Showing papers in "Logica Universalis in 2007"

A Universal Logic Approach to Adaptive Logics

Abstract: In this paper, adaptive logics are studied from the viewpoint of universal logic (in the sense of the study of common structures of logics). The common structure of a large set of adaptive logics is described. It is shown that this structure determines the proof theory as well as the semantics of the adaptive logics, and moreover that most properties of the logics can be proved by relying solely on the structure, viz. without invoking any specific properties of the logics themselves.

Cut-Free Ordinary Sequent Calculi for Logics Having Generalized Finite-Valued Semantics

Abstract: The paper presents a method for transforming a given sound and complete n-sequent proof system into an equivalent sound and complete system of ordinary sequents. The method is applicable to a large, central class of (generalized) finite-valued logics with the language satisfying a certain minimal expressiveness condition. The expressiveness condition decrees that the truth-value of any formula φ must be identifiable by determining whether certain formulas uniformly constructed from φ have designated values or not. The transformation preserves the general structure of proofs in the original calculus in a way ensuring preservation of the weak cut elimination theorem under the transformation. The described transformation metod is illustrated on several concrete examples of many-valued logics, including a new application to information sources logics.

A New Modal Lindström Theorem

Abstract: We prove new Lindstrom theorems for the basic modal propositional language, and for some related fragments of first-order logic. We find difficulties with such results for modal languages without a finite-depth property, high-lighting the difference between abstract model theory for fragments and for extensions of first-order logic. In addition we discuss new connections with interpolation properties, and the modal invariance theorem.

A Galois Connection

Abstract: The connection presented in this paper mirror-links two metamathematical structures, the finitary closure operators, and the compact consistency properties, in such a way that a specification of one structure induces a provably equivalent specification of the other.

Generalized Definitional Reflection and the Inversion Principle

Abstract: The term inversion principle goes back to Lorenzen who coined it in the early 1950s. It was later used by Prawitz and others to describe the symmetric relationship between introduction and elimination inferences in natural deduction, sometimes also called harmony. In dealing with the invertibility of rules of an arbitrary atomic production system, Lorenzen’s inversion principle has a much wider range than Prawitz’s adaptation to natural deduction. It is closely related to definitional reflection, which is a principle for reasoning on the basis of rule-based atomic definitions, proposed by Hallnas and Schroeder-Heister. After presenting definitional reflection and the inversion principle, it is shown that the inversion principle can be formally derived from definitional reflection, when the latter is viewed as a principle to establish admissibility. Furthermore, the relationship between definitional reflection and the inversion principle is investigated on the background of a universalization principle, called the ω-principle, which allows one to pass from the set of all defined substitution instances of a sequent to the sequent itself.

Modelling Inference in Argumentation through Labelled Deduction: Formalization and Logical Properties

Abstract: Artificial Intelligence (AI) has long dealt with the issue of finding a suitable formalization for commonsense reasoning. Defeasible argumentation has proven to be a successful approach in many respects, proving to be a con- fluence point for many alternative logical frameworks. Dierent formalisms have been developed, most of them sharing the common notions of argument and warrant. In defeasible argumentation, an argument is a tentative (de- feasible) proof for reaching a conclusion. An argument is warranted when it ultimately prevails over other conflicting arguments. In this context, defeasi- ble consequence relationships for modelling argument and warrant as well as their logical properties have gained particular attention. This article analyzes two non-monotonic inference operators Carg and Cwar intended for modelling argument construction and dialectical analysis (warrant), respectively. As a basis for such analysis we will use the LDSar framework, a unifying approach to computational models of argument using Labelled Deductive Systems (LDS). In the context of this logical framework, we show how labels can be used to represent arguments as well as argument trees, facilitating the definition and study of non-monotonic inference op- erators, whose associated logical properties are studied and contrasted. We contend that this analysis provides useful comparison criteria that can be extended and applied to other argumentation frameworks. Mathematics Subject Classification (2000). Primary 03B22; Secondary 03B42.

Information Algebras and Consequence Operators

Abstract: We explore a connection between different ways of representing information in computer science. We show that relational databases, modules, algebraic specifications and constraint systems all satisfy the same ten axioms. A commutative semigroup together with a lattice satisfying these axioms is then called an “information algebra”. We show that any compact consequence operator satisfying the interpolation and the deduction property induces an information algebra. Conversely, each finitary information algebra can be obtained from a consequence operator in this way. Finally we show that arbitrary (not necessarily finitary) information algebras can be represented as some kind of abstract relational database called a tuple system.

From Fibring to Cryptofibring. A Solution to the Collapsing Problem

Abstract: The semantic collapse problem is perhaps the main difficulty associated to the very powerful mechanism for combining logics known as fibring. In this paper we propose cryptofibred semantics as a generalization of fibred semantics, and show that it provides a solution to the collapsing problem. In particular, given that the collapsing problem is a special case of failure of conservativeness, we formulate and prove a sufficient condition for cryptofibring to yield a conservative extension of the logics being combined. For illustration, we revisit the example of combining intuitionistic and classical propositional logics.

A Global Glance on Categories in Logic

Abstract: We explore the possibility and some potential payoffs of using the theory of accessible categories in the study of categories of logics. We illustrate this by two case studies focusing on the category of finitary structural logics and its subcategory of algebraizable logics. Mathematics Subject Classification (2000). Primary 03B22; Secondary 18C35.

