Hiroakira Ono

Japan Advanced Institute of Science and Technology · School of Information Science

This is a short autobiography consisting of personal recollections and experiences in my academic life. It starts from my student days, and covers the time at the end of the 1990s when substructural logics began to attract a lot of attention of various researchers.

In this paper, residuated expansions of lattice-ordered structures are explored, in particular, of both lattice-ordered groupoids and lattices with implication. Here, a residuated expansion is an expansion in which the law of (left) residuation between fusion and implication holds. Thus, residuated expansions discussed here take the form of (left)...

Dynamic epistemic logic is a logic that is aimed at formally expressing how a person’s knowledge changes. We provide a cut-free labelled sequent calculus ($\textbf{GDEL}$) on the background of existing studies of Hilbert-style axiomatization $\textbf{HDEL}$ of dynamic epistemic logic and labelled calculi for public announcement logic. We first show...

This volume gathers selected papers presented at the Fourth Asian Workshop on Philosophical Logic, held in Beijing in October 2018. The contributions cover a wide variety of topics in modal logic (epistemic logic, temporal logic and dynamic logic), proof theory, algebraic logic, game logics, and philosophical foundations of logic. They also reflect...

Cut elimination for a given sequent system \(\mathbf L\) means that if a sequent is provable in \(\mathbf L\) then it is also provable in \(\mathbf L\) without using cut rule. Any proof P of \(\mathbf L\) is said to be cut-free when P contains any application of cut rule in it. Intuitively, any cut-free proof is a kind of a proof without detours, o...

In the following, we show how proof-theoretic arguments will work well in study of logical properties. Distinguishing features of proof-theoretic approach lie in its concrete and combinatorial aspects, which often yield information much more than semantical approach. Two major instruments for developing proof-theoretic study are cut elimination and...

Until now, we have discussed connections between particular logics and corresponding algebras, e.g., between classical logic and Boolean algebras, and also between intuitionistic logic and Heyting algebras.

The main goal of this chapter is to introduce several basic concepts in algebraic logic, i.e., Lindenbaum-Tarski algebras, locally finite algebras, finite embeddability property and canonical extensions. They are important algebraic tools for developing algebraic approach to logic.

Throughout Part I, we have been discussing sequent systems for particular logics, like classical logic and intuitionistic logic, and logical properties of these logics by proof-theoretic analysis of sequent systems for them. These results are sharp and deep, which are often obtained as consequences of cut elimination. On the other hand, cut elimina...

Semantical study of modal logics have been developed successfully already by using Kripke semantics. In the present chapter, we will discuss an algebraic approach to modal logics. Since our algebraic approach has many parallels with what we explained already in the previous chapters of Part II, it will be explained rather briefly. After introducing...

After giving preliminary remarks and a brief explanation of the scope of this book in Sect. 1.1, we will introduce two sequent systems \(\mathbf {LK}\) and \(\mathbf {LJ}\) for classical logic and intuitionistic logic, respectively.

In this chapter, we give a short introduction to residuated structures which are algebraic structures for substructural logics. Boolean algebras and also Heyting algebras are defined to be lattice structures with a binary relation \(\rightarrow \) which satisfy the law of residuation between \(\wedge \) and \(\rightarrow \), i.e., \(a \wedge b \le...

In this section, we will give a brief introduction to proof theory for two important branches of nonclassical logics, that is, modal logics and substructural logics. They are important because both of them include vast varieties of logics that have been actively studied.

Syntactic or symbolic approaches to logic began from the middle of nineteenth century. G. Boole attempted to express logical inference as an algebraic calculation in his book Boole 1854. It took several decades before Hilbert-style formal systems were introduced.

This book offers a concise introduction to both proof-theory and algebraic methods, the core of the syntactic and semantic study of logic respectively. The importance of combining these two has been increasingly recognized in recent years. It highlights the contrasts between the deep, concrete results using the former and the general, abstract ones...

We prove that certain natural sequent systems for bi-intuitionistic logic have the analytic cut property. In the process we show that the (global) subformula property implies the (local) analytic cut property, thereby demonstrating their equivalence. Applying a version of Maehara technique modified in several ways, we prove that bi-intuitionistic l...

Dynamic Epistemic Logic is a logic that is aimed at formally expressing how a person’s knowledge changes. We provide a cut-free labelled sequent calculus (\(\mathbf {GDEL}\)) on the background of existing studies of Hilbert-style axiomatization \(\mathbf {HDEL}\) by Baltag et al. (1989) and labelled calculi for Public Announcement Logic by Maffezio...

