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)

Some modal logics based on logics weaker than the classical logic have been studied by Fitch [4], Prior [7], Bull [1], [2], [3], Prawitz [6] etc. Here we treat modal logics based on the intuitionistic propositional logic, which call intuitionistic modal logics.

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)

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.

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)

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.

We develop a general algebraic and proof-theoretic study of substructural logics that may lack associativity, along with other structural rules. Our study extends existing work on substructural logics over the full Lambek Calculus [34], Galatos and Ono [18], Galatos et al. [17]). We present a Gentzen-style sequent system that lacks the structural rules of contraction, weakening, exchange and associativity, and can be considered a non-associative formulation of . Moreover, we introduce an equivalent Hilbert-style system and show that the logic associated (...) with and is algebraizable, with the variety of residuated lattice-ordered groupoids with unit serving as its equivalent algebraic semantics. Overcoming technical complications arising from the lack of associativity, we introduce a generalized version of a logical matrix and apply the method of quasicompletions to obtain an algebra and a quasiembedding from the matrix to the algebra. By applying the general result to specific cases, we obtain important logical and algebraic properties, including the cut elimination of and various extensions, the strong separation of , and the finite generation of the variety of residuated lattice-ordered groupoids with unit. (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.

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)

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.

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.

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.

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)

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)

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)

The aim of this paper is to examine the relationship between education and technological rationality from the perspective of the philosophy of education, and to show that while education is deeply related to technique, skills, or technology, it can never be reduced to technical knowledge, and that there are things in education that overflow technical knowledge. I will here ask why there is something in education that overflows technical knowledge — I will define it as knowledge of pathos — and (...) why it is necessary to redefine the meaning of education in terms of knowledge of pathos. In order to address this issue, I will first examine the relationship between technology and transcendence in the world of education in Kei Hachiya’s Education and Transcendence. Second, I will show that recent outstanding theoretical achievements of Japanese philosophers of education — Satoji Yano’s Education from the Perspectives of Gift and Exchange, Yasuo Imai’s Medial Perspective on Education, and Tsunemi Tanaka’s Clinical Theory of Human Becoming — are all located within the problematic area of ‘technology-pathos-education’. Finally, I would like to propose a ‘philosophy of education built on fragile parts’, with a view to redefining education through the knowledge of pathos, paying attention to the experience of pathei mathos and the opportunity of weakness in the philosophy of education. (shrink)