Logic Journal of IGPL - recent issues

Logics with the Qualitative Probability Operator

Ognjanovic, Z., Perovic, A., Raskovic, M. — 2008-03-17

The paper presents several strongly complete axiomatizations of qualitative probability within the framework of probabilistic logic. We show that in the proposed semantics qualitative probabilities are characterized by probability functions, so they also are comparative probabilities.

Heterogeneous Fibring of Deductive Systems Via Abstract Proof Systems

Cruz-Filipe, L., Sernadas, A., Sernadas, C. — 2008-03-17

Fibring is a meta-logical constructor that applied to two logics produces a new logic whose formulas allow the mixing of symbols. Homogeneous fibring assumes that the original logics are presented in the same way (e.g via Hilbert calculi). Heterogeneous fibring, allowing the original logics to have different presentations (e.g. one presented by a Hilbert calculus and the other by a sequent calculus), has been an open problem. Herein, consequence systems are shown to be a good solution for heterogeneous fibring when one of the logics is presented in a semantic way and the other by a calculus and also a solution for the heterogeneous fibring of calculi. The new notion of abstract proof system is shown to provide a better solution to heterogeneous fibring of calculi namely because derivations in the fibring keep the constructive nature of derivations in the original logics. Preservation of compactness and semi-decidability is investigated.

Sequent Calculi for Some Strict Implication Logics

Ishigaki, R., Kashima, R. — 2008-03-17

We introduce various sequent systems for propositional logics having strict implication, and prove the completeness theorems and the finite model properties of these systems.The cut-elimination theorems or the (modified) subformula properties are proved semantically.

Constructing Finite Least Kripke Models for Positive Logic Programs in Serial Regular Grammar Logics

Nguyen, L. A. — 2008-03-17

A serial context-free grammar logic is a normal multimodal logic L characterized by the seriality axioms and a set of inclusion axioms of the form _t->_s₁..._{s_k}. Such an inclusion axiom corresponds to the grammar rule t -> s₁... s_k. Thus the inclusion axioms of L capture a context-free grammar $$\mathcal{G}(L)$$. If for every modal index t, the set of words derivable from t using $$\mathcal{G}(L)$$ is a regular language, then L is a serial regular grammar logic.

In this paper, we present an algorithm that, given a positive multimodal logic program P and a set of finite automata specifying a serial regular grammar logic L, constructs a finite least L-model of P. (A model M is less than or equal to model M' if for every positive formula , if M then M' .) A least L-model M of P has the property that for every positive formula , P iff M . The algorithm runs in exponential time and returns a model with size 2^O(n³). We give examples of P and L, for both of the case when L is fixed or P is fixed, such that every finite least L-model of P must have size 2⁽ⁿ⁾. We also prove that if G is a context-free grammar and L is the serial grammar logic corresponding to G then there exists a finite least L-model of _s p iff the set of words derivable from s using G is a regular language.

On Ignorance and Contradiction Considered as Truth-Values

Dubois, D. — 2008-03-17

A critical view of the alleged significance of Belnap four-valued logic for reasoning under inconsistent and incomplete information is provided. The difficulty lies in the confusion between truth-values and information states, when reasoning about Boolean propositions. So our critique is along the lines of previous debates on the relevance of many-valued logics and especially of the extension of the Boolean truth-tables to more than two values as a tool for reasoning about uncertainty. The critique also questions the significance of partial logic.

Conferences

2008-03-17

Acknowledgements

2008-03-17

An Introduction to Basic Arithmetic

Ardeshir, M., Hesaam, B. — 2008-01-23

We study Basic Arithmetic BA, which is the basic logic BQC equivalent of Heyting Arithmetic HA over intuitionistic logic IQC, and of Peano Arithmetic PA over classical logic CQC. It turns out that The Friedman translation is applicable to BA. Using this translation, we prove that BA is closed under a restricted form of the Markov rule. Moreover, it is proved that all nodes of a finite Kripke model of BA are classical models of $${I\exists }_{1}^{+}$$, a fragment of PA with Induction restricted to the formulas made up of , and/or . We also study an interesting extension of BQC, called EBQC, which is the extension by the axiom schema -> . We show that this extension behaves very like to IQC, and the corresponding arithmetic, EBA shows similarities to HA.

Mathematics Subject Classification: Primary 03F30; secondary 03F50.

