Tarski's theorem essentially says that the Liar paradox is paradoxical in the minimal reflexive frame. We generalise this result to the Liar-like paradox $\lambda^\alpha$ for all ordinal $\alpha\geq 1$. The main result is that for any positive integer $n = 2^i(2j+1)$, the paradox $\lambda^n$ is paradoxical in a frame iff this frame contains at least a cycle the depth of which is not divisible by $2^{i+1}$; and for any ordinal $\alpha \geq \omega$, the paradox $\lambda^\alpha$ is paradoxical in a frame (...) iff this frame contains at least an infinite walk which has an arbitrarily large depth. We thus get that $\lambda^n$ has a degree of paradoxicality no more than $\lambda^m$ iff the multiplicity of 2 in the (unique) prime factorisation of $n$ is no more than that in the prime factorisation of $m$; and all tranfinite $\lambda^\alpha$ has the same degree of paradoxcality but has a higher degree of paradoxicality than any $\lambda^n$. (shrink)

A relativized version of Tarski's T-scheme is introduced as a new principle of the truth predicate. Under the relativized T-scheme, the paradoxical objects, such as the Liar sentence and Jourdain's card sequence, are found to have certain relative contradictoriness. That is, they are contradictory only in some frames in the sense that any valuation admissible for them in these frames will lead to a contradiction. It is proved that for any positive integer n, the n-jump liar sentence is contradictory in (...) and only in those frames containing at least an n-jump odd cycle. In particular, the Liar sentence is contradictory in and only in those frames containing at least an odd cycle. The Liar sentence is also proved to be less contradictory than Jourdain's card sequence: the latter must be contradictory in those frames where the former is so, but not vice versa. Generally, the relative contradictoriness is the common characteristic of the paradoxical objects, but different paradoxical objects may have different relative contradictoriness. (shrink)

Chinese culture is a lively and active organism. If we are to get to the real image of Chinese culture, we should take as a starting point the first passage of the Chung-yung [Doctrine of the Mean], which expounds its basic principles: "That which Heaven has decreed is called Nature, to follow that Nature is called tao [way] and to cultivate the tao is called instruction." This passage uses three phrases and one breath; these three phrases have a unified nature (...) which cannot be separated and which constitute one long sentence. Because of this, the words as a whole lead people to think of a richly vital and great tree which has the will of Heaven as its roots, human nature as its shoots, the tao as its trunk, and cultural instruction as its branches, leaves, and blossoms. As for the fruit, in terms of the individual, it is the achievement of a completely beautiful character, and in terms of the masses of the people, it is the ordering of the country and the pacification of the realm, the progression from the era of peace under the ancient kings to the age of the Grand Unity [ta-t'ung]. (shrink)

According to the revision theory of truth, the paradoxical sentences have certain revision periods in their valuations with respect to the stages of revision sequences. We find that the revision periods play a key role in characterizing the degrees of paradoxicality for Boolean paradoxes. We prove that a Boolean paradox is paradoxical in a digraph, iff this digraph contains a closed walk whose height is not any revision period of this paradox. And for any finitely many numbers greater than 1, (...) if any of them is not divisible by any other, we can construct a Boolean paradox whose primary revision periods are just these numbers. Consequently, the degrees of Boolean paradoxes form an unbounded dense lattice. The area of Boolean paradoxes is proved to be rich in mathematical structures and properties. (shrink)

This paper gives a definition of self-reference on the basis of the dependence relation given by Leitgeb (2005), and the dependence digraph by Beringer & Schindler (2015). Unlike the usual discussion about self-reference of paradoxes centering around Yablo's paradox and its variants, I focus on the paradoxes of finitary characteristic, which are given again by use of Leitgeb's dependence relation. They are called 'locally finite paradoxes', satisfying that any sentence in these paradoxes can depend on finitely many sentences. I prove (...) that all locally finite paradoxes are self-referential in the sense that there is a directed cycle in their dependence digraphs. This paper also studies the 'circularity dependence' of paradoxes, which was introduced by Hsiung (2014). I prove that the locally finite paradoxes have circularity dependence in the sense that they are paradoxical only in the digraph containing a proper cycle. The proofs of the two results are based directly on König's infinity lemma. In contrast, this paper also shows that Yablo's paradox and its nested variant are non-self-referential, and neither McGee's paradox nor the omega-cycle liar paradox has circularity dependence. (shrink)

