Both Ambrose St. John (1815–1875) and John Henry Newman (1801–1890), who were received into the Roman Catholic Church in 1845, became members of the Birmingham Oratory. Newman’s closest companion for over three decades, St. John’s death was extremely painful for Newman, not only because it was unexpected, but because of his devotion to Newman as well as his dedication to his spiritual duties. Along with presenting Newman’s narrative of the last few weeks of St. John’s life, this essay raises the (...) question: why did Newman write this “account.”. (shrink)

In the context of constructive reverse mathematics, we show that weak Kőnig's lemma () implies that every pointwise continuous function is induced by a code in the sense of reverse mathematics. This, combined with the fact that implies the Fan theorem, shows that implies the uniform continuity theorem: every pointwise continuous function has a modulus of uniform continuity. Our results are obtained in Heyting arithmetic in all finite types with quantifier‐free axiom of choice.

The strong continuity principle reads “every pointwise continuous function from a complete separable metric space to a metric space is uniformly continuous near each compact image.” We show that this principle is equivalent to the fan theorem for monotone \ bars. We work in the context of constructive reverse mathematics.

A function from Baire space to the natural numbers is called formally continuous if it is induced by a morphism between the corresponding formal spaces. We compare formal continuity to two other notions of continuity on Baire space working in Bishop constructive mathematics: one is a function induced by a Brouwer‐operation (i.e., inductively defined neighbourhood function); the other is a function uniformly continuous near every compact image. We show that formal continuity is equivalent to the former while it is strictly (...) stronger than the latter. The equivalence of formally continuous functions and those induced by Brouwer‐operations requires Countable Choice. (shrink)

Glivenko-type theorems for substructural logics are comprehensively studied in the paper [N. Galatos, H. Ono, Glivenko theorems for substructural logics over FL, Journal of Symbolic Logic 71 1353–1384]. Arguments used there are fully algebraic, and based on the fact that all substructural logics are algebraizable 279–308] and also [N. Galatos, P. Jipsen, T. Kowalski, H. Ono, Residuated Lattices: An Algebraic Glimpse at Substructural Logics, in: Studies in Logic and the Foundations of Mathematics, vol. 151, Elsevier, 2007] for the details). As (...) a complementary work to the algebraic approach developed in [N. Galatos, H. Ono, Glivenko theorems for substructural logics over FL, Journal of Symbolic Logic 71 1353–1384], we present here a concise, proof-theoretic approach to Glivenko theorems for substructural logics. This will show different features of these two approaches. (shrink)

In this paper, a theorem on the existence of complete embedding of partially ordered monoids into complete residuated lattices is shown. From this, many interesting results on residuated lattices and substructural logics follow, including various types of completeness theorems of substructural logics.

In this paper, a semantics for predicate logics without the contraction rule will be investigated and the completeness theorem will be proved. Moreover, it will be found out that our semantics has a close connection with Beth-type semantics.

In this paper, we will develop an algebraic study of substructural propositional logics over FLew, i.e. the logic which is obtained from intuitionistic logics by eliminating the contraction rule. Our main technical tool is to use residuated lattices as the algebraic semantics for them. This enables us to study different kinds of nonclassical logics, including intermediate logics, BCK-logics, Lukasiewicz’s many-valued logics and fuzzy logics, within a uniform framework.

This paper discusses Crawley completions of residuated lattices. While MacNeille completions have been studied recently in relation to logic, Crawley completions (i.e. complete ideal completions), which are another kind of regular completions, have not been discussed much in this relation while many important algebraic works on Crawley completions had been done until the end of the 70’s. In this paper, basic algebraic properties of ideal completions and Crawley completions of residuated lattices are studied first in their conncetion with the join (...) infinite distributivity and Heyting implication. Then some results on algebraic completeness and conservativity of Heyting implication in substructural predicate logics are obtained as their consequences. (shrink)

An intermediate predicate logicS + n (n>0) is introduced and investigated. First, a sequent calculusGS n is introduced, which is shown to be equivalent toS + n and for which the cut elimination theorem holds. In § 2, it will be shown thatS + n is characterized by the class of all linear Kripke frames of the heightn.

Epistemological investigation belonged to the core topics in Indian philosophical traditions, too. Right cognition had generally been regarded as one of the important means to emancipation (niḥśreyasa) since ancient times. To reach this religious goal, they keenly discussed the problems of what kinds of cognition we should accept as right or what kinds of objects a right cognition refers to. Specifically it is about the number and the nature of the means of right cognition that opinions differ from school to (...) school. The number ranges from one (perception) to six or even ten (perception, inference, comparison, testimony, implication, non-perception, equivalence, tradition, gesture, and intuition). The concept of each means of right cognition, too, varies greatly among schools. In this paper I take up the Nyāya System, a rationalistic school of Brahmanic philosophy. In Nyāya the inference is regarded as particularly important, but it never means that logical thinking dominates testimony or the authority of religious scriptures in the Nyāya System. On the contrary we find such cases as the religious authority seems to delimit the validity of inference. Some inferences are obstructed by an axiom established in the school, whereas others by a ristriction of Brahmanic tradition. In this manner they seemed to protect their whole system from followers of other Schools. By examining this topic I would like to throw a tiny light on the characteristic affinity between philosophy and religion in Indian thought. (shrink)

LetL be any modal or tense logic with the finite model property. For eachm, definer L (m) to be the smallest numberr such that for any formulaA withm modal operators,A is provable inL if and only ifA is valid in everyL-model with at mostr worlds. Thus, the functionr L determines the size of refutation Kripke models forL. In this paper, we will give an estimation ofr L (m) for some linear modal and tense logicsL.