This is a short survey of semantical study of cut elimination and subformula property in modal logics. Cut elimination is a basic proof-theoretic notion in sequent systems, and subformula property is the most important consequence of cut elimination. A special feature of our presentation is its unified semantical approach to them based on Kripke mo...

This contributed volume includes both theoretical research on philosophical logic and its applications in artificial intelligence, mostly employing the concepts and techniques of modal logic. It collects selected papers presented at the Second Asia Workshop on Philosophical Logic, held in Guangzhou, China in 2014, as well as a number of invited pap...

Uniform interpolation property of a given logic is a stronger form of Craig’s interpolation property where both pre-interpolant and post-interpolant always exist uniformly for any provable implication in the logic. It is known that there exist logics, e.g., modal propositional logic S4 , which have Craig’s interpolation property but do not have uni...

We introduce a class of algebras, called twist-structures, whose members are built as special squares of an arbitrary residuated lattice. We show how our construction relates to and encompasses results obtained by several authors on the algebraic semantics of non-classical logics. We define a logic that corresponds to our twist-structures and show...

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 exponential...

The paper deals with involutive FLe -monoids, that is, commutative residuated, partially-ordered monoids with an involutive negation. Involutive FLe -monoids over lattices are exactly involutive FLe -algebras, the algebraic counterparts of the substructural logic IUL. A cone representation is given for conic involutive FLe -monoids, along with a ne...

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...

In this paper, we will develop an algebraic study of substructural propositional logics over FL_ , i.e. the logic which is obtained from the 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 lo...

This article develops a comprehensive study of various types of interpolation properties and Beth definability properties (BDPs) for substructural logics, and their algebraic characterizations through amalgamation properties (APs) and epimorphisms surjectivity. In general, substructural logics are algebraizable but lack many of the basic logical pr...

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 (associative) substructural logics over the full Lambek Calculus (see, for example, Ono (2003) [34], Galatos and Ono (2006) [18], Galatos et al. (2007) [17]). We present...

Hájek's basic logic BL is an extension of the substructural logic FLew, or equivalently, Höhle's monoidal logic. Thus, fuzzy logics can be viewed as a special subclass of substructural logics. On the other hand, their close connections are often overlooked, since these two classes of logics have been motivated by different aims, and so introduced a...

Glivenko-type theorems for substructural logics (over FL) are comprehensively studied in the paper [N. Galatos, H. Ono, Glivenko theorems for substructural logics over FL, Journal of Symbolic Logic 71 (2006) 1353-1384]. Arguments used there are fully algebraic, and based on the fact that all substructural logics are algebraizable (see [N. Galatos,...

We will give a state-of-the-art survey of the study of substructural logics. Originally, substructural logics were introduced as logics which, when formulated as Gentzen-style systems, lack some of the three basic structural rules, i.e. contraction, weakening and exchange. For example, relevance logics and linear logic lack the weakening rule, many...

Keynote Addresses.- An Algebraic Approach to Substructural Logics - An Overview.- On Modeling of Uncertainty Measures and Observed Processes.- Statistics under Interval Uncertainty and Imprecise Probability.- Fast Algorithms for Computing Statistics under Interval Uncertainty: An Overview.- Trade-Off between Sample Size and Accuracy: Case of Static...

This paper gives algebraic characterizations of Hallden completeness (HC), and of Maksimova's variable separation property (MVP) and its deductive form. Though algebraic characterizations of these properties have been already studied for modal and superintuitionistic logics, e.g. in Wroński [12], Maksimova [7], [9], a deeper analysis of these prope...

The book is meant to serve two purposes. The first and more obvious one is to present state of the art results in algebraic research into residuated structures related to substructural logics. The second, less obvious but equally important, is to provide a reasonably gentle introduction to algebraic logic. At the beginning, the second objective is...

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 s...

Without Abstract

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 theor...

Without Abstract

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 sy...

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 follows, including various types of completeness theorems of substructural logics.

This is an introductory survey of substructural logics and of residuated lattices which are algebraic structures for substructural logics. Our survey starts from sequent systems for basic substructural logics and develops the proof theory of them. Then, residuated lattices are introduced as algebraic structures for substructural logics, and some re...

Let FL ew be the logic obtained from the intuitionistic propositional logic by deleting contraction rule if we formulate it in a sequent system. Sometimes, this logic is called intuitionistic affine logic. The class of logics over FL ew , i.e. logics stronger than or equal to FL ew , includes many interesting logics, e.g., intermediate logics, Łuka...

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 unde...

We give here a brief overview of completions of algebras, and of completeness results on both modal logics and substructural logics which are obtained by using completions. Our aim is to discuss completions of algebras and completeness theorems of logics of various kinds in a uniform way, and to try to find common features among them. As is shown i...

