Binary resolution over Boolean lattices

Binary resolution over complete residuated Stone lattices

“…Throughout the paper, we shall use the definitions and notation concerning L-valuations, L-interpretations, graded formulae, graded clauses, and substitutions given in Section 2 [2]. The only difference is that formulae are built up from atoms using the connectives ¬, negation, ∧, conjunction, &, strong conjunction, ∨, disjunction, and →, implication, which are interpreted by the respective operators − , ∧, * , ∨, and ⇒ on L. PropForm, Form L denotes the set of (propositional) formulae (of L), and analogously, PropForm L , Form L L the set of graded formulae, of the form (, c) where ∈ PropForm, Form L and c ∈ L is its truth degree.…”

“…The propositional and first-order binary L-resolution rules over Boolean lattices [2] can be generalised over residuated Stone lattices:…”

“…The formal definitions of (propositional) binary resolvent, the (propositional) binary L-resolution operator PropBC, BC, and of deduction, refutation by binary L-resolution are analogous to those given in Section 3 [2] (Definitions 3, 5); ∧ is replaced by * . In the following text, we shall focus on the first-order case, since the propositional one is obtainable from it as a special subcase; only nullary predicate symbols are considered.…”

“…In the following text, we shall focus on the first-order case, since the propositional one is obtainable from it as a special subcase; only nullary predicate symbols are considered. We list a few lemmata, which have the same arguments as Lemmata 6-8, Section 3 [2]. Proof.…”

“…This paper concerns binary resolution proof method. In [2], we have devised binary L-resolution over complete Boolean lattices. Binary L-resolution operates on graded clauses, pairs of the form (l 1 ∨ · · · ∨ l n , c) where l i are literals and c is a truth degree in L. Our main logic-theoretical motivation is to investigate binary L-resolution over so general algebraic structures that the refutational completeness can be still achieved.…”