Kierstead showed that every computable marriage problem has a computable matching under the assumption of computable expanding Hall condition and computable local finiteness for boys and girls. The strength of the marriage theorem reaches or if computable expanding Hall condition or computable local finiteness for girls is weakened. In contrast, the provability of the marriage theorem is maintained in even if local finiteness for boys is completely removed. Using these conditions, we classify the strength of variants of marriage theorems in (...) the context of reverse mathematics. Furthermore, we introduce another condition that also makes the marriage theorem provable in, and investigate the sequential and Weihrauch strength of marriage theorems under that condition. (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)

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.

We refine the arithmetical hierarchy of various classical principles by finely investigating the derivability relations between these principles over Heyting arithmetic. We mainly investigate some restricted versions of the law of excluded middle, De Morgan's law, the double negation elimination, the collection principle and the constant domain axiom.

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.

We aim at developing a systematic method of separating omniscience principles by constructing Kripke models for intuitionistic predicate logic $\mathbf {IQC}$ and first-order arithmetic $\mathbf {HA}$ from a Kripke model for intuitionistic propositional logic $\mathbf {IPC}$. To this end, we introduce the notion of an extended frame, and show that each IPC-Kripke model generates an extended frame. By using the extended frame generated by an IPC-Kripke model, we give a separation theorem of a schema from a set of schemata in (...) $\mathbf {IQC}$ and a separation theorem of a sentence from a set of schemata in $\mathbf {HA}$. We see several examples which give us separations among omniscience principles. (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)

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 propose a new game to strategically justify the constrained equal awards rule in claims problems. Our game is “procedurally fair” and “multilateral”. In addition, even if claimants cannot reach an agreement in any period, they can renegotiate in the next period. We show that, for each claims problem, the awards vector chosen by the constrained equal awards rule achieved at period 1 is the unique subgame perfect equilibrium outcome of the game.

This article inquires into a paradoxical position held by the concept of ‘nature’ in Derrida's thought. While a pivotal part of his project of deconstruction is devoted to a critique of the metaphysical privileging of nature over its others (technics, culture, and so on), the same project also aims at dismantling the hierarchical binary opposition of man/animal. Insofar as the term ‘animal’ or ‘animality’ to a large extent overlaps with nature, these two strands of his thought appear to stand in (...) a significant tension with each other. I explore this conceptual tension by re-examining Derrida's early critique of the notion of nature – specifically in his reading of Rousseau – in conjunction with his later engagement with the question of animality. As my reading shows, deviating from his early approach, his later work indicates the way in which animal others (and, by extension, all living and non-living others) are indeed constituted by supplementarity and yet, in their irreplaceable singularity, exceed the very structure of supplementarity. This opens up a new way of conceiving nature, no longer as originary presence nor simply as an effect of supplementary play, but in terms of the aporetic relation between supplementarity and what may be termed ‘the unsupplementable’. (shrink)

Climate change apparently calls for urgent action, but both individuals and governments have been slow to react. This is partly because, in addition to natural science, economics and moral philosop...

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.

We provide a fine‐grained analysis on the relation between König's lemma, weak König's lemma, and the decidable fan theorem in the context of constructive reverse mathematics. In particular, we show that double negated variants of König's lemma and weak König's lemma are equivalent to double negated variants of the general decidable fan theorem and the binary decidable fan theorem, respectively, over a nearly intuitionistic system containing a weak countable choice only. This implies that the general decidable fan theorem is not (...) equivalent to the binary decidable fan theorem in the absence of a strong countable choice. In addition, we introduce a principle which fills the gap between König's lemma and weak König's lemma, and show that König's lemma is equivalent to weak König's lemma plus this principle. We also solve several problems arising from the investigation. (shrink)

A simple and complete proof of strong normalization for first- and second-order intuitionistic natural deduction including disjunction, first-order existence and permutative conversions is given. The paper follows the Tait–Girard approach via computability predicates and saturated sets. Strong normalization is first established for a set of conversions of a new kind, then deduced for the standard conversions. Difficulties arising for disjunction are resolved using a new logic where disjunction is restricted to atomic formulas.