. It is shown that the only algebra that splits the lattice of subvarieties of the variety of residuated lattices is the two element boolean algebra.

We show that the variety of residuated lattices is generated by its finite simple members, improving upon a finite model property result of Okada and Terui. The reasoning is a blend of proof-theoretic and algebraic arguments.

It is known that in f!;g-fragment of the natural deduction system of intuitionistic propositional logic NJ balanced formulas have unique fij-normal proofs. A balanced formula is a formula such that 1. every propositional variable has at most one negative occurrence and 2. every propositional variable has at most one positive occurrence. It will be...

A minimal formula of a given logic L is a formula which is provable in L and is not a non-trivial substitution instance of other provable formulas in L. In [5], Y. Komori asked whether normal proofs of minimal formulas are unique in the implicational fragments of natural deduction systems for the intuitionistic logic and the logic BCK. It was alrea...

In this short note, we will discuss some problems related to noncommutative substructural logics. Here, by noncommutative substructural logics, we mean substructural logics which have neither exchange rules nor axioms for exchange, in general. So, they include some extensions of Lambek calculus and some of relevant logics, for which many works have...

In the present paper, the Kripke completeness of both minimum predicate and infinitary extensions of a variety of modal propositional logics is shown in a uniform way. Our approach taken here is similar to one taken by Jónsson and Tarski in 1951. The representation theorem of modal algebras based on the Rasiowa-Sikorski lemma is shown first. Since...

Maksimova's principle of variable separation says that if propositional formulas $A_1 \supset A_2$ and $B_1 \supset B_2$ have no propositional variables in common and if a formula $A_1\wedge B_1 \supset A_2\vee B_2$ is provable, then either $A_1 \supset A_2$ or $B_1 \supset B_2$ is provable. Results on Maksimova's principle until now are obtained m...

A b s t r a c t. The present paper is concerned with the cut elim- inability for some sequent systems of noncommutative substructural logics, i.e. substructural logics without exchange rule. Sequent sys- tems of several extensions of noncommutative logics FL and LBB0I, which is sometimes called T! W, will be introduced. Then, the cut elimination th...

this paper, we will give a short survey of results on decision problems and the finite model property of substructural logics. The paper is far from a complete list of these results, since a lot of results have been obtained already in some restricted classes of substructural logics, like relevant logics, and therefore it is impossible to cover all...

Extensions of the intuitionistic linear logic with knotted structural rules are discussed. Each knotted structural rule is a rule of inference in sequent calculi of the form: from Γ,A,⋯,A (ntimes)→C infer Γ,A,⋯,A (ktimes)→C, which is called the (n⇝k)-rule. It is a restricted form of the weakening rule when n<k, and of the contraction rule when n>k....

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. Her...

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 predi...

For each ordinal α > 0, L(α) 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 η(> 0), there exists a countab...

Gentzen-type sequent calculi usually contain three structural rules, i.e., exchange, contraction and weakening rules. In recent years, however, there have been various studies on logics that have not included some or any of these structural rules. The motives or purposes of these studies have been so diverse that sometimes close connections between...

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.

A variant of the Robinson property (ROB*) is introduced. The property ROB* is equivalent to the usual Robinson property and hence is equivalent also to the interpolation property, in any compact logic closed under the Boolean operations. On the other hand, it will be shown that ROB* is not always equal to the interpolation property, if a logic is n...

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 g...

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.

We will study syntactical and semantical properties of propositional logics weaker than the intuitionistic, in which the contraction rule (or, the exchange rule or the weakening rule, in some cases) does not hold. Here, the contraction rule means the rule of inference of the form if we formulate our logics in a Gentzen-type formal system. Some syn...

Pictures of functions of one variable in the natural number theory are introduced as patterns over {0,1} in the first quadrant. It is proved that the set of pictures of all computable functions of one variable is obtained as the bottom planes of three-dimensional arrays accepted by some nondeterministic automaton on ω3-tapes. To our surprise, it is...

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 gi...

It is shown that the first-order arithmetic A[P(x), 2x, x + 1] with two functions 2x, x + 1 and a monadic predicate symbol P(x) is undecidable, by using a kind of two-dimensional finite automata, called finite causal ω2-systems. From this immediately follows R.M. Robinson's result, which says that the monadic second-order theory with two functions...

We know that many parts of ordinary recursive function theory can be developed formally in a certain extension of the formal number theory (e.g. Peano arithmetic). But we encounter some difficulties when we want to deal with partial recursive functions, since in ordinary logical calculi only total functions and predicates can be treated. The most n...