There seems to be a consensus among philosophers who are interested in Russell's early philosophy of mathematics. They hold that we can best understand the process of its development on the assumption that Russell kept trying to reconcile a type-theoretical solution to the paradoxes with the doctrine of univocality of being. This assumption have worked well in so far as to reconstruct the history of Russell's endearvor from Principles of Mathematics (1903) through the invention of substitutional theory (1905-7). Taking Principia Mathematica (1910) into consideration, however, this assumption seems to fail. There are many questions left unanswerable concerning the relation between PM and his former position. In this paper, I will survey some of the recent findings in the Russell Archives and Gregory Landini's works based on these findings, and clarify the relevance they could have to the "unanswered questions" mentioned above.

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In this paper we intend to place the early Russell in the context of the refutation of idealism in the school of Meinong. We look into Mally's arguments against idealism, which have recourse to Russell's paradox, and Meinong's critique of them. Then we propose the hypothesis that the early Russell made up his thought in the Meinongian framework. From this point of view, we could point out that the origin of the paradox might be in the early Russell's criticism of Bradley's idealism. And Wittgenstein's resolution which could make Russell's theory of types dispensable might be compared to Mally's method of arguments which Meinong never adopts.

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It is a well-known story that Russell's discovery of his paradox shook the foundations of Frege's logical system for arithmetic. But there is another route to this paradox. Hilbert pointed out to Frege that he had already found other even more convincing contradictions which he communicated to Zermelo, thereby initiating Zermelo's independent discovery of Russell's paradox. In this paper, we follow this less familiar route and analyze three paradoxes, namely Hilbert's paradox, Zermelo's version of Russell's paradox and Schröder's paradox of 0 and 1. Furthermore, tradition in which these paradoxes were found is reconsidered. We examine Schröder's place in the foundational study and criticize an alleged dichotomy between the algebraic and logistic traditions.

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It is observed by Grishin that inconsistency of naive set theory can be avoided by restricting the logical law of contraction, as it is contraction that enables us to derive logical inconsistency from set-theoretic paradoxes such as Russell's paradox.
In this paper, we examine Grishin's contraction-free naive set theory to better understand Russell's paradox and the naive comprehension principle from a purely formal standpoint. We study both static-propositional and dynamic-procedural aspects of naive comprehension and argue that it could lead to an ideal formalization of (part of) mathematics, where both propositional knowledge (theorems) and procedural knowledge (algorithms) reside in harmony.

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The Cantorian set theory survived the Russell Paradox by means of axiomatizing the theory into the standard theory named ZF. ZF has a set theoretical counterpart named FA (Foundation Axiom) to the vicious-circle principle of Russell's ramified type theory. Despite the principle, circular objects and phenomena are ubiquitous in many applications fields of ZF. For modeling such circular things directly as circular sets, Aczel replaced FA with his Anti-Foundation Axiom (AFA) to allow non-well-founded sets in a strong way. The foundation of the new set theory is explained in details.

