From IF to BI: a tale of dependence and separation

We take a fresh look at the logics of informational dependence and independence of Hintikka and Sandu and Vaananen, and their compositional semantics due to Hodges. We show how Hodges' semantics can be seen as a special case of a general construction, which provides a context for a useful completeness theorem with respect to a wider class of models. We shed some new light on each aspect of the logic. We show that the natural propositional logic carried by the semantics is the logic of Bunched Implications due to Pym and O'Hearn, which combines intuitionistic and multiplicative connectives. This introduces several new connectives not previously considered in logics of informational dependence, but which we show play a very natural role, most notably intuitionistic implication. As regards the quantifiers, we show that their interpretation in the Hodges semantics is forced, in that they are the image under the general construction of the usual Tarski semantics; this implies that they are adjoints to substitution, and hence uniquely determined. As for the dependence predicate, we show that this is definable from a simpler predicate, of constancy or dependence on nothing. This makes essential use of the intuitionistic implication. The Armstrong axioms for functional dependence are then recovered as a standard set of axioms for intuitionistic implication. We also prove a full abstraction result in the style of Hodges, in which the intuitionistic implication plays a very natural r\^ole.Comment: 28 pages, journal versio

Implicational Logic, Relevance, and Refutability

The goal of this paper is to analyse Implicational Relevance Logic from the point of view of refutability. We also correct an inaccuracy in our paper “The RM paraconsistent refutation system” (DOI: http://dx.doi.org/10.12775/LLP.2009.005)

Efficient Algorithms on the Family Associated to an Implicational System

International audienceAn implication system (IS) on a finite set S is a set of rules called Σ -implications of the kind A →_Σ B, with A,B ⊆ S. A subset X ⊆ S satisfies A →_Σ B when ''A ⊆ X implies B ⊆ X'' holds, so ISs can be used to describe constraints on sets of elements, such as dependency or causality. ISs are formally closely linked to the well known notions of closure operators and Moore families. This paper focuses on their algorithmic aspects. A number of problems issued from an IS Σ (e.g. is it minimal, is a given implication entailed by the system) can be reduced to the computation of closures φ _Σ (X), where φ _Σ is the closure operator associated to Σ . We propose a new approach to compute such closures, based on the characterization of the direct-optimal IS Σ _do which has the following properties: \beginenumerate ıtemit is equivalent to Σ ıtemφ _Σ _do(X) (thus φ _Σ (X)) can be computed by a single scanning of Σ _do-implications ıtemit is of minimal size with respect to ISs satisfying 1. and 2. \endenumerate We give algorithms that compute Σ _do, and from Σ _do closures φ _Σ (X) and the Moore family associated to φ _Σ

S (for Syllogism) Revisited: "The Revolution Devours its Children"

In 1978, the authors began a paper, “S (for Syllogism),” henceforth [S4S], intended as a philosophical companion piece to the technical solution [SPW] of the Anderson-Belnap P–W problem. [S4S] has gone through a number of drafts, which have been circulated among close friends. Meanwhile other authors have failed to see the point of the semantics which we introduced in [SPW]. It will accordingly be our purpose here to revisit that semantics, while giving our present views on syllogistic matters past, present and future, especially as they relate to not begging the question via such dubious theses as A →’ A. We shall investigate in particular a paraconsistent attitude toward such theses

Logics of Finite Hankel Rank

We discuss the Feferman-Vaught Theorem in the setting of abstract model theory for finite structures. We look at sum-like and product-like binary operations on finite structures and their Hankel matrices. We show the connection between Hankel matrices and the Feferman-Vaught Theorem. The largest logic known to satisfy a Feferman-Vaught Theorem for product-like operations is CFOL, first order logic with modular counting quantifiers. For sum-like operations it is CMSOL, the corresponding monadic second order logic. We discuss whether there are maximal logics satisfying Feferman-Vaught Theorems for finite structures.Comment: Appeared in YuriFest 2015, held in honor of Yuri Gurevich's 75th birthday. The final publication is available at Springer via http://dx.doi.org/10.1007/978-3-319-23534-9_1

Lattices with non-Shannon Inequalities

We study the existence or absence of non-Shannon inequalities for variables that are related by functional dependencies. Although the power-set on four variables is the smallest Boolean lattice with non-Shannon inequalities there exist lattices with many more variables without non-Shannon inequalities. We search for conditions that ensures that no non-Shannon inequalities exist. It is demonstrated that 3-dimensional distributive lattices cannot have non-Shannon inequalities and planar modular lattices cannot have non-Shannon inequalities. The existence of non-Shannon inequalities is related to the question of whether a lattice is isomorphic to a lattice of subgroups of a group.Comment: Ten pages. Submitted to ISIT 2015. The appendix will not appear in the proceeding

Relevance through topical unconnectedness: Ackermann and Plumwood’s motivational ideas on entailment

Ackermann’s motivational spin on his theory of rigorous implication is analyzed and it is shown to contain en equivalent idea to Plumwood’s notion of suppression freedom. The formal properties these ideas back turn out to be properly weaker than Belnap’s variable sharing property, but it is shown that they can be strengthen in various ways. Some such strengthenings, it is shown, yield properties which are equivalent to Belnap’s, and thus provide for new ways of motivating Belnap’s fundamental relevance principle