On conjectures in orthocomplemented lattices

On the reducibility of hypotheses and consequences

“…[1,2]), and the mathematical model for conjectures in ortholattices introduced in [8]. Let us first recall the definition of an ortholattice: Definition 1.…”

“…Although consequences and hypotheses, as well as several types of non-monotonic reasoning, deserved a good deal of attention by logicians, philosophers, computer scientists, and probabilists, no attempt of formalizing the concept of conjecture appeared before [8] was published. In the framework of an ortholattice, conjectures are defined in [8] as those elements non-contradictory with a given set of (non-incompatible) premises reflecting the available information.…”

“…In the framework of an ortholattice, conjectures are defined in [8] as those elements non-contradictory with a given set of (non-incompatible) premises reflecting the available information. That is, conjectures are those elements in the ortholattice that are ''possible", once the conjunction of the premises is known.…”

“…Paper [8] was followed by papers [7,4,10], as well as by Ying and Wang [11] and Qiu [5] where the case of residuated lattices was taken into account. The case of fuzzy set theories, which are never ortholattices, was considered in [3], and [9] deals with a generalization, by means of a previous Tarski's operator of consequences, of the way conjectures were introduced in [8], and newly allowing to consider hypotheses and speculations as particular conjectures.…”

“…For the time being only ortholattices (with the particular cases of orthomodular and boolean lattices) and residuated lattices have been considered. The present paper is set in the framework of ortholattices, continuing what was established in [8] for the case of hypotheses, aiming to provide a deeper insight on the structure of both hypotheses and consequences. In order to do so, we study what we call their reducibility, that is, the possibility for an hypothesis (respectively a consequence) to be obtained as the meet (respectively the join) of the premises' infimum with some other element of the lattice.…”