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We develop a series of small infinitary epistemic logics to study deductive inference involving intra-/interpersonal beliefs/knowledge such as common knowledge, common beliefs, and infinite regress of beliefs. Specifically, propositional epistemic logics GL (Lα) are presented for ordinal α up to a given αo (αo ≥ ω) so that GL(L0) is finitary KDn with n agents and GL(Lα) (α ≥ 1) allows conjunctions of certain countably infinite formulae. GL(Lα) is small in that the language is countable and can be constructive. The set of formulae Lα is increasing up to α = ω but stops at ω We present Kripke-completeness for GL(Lα) for each α ≤ ω, which is proved using the Rasiowa–Sikorski lemma and Tanaka–Ono lemma. GL(Lα) has a sufficient expressive power to discuss intra-/interpersonal beliefs with infinite lengths. As applications, we discuss the explicit definability of Axioms T (truthfulness), 4 (positive introspection), 5 (negative introspection), and of common knowledge in GL(Lα) Also, we discuss the rationalizability concept in game theory in our framework. We evaluate where these discussions are done in the series GL(Lα), α ≤ ω.

Epistemic logic of shallow depths has been developed by Kaneko–Suzuki using classical logic. In this paper, we adopt intuitionistic logic as the base logic to limit each player’s reasoning to constructive one. The resulting logical system is denoted by IG\(_{EF}\), and we introduce epistemic possible world semantics for IG\(_{EF}\). We show the soundness and completeness of IG\(_{EF}\) with respect to this semantics for it. We also show some applications to game theoretical decision making.

The energy of the lowest π molecular orbital in a planar boron cluster can be estimated from the connectivity of constituent boron atoms, or the mean valency of the boron atoms. Low-energy π molecu...

We investigate the expressive power and computational properties of two different types of languages intended for speaking about distances. First, we consider a first-order language FM the two-variable fragment of which turns out to be undecidable in the class of distance spaces validating the triangular inequality as well as in the class of all metric spaces. Yet, this two-variable fragment is decidable in various weaker classes of distance spaces. Second, we introduce a variable-free modal language MS that, when interpreted in metric spaces, has the same expressive power as the two-variable fragment of FM. We determine natural and expressive fragments of MS which are decidable in various classes of distance spaces validating the triangular inequality, in particular, the class of all metric spaces.

One criterion of constructive logics is the disjunction, property (DP). The Hallden-completeness is a weak DP, and is related to the relevance principle and variable separation. This concept is well-understood in the case of propositional logics. We extend this notion to predicate logics. Then three counterparts naturally arise. We discuss relationships between these properties and meet-irreducibility in the lattice of logics.

We prove that the (quasi-) identities in the lattice of subuniverses definable by lattice terms hold also in the lattice of fuzzy subalgebras for an arbitrary algebra. This means that the set of (quasi-) identities true in the lattice of subuniverses is equal to the set of those true in the lattice of fuzzy subalgebras for any given algebra. Moreover, we mention the case when there occur infinitary join operations. For this purpose, we define the notion of weights over an abstract algebra, and show that there is a canonical one-to-one correspondence between the lattice of fuzzy subalgebras and the lattice of weights. By using the notion of weights, we give a concrete description of the fuzzy subalgebra generated by an arbitrary fuzzy subset.

In [STU 00, KUT 03] we introduced a family of ‘modal' languages intended for talking about distances. These languages are interpreted in ‘distance spaces' which satisfy some (or all) of the standard axioms of metric spaces. Among other things, we singled out decidable logics of distance spaces and proved expressive completeness results relating classical and modal languages. The aim of this paper is to axiomatize the modal fragments of the semantically defined distance logics of [KUT 03] and give a new proof of their decidability.

In order to capture the concept of common knowledge, various extensions of multi-modal epistemic logics, such as fixed-point ones and infinitary ones, have been proposed. Although we have now a good list of such proposed extensions, the relationships among them are still unclear. The purpose of this paper is to draw a map showing the relationships among them. In the propositional case, these extensions turn out to be all Kripke complete and can be comparable in a meaningful manner. F. Wolter showed that the predicate extension of the Halpern-Moses fixed-point type common knowledge logic is Kripke incomplete. However, if we go further to an infinitary extension, Kripke completeness would be recovered. Thus there is some gap in the predicate case. In drawing the map, we focus on what is happening around the gap in the predicate case. The map enables us to better understand the common knowledge logics as a whole.

In so-called Kripke-type models, each sentence is assigned either to true or to false at each possible world. In this setting, every possible world has the two-valued Boolean algebra as the set of truth values. Instead, we take a collection of algebras each of which is attached to a world as the set of truth values at the world, and obtain an extended semantics based on the traditional Kripke-type semantics, which we call here the algebraic Kripke semantics. We introduce algebraic Kripke sheaf semantics for super-intuitionistic and modal predicate logics, and discuss some basic properties. We can state the Godel-McKinsey-Tarski translation theorem within this semantics. Further, we show new results on super-intuitionistic predicate logics. We prove that there exists a continuum of super-intuitionistic predicate logics each of which has both of the disjunction and existence properties and moreover the same propositional fragment as the intuitionistic logic.

A possible world structure consist of a set W of possible worlds and an accessibility relation R. We take a partial function r(·,·) to the unit interval [0, 1] instead of R and obtain a Kripke frame with graded accessibility r Intuitively, r(x, y) can be regarded as the reliability factor of y from x We deal with multimodal logics corresponding to Kripke frames with graded accessibility in a fairly general setting. This setting provides us with a framework for fuzzy possible world semantics. The basic propositional multimodal logic gK (grated K) is defined syntactically. We prove that gK is sound and complete with respect to this semantics. We discuss some extensions of gK including logics of similarity relations and of fuzzy orderings. We present a modified filtration method and prove that gK and its extensions introduced here are decidable.

