Structural rule

Abstract: We establish decidability for the infinitely many axiomatic extensions of the commutative Full Lambek logic with weakening FLew (i.e. IMALLW) that have a cut-free hypersequent proof calculus (specifically: every analytic structural rule extension). Decidability for the corresponding extensions of its contraction counterpart FLec was established recently but their computational complexity was left unanswered. In the second part of this paper, we introduce just enough on length functions for well-quasi-orderings and the fast-growing complexity classes to obtain complexity upper bounds for both the weakening and contraction extensions. A specific instance of this result yields the first complexity bound for the prominent fuzzy logic MTL (monoidal t-norm based logic) providing an answer to a long-standing open problem.

Abstract: Metacompleteness is used to prove properties such as the disjunction property and the existence property in the area of relevant logics. On the other hand, the disjunction property of several basic propositional substructural logics over FL has been proved using the cut elimination theorem of sequent calculi and algebraic characterization. The present paper shows that Meyer's metavaluational technique and Slaney's metavaluational technique can be applied to basic predicate intuitionistic substructural logics and basic predicate involutive substructural logics, respectively. As a corollary of metacompleteness, the disjunction property, the existence property, and the admissibility of certain rules in such logics can be proved.

Abstract: This chapter returns to some of the foundational issues discussed in the introduction and first chapter, particularly regarding the justification of logical rules. In Chaps. 1 and 2, I argued that standard inferentialist accounts do not suffice to provide justification of basic logical rules. In response, an analysis of logical inference as acts taking place in dialogical situations is provided, by taking interactions to be structured around moves that may be defined as coherent under certain circumstances. This is used to underpin a novel account of the proof-theoretic notion of harmony as a way of balancing moves. I show that this leads to a constructive logic over both proofs and refutations, where logical rules are justfied internally, at the termination of dialogue. This means that, rather than start with logical rules a priori, the rules themselves are derivative of dialogical balance, so we can reconstruct certain logical rules subsequent to interaction. I finish by discussing the relationships between the formal structure presented here, and related structures.

Abstract: Knowledge is an essential part of most Semantic Web applications and ontology, which is a formal explicit description of concepts or classes in a domain of discourse, is the most important part of the knowledge. There are some problems with extracting rules from text. First, which words of the Web page are rule components and which types of rule components are they? Second, how can we compose rules with the rule components? There are numerous possible combinations of making rules. Our idea for solving these problems is using rules of similar sites in limited situations under a couple of assumptions. We proposed two main steps of rule acquisition, which consists of rule component identification and rule composition with the identified rule components. In other words, we identify rule components such as variables and values in Web pages by using RuleToOnto in the first step, and we combine the variables to compose rules in the second step.