An orthogonality-based classification of conjectures in ortholattices

Abstract

A mathematical model for conjectures (including hypotheses, consequences and speculations), was recently introduced, in the context of ortholattices, by Trillas, Cubillo and Castiñeira (Artificial Intelligence 117, 2000, 255-257). The aim of the present paper is to further clarify the structure of this model by studying its relationships with one of the most important ortholattices' relation, the orthogonality relation. The particular case of orthomodular lattices -the framework for both Boolean and quantum logics- is specifically taken into account.

[Reviewed by Vinayak V. Joshi]- -In the present paper, the authors prove results regarding the orthogonality of the elements of the set of strict conjectures with $p_{\wedge}$ in ortholattices; their results may be summarized as follows:

1. The only conjectures which are always left-orthogonal to $p_{\wedge}$ are consequences, whereas hypotheses are the only ones which are always right-orthogonal. But in the case of orthomodular lattices, consequences and hypotheses both are orthogonal to $p_{\wedge}$.

2. In general, it is possible to find conjectures of any kind (consequences, hypotheses or speculations of the two types) that are both left- and right-orthogonal to $p_{\wedge}$.

3. A conjecture that is neither left- nor right-orthogonal to $p_{\wedge}$ is necessarily a speculation.