussing the Russell paradox, Russell and Frege discussed the paradox of propositions considered informally in Appendix B of Russell’s Principles of Mathematics. It seems that the proposition, p, stating the logical product of the class w, namely, the class of all propositions stating the logical product of a class they are not in, is in w if and only if it is not. Frege believed that this paradox was avoided within his philosophy (...) due to his distinction between sense (Sinn) and reference (Bedeutung). However, I show that while the paradox as Russell formulates it is ill-formed with Frege’s extant logical system, if Frege’s system is expanded to contain the commitments of his philosophy of language, an analogue of this paradox is formulable. This and other concerns in Fregean intensional logic are discussed, and it is discovered that Frege’s logical system, even without its naive class theory embodied in its infamous Basic Law V, leads to inconsistencies when the theory of sense and reference is axiomatized therein. (shrink)

Demonstratives seem to have posed a severe difficulty for Frege’s philosophy of language, to which his doctrine of incommunicable senses was a reaction. In “The Thought,” Frege briefly discusses sentences containing such demonstratives as “today,” “here,” and “yesterday,” and then turns to certain questions that he says are raised by the occurrence of “I” in sentences (T, 24-26). He is led to say that, when one thinks about oneself, one grasps thoughts that others cannot grasp, that cannot be communicated. However, (...) nothing could be more out of the spirit of Frege’s account of sense and thought than an incommunicable, private thought. In the first part of the paper, I explain the problem demonstratives pose for Frege, and explore three ways he might have dealt with it. I argue that none of these ways provides Frege with a solution to his problem consistent with his philosophy of language. The first two are plausible as solutions, but contradict his identification of the sense expressed by a sentence with a thought. The third preserves the identification, but is implausible. In the second part, I suggest that Frege was led to his doctrine of incommunicable senses as a result of some appreciation of the difficulties his account of demonstratives faces, for these come quickly to the surface when we think about “I.” I argue that incommunicable senses won’t help. I end by trying to identify the central problem with Frege’s approach, and sketching an alternative. (shrink)

What is involved in the meaning of our expressions? Frege suggested that there is an aspect of an expression.

This book is in the Studies in Philosophy: Outstanding Dissertations series. Its central theme is that Frege’s concept-notation is inadequate because it does not formalize his semantic theory after the introduction of the sense-reference distinction in 1891. This failing, according to Klement, opens Frege up to a number of philosophical and logical challenges that can be met only by completing the project of showing “how Frege’s mature semantic views would be incorporated into his mature logical system”, a project which, Klement (...) plausibly argues, Frege would have endorsed. This involves: allowing the concept-notation to represent explicitly what is expressed in natural language with indirect constructions such as “said that...” or “believes that... ”; allowing principles to be formulated over this notion which capture inferences involving indirect constructions such as: The Morning Star is a planet, Gottlob believes the Morning Star is a planet, so, Gottlob believes something true; being able to respond to certain objections to Frege’s views, some first bruited by Russell and in later correspondence, others involving the identity conditions on intensional entities and “quantifying in” ; making the concept-notation reflect Frege’s ontology; and evaluating the consistency of Frege’s ontology. The chapters of this book are devoted to fulfilling this project “based upon a careful reading of Frege’s own works”, and their contents are mostly self-explanatory. Chapter 2 sets out the formal apparatus of the Grundgeseze, even the bits Frege left out. Chapter 3 is the most philosophical. Senses are construed as interpersonally accessible information uniquely true of the references they determine—senses are objects and functions in the “third realm.” Singular terms are associated with an impossibly rich hierarchy of senses as objects. Functors, including predicates, are associated with senses as functions. Sentences are associated with thoughts as objects, composed of constituent sense-functions, and in many cases, also sense-objects. Robust compositionality at the level of sense raises the question of the nature of the relation between constituents and the whole—is it functional, or not? Compositionality potentially conflicts with a number of other claims, and Klement’s conclusion is reasonable: “the context principle, the priority thesis, and the possibility of there existing multiple decompositions for the same Gedanke, do not pose any great difficulty for the compositionality principle or the understanding of Gedanke as having determinate inner structures”. Chapter 4 sets out Church’s original system and investigates reformulations suggested by Church himself, Kaplan, and others. Chapter 5 presents Klement’s formalization of an intensional logic of sense and reference, based on the results of chapters 2 and 4, and sufficient for indirect speech and quantifying-in, but containing a contradiction. Chapter 6 takes up arguments of early Russell, Searle, and Bergmann. Chapter 7 addresses strategies for repairing the inconsistencies of chapter 5, but in a Fregean spirit: dropping some realms of entities, restricting their number, or modifying our conception of their nature. (shrink)

This book explores an important central thread that unifies Russell's thoughts on logic in two works previously considered at odds with each other, the Principles of Mathematics and the later Principia Mathematica. This thread is Russell's doctrine that logic is an absolutely general science and that any calculus for it must embrace wholly unrestricted variables. The heart of Landini's book is a careful analysis of Russell's largely unpublished "substitutional" theory. On Landini's showing, the substitutional theory reveals the unity of Russell's (...) philosophy of logic and offers new avenues for a genuine solution of the paradoxes plaguing Logicism. (shrink)

Russell's "new contradiction" about "the totality of propositions" has been connected with a number of modal paradoxes. M. Oksanen has recently shown how these modal paradoxes are resolved in the set theory NFU. Russell's paradox of the totality of propositions was left unexplained, however. We reconstruct Russell's argument and explain how it is resolved in two intensional logics that are equiconsistent with NFU. We also show how different notions of possible worlds are represented in these intensional logics.