Functorial Duality for Ortholattices and De Morgan Lattices

Abstract: Relational semantics for nonclassical logics lead straightforwardly to topological representation theorems of their algebras. Ortholattices and De Morgan lattices are reducts of the algebras of various nonclassical logics. We define three new classes of topological spaces so that the lattice categories and the corresponding categories of topological spaces turn out to be dually isomorphic. A key feature of all these topological spaces is that they are ordered relational or ordered product topologies.

Recovering a Logic from Its Fragments by Meta-Fibring

Abstract: In this paper we address the question of recovering a logic system by combining two or more fragments of it. We show that, in general, by fibring two or more fragments of a given logic the resulting logic is weaker than the original one, because some meta-properties of the connectives are lost after the combination process. In order to overcome this problem, the categories Mcon and Seq of multiple-conclusion consequence relations and sequent calculi, respectively, are introduced. The main feature of these categories is the preservation, by morphisms, of meta-properties of the consequence relations, which allows, in several cases, to recover a logic by fibring of its fragments. The fibring in this categories is called meta−fibring. Several examples of well-known logics which can be recovered by meta-fibring its fragments (in opposition to fibring in the usual categories) are given. Finally, a general semantics for objects in Seq (and, in particular, for objects in Mcon) is proposed, obtaining a category of logic systems called Log. A general theorem of preservation of completeness by fibring in Log is also obtained.

Structuralist logic: Implications, inferences, and consequences

Abstract: On a structuralist account of logic, the logical operators, as well as modal operators are defined by the specific ways that they interact with respect to implication. As a consequence, the same logical operator (conjunction, negation etc.) can appear to be very different with a variation in the implication relation of a structure. We illustrate this idea by showing that certain operators that are usually regarded as extra-logical concepts (Tarskian algebraic operations on theories, mereological sum, products and negates of individuals, intuitionistic operations on mathematical problems, epistemic operations on certain belief states) are simply the logical operators that are deployed in different implication structures. That makes certain logical notions more omnipresent than one would think.

Abstract logics, logic maps, and logic homomorphisms

Abstract: What is a logic? Which properties are preserved by maps between logics? What is the right notion for equivalence of logics? In order to give satisfactory answers we generalize and further develop the topological approach of [4] and present the foundations of a general theory of abstract logics which is based on the abstract concept of a theory. Each abstract logic determines a topology on the set of theories. We develop a theory of logic maps and show in what way they induce (continuous, open) functions on the corresponding topological spaces. We also establish connections to well-known notions such as translations of logics and the satisfaction axiom of institutions [5]. Logic homomorphisms are maps that behave in some sense like continuous functions and preserve more topological structure than logic maps in general. We introduce the notion of a logic isomorphism as a (not necessarily bijective) function on the sets of formulas that induces a homeomorphism between the respective topological spaces and gives rise to an equivalence relation on abstract logics. Therefore, we propose logic isomorphisms as an adequate and precise notion for equivalence of logics. Finally, we compare this concept with another recent proposal presented in [2].

General Logic-Systems and Finite Consequence Operators

Abstract: In this paper, the significance of using general logic-systems and finite consequence operators defined on non-organized languages is discussed. Results are established that show how properties of finite consequence operators are independent from language organization and that, in some cases, they depend only upon one simple language characteristic. For example, it is shown that there are infinitely many finite consequence operators defined on any non-organized infinite language L that cannot be generated from any finite logic-system. On the other hand, it is shown that for any nonempty language L, a set map $$ C:{\user1{\mathcal{P}}}(L) \to {\user1{\mathcal{P}}}(L) $$ is a finite consequence operator if and only if it is defined by a general logic-system. Simple logic-system examples that determine specific consequence operator properties are given.

Structuring the Universe of Universal Logic

Abstract: How, why and what for we should combine logics is perfectly well explained in a number of works concerning this issue. But the interesting question seems to be the nature and the structure of the general universe of possible combinations of logical systems. Adopting the point of view of universal logic in the paper the categorical constructions are introduced which along with the coproducts underlying the fibring of logics describe the inner structure of the category of logical systems. It is shown that categorically the universe of universal logic turns out to be a topos and a paraconsistent complement topos.

Some Restricted Lindenbaum Theorems Equivalent to the Axiom of Choice

Abstract: Dzik [2] gives a direct proof of the axiom of choice from the generalized Lindenbaum extension theorem LET. The converse is part of every decent logical education. Inspection of Dzik’s proof shows that its premise let attributes a very special version of the Lindenbaum extension property to a very special class of deductive systems, here called Dzik systems. The problem therefore arises of giving a direct proof, not using the axiom of choice, of the conditional \( let \Rightarrow LET \). A partial solution is provided.

Some Multi-Conclusion Modal Paralogics

Abstract: I give a systematic presentation of a fairly large family of multiple-conclusion modal logics that are paraconsistent and/or paracomplete. After providing motivation for studying such systems, I present semantics and tableau-style proof theories for them. The proof theories are shown to be sound and complete with respect to the semantics. I then show how the “standard” systems of classical, single-conclusion modal logics fit into the framework constructed.