By formalizing Berry's paradox, Vopěnka, Chaitin, Boolos and others proved the incompleteness theorems without using the diagonal argument. In this paper, we shall examine these proofs closely and show their relationships. Firstly, we shall show that we can use the diagonal argument for proofs of the incompleteness theorems based on Berry's paradox. Then, we shall show that an extension of Boolos' proof can be considered as a special case of Chaitin's proof by defining a suitable Kolmogorov complexity. We shall show (...) also that Vopěnka's proof can be reformulated in arithmetic by using the arithmetized completeness theorem. (shrink)

We shall prove the second incompleteness theorem via Kolmogorov complexity.

Within a weak subsystem of second-order arithmetic , that is -conservative over , we reformulate Kreisel's proof of the Second Incompleteness Theorem and Boolos' proof of the First Incompleteness Theorem.

Tanabe Hajime, another pole of the so-called Kyoto-School of Philosophy of modern Japan, attempts to construct a dialectical, triadic logic of genus, species and individual as a creative synthesis between Eastern and Western philosophy. Although the formal pattern of his method is influenced by the Hegelian dialectic, the way of his thinking is rather prevailed by Kantian dualism. This makes a sharp contrast to his mentor Nishida Kitaro, whose logic of Topos or Place qua Absolute Nothingness is criticized as all-embracing (...) and static in character by him. The difference between them might be parallel to that of Greek and Latin theology concerning the Trinity. Tanabe never presupposes any preexistent entity as the primordial One in the eternal dimension, but rather maintains the individuality as the free subjective agent in the field of history. The dichotomy between the universal and the individual is overcome in and through the mediation of the third term— the species — as the negatively self-realized, specific form of the genus. The species, however, turns out to be the self-estrangement, when it loses the perpetually negative mediation of the free subjective activity of the individual. (shrink)

Some formal properties of enriched systems of Lambek calculus with analogues of conjunction and disjunction are investigated. In particular, it is proved that the class of languages recognizable by the Lambek calculus with added intersective conjunction properly includes the class of finite intersections of context-free languages.

In this paper, I show that the availability of what some authors have called the weak reading and the strong reading of donkey sentences with relative clauses is systematically related to monotonicity properties of the determiner. The correlation is different from what has been observed in the literature in that it concerns not only right monotonicity, but also left monotonicity (persistence/antipersistence). I claim that the reading selected by a donkey sentence with a double monotone determiner is in fact the one (...) that validates inference based on the left monotonicity of the determiner. This accounts for the lack of strong reading in donkey sentences with MON determiners, which have been neglected in the literature. I consider the relevance of other natural forms of inference as well, but also suggest how monotonicity inference might play a central role in the actual process of interpretation. The formal theory is couched in dynamic predicate logic with generalized quantifiers. (shrink)

We define a liar-type paradox as a consistent proposition in propositional modal logic which is obtained by attaching boxes to several subformulas of an inconsistent proposition in classical propositional logic, and show several famous paradoxes are liar-type. Then we show that we can generate a liar-type paradox from any inconsistent proposition in classical propositional logic and that undecidable sentences in arithmetic can be obtained from the existence of a liar-type paradox. We extend these results to predicate logic and discuss Yablo’s (...) Paradox in this framework. Furthermore, we define explicit and implicit self-reference in paradoxes in the incompleteness phenomena. (shrink)

On sait que Platon fait grand cas des êtres intelligibles en instaurant la théorie des Idées. Mais il n’est pas approprié de le considérer comme penseur qui néglige le rôle de l’apparence sensible. Ce philosophe demeure très sensible à la modalité par laquelle les phénomènes apparaissent dans le champ de notreperception. En distinguant deux types d’apparence : image et simulacre, il donne à l’image le rôle d’intermédiaire actif entre le sensible et l’intelligible. L’examen des modalités des actions humaines qui reçoivent (...) et créent des images, nous aide à voir comment Platon intègre la fonction des images dans la structure de sa philosophie. La réception et la création des images sont étroitement reliées et synthétisées pour former le système original de Platon, système dans lequel le fondement de la théorie des Idées relève d’un certain dynamisme de l’action humaine. Dans la recherche relative à la modalité de réception et de création des images, je porte mon attention sur l’usage platonicien du terme tupos qui exprime le principe de la conformité de l’image et de son modèle.Notamment dans la théorie de la formation des futurs gardiens de la cité idéale, le tupos fonctionne, comme un principe qui réglemente et oriente à plusieurs niveaux les activités de la mimèsis et de la poièsis qui portent sur les images. La réflexion sur cette fonction conductrice. (shrink)

We give a proof of Gödel's first incompleteness theorem based on Berry's paradox, and from it we also derive the second incompleteness theorem model-theoretically.