Congruences on Dynamic Algebras

Pinto, S. M., Oliveira-Martins, T., Pinto, M. C. — 2008-01-23

The lattice Cong$$\mathcal{D}$$ of all dynamic congruences on a given dynamic algebra $$\mathcal{D}$$ is presented. Whenever $$\mathcal{D}$$ is separable with zero we define dynamic ideal on $$\mathcal{D}$$, given rise to the lattice Ide$$\mathcal{D}$$. The notions of kernel of a dynamic congruence and the congruence generated by a dynamic ideal are introduced to describe a Galois connection between Cong$$\mathcal{D}$$ and Ide$$\mathcal{D}$$. We study conditions under which a dynamic congruence is determined by its kernel.

The Basic Constructive Logic for a Weak Sense of Consistency defined with a Propositional Falsity Constant

Robles, G., Mendez, J. M. — 2008-01-23

The logic B_Kc1 is the basic constructive logic in the ternary relational semantics (without a set of designated points) adequate to consistency understood as the absence of the negation of any theorem. Negation is introduced in B_Kc1 with a negation connective. The aim of this paper is to define the logic B_Kc1F. In this logic negation is introduced via a propositional falsity constant. We prove that B_Kc1 and B_Kc1F are definitionally equivalent.

Term Definable Classes of Boolean Functions and Frame Definability in Modal Logic

Couceiro, M., Hella, L., Kivela, J. — 2008-01-23

We establish a connection between term definability of Boolean functions and definability of finite modal frames. We introduce a bijective translation between functional terms and uniform degree-1 formulas and show that a class of Boolean functions is defined by functional terms if and only if the corresponding class of Scott-Montague frames is defined by the translations of these functional terms, and vice versa. As a special case, we get that the clone ₁ of all conjunctions corresponds to the class of all Kripke frames. We also characterize some classes of Scott-Montague frames corresponding to subclones of ₁ by restricting the class of Kripke frames in a natural way. Furthermore, by modifying Kripke semantics, we extend our results to correspondences between linear clones and classes of Kripke frames equipped with modified semantics.

Ternary Exclusive Or

Pelletier, F. J., Hartline, A. — 2008-01-23

Ternary exclusive or is the (two valued) truth function that is true just in case exactly one of its three arguments is true. This is an interesting truth function, not definable in terms of the binary exclusive or alone, although the binary case is definable in terms of the ternary case. This article investigates the types of truth functions that can be defined by ternary exclusive or, and relates these findings to the seminal work of Emil Post.

Maximum Entropy Inference with Quantified Knowledge

Barnett, O., Paris, J. — 2008-01-23

We investigate uncertain reasoning with quantified sentences of the predicate calculus treated as the limiting case of maximum entropy inference applied to finite domains.

An Application of Model Theory to Semimodules

Zayed, M. — 2008-01-23

In this note, we prove that the theory T of cancellative semimodules over a semiring R has the amalgamation property. If R is an entire cancellative zerosumfree semiring, then T has no model-companion. In particular, the theory of commutative additively cancellative monoids forms an example of a non-companionable theory.

Acknowledgements

2008-01-23

Foreword

2007-12-03

An Inductive Theorem on the Correctness of General Recursive Programs

Bohorquez, J. A. — 2007-12-03

We prove a relatively simple inductive theorem (analogous to Floyd and Dijkstra's Invariance Theorem for iterative programs) to verify the correctness of an ample class of non-deterministic general recursive programs. This result is based on Hoare and Jifeng's relational semantics. By considering the structure of its code and specification, we propose regularity conditions on the predicate transformer associated to the fixed-point equation of a general (non deterministic) recursive program to prove it correct by induction on a well founded ordering of a covering of the domain of its corresponding input-output relation. All fixed point solutions associated to a predicate transformer satisfying these regularity conditions coincide when restricted to the domain of its least fixed point solution. We find these conditions non unduly restrictive, since continuous operators defining deterministic programs as their corresponding least fixed-point solutions must fulfill them. We couch deterministic programs (viewed as least solutions of fixed-point equations) in a restriction of Hoare and Jifeng's $$\mathcal{P}$$ programming language of (partial) finitary relations into the greatest solutions of fixed-point equations expressed in terms of "total finitary" relations of an adequate restriction of their $$\mathcal{D}$$ language

The Rules of Logic Composition for the Bayesian Epistemic e-Values

Borges, W., Stern, J. M. — 2007-12-03

In this paper, the relationship between the e-value of a complex hypothesis, H, and those of its constituent elementary hypotheses, H^j, j = 1... k, is analyzed, in the independent setup. The e-value of a hypothesis H, ev(H), is a Bayesian epistemic, credibility or truth value defined under the Full Bayesian Significance Testing (FBST) mathematical apparatus. The questions addressed concern the important issue of how the truth value of H, and the truth function of the corresponding FBST structure M, relate to the truth values of its elementary constituents, H^j, and to the truth functions of their corresponding FBST structures M^j, respectively.

