An algebraic proof is presented for the finite strong standard completeness of the Involutive Uninorm Logic with Fixed Point ($${{\mathbf {IUL}}^{fp}}$$ IUL fp ). It may provide a first step towards settling the standard completeness problem for the Involutive Uninorm Logic ($${\mathbf {IUL}}$$ IUL, posed in G. Metcalfe, F. Montagna. (J Symb Log 72:834–864, 2007)) in an algebraic manner. The result is proved via an embedding theorem which is based on the structural description of the class of odd (...) involutive FL$$_e$$ e -chains which have finitely many positive idempotent elements. (shrink)

Axiomatizations are presented for fuzzy logics characterized by uninorms continuous on the half-open real unit interval [0,1), generalizing the continuous t-norm based approach of Hájek. Basic uninorm logic BUL is defined and completeness is established with respect to algebras with lattice reduct [0,1] whose monoid operations are uninorms continuous on [0,1). Several extensions of BUL are also introduced. In particular, Cross ratio logic CRL, is shown to be complete with respect to one special uninorm. A Gentzen-style hypersequent calculus (...) is provided for CRL and used to establish co-NP completeness results for these logics. (shrink)

This paper presents several results on the non-commutative fuzzy logic HpsUL, a Hilbert system whose corresponding algebraic semantics is the variety of bounded representable residuated lattices. In particular, we prove that HpsUL is not complete with respect to algebras based on the real unit interval, which answers the question posed by Metcalfe, Olivetti and Gabbay and shows that HpsUL is not the logic of pseudo-uninorms and their residua.

An involutive micanorm-based logic with a weak form of associativity is introduced and its finite standard completeness is addressed. More precisely, we first introduce the logic WAuIBUL as a [0, u]-continuous wau-uninorm analogue of the involutive logic IBUL. We next discuss its algebraic semantics. We then introduce involutive wau-uninorms as involutive uninorms with weak u-associativity in place of associativity and deal with related properties. We last provide finite strong standard completeness for WAuIBUL using a construction of Yang-style. -/- 약한 (...) 형식의 결합 원리를 만족하는 누승적인 미카놈에 기반한 논리 체계를 소개하고 그 체계의 유한 표준 완전성을 다룬다. 보다 구체적으로 논리 체계 WAuIBUL을 누승적 유니놈 논리 IUML의 [0, u]-연속인 wau-유니놈 일반화로 먼저 소개한다. 그리고 이 체계의 대수적 완전성을 다룬다. 다음으로 결합 원리 대신에 약한 u-결합 원리를 만족하는 누승적 유니놈으로 누승적 wau-유니놈을 소개하고 그 성질을 다룬다. 마지막으로 유한 집합 위에서 WAuIBUL의 표준 완전성을 보인다. (shrink)

This paper proposes a new topic in substructural logic for use in research joining the fields of relevance and fuzzy logics. For this, we consider old and new relevance principles. We first introduce fuzzy systems satisfying an old relevance principle, that is, Dunn’s weak relevance principle. We present ways to obtain relevant companions of the weakening-free uninorm systems introduced by Metcalfe and Montagna and fuzzy companions of the system R of relevant implication and its neighbors. The algebraic structures corresponding (...) to the systems are then defined, and completeness results are provided. We next consider fuzzy systems satisfying new relevance principles introduced by Yang. We show that the weakening-free uninorm systems and some extensions and neighbors of R satisfy the new relevance principles. (shrink)

Substructural fuzzy logics are substructural logics that are complete with respect to algebras whose lattice reduct is the real unit interval [0.1]. In this paper, we introduce Uninorm logic UL as Multiplicative additive intuitionistic linear logic MAILL extended with the prelinearity axiom ((A → B) ∧ t) ∨ ((B → A) ∧ t). Axiomatic extensions of UL include known fuzzy logics such as Monoidal t-norm logic MTL and Gödel logic G, and new weakening-free logics. Algebraic semantics for these logics (...) are provided by subvarieties of (representable) pointed bounded commutative residuated lattices. Gentzen systems admitting cut-elimination are given in the framework of hypersequents. Completeness with respect to algebras with lattice reduct [0, 1] is established for UL and several extensions using a two-part strategy. First, completeness is proved for the logic extended with Takeuti and Titani's density rule. A syntactic elimination of the rule is then given using a hypersequent calculus. As an algebraic corollary, it follows that certain varieties of residuated lattices are generated by their members with lattice reduct [0, 1]. (shrink)

In this paper, we show that the finite model property fails for certain non‐integral semilinear substructural logics including Metcalfe and Montagna's uninorm logic and involutive uninorm logic, and a suitable extension of Metcalfe, Olivetti and Gabbay's pseudo‐uninorm logic. Algebraically, the results show that certain classes of bounded residuated lattices that are generated as varieties by their linearly ordered members are not generated as varieties by their finite members.

이 논문은 약한 결합 형식을 갖는 두 준선형 논리와 그것들의 누승적 확장 논리 사이의 동치 관계와 이러한 논리 체계 중 한 논리 체계의 표준 완전성을 다룬다. 좀더 구체적으로 먼저 논리 IWAUBUL을 약한 결합 형태를 갖는 준선형 논리 WAUBUL의 누승적 확장으로 소개하고 이 체계의 대수적 의미론을 다룬다. 특히 우리는 IWAUBUL이 WAUIBUL 체계와 동치라는 것을 보인다. 다음으로 우리는 관련된 약한 결합 힝식을 만족하는 누승적 미카놈을 소개한다. 마지막으로 이 논리를 위한 대수적 조건들에 바탕을 둔 표준 완전성을 다룬다.