The issue of justice after catastrophe is an enormous challenge to contemporary theories of distributive justice. In the past three decades, the controversy over distributive justice has centered on the ideal of equality. One of intensely debated issues concerns what is often called the “equality of what,” on which there are three primary views: welfarism, resourcism, and the capabilities approach. Another major point of dispute can be termed the “equality or another,” about which three positions debate: egalitarianism, prioritarianism, and sufficientarianism. (...) On these topics of distributive justice, authors are concerned with the current difference between the better-off and the worse-off or the present situation of the badly-off. By contrast, it is essential to take account of the past distribution of well-being as well as the present situation in order to explore questions of post-catastrophe justice. Without looking at the pre-disaster distribution of income, preference satisfaction, or basic capabilities among affected people, no present assessment of the damage caused by the disaster could be correct and no proposed remedy adequate. It is true that luck egalitarians assess the current distribution among people by referring to the decision that each individual made. Yet they pay scant attention to the situation in which each one stayed in the past. Therefore, we can legitimately say that most theorists of distributive justice, including luck egalitarians, have failed to give consideration to the past state of each person. -/- To fill this gap in the literature, the present article explores philosophical questions that arise when we take account of each person’s past and present situations in discussing distributive justice regarding public compensation and assistance to survivors and families of victims of natural and industrial disasters. In addressing these novel questions, I develop and refine various concepts, ideas, and arguments that have been presented in the study of distributive justice in normal settings. I tackle two tasks, the first of which is to explore the foundation and scope of luck egalitarianism. Despite the moral appeal it has in many cases, luck egalitarianism has attracted the so-called harshness objection. Some luck egalitarians attempt to avoid this objection in a pragmatic way by combining the luck egalitarian doctrine with the principle of basic needs satisfaction. However, they do not provide any systematic rationale for this combination. In contrast with such pragmatic responses, I seek to offer a principled argument for holding individuals responsible for their choices only when their basic needs are met, by invoking the ideas of respect for human voluntariness and rescue of human vulnerability. Based on this argument, I propose a form of responsibility-sensitive theory, which considers the pre-disaster distribution of well-being as a default position. The second task I take on is to refine sufficientarianism in the context of post-catastrophe justice. Luck egalitarianism with boundaries set by the basic needs principle seems to indicate the potential for sufficientarianism. But major proponents of this view conceive the welfarist assumption, a considerably high standard of well-being, and the controversial treatment of persons staying below the threshold, all of which seem problematic in the post-disaster situation. I try to construct a new version of sufficientarianism by replacing these current features with more robust ones. (shrink)

Akama et al. [1] systematically studied an arithmetical hierarchy of the law of excluded middle and related principles in the context of first-order arithmetic. In that paper, they first provide a prenex normal form theorem as a justification of their semi-classical principles restricted to prenex formulas. However, there are some errors in their proof. In this paper, we provide a simple counterexample of their prenex normal form theorem [1, Theorem 2.7], then modify it in an appropriate way which still serves (...) to largely justify the arithmetical hierarchy. In addition, we characterize a variety of prenex normal form theorems by logical principles in the arithmetical hierarchy. The characterization results reveal that our prenex normal form theorems are optimal. For the characterization results, we establish a new conservation theorem on semi-classical arithmetic. The theorem generalizes a well-known fact that classical arithmetic is $\Pi _2$ -conservative over intuitionistic arithmetic. (shrink)

In this collection on the Kyoto School of Philosophy, the author offers the reader Tanabe's religious philosophy, but also, and for the first time, his ...

It is well known that Gödel’s incompleteness theorems hold for ∑1-definable theories containing Peano arithmetic. We generalize Gödel’s incompleteness theorems for arithmetically definable theories. First, we prove that every ∑n+1-definable ∑n-sound theory is incomplete. Secondly, we generalize and improve Jeroslow and Hájek’s results. That is, we prove that every consistent theory having ∏n+1set of theorems has a true but unprovable ∏nsentence. Lastly, we prove that no ∑n+1-definable ∑n-sound theory can prove its own ∑n-soundness. These three results are generalizations of Rosser’s (...) improvement of the first incompleteness theorem, Gödel’s first incompleteness theorem, and the second incompleteness theorem, respectively. (shrink)

Every language recognized by the Lambek calculus with brackets is context-free. This is shown by combining an observation by Jäger with an entirely straightforward adaptation of the method Pentus used for the original Lambek calculus. The case of the variant of the calculus allowing sequents with empty antecedents is slightly more complicated, requiring a restricted use of the multiplicative unit.