Logical and Philosophical Remarks on Quasi-Set Theory

Da Costa, N. C. A., Krause, D. — 2007-12-03

Quasi-set theory is a theory for dealing with collections of indistinguishable objects. In this paper we discuss some logical and philosophical questions involved with such a theory. The analysis of these questions enable us to provide the first grounds of a possible new view of physical reality, founded on an ontology of non-individuals, to which quasi-set theory may constitute the logical basis.

Large Cardinals and Topology: a Short Retrospective and Some New Results

Da Silva, S. G. — 2007-12-03

This paper is an enlarged version of the short talk delivered at XIV Brazilian Logic Conference (XIV EBL 2006, Itatiaia, Rio de Janeiro). The author's purpose is to present some applications of large cardinals in general topology, pointing out that there are several topological problems that cannot be settled without dealing with inaccessible cardinals. Various "classical examples" are mentioned, together with recent results. In the last section a new result is presented: it is shown that the existence of a separable space with an uncountable closed discrete subset satisfying a certain relative version of countable paracompactness implies the existence of inner models with measurable cardinals.

Fast-Growing Functions and the P vs. NP Question

Doria, F. A. — 2007-12-03

Out of a folklore–like fact about the counterexample function to P=NP, a function which grows about as fast as the Busy Beaver function, we review the consequences of our (with da Costa) exotic formalization for P=NP and then speculate about possible ways to extend our work.

Using the Internal Logic of a Topos to Model Search Spaces for Problems

do Amaral, F. N., Haeusler, E. H. — 2007-12-03

We present a structural model for (meta)heuristic search strategies for solving computational problems. The model is defined through the use of topos-theoretical tools and techniques, which provide an appropriate internal logic (with the language of local set theory) where objects of interest can be represented.

Fibring in the Leibniz Hierarchy

Fernandez, V. L., Coniglio, M. E. — 2007-12-03

This article studies preservation of certain algebraic properties of propositional logics when combined by fibring. The logics analyzed here are classified in protoalgebraic, equivalential and algebraizable. By introducing new categories of algebrizable logics and of deductivizable quasi-varieties, it is stated an isomorphism between these categories. This constitutes an alternative to a similar result found in the literature.

On Minimal Models

Ferreira, F. M., Martins, A. T. — 2007-12-03

We investigate some logics which use the concept of minimal models in their definition. Minimal objects are widely used in Logic and Computer Science. They are applied in the context of Inductive Definitions, Logic Programming and Artificial Intelligence. An example of logic which uses this concept is the MIN(FO) logic due to van Benthem [20]. He shows that MIN(FO) is equivalent to the Least Fixed Point logic (LFP) in expressive power. In [6], we extended MIN(FO) to the MIN Logic and proved it is equivalent to second-order logic in expressive power. Here, we exhibit a fragment of MIN, the MIN logic, which is more expressive than LFP, less expressive than MIN and closed under boolean connectives and first-order quantification. In order to do this, in the Section 2, we prove that the Downward Löwenheim-Skolem Theorem holds for arbitrary countable sets of LFP-formulas by showing that every infinite structure has a countable LFP-substructure. The method may be used to generalize this theorem to any set of LFP-formulas. We also analyse the expressive power of the Nested Abnormality Theories (NATs) of Lifschitz, another formalism based on minimal models used in Artificial Intelligence, and we demonstrate that for each second-order theory there is a NAT which is a conservative extension of . We give a translation from second-order sentences into such NATs which is linear in the size of the sentence in prenex normal form. Finally, we establish a hierarchy of expressiveness of these logics that deal with the concept of minimal models.

Pure Hilbert Algebras with Infimum

Figallo, A. — 2007-12-03

In [6], iH-algebras were introduced in order to indicate an equational version of the class of Hilbert algebras where each pair of elements has infimum. These authors also proved that this variety has the class of Curry's implicative semilattices ([9]) as a proper subvariety. On the other hand, in [4] a special class of Hilbert algebras associated with ordered sets, which they called order algebras, were investigated. These algebras were also studied in [1] under the name of pure Hilbert algebras.

Bearing in mind the above results, in this paper we introduce the notion of pure Hilbert algebras with infimum (or ipH-algebras, for short). Furthermore, we characterize the lattice of ipH-congruences and we determine the subdirectly irreducible ipH-algebras. Besides, we prove that subdirectly irreducible ipH-algebras are also subdirectly irreducible iH-algebras.

