Maximality in finite-valued Lukasiewicz logics defined by order filters

2018/03/26 by Marcelo E. Coniglio, Francesc Esteva, Coniglio, Marcelo E. +5
Computer Science · #03B53 #06D35 #Advanced Algebra and Logic #FOS: Mathematics #Formal Methods in Verification #Logic (math.LO)

paper · pdf · doi:10.48550/arxiv.1803.09815

openalex publication_date 2018/03/26 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this paper we consider the logics L_nⁱ obtained from the (n+1)-valued Lukasiewicz logics L_n+1 by taking the order filter generated by i/n as the set of designated elements. In particular, the conditions of maximality and strong maximality among them are analysed. We present a very general theorem which provides sufficient conditions for maximality between logics. As a consequence of this theorem it is shown that L_nⁱ is maximal w.r.t. CPL whenever n is prime. Concerning strong maximality between the logics L_nⁱ (that is, maximality w.r.t. rules instead of axioms), we provide algebraic arguments in order to show that the logics L_nⁱ are not strongly maximal w.r.t. CPL, even for n prime. Indeed, in such case, we show there is just one extension between L_nⁱ and CPL obtained by adding to L_nⁱ a kind of graded explosion rule. Finally, using these results, we show that the logics L_nⁱ with n prime and i/n < 1/2 are ideal paraconsistent logics.

Citations

• On very true

Related

• A proof for completeness of Łukasiewicz logic
• An algebraic study of the first order version of some implicational fragments of the three-valued Lukasiewicz logic
• The unification type of Lukasiewicz logic with a bounded number of variables
• Strong standard completeness theorems for S5-modal Lukasiewicz logics
• On the expressive power of Lukasiewicz's square operator