In this paper, we regard fuzzy points as data and study to construct fuzzy objects from the set of given data on an arbitrary group. Consulting the method of least squares, we define optimal fuzzy subgroups for the set of data and show that one of them is obtained as a fuzzy subgroup by a set of some modified data.

Wajsberg and Jankov provided us with methods of constructing a continuum of logics. However, their methods are not suitable for super-intuitionistic and modal predicate logics. The aim of this paper is to present simple ways of modification of their methods appropriate for such logics. We give some concrete applications as generic examples. Among others, we show that there is a continuum of logics (1) between the intuitionistic predicate logic and the logic of constant domains, (2) between a predicate extension ofS4 andS4 with the Barcan formula. Furthermore, we prove that (3) there is a continuum of predicate logics with equality whose “equality-free fragment” is just the intuitionistic predicate logic.

We introduce here the notion of weights over an arbitrary group. We have shown that all fuzzy subgroups can be characterized by making use of weights. Furthermore, we have given a concrete description of the fuzzy subgroups of a wide variety of groups, using the notion of weights.

Some properties of Kripke-sheaf semantics for super-intuitionistic predicate logics are shown. The concept ofp-morphisms between Kripke sheaves is introduced. It is shown that if there exists ap-morphism from a Kripke sheaf κ1 into κ2 then the logic characterized by κ1 is contained in the logic characterized by κ2. Examples of Kripke-sheaf complete and finitely axiomatizable super-intuitionistic (and intermediate) predicate logics each of which is Kripke-frame incomplete are given. A correction to the author's previous paper “Kripke bundles for intermediate predicate logics and Kripke frames for intuitionistic modal logics” (Studia Logica, 49(1990), pp. 289–306 ) is stated.

Shehtman and Skvortsov introduced Kripke bundles as semantics of non-classical first-order predicate logics. We show the structural equivalence between Kripke bundles for intermediate predicate logics and Kripke-type frames for intuitionistic modal prepositional logics. This equivalence enables us to develop the semantical study of relations between intermediate predicate logics and intuitionistic modal propositional logics. New examples of modal counterparts of intermediate predicate logics are given.

Kaneko-Suzuki developed epistemic logics of shallow depths with multiple players for investigations of game theoretical problems. By shallow depth, we mean that nested occurrences of belief operators of players in formulae are restricted, typically to be of finite depths, by a given epistemic structure. In this paper, we develop various methods of surgical operations (cut and paste) of epistemic world models. An example is a bouquet-making, i.e., tying several models into a bouquet. Another example is to engraft a model to some branches of another model. By these methods, we obtain various meta-theorems on semantics and syntax on epistemic logics. To illustrate possible uses of our meta-theorems, we present one game theoretical theorem, which is also a meta-theorem in the sense of logic.

Summary. Individual decision making is based on predictions about other players’ choices as well as on valuations of reactions to predictions. In this sense, a player has a prediction-decision criterion for decision making. We develop a theory of prediction-decision criteria, which enables us to capture new phenomena on individual decision making in games. The decision making situation is described in the epistemic logic GLEF of shallow depths. There, each player considers his and other players’ decision making down to some shallow depths. It is a point of our theory to investigate inferential complexities of interpersonal introspections. In particular, we can discuss a minimal epistemic inferential structure for prediction-decision making. We will find parallel structures in decision making and prediction making, which is called an inner parallelism. The climax of the paper is the consideration of inner parallelisms of prediction-decision making.

We introduce a family of languages intended for representing knowledge and reasoning about metric (and more general distance) spaces. While the simplest language can speak only about distances between individual objects and Boolean relations between sets, the more expressive ones are capable of capturing notions such as 'somewhere in (or somewhere out of) the sphere of a certain radius', 'everywhere in a certain ring', etc. The computational complexity of the satisfiability problem for formulas in our languages ranges from NP-completeness to undecidability and depends on the class of distance spaces in which they are interpreted. Besides the class of all metric spaces, we consider, for example, the spaces R × R and N × N with their natural metrics.

The interactions between 4 anionic surfactants and 3 azo oil dyes have been studied by a spectrophotometer. The spectrographic behavior of these solutions was dependent on the kinds of hydrophilic groups in each surfactant and the kind of auxochrome in the azo oil dyes. In particular, a sulfate and two sulfonates including strongly hydrophilic groups interacted strongly with the oil dyes (4-NH2) which have an amino group at 4-position of naphthalene. At lower concentrations, an isosbestic point was recognized; this occurred because of the formation of the hydrophobic dye–surfactant complex (1:1). In the neighborhood of the cmc region, the spectrum with a peak at 520 nm changed with temperature into another one with a 440 nm band; this is the so-called thermochromism. This may be attributed to the fact that, in the presence of surfactant micelles, the 4-NH2 exhibits an azo form at higher temperature, but a resonance stabilized azonium form is found at lower temperature.

Fluorescence excitation spectra of weakly bound complexes formed between benzene and various solvent molecules prepared in a supersonic free jet have been observed. The intermolecular interaction and the structure of the complexes have been discussed.

Modal counterparts of intermediate predicate logics will be studied by means of algebraic devise. Our main tool will be a construction of algebraic semantics for modal logics from algebraic frames for predicate logics. Uncountably many examples of modal counterparts of intermediate predicate logics will be given.