Monadic Distributive Lattices

Figallo, A. V., Pascual, I., Ziliani, A. — 2007-12-03

The purpose of this paper is to investigate the variety of algebras, which we call monadic distributive lattices, as a natural generalization of monadic Heyting algebras [16]. It is worth mentioning that the latter is a proper subvariety of the first one, as it is shown in a simple example. Our main interest is the characterization of simple and subdirectly irreducible monadic distributive lattices. In order to do this, a duality theory for these algebras is developed. The duality enables us to describe the lattice of congruences on monadic distributive lattices. Finally, our attention is focused upon the relationship between the category of dual spaces associatted with these algebras and the category of perfect Ono frames considered by Bezhanishvili in order to represent monadic Heyting algebras.

Equal Rights for the Cut: Computable Non-analytic Cuts in Cut-based Proofs

Finger, M., Gabbay, D. — 2007-12-03

This work studies the structure of proofs containing non-analytic cuts in the cut-based system, a sequent inference system in which the cut rule is not eliminable and the only branching rule is the cut. Such sequent system is invertible, leading to the KE-tableau decision method. We study the structure of such proofs, proving the existence of a normal form for them in the form of a comb-tree proof.

We then concentrate on the problem of efficiently computing non-analytic cuts. For that, we study the generalisation of techniques present in many modern theorem provers, namely the techniques of conflict-driven formula learning.

On Positive Relational Calculi

de Freitas, R., Veloso, P. A.S., Veloso, S. R.M., Viana, P. — 2007-12-03

We discuss the question of inclusions between positive relational terms and some of its aspects, using the form of a dialogue. Two possible approaches to the problem are emphasized: natural deduction and graph manipulations. Both provide sound and complete calculi for proving the valid inclusions, supporting nice strategies to obtain proofs in normal form, but the latter appears to present several advantages, which are discussed.

Language and Logical Pluralism: Some Aspects of a Wittgensteinian Perspective on the Nature of Logic

Gomez, S. — 2007-12-03

This essay examines the importance of some aspects of Wittgenstein's post-Tractatus work in the realm of discussions on the nature of logic. The first part considers a relationship between certain conceptions of language and certain positions on the nature of logical laws and logical pluralism. Supposing the rejection of mentalism in the field of meaning leads to a rejection of psychologism, it presents some alternatives different from psychologism, based on non mentalistic theories of meaning. One is the Platonistic Fregean approach to language and logic, the other is Carnap's formalist view on both topics. The second part concentrates on Wittgenstein's non mentalistic and non Platonistic proposals about language and his defense of the logical pluralism proposed by psychologists. It compares two periods on Wittgenstein's work after Tractatus -the periods of ‘calculus conception’ and ‘languange games conception’- and it shows how characteristic notions of Wittgenstein's later conception of language, like ‘use’, ‘language games’, and ‘forms of life’, work on the characterization of logic and specially on the kind of logical pluralism that the author seems to defend in his last period. In doing so, this essay offers an approach to some of the author's considerations about contradictions and the possibility of the existence of a calculus that includes them. This approach emphasizes on the idea of applicability (or use of a linguistic expression) introduced by the author in some of his last works, and in some examples of functional contradictions that can help to understand and complement that idea.

Why and How Platonism?

Rosado Haddock, G. E. — 2007-12-03

Probably the best arguments for Platonism are those directed against its rival philosophies of mathematics. Frege's arguments against formalism, Gödel's arguments against constructivism and those against the so-called syntactic view of mathematics, and an argument of Hodges against Putnam are expounded, as well as some arguments of the author. A more general criticism of Quine's views follows. The paper ends with some thoughts on mathematics as a sort of Platonism of structures, as conceived by Husserl and essentially endorsed by the author.

On Extensions of Elementary Submodels by Forcing

Junqueira, L. R., Larson, P., Passos, M. D. — 2007-12-03

We study when a partial order $$\mathbb{P}$$ preserves certain properties of an elementary submodel (of a structure of the form H), including countable closure and -covering. In particular, we discuss the existence of -covering elementary submodels of different sizes.

