Kevin Klement

Certain consequentialists have responded to deontological worries regarding personal projects or options and agent-centered restrictions or constraints by pointing out that it is consistent with consequentialist principles that people develop within themselves, dispositions to act with such things in mind, even if doing so does not lead to the best consequences on every occasion. This paper argues that making this response requires shifting the focus of moral evaluation off of evaluation of individual actions and towards evaluation of whole character traits and patterns of behavior. However, this weds consequentialism to a sort of psychological determinism, with which it is incompatible, because it makes any sort of assessment in terms of right or wrong incoherent. The paper concludes by sketching an ethical theory that abandons right and wrong as its organizing concepts, but nevertheless preserves much of the spirit of consequentialism. Relationships between whole patterns of behavior and the production of good and bad consequences can be studied and analyzed, and this information can be used to improve society and ourselves, without it being required that any individual persons acts or dispositions to behave be evaluated as right or wrong.

Russell corresponded with Sir Michael Dummett (1925–2011) between 1953 and 1963 while the latter was working on a book on Frege, eventually published as Frege: Philosophy of Language (1973). In their letters they discuss Russell’s correspondence with Frege, translating it into English, as well as Frege’s attempted solution to Russell’s paradox in the appendix to vol. 2 of his Grundgesetze der Arithmetik. After Dummett visited the University of Münster to view Frege’s Nachlaß, he sent reports back to Russell concerning both the philosophical materials Frege left behind, as well as information from Frege’s journal revealing his anti-semitic political opinions. Their interaction contains interpretive conjectures and insights on Dummett’s side, and some dark humor on Russell’s.

Analytic philosophy has been perhaps the most successful philosophical movement of the twentieth century. While there is no one doctrine that defines it, one of the most salient features of analytic philosophy is its reliance on contemporary logic, the logic that had its origin in the works of George Boole and Gottlob Frege and others in the mid‐to‐late nineteenth century. Boolean algebra, the heart of Boole's contributions to logic, has also come to represent a cornerstone of modern computing. Frege had broad philosophical interests, and his writings on the nature of logical form, meaning and truth remain the subject of intense theoretical discussion, especially in the analytic tradition. Frege's works, and the powerful new logical calculi developed at the end of the nineteenth century, influenced many of its most seminal figures, such as Bertrand Russell, Ludwig Wittgenstein and Rudolf Carnap.

In he Problems of Philosophy and other works of the same period, Russell claims that every proposition must contain at least one universal. Even fully general propositions of logic are claimed to contain “abstract logical universals”, and our knowledge of logical truths claimed to be a species of a priori knowledge of universals. However, these views are in considerable tension with Russell’s own philosophy of logic and mathematics as presented in Principia Mathematica. Universals generally are qualities and relations, but if, for example, PM’s disjunction (∨) is a relation, what is it a relation between? There is no obvious answer to this given Russell’s other philosophical commitments at this time, although hints are left in some of the pre-PM manuscripts. In this paper, I explore this tension in Russell's philosophy and relate it to developments both before and after Problems.

Russell's paradox represents either of two interrelated logical antinomies. The most commonly discussed form is a contradiction arising in the logic of sets or classes. Some classes (or sets) seem to be members of themselves, while some do not. The class of all classes is itself a class, and so it seems to be in itself. The null or empty class, however, must not be a member of itself. However, suppose that we can form a class of all classes (or sets) that, like the null class, are not included in themselves. The paradox arises from asking the question of whether this class is in itself. It is if and only if it is not. The other form is a contradiction involving properties. Some properties seem to apply to themselves, while others do not. The property of being a property is itself a property, while the property of being a cat is not itself a cat. Consider the property that something has just in case it is a property (like that of being a cat ) that does not apply to itself. Does this property apply to itself? Once again, from either assumption, the opposite follows. The paradox was named after Bertrand Russell, who discovered it in 1901.