The unwinding that Cook, 767–774 2004) proposed is a simple but powerful method of generating new paradoxes from known ones. This paper extends Cook’s unwinding to a larger class of paradoxes and studies further the basic properties of the unwinding. The unwinding we study is a procedure, by which when inputting a Boolean modal net together with a definable digraph, we get a set of sentences in which we have a ‘counterpart’ for each sentence of the Boolean modal net and (...) each point of the digraph. What is more, whenever a sentence of the Boolean modal net says another sentence is necessary, then the counterpart of the first sentence at a point correspondingly says the counterparts of the second one at all accessible points of that point are all true. The output of the procedure is called ‘the unwinding of a Boolean modal net on a definable digraph’. We prove that the unwinding procedure preserves paradoxicality: a Boolean modal net is paradoxical on a definable digraph, iff the unwinding of it on this digraph is also paradoxical. Besides, the dependence digraph for the unwinding of a Boolean modal net on a definable digraph is proved to be isomorphic to the unwinding of the dependence digraph for the Boolean modal net on the previous definable digraph. So the unwinding of a Boolean modal net on a digraph is self-referential, iff the Boolean modal net is self-referential and the digraph is cyclic. Thus, on the one hand, the unwinding of any Boolean modal net on an acyclic digraph is non-self-referential. In particular, the unwinding of any Boolean modal net on \ is non-self-referential. On the other hand, if a Boolean modal net is paradoxical on a locally finite digraph, the unwinding of it on that digraph must be self-referential. Hence, starting from a Boolean modal paradox, the unwinding can output a non-self-referential paradox only if the digraph is not locally finite. (shrink)

Lenin, the great teacher of the proletariat, was deeply concerned with both the development of socialist literature and the problem of the correct critical acceptance of the literary heritage by the proletariat. Lenin frequently spoke of the works of Pushkin, Lermontov, Gogol, Tolstoy, Turgenev, Nekrasov, Shchedrin, Chekov, and others. Among the western European classical authors he particularly respected Shakespeare, Goethe, Heine, Hugo, Dickens, and Zola. In the brilliant classical writings which Lenin has left us he frequently made use of artistic (...) images, first molded in classical literary works, with which the people were familiar in order to expound problems of a theoretical nature. Furthermore he used them as tools for carrying on ideological education for the people. For example, in his writings Materialism and Empiriocriticism, and "The Marxist Critics and the Land Question," he used the image of Voroshilov, the false man of learning in Turgenev's novel Smoke, to elucidate a problem; he used the satirical images of bourgeois liberals from Shchedrin to reveal the basic nature of the enslavement and oppression of the people by the reactionary party, and so on. In addition, Lenin has left us in his works many incisive views directly related to the problem of the critical acceptance of the literacy heritage. They constitute an indivisible part of Lenin's theoretical estate. To study new Lenin's theories on the problem of the critical acceptance of the literacy heritage can be of great help to us today for our own correct treatment of the classical literacy heritage. (shrink)

The “wisdom of the crowd” phenomenon refers to the finding that the aggregate of a set of proposed solutions from a group of individuals performs better than the majority of individual solutions. Most often, wisdom of the crowd effects have been investigated for problems that require single numerical estimates. We investigate whether the effect can also be observed for problems where the answer requires the coordination of multiple pieces of information. We focus on combinatorial problems such as the planar Euclidean (...) traveling salesperson problem, minimum spanning tree problem, and a spanning tree memory task. We develop aggregation methods that combine common solution fragments into a global solution and demonstrate that these aggregate solutions outperform the majority of individual solutions. These case studies suggest that the wisdom of the crowd phenomenon might be broadly applicable to problem-solving and decision-making situations that go beyond the estimation of single numbers. (shrink)

