The logics of formal inconsistency (LFIs) are paraconsistent logics which permit us to internalize the concepts of consistency or inconsistency inside our object language, introducing new operators to talk about them, and allowing us, in principle, to logically separate the notions of contradictoriness and of inconsistency. We present the formal definitions of these logics in the context of General Abstract Logics, argue that they in fact represent the majority of all paraconsistent logics existing up to this point, if not the most exceptional ones, and we single out a subclass of them called C-systems, as the LFIs that are built over the positive basis of some given consistent logic. Given precise characterizations of some received logical principles, we point out that the gist of paraconsistent logic lies in the Principle of Explosion, rather than in the Principle of Non-Contradiction, and we also sharply distinguish these two from the Principle of Non-Triviality, considering the next various weaker formulations of explosion, and investigating their interrelations. Subsequently, we present the syntactical formulations of some of the main C-systems based on classical logic, showing how several well-known logics in the literature can be recast as such a kind of C-systems, and carefully study their properties and shortcomings, showing for instance how they can be used to faithfully

1.1 Contradictoriness and inconsistency, consistency and non-contradictoriness In traditional logic, contradictoriness (the presence of contradictions in a theory or in a body of knowledge) and triviality (the fact that such a theory

We study algebras whose elements are relations, and the operations are natural "manipulations" of relations. This area goes back to 140 years ago to works of De Morgan, Peirce, Schroder (who expanded the Boolean tradition with extra operators to handle algebras of binary relations). Well known examples of algebras of relations are the varieties RCAn of cylindric algebras of n-ary relations, RPEAn of polyadic equality algebras of n-ary relations, and RRA of binary relations with composition. We prove that any axiomatization, say E, of RCAn has to be very complex in the following sense: for every natural number k there is an equation in E containing more than k distinct variables and all the operation symbols, if 2 ! n ! !. Completely analogous statement holds for the case n !. This improves Monk's famous non-finitizability theorem for which we give here a simple proof. We prove analogous nonfinitizability properties of the larger varieties SNrnCA n+k . We prove that the complementa...

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Object-oriented class libraries offer the potential for individual researchers to manage the large bodies of code generated in the experimental development of complex interactive systems. This article analyzes the structure of such a class library that supports the rapid prototyping of a wide range of systems including collaborative networking, shared documents, hypermedia, machine learning, knowledge acquisition and knowledge representation, and various combinations of these technologies. The overall systems architecture is presented in terms of a heterogeneous collection of systems providing a wide range of application functionalities. Examples are given of group writing, multimedia and knowledge-based systems which are based on combining these functionalities. The detailed design issues of the knowledge representation server component of the system are analyzed in terms of requirements, current state-of-the-art, and the underlying theoretical principles that lead to an effective obj...

Abstract. The aim of the paper is discussion of connections between the three kinds of objects named in the title. In a sense, it is a survey of such connections; however, some new directions are also considered. This relates, especially, to sections 3, 4 and 5, where we consider a field that could be understood as an universal algebraic geometry. This geometry is parallel to universal algebra. In the monograph [51] algebraic logic was used for building up a model of a database. Later on, the structures arising there turned out to be useful for solving several problems from algebra. This is the position which the present paper is written from.

Fibring is de ned as a mechanism for combining logics with a rstorder base, at both the semantic and deductive levels. A completeness theorem is established for a wide class of such logics, using a variation of the Henkin method that takes advantage of the presence of equality and inequality in the logic. As a corollary, completeness is shown to be preserved when bring logics in that class. A modal rst-order logic is obtained as a bring where neither the Barcan formula nor its converse hold.

Abstract. In informal mathematical usage we often reason using languages with binding. We usually find ourselves placing capture-avoidance constraints on where variables can and cannot occur free. We describe a logical derivation system which allows a direct formalisation of such assertions, along with a direct formalisation of their constraints. We base our logic on equality, probably the simplest available judgement form. In spite of this, we can axiomatise systems of logic and computation such as first-order logic or the lambda-calculus in a very direct and natural way. We investigate the theory of derivations, prove a suitable semantics sound and complete, and discuss existing and future research. 1

. The two main directions pursued in the present paper are the following. The first direction was (perhaps) started by Pigozzi in 1969. In [Mak 91] and [Mak 79] Maksimova proved that a normal modal logic (with a single unary modality) has the Craig interpolation property iff the corresponding class of algebras has the superamalgamation property. In this paper we extend Maksimova's theorem to normal multi-modal logics with arbitrarily many, not necessarily unary modalities, and to not necessarily normal multi-modal logics with modalities of ranks smaller than 2. To extend the characterization beyond multi-modal logics, we look at arbitrary algebraizable logics. We will introduce an algebraic property equivalent with the Craig interpolation property in algebraizable (and in strongly nice) logics, and prove that the superamalgamation property implies the Craig interpolation property. The problem of extending the characterization result to non-normal non-unary modal logics will be discus...

�������� � In this paper we mainly deal with first-order languages without equality and introduce a weak form of equality predicate, the so-called Leibniz equality. This equality is characterized algebraically by means of a natural concept of congruence; in any structure, it turns out to be the maximum congruence of the structure. We show that first-order logic without equality has two distinct complete semantics (full semantics and reduced semantics) related by a reduction operator. The last and main part of the paper contains a series of Birkhoff-style theorems characterizing certain classes of structures defined without equality, not only full classes but also reduced ones. 1. Preliminaries 1.1. Basic Notation and Terminology. Let the triple L = 〈F, R, ρ 〉 be a first order language; F and R denote pairwise disjoint sets of function and relation symbols of L respectively (R must be nonempty), and ρ is the arity function from F ∪ R into the set of natural numbers. We use capital Gothic letters A, B, C,..., with appropriate subscripts, to represent structures over L, also called L-structures. In order to be consistent with the notation, we denote by A the universe of A, and by FA and RA the interpretations on A of the collections of function and relation symbols of L respectively, i.e., FA = {f A: f ∈ F} and RA = {r A: r ∈ R}. The corresponding boldface letter A is used to denote the underlying algebra 〈A, FA 〉 of A, and we normally write f A instead of f A. Lowercase boldface letters a,b,... are used to indicate members of the cartesian product of some family of sets. So, if A is an L-structure, a = 〈a1,..., an 〉 belongs to A n, f ∈ F and r ∈ R, and h is any mapping with domain A, then f A a, a ∈ r A and ha are short-hand for f A a1... an, 〈a1,..., an 〉 ∈ r A and 〈ha1,..., han〉, respectively. By an L-algebra we mean the underlying algebra of any L-structure; of course, if the set of function symbols is empty, an L-algebra simply means an arbitrary