Bertrand Russell was one of the best-known proponents of logicism: the theory that mathematics reduces to, or is an extension of, logic. Russell argued for this thesis in his 1903 The Principles of Mathematics and attempted to demonstrate it formally in Principia Mathematica (PM 1910–1913; with A. N. Whitehead). Russell later described his work as a further “regressive” step in understanding the foundations of mathematics made possible by the late 19th century “arithmetization” of mathematics and Frege’s logical definitions of arithmetical concepts. The logical system of PM sought to improve on earlier attempts by solving the contradictions found in, e.g., Frege’s system, by employing a theory of types. In this article, I also consider and critically evaluate the most common objections to Russell’s logicism, including the claim that it is undermined by Gödel’s incompleteness results, and Putnam's charge of “if-thenism”. I suggest that if we are willing to accept a slightly revisionist account of what counts as a mathematical truth, these criticisms do not obviously refute Russell’s claim to have established that mathematical truths generally are a species of logical truth.

Frege developed the theory of sense and reference while composing his Grundgesetze and considering its philosophical implications. The Grundgesetze is thus the most important test case for the application of this theory of meaning. I argue that evidence internal and external to the Grundgesetze suggests that he thought of senses as having a structure isomorphic to the Grundgesetze expressions that would be used to express them, which entails a theory about the identity conditions of senses that is relatively fine-grained, though still coarser than some other commentators have suggested. While this interpretation does not make Frege’s ontological commitment to the denizens of a “third realm” as profligate as some have alleged, it is sufficiently bloated to lead to Cantorian paradoxes and diagonal contradictions independent of his Basic Law V.

Logicism is the view that mathematical truths are logical truths. But a logical truth is commonly thought to be one with a universally valid form. The form of “7 > 5” would appear to be the same as “4 > 6”. Yet one is a mathematical truth, and the other not a truth at all. To preserve logicism, we must maintain that the two either are different subforms of the same generic form, or that their forms are not at all what they appear. The historical record shows that Russell pursued both these options, but that the struggle with the logical paradoxes pushed him away from the first kind of response and toward the second. An object cannot itself have a kind of inner logical complexity that makes a proposition have a different logical form merely in virtue of being about it, nor can their representatives in logical forms be single things different for different forms, at least not without postulating too many such objects and thereby creating Cantorian diagonal paradoxes. There are only apparent objects which are actually fragments of logical forms, different in different cases.

Gottlob Frege (1848-1925) was a German logician, mathematician and philosopher who played a crucial role in the emergence of modern logic and analytic philosophy. Frege's logical works were revolutionary, and are often taken to represent the fundamental break between contemporary approaches and the older, Aristotelian tradition. He invented modern quantificational logic, and created the first fully axiomatic system for logic, which was complete in its treatment of propositional and first-order logic, and also represented the first treatment of higher-order logic. In the philosophy of mathematics, he was one of the most ardent proponents of logicism, the thesis that mathematical truths are logical truths, and presented influential criticisms of rival views such as psychologism and formalism. His theory of meaning, especially his distinction between the sense and reference of linguistic expressions, was groundbreaking in semantics and the philosophy of language. He had a profound and direct influence on such thinkers as Russell, Carnap and Wittgenstein. Frege is often called the founder of modern logic, and he is sometimes even heralded as the founder of analytic philosophy.

The Russell-Myhill Antinomy, also known as the Principles of Mathematics Appendix B Paradox, is a contradiction that arises in the logical treatment of classes and "propositions", where "propositions" are understood as mind-independent and language-independent logical objects. If propositions are treated as objectively existing objects, then they can be members of classes. But propositions can also be about classes, including classes of propositions. Indeed, for each class of propositions, there is a proposition stating that all propositions in that class are true. Propositions of this form are said to "assert the logical product" of their associated classes. Some such propositions are themselves in the class whose logical product they assert. For example, the proposition asserting that all-propositions-in-the- class-of-all-propositions-are-true is itself a proposition, and therefore it itself is in the class whose logical product it asserts. However, the proposition stating that all-propositions-in-the-null-class-are-true is not itself in the null class. Now consider the class w, consisting of all propositions that state the logical product of some class m in which they are not included. This w is itself a class of propositions, and so there is a proposition r, stating its logical product. The contradiction arises from asking the question of whether r is in the class w. It seems that r is in w just in case it is not. This antinomy was discovered by Bertrand Russell in 1902, a year after discovering a simpler paradox usually called Russell's paradox ". It was discussed informally in Appendix B of his 1903 Principles of Mathematics . In 1958, the antinomy was independently rediscovered by John Myhill, who found it to plague the "Logic of Sense and Denotation" developed by Alonzo Church.

