Factor semantics forn-valued logics

In this note we prove that some familiar systems of finitely many-valued logics havefactor semantics, and establish necessary conditions for a system of many-valued logic having semantics of this kind.     

The analysis of semigroups of multirelational systems

A finite family of binary relations on a finite set, termed here a relational system, generates a finite semigroup under the operation of relational composition. The relationship between simplifications of the semigroup of a relational system in the form of homomorphic images, and simplifications of the relational system itself is examined. First of all, the list of relational conditions establishing a relationship between a homomorphic image of the semigroup of a relational system and a simplified, or derived, version of that relational system, is reviewed and extended. Then a definition of empirical relationship is introduced (the Correspondence Definition) and it is shown how, in conjunction with a factorization procedure for finite semigroups (P. E. Pattison & W. K. Bartlett, Journal of Mathematical Psychology, 1982, in press), it leads to a systematic and efficient analysis for a relational system. Applications of the procedure to an empirical blockmodel and to a class of simple relational systems are presented.     

Contextual Deduction Theorems

Logics that do not have a deduction-detachment theorem (briefly, a DDT) may still possess a contextual DDT??a syntactic notion introduced here for arbitrary deductive systems, along with a local variant. Substructural logics without sentential constants are natural witnesses to these phenomena. In the presence of a contextual DDT, we can still upgrade many weak completeness results to strong ones, e.g., the finite model property implies the strong finite model property. It turns out that a finitary system has a contextual DDT iff it is protoalgebraic and gives rise to a dually Brouwerian semilattice of compact deductive filters in every finitely generated algebra of the corresponding type. Any such system is filter distributive, although it may lack the filter extension property. More generally, filter distributivity and modularity are characterized for all finitary systems with a local contextual DDT, and several examples are discussed. For algebraizable logics, the well-known correspondence between the DDT and the equational definability of principal congruences is adapted to the contextual case.     

Nondeterministic three-valued logic: Isotonic and guarded truth-functions

Nondeterministic programs occurring in recently developed programming languages define nondeterminate partial functions. Formulas (Boolean expressions) of such nondeterministic languages are interpreted by a nonempty subset of {T (“true”), F (“false”), U (“undefined)}. As a semantic basis for the propositional part of a corresponding nondeterministic three-valued logic we study the notion of a truth-function over {T, F, U} which is computable by a nondeterministic evaluation procedure. The main result is that these truth-functions are precisely the functions satisfying four basic properties, called \( \subseteq \) -isotonic, \( \subseteq \) ^?-isotonic, hereditarily guarded, and hereditarily guard-using, and that a function satisfies these properties iff it is explicitly definable (in a certain normal form) from “if..then..else..fi”, binary choice, and constants.     

A General Setting for Dedekind's Axiomatization of the Positive Integers

A Dedekind algebra is an ordered pair (B, h), where B is a non-empty set and h is a similarity transformation on B. Among the Dedekind algebras is the sequence of the positive integers. From a contemporary perspective, Dedekind established that the second-order theory of the sequence of the positive integers is categorical and finitely axiomatizable. The purpose here is to show that this seemingly isolated result is a consequence of more general results in the model theory of second-order languages. Each Dedekind algebra can be decomposed into a family of disjoint, countable subalgebras called the configurations of the algebra. There are ?₀ isomorphism types of configurations. Each Dedekind algebra is associated with a cardinal-valued function on ω called its configuration signature. The configuration signature counts the number of configurations in each isomorphism type that occurs in the decomposition of the algebra. Two Dedekind algebras are isomorphic iff their configuration signatures are identical. The second-order theory of any countably infinite Dedekind algebra is categorical, and there are countably infinite Dedekind algebras whose second-order theories are not finitely axiomatizable. It is shown that there is a condition on configuration signatures necessary and sufficient for the second-order theory of a Dedekind algebra to be finitely axiomatizable. It follows that the second-order theory of the sequence of the positive integers is categorical and finitely axiomatizable.     

Equational Bases for Joins of Residuated-lattice Varieties

Given a positive universal formula in the language of residuated lattices, we construct a recursive basis of equations for a variety, such that a subdirectly irreducible residuated lattice is in the variety exactly when it satisfies the positive universal formula. We use this correspondence to prove, among other things, that the join of two finitely based varieties of commutative residuated lattices is also finitely based. This implies that the intersection of two finitely axiomatized substructural logics over FL ⁺ is also finitely axiomatized. Finally, we give examples of cases where the join of two varieties is their Cartesian product.     

On detachment-substitutional formalization in normal modal logics

The aim of this paper is to propose a criterion of finite detachment-substitutional formalization for normal modal systems. The criterion will comprise only those normal modal systems which are finitely axiomatizable by means of the substitution, detachment for material implication and Gödel rules.Some results of this paper were announced in the abstract [2].Allatum est die 10 Junii 1976     

On the Complexity of Nonassociative Lambek Calculus with Unit

Nonassociative Lambek Calculus (NL) is a syntactic calculus of types introduced by Lambek [8]. The polynomial time decidability of NL was established by de Groote and Lamarche [4]. Buszkowski [3] showed that systems of NL with finitely many assumptions are decidable in polynomial time and generate context-free languages; actually the P-TIME complexity is established for the consequence relation of NL. Adapting the method of Buszkowski [3] we prove an analogous result for Nonassociative Lambek Calculus with unit (NL1). Moreover, we show that any Lambek grammar based on NL1 (with assumptions) can be transformed into an equivalent context-free grammar in polynomial time.     