Along the line of Hirst‐Mummert and Dorais, we analyze the relationship between the classical provability of uniform versions Uni(S) of Π2‐statements S with respect to higher order reverse mathematics and the intuitionistic provability of S. Our main theorem states that (in particular) for every Π2‐statement S of some syntactical form, if its uniform version derives the uniform variant of over a classical system of arithmetic in all finite types with weak extensionality, then S is not provable in strong semi‐intuitionistic systems (...) including bar induction in all finite types but also nonconstructive principles such as Kőnig's lemma and uniform weak Kőnig's lemma. Our result is applicable to many mathematical principles whose sequential versions imply. (shrink)

We systematically study the interrelations between all possible variations of \(\Delta ^0_1\) variants of the law of excluded middle and related principles in the context of intuitionistic arithmetic and analysis.

A minimal formula is a formula which is minimal in provable formulas with respect to the substitution relation. This paper shows the following: (1) A β-normal proof of a minimal formula of depth 2 is unique in NJ. (2) There exists a minimal formula of depth 3 whose βη-normal proof is not unique in NJ. (3) There exists a minimal formula of depth 3 whose βη-normal proof is not unique in NK.

We systematically study several principles and give a principle which is weaker than disjunctive Markov’s principle. We also show that the principle is underivable and strictly weaker than MP∨ in certain extensions of the system EL of elementary analysis.

We systematically study conservation theorems on theories of semi-classical arithmetic, which lie in-between classical arithmetic $\mathsf {PA}$ and intuitionistic arithmetic $\mathsf {HA}$. Using a generalized negative translation, we first provide a structured proof of the fact that $\mathsf {PA}$ is $\Pi _{k+2}$ -conservative over $\mathsf {HA} + {\Sigma _k}\text {-}\mathrm {LEM}$ where ${\Sigma _k}\text {-}\mathrm {LEM}$ is the axiom scheme of the law-of-excluded-middle restricted to formulas in $\Sigma _k$. In addition, we show that this conservation theorem is optimal in the (...) sense that for any semi-classical arithmetic T, if $\mathsf {PA}$ is $\Pi _{k+2}$ -conservative over T, then ${T}$ proves ${\Sigma _k}\text {-}\mathrm {LEM}$. In the same manner, we also characterize conservation theorems for other well-studied classes of formulas by fragments of classical axioms or rules. This reveals the entire structure of conservation theorems with respect to the arithmetical hierarchy of classical principles. (shrink)

A critical examination of Japanese literary and art theories.

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)

The uniform continuity theorem states that every pointwise continuous real‐valued function on the unit interval is uniformly continuous. In constructive mathematics, is strictly stronger than the decidable fan theorem, but Loeb [17] has shown that the two principles become equivalent by encoding continuous real‐valued functions as type‐one functions. However, the precise relation between such type‐one functions and continuous real‐valued functions (usually described as type‐two objects) has been unknown. In this paper, we introduce an appropriate notion of continuity for a modulus (...) of a continuous real‐valued function on [0, 1], and show that real‐valued functions with continuous moduli are exactly those functions induced by Loeb's codes. Our characterisation relies on two assumptions: (1) real numbers are represented by regular sequences (equivalently Cauchy sequences with explicitly given moduli); (2) the continuity of a modulus is defined with respect to the product metric on the regular sequences inherited from the Baire space. Our result implies that is equivalent to the statement that every pointwise continuous real‐valued function on [0, 1] with a continuous modulus is uniformly continuous. We also show that is equivalent to a similar principle for real‐valued functions on the Cantor space. These results extend Berger's [2] characterisation of for integer‐valued functions on and unify some characterisations of in terms of functions having continuous moduli. (shrink)