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In this paper we show some logical presumptions for the contradiction-form to really mean contradiction. We first give an introductory note that the same argument-form of Russell paradox could be interpreted to derive a contradiction (as Russell did) and to derive some positive non-contradictory results (such as Gödel's lemma on incompleteness and Cantor's lemma on cardinality), depending on the context. This surprisingly suggests that a logical argument of a contradiction itself is rather independent of interpreting it as contradiction or non-contradiction. In the main section (Section 2) we investigate further in the hidden logical assumptions underlying a usual derivation of contradiction (such as the last step from the Russell argument to conclude a contradiction). We show the logical form of contradiction does not always mean a contradiction in a deep structure level of logic. We use the linear logical analysis for this claim. Linear logic, in the author's opinion, provides fundamental logical structures of the traditional logics (such as classical logic and intuitionistic logic). The each traditional logical connectives split into two different kinds of connective, corresponding to the fundamental distinction, parallel or choice, of the fundamental level of logic; more precisely the connectives related to parallel-assertings and the connectives related to choice-assertions. We claim that (1) the law of contradiction is indisputable for the parallel-connectives, but (2) the law of contradiction is not justified for the choice-connectives. (In fact, the law of contradiction has the same meaning as other Aristotlean laws (the indentity, the excluded middle) from the view point of the duality principle in linear logic, and the disputability of the law of contradiction is exactly the same as the disputability of the law of excluded middle, in the linear logic level.) Here, although (1) admits the law of contradiction, the meaning of contradiction is quite diferrent and, in the author opinion, more basic than the traditional sense of contradiction. (2) tells us that the disputability of the law of contradiction for the choice-connectives is equivalent to the disputability of the law of excluded middle. However, this disputability is more basic than the traditional logicist-intuitionist issue on the excluded middle, since admitting the traditional law of excluded middle (from the classical or logicist viewpoint) is compatible with this disputability of the excluded middle (and equivalently the law of contradiction) with respect to the choice-connectives of the linear logic. Then, the traditional logics (classical and intuitionistic logics) are perfectly constituted from this fundamental level of logic by the use of reconstructibity or re-presentation operator, (which is the linear-logical modal operators). With the use of modal operator the originally splitted two groups of logical connectives merge into a single group, which makes the traditional logical connectives. (The use of slightly different modalities results in the difference between the traditional classical logic and the traditional intuitionistic logic.) With the use of modal operator, the contradiction-form becomes to get the traditional sense of contradiction. This situation shows that the traditional sense of contradiction presumes re-presentation or reconstruction of the inference-resources, which is now explicit by the use of linear logical modal operator(s), and which also makes possible the denotational or objectivity interpretation of logical language. The merge of the two different aspects (the parallel-connectives and the choice-connectives) into one, by the presence of the modal operators, also eliminates the original conflict (on the indisputability of the law of contradiction in the parallel-connectives side and the disputability of that in the choice-connectives side.)

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Mathematical structures are identified with classes (in naive set theories, which were based on Comprehension Principle), or with sets (in axiomatic set theories, which adopted the principle in its restricted form). By analyzing abstraction operators and the set-theoretical diagonal arguments, the author indicates both classes and sets could be best regarded as certain self-applicable functions, treated as objects on their own. On the other hand, contemporary higher-order type theories, guided by the Curry-Howard Isomorphism, identify mathematical structures with propositions. According to this conception, formal derivation of a judgment counts both as the proof of a proposition and as the construction of a structure. The author examines its significance to the study of how language contributes to the construction of mathematical structures.

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This paper is an attempt for clarifying what made Hume adapt such an obscure notion as sympathy in establishing the system of passions. In order to answer this question, it is necessary to see that his basic strategy in the Treatise is to hold the analogy between the two systems of the understanding and passions, and that sympathy is intended as the phenomenon in which this analogy is explicitly demonstrated.

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This work proposes a new semantics named "context semantics", which interprets predicate modal logic in which the modality symbols means logical validity. Although the possible world semantics is the most well-known method for semantics of modal logic, it is not so useful or so essential in studying the predicate modal logic. Especially, the transworld identification always makes serious problem. In order to avoid the problem, we propose the new semantics. Our semantics interprets a formula with finite information. This point is the most essential difference between our semantics and possible world semantics.

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Scientific epistemology, which showed innovative developments in the last century, has been expected to involve various implications. Among other things, penetrating criticism of absolute truth to scientific theories is an important one. The concepts developed in relation with this knowledge skepticism include falsifiability, holism and theory ladenness. It looks, however, still sparse to develop their implications to research designs employed in usual research programs. This article attempts to provide five important requirements for obtaining evidences of high quality, and to discuss the relationship of these requirements with scientific epistemology. Special emphases are placed on how to cooperate for obtaining many individual scientific findings and also on how to take account of relativism in scientific researches.

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D. Lewis said that there are three solutions for the problem of temporary intrinsics, namely, relationalism, presentism, and four-dimensionalism. Lewis also argued that the only tenable one is four-dimensionalism. But advocates of the other solutions reject it. Although objection is possible, it seems that Lewis' argument is not enough to show that four-dimensionalism is the best. However, it doesn't mean that four-dimensionalism is not worth considering. For relationalism and presentism are inconsistent. Moreover, relationalism must deny.ordinary talk and presentism doesn't contradict with four-dimensionalism. So it is concluded that four-dimensionalist doesn't have to throw away his position.

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