From IF to BI: a tale of dependence and separation

TL;DR

This paper explores the semantics of informational dependence and independence, revealing their connection to Bunched Implications and introducing new logical connectives, with implications for understanding dependence in logical systems.

Contribution

It demonstrates how Hodges' semantics generalizes to a broader class of models, introduces new connectives like intuitionistic implication, and relates dependence predicates to simpler forms within a unified logical framework.

Findings

Hodges' semantics is a special case of a general construction.

The logic of Bunched Implications naturally arises from the semantics.

Dependence predicates can be defined using simpler predicates and intuitionistic implication.

Abstract

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.