The Logic of Pseudo-Uninorms and Their Residua

Abstract

Our method for density elimination is generalized to the non-commutative substructurallogic GpsUL*‡. Then, the standard completeness of HpsUL*‡ follows as a lemma by virtue of previouswork by Metcalfe and Montagna. This result shows that HpsUL‡* is the logic of pseudo-uninormsand their residua and answered the question posed by Prof. Metcalfe, Olivetti, Gabbay and Tsinakis.

A Proof of the Standard Completeness for the Involutive Uninorm Logic

“…Recently, we have generalized our method described in this paper to the non-commutative substructural logic GpsUL * in [20]. This result shows that GpsUL * is the logic of pseudo-uninorms and their residua and answered the question posed by Metcalfe, Olivetti, Gabbay and Tsinakis in [21,22].…”

Comparing Calculi for First-Order Infinite-Valued Łukasiewicz Logic and First-Order Rational Pavelka Logic

“…Before the present work, density elimination proofs were known for some (classes of) hypersequent calculi, though for logics different from Ł∀ and RPL∀; see [3], [1], [22], [11], [23], [8], [4], [5], [6], [24], [7], [28], and [29] (in chronological order). In all these works except [1], such proofs use the cut rule even if no application of it is in an initial formal proof.…”