Games on Trees and Syntactical Complexity of Formulas

Krynicki, M., Torres, J. M. T. — 2007-12-03

Ehrenfeucht-Fraïssé games have been introduced as a means of characterizing the relation of elementary equivalence between structures, or relational database instances in first order logic (FO), or equivalently Relational Calculus. In~the usual Ehrenfeucht-Fraïssé games the rules are determined by a linear ordering of a fixed lenght or, equivalently, by a special kind of tree – a chain of a fixed length –, where each point of that ordering or node of that tree corresponds to a quantification operation. Here we consider Ehrenfeucht-Fraïssé games whose rules are determined by arbitrary trees such that their nodes correspond either to quantification operations ("q–nodes") or to connective operations ("c–nodes"). By playing games on trees, we can refine the class of sentences which are considered in a given game, since a tree represents a particular class of sentences. We define and study several variations of tree games, for first and second order logic (SO). We give a sufficient condition for FO and SO equivalence restricted to formulae with up to n connectives, and hence also a sufficient condition for the non expressibility of a given query in those logics with formulae whose number of logical connectives is less than a given integer. We also give a sufficient condition to prove simultaneous lower bounds in both the number of connectives and in the quantifier types needed to express a given property in FO. If we consider only quantifier types, we get a characterization of the relation of preservation of sentences in the fragment of FO with the given set of quantifier types. We also study tree games for _n and _n formulae. To illustrate the use of our games we use them to prove lower bounds in the connective size for several FO queries, like size of a database, size of a clique in a graph, size of a unary relation, transitive property in a graph, and degree of a node in a graph. Regarding SO, we prove lower bounds for quantifier rank for the parity query.

Finally, we give a precise characterization of the logic whose elementary equivalence is characterized by a given tree game, as well as several equivalent characterizations of the existence of a winning strategy for Duplicator in the classical Ehrenfeucht-Fraïssé game.

The Logic of Classes

Lopez-Escobar, E. G. K. — 2007-12-03

An extension of the Quantified Propositional Calculus¹ obtained by the addition of two binary propositional functions is put forward as an inheritor of E. Schröder's "Algebra der Logik". The formal system is itself not new, in fact it forms part of A. P. Morse's "A Theory of Sets"; although the latter is considered as a first-order system (of a rather non-standard type).

Since the additional propositional functions are not invariant under the logical biconditional, this system–and many others naturally obtained from it–give us a collection of examples of non-standard, but mathematically meaningful, propositional systems.

Fibred and Indexed Categories for Abstract Model Theory

Martini, A., Wolter, U., Haeusler, E. H. — 2007-12-03

Indexed and Fibred category theory have a long tradition in computer science as a language to formalize different presentations of the notion of a logic, as for instance, in the theory of institutions and general logics, and as unifying models of (categorical) logic and type theory as well. Here we introduce the notions of indexed and fibred frames and construct a rich mathematical workspace where many relevant and useful concepts of logics can be elegantly modelled. To demonstrate the applicability of these tools, essential ideas around the theory of institutions are recasted and described.

Restricted Classical Modal Logics

Mortari, C. A. — 2007-12-03

We consider a family of noncongruential modal logics obtained by restricting the smallest classical modal logic E and some of its extensions. We show that these logics are also properly contained in Lemmon's S0.5; semantics for them are adapted from Cresswell's semantics for S0.5 and from neighborhood semantics for classical modal logics. Some extensions of these logics by means of usual modal logical axioms are also considered, and determination results proved. As a further example, we also show how to obtain restricted versions of some of Chellas and Segerberg's prenormal modal logics.

Criteria of Identity and their Logical Form

Pedroso, M. M. — 2007-12-03

The goal of this paper is to wonder whether there is some pattern preserved throughout every criterion of identity. Two solutions will be examined: Leibniz's Law and Lowe's criterion of identity. In particular, it will be defended the following theses about them:

they are circular definitions;

their circularity, by itself, does not make them untenable accounts.

A Note on Gentzen's LJ and NJ Systems Isomorphism

De Campos Sanz, W. — 2007-12-03

In this paper we are going to examine intuitionistic sequent calculus and its negation rules. We state new negation rules defining, in this way, a new sequent system. It will be used to clarify Gentzen's NJ and LJ systems isomorphism. These new negation rules are a direct reading of new natural deduction negation rules obtained by a slight modification of NJ rules. We also show that the new system is equivalent to LJ and that the Hauptsatz holds for it.

Natural Deduction for 'Generally'

Vana, L. B., Veloso, P. A. S., Veloso, S. R. M. — 2007-12-03

Logics for ‘generally’ (LG’s) were introduced for handling assertions with vague notions (e.g. ‘generally’, ‘most’, ‘several’), which occur often in ordinary language and in science. LG’s provide a framework for distinct notions of ‘generally’: one builds a specific logic for the notion one has in mind. We introduce deductive systems, in natural deduction style, for LG’s and show that these systems are normalizable.

Acknowledgements

2007-12-03