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.
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)
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)
A semantical proof of Craig's interpolation theorem for the intuitionistic predicate logic and some intermediate prepositional logics will be given. Our proof is an extension of Henkin's method developed in [4]. It will clarify the relation between the interpolation theorem and Robinson's consistency theorem for these logics and will enable us to give a uniform way of proving the interpolation theorem for them.
Substructural logics have received a lot of attention in recent years from the communities of both logic and algebra. We discuss the algebraization of substructural logics over the full Lambek calculus and their connections to residuated lattices, and establish a weak form of the deduction theorem that is known as parametrized local deduction theorem. Finally, we study certain interpolation properties and explain how they imply the amalgamation property for certain varieties of residuated lattices.
It is well known that classical propositional logic can be interpreted in intuitionistic propositional logic. In particular Glivenko's theorem states that a formula is provable in the former iff its double negation is provable in the latter. We extend Glivenko's theorem and show that for every involutive substructural logic there exists a minimum substructural logic that contains the first via a double negation interpretation. Our presentation is algebraic and is formulated in the context of residuated lattices. In the last part (...) of the paper, we also discuss some extended forms of the Kolmogorov translation and we compare it to the Glivenko translation. (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.
We will give here a purely algebraic proof of the cut elimination theorem for various sequent systems. Our basic idea is to introduce mathematical structures, called Gentzen structures, for a given sequent system without cut, and then to show the completeness of the sequent system without cut with respect to the class of algebras for the sequent system with cut, by using the quasi-completion of these Gentzen structures. It is shown that the quasi-completion is a generalization of the MacNeille completion. (...) Moreover, the finite model property is obtained for many cases, by modifying our completeness proof. This is an algebraic presentation of the proof of the finite model property discussed by Lafont [12] and Okada-Terui [17]. (shrink)
This paper shows that both implicational logicsBCK andBCIW have the finite model property. The proof of the finite model property forBCIW, which is equal to the relevant logicR , was originally given by the first author in his unpublished paper [6] in 1973. The finite model property forBCK can be obtained by modifying the proof of that forBCIW. Here, both of these proofs will be given in a unified form and (...) the difference between them will be clarified. Further discussions will be given in the last section. (shrink)
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.
This paper shows a role of the contraction rule in decision problems for the logics weaker than the intuitionistic logic that are obtained by deleting some or all of structural rules. It is well-known that for such a predicate logic L, if L does not have the contraction rule then it is decidable. In this paper, it will be shown first that the predicate logic FLec with the contraction and exchange rules, but without the weakening rule, is undecidable while the (...) propositional fragment of FLec is decidable. On the other hand, it will be remarked that logics without the contraction rule are still decidable, if our language contains function symbols. (shrink)
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)
For each ordinal $\alpha > 0, L(\alpha)$ is the intermediate predicate logic characterized by the class of all Kripke frames with the poset α and with constant domain. This paper will be devoted to a study of logics of the form L(α). It will be shown that for each uncountable ordinal of the form α + η with a finite or a countable $\eta (> 0)$ , there exists a countable ordinal of the form β + η such that L(α (...) + η) = L(β + η). On the other hand, such a reduction of ordinals to countable ones is impossible for a logic L(α) if α is an uncountable regular ordinal. Moreover, it will be proved that the mapping L is injective if it is restricted to ordinals less than ω ω , i.e. α ≠ β implies L(α) ≠ L(β) for each ordinal $\alpha,\beta. (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.
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.
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.
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)
Abstract Some aspects of the coverage of bioethical issues in Japanese (11) and German (10 series) biology textbooks for lower secondary school have been investigated, concentrating on the treatment of environmental issues. It was found that German textbooks devote more space to these problems than the Japanese ones and that the style of presentation in German books is aimed at appealing to the emotions of the pupils, whereas that of the Japanese ones is a more traditional scientific one. The inclusion (...) of ethical view points in biology teaching is discussed in this context. (shrink)