There is ample justification for characterizing imperial China as a "Confucian State" (or "Confucian Society") as many students of Chinese history do. Such characterization is justified by the fact that Confucianism had contributed much to shaping and sustaining the imperial system from the Han dynasty to the Ch'ing. But it should be pointed out that Legalism had also played a crucial part in the development of that system and that, insofar as the above-mentioned characterization ignores the Legalist role, it is (...) a misnomer. The purpose of the present article is to call attention to the Legalist contribution to autocratic government, to counterbalance the exclusive importance that has often been attached to Confucianism. For the sake of convenience, I shall use the term "Legalist" roughly in the same loose way as the term "Confucian" is used in "Confucian State." Only a few rather obvious, if not altogether superficial, points will be made, which, I hope, might lead to further discussions. (shrink)

First published in 2000. Routledge is an imprint of Taylor & Francis, an informa company.

Brentano hat in seinen Analysen der Wahrnehmung zwei wichtige Punkte hervorgehoben: (a) daß die innere Wahrnehmung nur ein Bewußtsein "nebenbei" sei; und (b) daß die äußere Wahrnehmung ein räumlich Ausgedehntes (und nicht eine Idee) zum Objekt habe. Er ging aber nicht weit genug, sondern blieb dem Kartesianismus verhaftet, indem er die innere Wahrnehmung immer noch ein Erkennen nannte, und andererseits vom Objekt der äußeren Wahrnehmung sagte, daß es in Wahrheit gar nicht bestehe. Wenn man aber weiter geht und zugesteht, daß (...) die sogenannte innere Wahrnehmung gar kein eigentliches Erkennen ist, und daß es im Gegenteil die äußere Wahrnehmung ist, welche die Wirklichkeit erkennt, dann wird es unmöghch, sich in der Erkenntnistheorie mit Descartes auf unmittelbare untrügliche Evidenz zu berufen. Descartes bleibt aber insofern im Recht, als es trotz allem sinnvoll bleibt, zur Rechtfertigung von Erkenntnissen auf die sogenannte innere Wahrnehmung zu verweisen. (shrink)

Studies of occupational sex segregation rely on the sociocultural model to explain why some occupations are numerically dominated by women and others by men. This model argues that occupational sex segregation is driven by norms about gender-appropriate work, which are frequently conceptualized as gender-typed skills: work-related tasks, abilities, and knowledge domains that society views as either feminine or masculine. The sociocultural model thus explains the primary patterns of occupational sex segregation, which conform to these norms: Requirements for feminine skills increase (...) with women’s representation in the occupation. However, the model does not adequately explain cases of segregation that deviate from these norms or investigate the ways in which feminine and masculine skills co-occur in occupations. The present study fills these gaps by evaluating two previously untested explanations for deviations from the sociocultural model. The findings show that requirements for physical strength increase with women’s representation in professional occupations because physical strength skills co-occur with substantially higher requirements for feminine skills that involve helping and caring for others. These results indicate that the sociocultural model, and more generally explanations for how gender norms drive occupational sex segregation, can be improved by examining patterns of gender-typed skill co-occurrence. (shrink)

Algorithmically driven online dating platforms today promise the ability to sort through relevant data and identify one’s ideal amorous matches effectively. Yet the appeal of technological and scientific solutions to the messy problem of finding partners is hardly new. This introduction to the Focus section “It’s a Match!” argues that the history of amorous matching has long been part and parcel of the history of science, in particular the social sciences. Taking matching as an “applied science of social harmony,” the (...) authors argue that concern over more reliable techniques for determining the suitability of partners has formed an essential part of both the maintenance of social order and the shaping of subjectivities, enabling discourses of informed choice and the rational management of the passions, while also reinforcing and subverting structures of age, gender, race, and sexuality. (shrink)