It is proved that for any k, the class of classical categorial grammars that assign at most k types to each symbol in the alphabet is learnable, in the Gold (1967) sense of identification in the limit from positive data. The proof crucially relies on the fact that the concept known as finite elasticity in the inductive inference literature is preserved under the inverse image of a finite-valued relation. The learning algorithm presented here incorporates Buszkowski and Penn's (1990) algorithm for (...) determining categorial grammars from input consisting of functor-argument structures. (shrink)

Second-order abstract categorial grammars (de Groote in Association for computational linguistics, 39th annual meeting and 10th conference of the European chapter, proceedings of the conference, pp. 148–155, 2001) and hyperedge replacement grammars (Bauderon and Courcelle in Math Syst Theory 20:83–127, 1987; Habel and Kreowski in STACS 87: 4th Annual symposium on theoretical aspects of computer science. Lecture notes in computer science, vol 247, Springer, Berlin, pp 207–219, 1987) are two natural ways of generalizing “context-free” grammar formalisms for string and tree (...) languages. It is known that the string generating power of both formalisms is equivalent to (non-erasing) multiple context-free grammars (Seki et al. in Theor Comput Sci 88:191–229, 1991) or linear context-free rewriting systems (Weir in Characterizing mildly context-sensitive grammar formalisms, University of Pennsylvania, 1988). In this paper, we give a simple, direct proof of the fact that second-order ACGs are simulated by hyperedge replacement grammars, which implies that the string and tree generating power of the former is included in that of the latter. The normal form for tree-generating hyperedge replacement grammars given by Engelfriet and Maneth (Graph transformation. Lecture notes in computer science, vol 1764. Springer, Berlin, pp 15–29, 2000) can then be used to show that the tree generating power of second-order ACGs is exactly the same as that of hyperedge replacement grammars. (shrink)

Poppe r e nt i e nde l a r a c i ona l i da d e ntérminos de nuestra actitud intelectual.Nuestra racionalidad no es una facultad ni un don intelectual.No es al go dado a un i ndi vi duo, deacuerdo con él. Es una actitud que hemosadquirido de nuestra relación intelectualcon otros. Popper no usa “racionalismo”como un término filosófico que significaintelectualismo en oposición a empirismo.El artículo muestra claramente que Popperent i ende el r aci onal i (...) smo no como l agr a nde z a de l a r a z ón, s i no c omo l acomprensión de uno del límite de la razón.Sin embargo, surge una pregunta; ¿qué tanbien ha comprendido tal racionalista ellímite de su propio racionalismo? ¿Cómodebemos entender este límite? Popperafi rma que un raci onal i sta ha notadoel límite del racionalismo al prever elcarácter auto-destructivo del racionalismoc ompr e he ns i vo. A f i n de ha c e r má scomprensible su afirmación tenemos queexplicar el racionalismo comprehensiveque Popper aprehendió.Popper understands the rationality interms of our intellectual attitude. Ourrationality is neithera faculty nor an intellectual gift. It isnot something given to an individual,according to him. It is an attitude thatwe have acquired from our intellectualintercourse with others. Popper does notuse “rationalism” as a philosophical termthat means intellectualism in oppositionto empiricism.This paper clearly shows that Popperunde r s t a nds r a t i o na l i s m no t a s t heal mi ght i nes s of r eas on, but as one’ srealization of the limit of reason. However,there arises a question; how has such arationalist realized the limit of his ownrationalism? How we should understandthis limit? Popper asserts that a rationalisthas noticed the limit of rationalism byenvisaging the self-defeating character ofthe comprehensive rationalism. In order tomake more understandable his assertion,we must at first explain the comprehensiverationalism apprehended by Popper. (shrink)

In the past decade, a growing number of authors, notably Thomas Pogge, have maintained that citizens in economically advanced societies are responsible for extreme and extensive poverty in the developing world. Iris Marion Young proposed the social connection model of responsibility, which asserts that these citizens participate in networks that give rise to global structural injustices. While Pogge’s argument for the existence of citizens’ responsibility has been the subject of widespread debate, few efforts have been made to scrutinise the solidity (...) of Young’s perspective. To plug this gap in the literature, this paper assesses the pertinence of Young’s view. A more traditional view than those of Pogge and Young considers poverty as indicating a lack of respect for the human rights of those living in less-developed countries. Rights theorists of global justice, however, have paid scant attention to philosophical observations concerning redistribution within the borders of a society. To remedy this shortcoming, this paper endeavours to develop the theory that citizens in affluent societies bear a duty correlative to the subsistence right of the global needy, by exploring sufficientarianism, which is one of the primary views on domestic redistribution. To begin with, I make a distinction between the responsibility-based theory and the right-based theory of global justice. This is followed by a close examination of Young’s social connection model as a significant version of the former position. I then offer a right-based argument that invokes the sufficientarian idea of the human right to live above the threshold of safe and healthy subsistence. (shrink)