Zdá se, že není nic přirozenějšího, než se spolu s Russellem domnívat, že „máme-li smysluplně hovořit a ne pouze vydávat zvuky, musíme slovům, která užíváme, dávat nějaký význam; a významem, který svým slovům dáváme, musí být něco, s čím jsme přišli do styku“. Naše slova přece musí, aby byla skutečně smysluplná, něco představovat! Od toho se odvíjí běžná poučka, která nám říká, že slova jazyka jsou symboly, to jest (podle Encyklopedie Britannica), „prvky komunikace, které mají představovat osobu, předmět, skupinu, proces (...) nebo ideu“. Problém je ovšem v tom, že není zdaleka zřejmé, co to vůbec znamená něco představovat; a co to tedy znamená být symbolem. V běžném jazyce hovoříme o představování například tehdy, když říkáme, že herec na jevišti divadla představuje dánského prince Hamleta, nebo že krabička sirek, kterou použijeme namísto ztracené šachové figurky, představuje černou věž. Jak vůbec může dojít k tomu, aby něco (nebo někdo) představovalo něco (nebo někoho) jiného? Jednou ze možností jistě je, že to někdo vyhlásí a jiní to přijmou. V programu divadla se například napíše, že se hraje Hamlet, diváci si to přečtou a vědí, že člověk, který pobíhá po jevišti s lebkou, představuje onoho dánského prince. Člověk, který zjistí, že mu chybí šachová figurka, vezme krabičku sirek a prohlásí „Tato krabička bude představovat černou věž“. To je čirá konvence: lidé se o tom, že něco bude představovat něco jiného, jednoduše dohodnou. K takové dohodě sice není potřeba, aby s ní ti, kdo ji přijímají, nahlas vyslovovali souhlas; je k ní nicméně potřeba, aby ji někdo vyhlásil a někdo jiný jeho vyhlášení porozuměl a přijal ho. Z toho ovšem plyne, že o takto konvenční druh představování se jazyk opírat nemůže; alespoň ne obecně. Brání tomu fakt, že k ustanovení takové konvence už jazyk potřebujeme – potřebujeme tedy již nějaká slova, která něco 'představují', mít. Když již nějaký jazyk máme, není problém zavést konvencí další jazyk – jak je to ale s tím prvním jazykem? (Nebylo by možné, abychom konvenci ustanovili za pomoci nějakých pouze 'předjazykových' komunikačních prostředků? Nemůžeme konvenci, na jejímž základě nějaký typ zvuku představuje velryby, ustanovit třebs pomocí pouhého ukazování na velryby? Problém je zřejmě v tom, že rámec, který by byl potřeba k tomu, aby mohlo být to či ono gesto interpretováno jako ukázání, které ustanovuje, co bude daný zvuk představovat, by musel sestávat z tak komplexních komunikčaních praktik, že je opět stěží představitelný jinak než v podobě jazyka.) Samozřejmě, že konvence není tou jedinou cestou, jak může dojít k tomu, že něco představuje něco jiného.. (shrink)
We prove that all semisimple varieties of FL ew-algebras are discriminator varieties. A characterisation of discriminator and EDPC varieties of FL ew-algebras follows. It matches exactly a natural classification of logics over FL ew proposed by H. Ono.
In [Ono 1987] H. Ono put the question about axiomatizing the intermediate predicate logicLFin characterized by the class of all finite Kripke frames (Problem 4,P41). It was established in [Skvortsov 1988] thatLFin is not recursively axiomatizable. One can easily show that for any finite posetM, the predicate logic characterized byM is recursively axiomatizable, and its axiomatization can be constructed effectively fromM. Namely, the set of formulas belonging to this logic is recursively enumerable, since it is embeddable in the two-sorted (...) classical predicate calculusCPC 2 (the definition of the truth in a Kripke model may be expressed by a formula ofCPC 2). Thus the logicLFin is II 2 0 -arithmetical.Here we give a more explicit II 2 0 -description ofLFin: it is presented as the intersection of a denumerable sequence of finitely axiomatizable Kripke-complete logics. Namely, we give an axiomatization of the logicLB n P m + characterized by the class of all posets of the finite height m and the finite branching n. A finite axiomatization of the predicate logicLP m + characterized by the class of all posets of the height m is known from [Yokota 1989] (this axiomatics is essentially first-order; the standard propositional axiom of the height m is not sufficient [Ono 1983]). We prove thatLB n P m + =(LP m + +B n),B n being the propositional axiom of the branching n (see [Gabbay, de Jongh 1974]). (shrink)
This paper deals with the varieties of monadic Heyting algebras, algebraic models of intuitionistic modal logic MIPC. We investigate semisimple, locally finite, finitely approximated and splitting varieties of monadic Heyting algebras as well as varieties with the disjunction and the existence properties. The investigation of monadic Heyting algebras clarifies the correspondence between intuitionistic modal logics over MIPC and superintuitionistic predicate logics and provides us with the solutions of several problems raised by Ono [35].
