Online research in philosophy

This article deals with the relation between a theory of law and a theory of legal reasoning. Starting from a close reading of Chapter VII of H. L. A. Hart's The Concept of Law, it claims that a theory of law like Hart's requires a particular theory of legal reasoning, or at least a theory of legal reasoning with some particular characteristics. It then goes on to say that any theory of legal reasoning that satisfies those requirements is highly implausible, and tries to show that this is the reason why not only Hart, but also writers like Neil MacCormick and Joseph Raz have failed to offer a theory of legal reasoning that is compatible with legal positivism as a theory of law. They have faced a choice between an explanation of legal reasoning that is incompatible with the core of legal positivism or else strangely sceptical, insofar as it severs the link between general rules and particular decisions that purport to apply them.

Curry's paradox, so named for its discoverer, namely Haskell B. Curry, is a paradox within the family of so-called paradoxes of self-reference (or paradoxes of circularity). Like the liar paradox (e.g., ‘this sentence is false’) and Russell's paradox , Curry's paradox challenges familiar naive theories, including naive truth theory (unrestricted T-schema) and naive set theory (unrestricted axiom of abstraction), respectively. If one accepts naive truth theory (or naive set theory), then Curry's paradox becomes a direct challenge to one's theory of logical implication or entailment. Unlike the liar and Russell paradoxes Curry's paradox is negation-free; it may be generated irrespective of one's theory of negation. An intuitive version of the paradox runs as follows.

This collection of original papers from distinguished legal theorists offers a challenging assessment of the nature and viability of legal positivism, a branch of legal theory which continues to dominate contemporary legal theoretical debates. To what extent is the law adequately described as autonomous? Should law claim autonomy? These and other questions are addressed by the authors in this carefully edited collection, and it will be of interest to all lawyers and scholars interested in legal philosophy and legal theory.

In Appendix B of Russell's The Principles of Mathematics occurs a paradox, the paradox of propositions, which a simple theory of types is unable to resolve. This fact is frequently taken to be one of the principal reasons for calling ramification onto the Russellian stage. The paper presents a detaiFled exposition of the paradox and its discussion in the correspondence between Frege and Russell. It is argued that Russell finally adopted a very simple solution to the paradox. This solution had nothing to do with ramified types but marked an important shift in his theory of propositions.

In Appendix B of Russell's The Principles of Mathematics occurs a paradox, the paradox of propositions, which a simple theory of types is unable to resolve. This fact is frequently taken to be one of the principal reasons for calling ramification onto the Russellian stage. The paper presents a detaiFled exposition of the paradox and its discussion in the correspondence between Frege and Russell. It is argued that Russell finally adopted a very simple solution to the paradox. This solution had nothing to do with ramified types but marked an important shift in his theory of propositions.

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.

Russell’s way out of his paradox via the impredicative theory of types has roughly the same logical power as Zermelo set theory - which supplanted it as a far more flexible and workable axiomatic foundation for mathematics. We discuss some new formalisms that are conceptually close to Russell, yet simpler, and have the same logical power as higher set theory - as represented by the far more powerful Zermelo-Frankel set theory and beyond. END.

A coherent theory of relations was a critical part of Russell’s metaphysics. In Appearance and Reality Bradley posed a problem that sits squarely in the way of any doctrine of “external” relations. Russell, determined to advance such a doctrine, tried several times to find a way around the paradox and apparently believed he had succeeded by making use of one of his inventions, the theory of logical types.Gilbert Ryle and Alan Donagan have advanced an argument that I read, over the objections of its authors, as a special case of Bradley’s. In this paper I argue that the ad hoc solution suggested by Donagan to the special problem is one that Russell had already indicated a willingness to accept but that the general problem of the paradox remains.What finally prevents Russell from solving the paradox is a combination of his refusal to abandon the claim that relations are constituents of facts and the necessity of distinguishing a relational fact from its converse. Following some hints that Russell left, I do some reconstruction, showing how the theory of types would (and should) have been applied had Russell followed through on his own insights. The result, I suggest, is a truly Russellian theory that escapes Bradley’s paradox.

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.

Influenced by G. E. Moore, Russell broke with Idealism towards the end of 1898; but in later years he characterized his meeting Peano in August 1900 as ?the most important event? in ?the most important year in my intellectual life?. While Russell discovered his paradox during his post-Peano period, the question arises whether he was already committed, during his pre-Peano Moorean period, to assumptions from which his paradox may be derived. Peter Hylton has argued that the pre-Peano Russell was thus vulnerable to (at least one version of) Russell's paradox and hence that the paradox exposes a pre-existing difficulty in Russell's Moorean philosophy. Contrary to Hylton, I argue that the Moorean Russell adhered to views which insulated him against the paradox. Further, I argue that Russell became vulnerable to his paradox as a result of changes in his Moorean position occasioned, first, by his acceptance of Cantor's theory of the transfinite, and, second, by his correspondence with Frege. I conclude with some general comments regarding Russell's acceptance of naïve set theory.