Merging inference and conjecture by information

Abstract

The intuitive notion of a binary relation on information-bearers, comparingthem with respect to their closeness to the available information, is oftenconstrued in terms of comparing their symmetric difference with, orcompositional similarity to, the available information. This happens forinstance in some treatments of verisimilitude. We expound an abstractmathematical rendering of the relevant data-dependent relation in theframework of Boolean algebras. For every element t of a Boolean algebra B we construct the t-modulated Boolean algebra Btin which the order relation represents `is at most as compatible with t as' or `is at best as similar to t as'. In the case of Lindenbaum-Tarskialgebras, t expresses the available information, and the compatibilityrelation turns out to be an entwinement of inferential and conjecturalrelations. It is just classical entailment when no information is available(i.e., when t is logically true) and becomes more boldly abductive themore information is available. The rich algebraic structures of a Boolean algebra –- including its Boolean group structures –- play a significant role in this combination of deduction and abduction and also induce cautious anddaring variants of the compatibility relation. Links with the literature onverisimilitude, abduction, and related topics are indicated.