We present a sequent calculus for the modal logic S4, and building on some relevant features of this system (the absence of contraction rules and the confinement of weakenings into axioms and modal rules) we show how S4 can easily be translated into full propositional linear logic, extending the Grishin-Ono translation of classical logic into linear logic. The translation introduces linear modalities (exponentials) only in correspondence with S4 modalities. We discuss the complexity of the decision problem for several classes of (...) linear formulas naturally arising from the proposed translations. (shrink)
Recently Lafont [6] showed the finite model property for the multiplicative additive fragment of linear logic (MALL) and for affine logic (LLW), i.e., linear logic with weakening. In this paper, we shall prove the finite model property for intuitionistic versions of those, i.e. intuitionistic MALL (which we call IMALL), and intuitionistic LLW (which we call ILLW). In addition, we shall show the finite model property for contractive linear logic (LLC), i.e., linear logic with contraction, and for its intuitionistic version (ILLC). (...) The finite model property for related substructural logics also follow by our method. In particular, we shall show that the property holds for all of FL and GL - -systems except FL c and GL - c of Ono [11], that will settle the open problems stated in Ono [12]. (shrink)
The aim of this paper is to show that the implicational fragment BKof the intuitionistic propositional calculus (IPC) without the rules of exchange and contraction has the finite model property with respect to the quasivariety of left residuation algebras (its equivalent algebraic semantics). It follows that the variety generated by all left residuation algebras is generated by the finite left residuation algebras. We also establish that BKhas the finite model property with respect to a class of structures that constitute a (...) Kripke-style relational semantics for it. The results settle a question of Ono and Komori [OK85]. (shrink)
The intermediate logics have been classified into slices (cf. Hosoi [1]), but the detailed structure of slices has been studied only for the first two slices (cf. Hosoi and Ono [2]). In order to study the structure of slices, we give a method of a finer classification of slices & n (n 3). Here we treat only the third slice as an example, but the method can be extended to other slices in an obvious way. It is proved that each (...) subslice contains continuum of logics. A characterization of logics in each subslice is given in terms of the form of models. (shrink)
Medvedev's intermediate logic (MV) can be defined by means of Kripke semantics as the family of Kripke frames given by finite Boolean algebras without units as partially ordered sets. The aim of this paper is to present a proof of the theorem: For every set of connectives such that the-fragment ofMV equals the fragment of intuitionistic logic. The final part of the paper brings the negative solution to the problem set forth by T. Hosoi and H. Ono, namely: is an (...) intermediate logic based on the axiom (abc) (ab)(a c) separable? (shrink)
In this paper we prove the equivalence between the Gentzen system G LJ*\c , obtained by deleting the contraction rule from the sequent calculus LJ* (which is a redundant version of LJ), the deductive system IPC*\c and the equational system associated with the variety RL of residuated lattices. This means that the variety RL is the equivalent algebraic semantics for both systems G LJ*\c in the sense of [18] and [4], respectively. The equivalence between G LJ*\c and IPC*\c is a (...) strengthening of a result obtained by H. Ono and Y. Komori [14, Corollary 2.8.1] and the equivalence between G LJ*\c and the equational system associated with the variety RL of residuated lattices is a strengthening of a result obtained by P.M. Idziak [13, Theorem 1].An axiomatization of the restriction of IPC*\c to the formulas whose main connective is the implication connective is obtained by using an interpretation of G LJ*\c in IPC*\c. (shrink)
We prove the finite model property (fmp) for BCI and BCI with additive conjunction, which answers some open questions in Meyer and Ono [11]. We also obtain similar results for some restricted versions of these systems in the style of the Lambek calculus [10, 3]. The key tool is the method of barriers which was earlier introduced by the author to prove fmp for the product-free Lambek calculus [2] and the commutative product-free Lambek calculus [4].
Badania dotyczące pojawiania się myśli oderwanych od zadania wykonywanego przez podmiot pokazują, że takich myśli jest tym mniej, im większe wymagania stawia ono systemowi poznawczemu, jeśli chodzi o zaangażowanie procesów zarządczych, złożoność, trudność bądź częstość wykonywanych operacji. Jednym z możliwych wyjaśnień tych zależności jest hipoteza biernej regulacji zasobowej: aktywność umysłowa oderwana od zadania rozwija się w takim zakresie, w jakim procesy realizujące zadanie pozostawiają wolne zasoby niezbędne do jej rozwoju. Analizy przedstawione w artykule motywowane są pytaniem o teoretyczną konkretyzację tej (...) idei. Jaka jest natura zasobowych ograniczeń, które mają uniemożliwiać równoczesne rozwijanie się aktywności umysłowej dotyczącej zadania i oderwanej od niego? Na jakim etapie rozwoju aktywności umysłowej prowadzącej do pojawienia się myśli oderwanej od zadania występuje w związku z tymi ograniczeniami zasobowa blokada? Pod tym kątem rozważane są cztery współczesne znaczące ujęcia „roboczej” części umysłu: modele pamięci roboczej Baddeleya i Cowana oraz architektury kognitywne ACT-R i CAPS. Chociaż w tych teoriach są uwzględnione ograniczenia dotyczące przetwarzania lub aktywnego przechowywania, z którymi może wiązać się bierna regulacja zasobowa, to ważne pytania dotyczące mechanizmów takiej regulacji pozostają bez odpowiedzi. Wyjaśnienie odwołujące się do idei biernej regulacji zasobowej stawia przed modelami funkcjonowania umysłu nowe wyzwania. (shrink)
Artykuł dotyczy zagadnienia znanego pod nazwą „problemu prostych umysłów” tak, jak klaruje się ono w zestawieniu czterech doniosłych głosów w debacie na temat możliwości przypisywania zwierzętom życia mentalnego bez przypisywania im zdolności do posługiwania się językiem. Głosy te należą do: Donalda Davidsona, Johna McDowella, Petera Carruthersa oraz Jose L. Bermúdeza. Dwaj pierwsi autorzy bronią przekonaniowo-pragnieniowego modelu myślenia, w którym decydującą rolę pełni zdolność do posługiwania się językiem. Dwaj pozostali akceptację modelu przekonaniowo-pragnieniowego łączą z argumentem przeciwko wiązaniu myśli z językiem. Analizując (...) szczegółowo argumenty obu stanowisk przychylam się do rozwiązania proponowanego w ramach podejścia sformułowanego w oparciu o ustalenia Carruthersa i Bermúdeza. (shrink)