The present paper deals with the predicate version MTL of the logic MTL by Esteva and Godo. We introduce a Kripke semantics for it, along the lines of Ono''s Kripke semantics for the predicate version of FLew (cf. [O85]), and we prove a completeness theorem. Then we prove that every predicate logic between MTL and classical predicate logic is undecidable. Finally, we prove that MTL is complete with respect to the standard semantics, i.e., with respect to Kripke frames on the (...) real interval [0,1], or equivalently, with respect to MTL-algebras whose lattice reduct is [0,1] with the usual order. (shrink)

This study explores the individual-level activities of legislators in parliament, which have been largely ignored in the literature on parliamentary democracies. Individual legislators are extensively involved in parliamentary activities such as drafting private members’ bills and posing questions, even though these activities have only been considered to play marginal roles in parliamentary democracies. Moreover, their engagement varies significantly. By using unique data from Japan, this study demonstrates that the personal attributes of legislators affect their choice of parliamentary activities. Under electoral (...) systems with intra-party competition, legislators use parliamentary activities as an important means to inform their constituents about what they can do for them and how they differ from other legislators. In elections, candidates cultivate personal votes by exploiting the image drawn from their personal attributes and, once elected, they behave in accordance with their attributes in order to maintain their electoral ground. Thus, they devote themselves to different activities in parliament. The data analyzed here support this argument. The results of empirical analyses show that legislators with local-level political experience engage in particularistic pork-barrel activities that will benefit their local interests, while legislators with legal-work experience allocate their time and energy to general policy-making activities that will enhance their public image and visibility as legal experts. (shrink)

John Henry Newman has rightly been hailed as a giant in the Catholic intellectual tradition. His contributions to theology, literature, and education have been studied at length; however, his contribution to philosophy has not received appropriate attention. This essay 1) explores Newman’s unique philosophical insights in terms of the phenomenological tradition of Edmund Husserl; 2) analyzes the transcendental approach of certain British scientists—notably Ronald Knox and Charles Darwin; and 3) discusses how Newman might be considered a phenomenologist.

We examined the effects of unconscious priming on temporal-interval production. In Experiment 1, participants were instructed to keep visual displays on a screen for 2500 ms intervals. Half of the displays were repeated across blocks throughout the entire experiment, and the others were newly generated from trial to trial. The displays consisted of patterns so complex that the participants could not intentionally memorize them. The results showed that significantly more time elapsed for old displays than for new displays before participants (...) indicated that a 2500 ms interval had elapsed. Experiment 2 replicated this effect and excluded an alternative account based on perceived pattern complexity. The effect of repetitive presentation was obtained despite the fact the participants did not recognize the repetition, suggesting that unconscious priming increased temporal production. These results suggest that time perception is affected by an unconscious process. (shrink)

In this paper we will discuss constraints on the number of (non-dummy) players and on the distribution of votes such that local monotonicity is satisfied for the Public Good Index. These results are compared to properties which are related to constraints on the redistribution of votes (such as implied by global monotonicity). The discussion shows that monotonicity is not a straightforward criterion of classification for power measures.

立法・法改正論争が盛んな現代日本の法理論の背後にあるものを理解するため、幕末から新憲法までの法思想の道筋を辿る。.

Along the same line as that in Ono (Ann Pure Appl Logic 161:246–250, 2009), a proof-theoretic approach to Glivenko theorems is developed here for substructural predicate logics relative not only to classical predicate logic but also to arbitrary involutive substructural predicate logics over intuitionistic linear predicate logic without exponentials QFLe. It is shown that there exists the weakest logic over QFLe among substructural predicate logics for which the Glivenko theorem holds. Negative translations of substructural predicate logics are studied by using (...) the same approach. First, a negative translation, called extended Kuroda translation is introduced. Then a translation result of an arbitrary involutive substructural predicate logics over QFLe is shown, and the existence of the weakest logic is proved among such logics for which the extended Kuroda translation works. They are obtained by a slight modification of the proof of the Glivenko theorem. Relations of our extended Kuroda translation with other standard negative translations will be discussed. Lastly, algebraic aspects of these results will be mentioned briefly. In this way, a clear and comprehensive understanding of Glivenko theorems and negative translations will be obtained from a substructural viewpoint. (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)