Density Elimination for Semilinear Substructural Logics

Abstract

We present a uniform method of density elimination for several semilinear substructural logics. Especially, the density elimination for the involutive uninorm logic IUL is proved. Then the standard completeness of IUL follows as a lemma by virtue of previous work by Metcalfe and Montagna.

The Logic of Pseudo-Uninorms and their Residua

“…Now, starting with G G * and its proof τ * , we construct a proof τ of G in GpsUL Ω such that each sequent of G is a copy of some sequent of G. Then ⊢ GpsUL Ω D(G ) by Theorem 3.8 and Lemma 3.16. Then ⊢ GpsUL * D 0 (G 0 ) by Lemma 9.1 in [6].…”

“…We overcome this difficulty by introducing a restricted subsystem GpsUL Ω of GpsUL * . GpsUL Ω is a generalization of GIUL Ω , which we introduced in [6] in order to solve a longstanding open problem, i.e., the standard completeness of IUL. Two new manipulations, which we call the derivation-splitting operation and derivation-splicing operation, are introduced in GpsUL Ω .…”

“…The third difficulty we encounter is that the conditions of applying the restricted external contraction rule (EC Ω ) become more complex in GpsUL Ω because new derivation-splitting operations make the conclusion of the generalized density rule to be a set of hypersequents rather than one hypersequent. We continue to apply derivation-grafting operations in the separation algorithm of the multiple branches of GIUL Ω in [6] but we have to introduce a new construction method for GpsUL Ω by induction on the height of the complete set of maximal (pEC)-nodes other than on the number of branches.…”

“…Let τ be a proof of G 0 in GpsUL * * by Theorem 2.8. Starting with τ, we construct a proof τ * of G G * in GL cf Ω by a preprocessing of τ described in Section 4 in [6]. In Step 1 of preprocessing of τ, a proof τ ′ is constructed by replacing inductively all applications of (∧ r ) and (∨ l ) in τ with (∧ rw ) and (∨ lw ) followed by an application of (EC), respectively.…”

“…In [6], G is constructed by eliminating (pEC)-sequents in G G * one by one. In order to control the process, we introduce the set I = {H c i1 , ⋯, H c im } of maximal (pEC)-nodes of τ * (See Definition 4.2) and the set I of the branches relative to I and construct G I such that G I doesn't contain the contraction sequents lower than any node in I, i.e., S c j ∈ G I implies H c j H c i for all H c i ∈ I.…”