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We present an overview of the different frameworks and structures that have been proposed during the last century in order to develop a general theory of logics. This includes Tarski’s consequence operator, logical matrices, Hertz’s Satzsysteme, Gentzen’s sequent calculus, Suszko’s abstract logic, algebraic logic, da Costa’s theory of valuation and universal logic itself.

We suggest abstract model theory as a framework for universal logic. For this end we present basic concepts of abstract model theory in a general form which covers both classical and non-classical logics. This approach aims at unifying model-theoretic results covering as large a variety of examples as possible, in harmony with the general aim of universal logic.

In this paper we develop a topological approach to a theory of model-theoretical abstract logics. A model-theoretical abstract logic is given by a set of expressions (formulas), a class of interpretations (models) and a satisfaction relation between interpretations and expressions. Notions such as theory, consequence, etc. are derived in a natural way. We define topologies on the space of (prime) theories and on the space of models. Then structural properties of a logic are mirrored in the respective topological spaces and can be studied now by topological means. We introduce the notion of logic-homomorphism, a map between model-theoretical abstract logics that preserve structural (topological) properties. We study in detail conditions under which logic-homomorphisms determine continuous or/and open functions between the respective topological spaces. We define a logic-isomorphism as a bijective logic-homomorphism. Such a map forces a homeomorphism on the corresponding topological spaces.

Moreover, we show that certain maps between logics lead to a condition which has the same form as the satisfaction axiom of institutions. This promising result may serve in future research to establish a connection between our approach and the well-known category-theoretical concept of institution.

The aim of this paper is to develop the theory of the selfextensional logics with an implication for which it holds the deduction-detachment theorem, as presented in [], but avoiding the use of Gentzen-systems to prove the main results as much as possible.

Self-deductibility is the Stoic version of the law of identity : if , then . After a discussion on its role, we suggest a natural system of axioms and rules for a logic in which this law is not valid, based on a simple model where proofs are families of strictly injective maps. Finally we develop some general theory of taxonomies (i.e. “categories without identities”) and place this particular example into a more general algebraic picture.

When can we say that two distinct logical systems are, nevertheless, essentially the “same”? In this paper we discuss the notion of “sameness” between logical systems, bearing in mind the expressive power of their associated spaces of theories, but without neglecting their syntactical dimension. Departing from a categorial analysis of the question, we introduce the new notion of between logical systems. We use several examples to illustrate our proposal and to support its comparison to other proposals in the literature, namely homeomorphisms [], and translational equivalence (or synonymity) [].

This paper builds on the theory of institutions, a version of abstract model theory that emerged in computer science studies of software specification and semantics. To handle proof theory, our institutions use an extension of traditional categorical logic with sets of sentences as objects instead of single sentences, and with morphisms representing proofs as usual. A natural equivalence relation on institutions is defined such that its equivalence classes are logics. Several invariants are defined for this equivalence, including a Lindenbaum algebra construction, its generalization to a Lindenbaum category construction that includes proofs, and model cardinality spectra; these are used in some examples to show logics inequivalent. Generalizations of familiar results from first order to arbitrary logics are also discussed, including Craig interpolation and Beth definability.

I will discuss the two problems of how to define identity between logics and how to define identity between proofs. For the identity of logics, I propose to simply use the notion of preorder equivalence. This might be considered to be folklore, but is exactly what is needed from the viewpoint of the problem of the identity of proofs: If the proofs are considered to be part of the logic, then preorder equivalence becomes equivalence of categories, whose arrows are the proofs. For identifying these, the concept of proof nets is discussed.

We show by way of example how one can provide in a lot of cases simple modular semantics for rules of inference, so that the semantics of a system is obtained by joining the semantics of its rules in the most straight-forward way. Our main tool for this task is the use of finite matrices, which are multi-valued structures in which the value assigned by a valuation to a complex formula can be chosen non-deterministically out of a certain nonempty set of options. The method is applied in the area of logics with a formal consistency operator (known as LFIs), allowing us to provide in a modular way effective, finite semantics for thousands of different LFIs.

The Polish logician Roman Suszko has extensively pleaded in the 1970s for a restatement of the notion of many-valuedness. According to him, as he would often repeat, “there are but two logical values, true and false.” As a matter of fact, a result by Wójcicki-Lindenbaum shows that any tarskian logic has a many-valued semantics, and results by Suszko-da Costa-Scott show that any many-valued semantics can be reduced to a two-valued one. So, why should one even consider using logics with more than two values? Because, we argue, one has to decide how to deal with bivalence and settle down the trade-off between logical 2-valuedness and truth-functionality, from a pragmatical standpoint.

This paper will illustrate the ups and downs of a two-valued reduction of logic. Suszko’s reductive result is quite non-constructive. We will exhibit here a way of effectively constructing the two-valued semantics of any logic that has a truth-functional finite-valued semantics and a sufficiently expressive language. From there, as we will indicate, one can easily go on to provide those logics with adequate canonical systems of sequents or tableaux. The algorithmic methods developed here can be generalized so as to apply to many non-finitely valued logics as well — or at least to those that admit of computable quasi tabular two-valued semantics, the so-called dyadic semantics.