Several interesting themes emerge from G. E. Moore’s previously unpub­lished review of _The Principles of Mathematics_. These include a worry concerning whether mathematical notions are identical to purely logical ones, even if coextensive logical ones exist. Another involves a conception of infinity based on endless series neglected in the Principles but arguably involved in Zeno’s paradox of Achilles and the Tortoise. Moore also questions the scope of Russell’s notion of material implication, and other aspects of Russell’s claim that mathematics reduces to logic.

This is not a research project so much as a kind of “personal manifesto” on meta-ethics, or my personal take on how to best think about and improve morality. Since my take on “morality” is not necessarily meant to be compatible with current or past understandings, I am amenable to calling it “schmorality” instead. I argue that (sch)morality can be taken to be teleological by definition, but that the objects of comparison for what produces the best results value-wise need not be taken as actions, but rather the holistic approaches to living and decision making themselves. The position therefore does not yield “consequentialism” in the usual sense. I further explain how this position is compatible with the emergence of something like deontological constraints from within a teleological approach to (sch)morality, and speculate about further likely consequences of this approach.

We can at last release our breath: the long awaited Russell volume in the popular Cambridge Companion series has finally arrived. It contains fifteen chapters written by well known Russell scholars dealing with a wide array of Russelliana, along with a quite extensive introductory essay by the volume editor. It is not difficult to see what took so long. Russell’s corpus, even considering only his philosophical writings, outstrips in both breadth and volume almost all the other figures covered in the Cambridge Companion series. A further complication in Russell’s case is his characteristic habit of so frequently changing his mind even about fundamental issues. Dealing with such a vast amount of information must have required a tremendous amount of sustained collaboration. Obviously, the volume could not cover everything; but the editor and authors have done a tremendous job selectively choosing topics and themes within Russell’s philosophical work to focus on. While falling short of perfection, the result is a collection of pieces that together provide the sort of sophisticated introduction to a complex philosopher that is able to make his work accessible to relative beginners without disguising the subtlety, complexity and still controversial nature of his views.

Gottlob Frege's theories of meaning, and, in particular, his distinction between sense and denotation were developed as part and parcel of his views in logic and the philosophy of arithmetic. Nevertheless, the logical calculus developed in his Grundgesetze der Arithmetik does not fully reflect his semantic views. It provides no method for transcribing the so-called "oblique" contexts of ordinary language, and does not reflect his metaphysical commitment to the "third realm" of sense. The dissertation highlights ways in which Frege's views cannot be fully evaluated until this gap in his logical language is filled. It then fills this gap by presenting a expansion of Frege's logical system in line with his mature views in the philosophy of language. Along the way, a number of exegetical issues with regard to Frege's understanding of both logic and semantics are discussed, and a new interpretation of the nature of senses emerges. Previous attempts at developing the logic of sense and denotation, such as those of Church and others, are discussed but are concluded to reflect inadequately the views of the historical Frege. ;However, once an accurate account of the logic of sense and denotation is in place, new and hitherto unnoticed problems with Frege's philosophical position are revealed. For example, it is shown that contradictions stemming from certain new semantical and Cantorian antinomies are demonstrable in the expanded system, and that some of them are independent of the inconsistent class theory already present in the extant system. Through a comparison with philosophers whose semantic views are in some ways similar, the source of the difficulties in the logic of sense and denotation is traced to a set of mutually untenable metaphysical commitments in Frege's philosophy of language. A number of possible revisions to Frege's semantic theories are considered, and it is concluded that while it may be possible to salvage a broadly Fregean theory of meaning and incorporate it within a logical calculus, the theory must abandon at least some of the core elements within Frege's robust ontology of abstract objects