Protoalgebraic logics

There exist important deductive systems, such as the non-normal modal logics, that are not proper subjects of classical algebraic logic in the sense that their metatheory cannot be reduced to the equational metatheory of any particular class of algebras. Nevertheless, most of these systems are amenable to the methods of universal algebra when applied to the matrix models of the system. In the present paper we consider a wide class of deductive systems of this kind called protoalgebraic logics. These include almost all (non-pathological) systems of prepositional logic that have occurred in the literature. The relationship between the metatheory of a protoalgebraic logic and its matrix models is studied. The following results are obtained for any finite matrix model U of a filter-distributive protoalgebraic logic : (I) The extension _U of is finitely axiomatized (provided has only finitely many inference rules); (II) _U has only finitely many extensions.     

Characterizing Equivalential and Algebraizable Logics by the Leibniz Operator

In [14] we used the term finitely algebraizable for algebraizable logics in the sense of Blok and Pigozzi [2] and we introduced possibly infinitely algebraizable, for short, p.i.-algebraizable logics. In the present paper, we characterize the hierarchy of protoalgebraic, equivalential, finitely equivalential, p.i.-algebraizable, and finitely algebraizable logics by properties of the Leibniz operator. A Beth-style definability result yields that finitely equivalential and finitely algebraizable as well as equivalential and p.i.-algebraizable logics can be distinguished by injectivity of the Leibniz operator. Thus, from a characterization of equivalential logics we obtain a new short proof of the main result of [2] that a finitary logic is finitely algebraizable iff the Leibniz operator is injective and preserves unions of directed systems. It is generalized to nonfinitary logics. We characterize equivalential and, by adding injectivity, p.i.-algebraizable logics.     

Finite Basis Theorem for Filter-distributive Protoalgebraic Deductive Systems and Strict Universal Horn Classes

We show that a finitely generated protoalgebraic strict universal Horn class that is filter-distributive is finitely based. Equivalently, every protoalgebraic and filter-distributive multidimensional deductive system determined by a finite set of finite matrices can be presented by finitely many axioms and rules.     

The Lattice of Subvarieties of {\sqrt{\prime}} quasi-MV Algebras

In the present paper we continue the investigation of the lattice of subvarieties of the variety of ${\sqrt{\prime}}$ quasi-MV algebras, already started in [6]. Beside some general results on the structure of such a lattice, the main contribution of this work is the solution of a long-standing open problem concerning these algebras: namely, we show that the variety generated by the standard disk algebra D _r is not finitely based, and we provide an infinite equational basis for the same variety.     

Representations of monadic <Emphasis Type="Italic">MV</Emphasis> -algebras

Representations of monadic MV -algebra, the characterization of locally finite monadic MV -algebras, with axiomatization of them, definability of non-trivial monadic operators on finitely generated free MV -algebras are given. Moreover, it is shown that finitely generated m-relatively complete subalgebra of finitely generated free MV -algebra is projective.     

Finite additivity,another lottery paradox and conditionalisation

In this paper I argue that de Finetti provided compelling reasons for rejecting countable additivity. It is ironical therefore that the main argument advanced by Bayesians against following his recommendation is based on the consistency criterion, coherence, he himself developed. I will show that this argument is mistaken. Nevertheless, there remain some counter-intuitive consequences of rejecting countable additivity, and one in particular has all the appearances of a full-blown paradox. I will end by arguing that in fact it is no paradox, and that what it shows is that conditionalisation, often claimed to be integral to the Bayesian canon, has to be rejected as a general rule in a finitely additive environment.     

On the Proof Theory of the Modal mu-Calculus

We study the proof-theoretic relationship between two deductive systems for the modal mu-calculus. First we recall an infinitary system which contains an omega rule allowing to derive the truth of a greatest fixed point from the truth of each of its (infinitely many) approximations. Then we recall a second infinitary calculus which is based on non-well-founded trees. In this system proofs are finitely branching but may contain infinite branches as long as some greatest fixed point is unfolded infinitely often along every branch. The main contribution of our paper is a translation from proofs in the first system to proofs in the second system. Completeness of the second system then follows from completeness of the first, and a new proof of the finite model property also follows as a corollary. Presented by Heinrich Wansing     

Multiple and iterated contraction reduced to single-step single-sentence contraction

Multiple contraction (simultaneous contraction by several sentences) and iterated contraction are investigated in the framework of specified meet contraction (s.m.c.) that is extended for this purpose. Multiple contraction is axiomatized, and so is finitely multiple contraction (contraction by a finite set of sentences). Two ways to reduce finitely multiple contraction to contraction by single sentences are introduced. The reduced operations are axiomatically characterized and their properties are investigated. Furthermore, it is shown how iterated contraction can be reduced to single-step, single-sentence contraction. However, in this framework the outcome of iterated contraction depends unavoidably on the order in which the inputs are received. This order-dependence makes it impossible to treat two inputs on an equal footing. Therefore it is often preferable to perform changes involving several pieces of information as multiple rather than iterated change.     

Checking quasi-identities in a finite semigroup may be computationally hard

We exhibit a 10-element semigroup Q such that the question “Does a given quasi-identity hold in Q?” is co-NP-complete while the question “Does a given identity hold in Q?” can be answered in linear time.     

Two core systems of numerical representation in infants

Two nonverbal representation systems, the analog magnitude system (AMS) and the object tracking system (OTS), have been proposed to explain how humans and nonhuman animals represent numerosities. There has long been debate about which of the two systems is responsible for representing small numerosities (<4). This review focuses on findings with human infants to inform that debate. We argue that the empirical data cannot all be explained by a single system, and in particular, infants’ failures to compare small and large numerosities – the boundary effect – undermines the claim that the AMS can account for infants’ numerical abilities in their entirety. We propose that although the two systems coexist throughout the lifespan, competition between the systems is primarily a developmental phenomenon. Potential factors that drive the engagement of each system in infancy, such as stimulus features and task demands, are discussed, and directions for future research are suggested.