Hypersequent and Labelled Calculi for Intermediate Logics

Abstract

Abstract. Hypersequent and labelled calculi are often viewed as antagonist formalisms to define cut-free calculi for non-classical logics. We focus on the class of intermediate logics to investigate the methods of turning Hilbert axioms into hypersequent rules and frame conditions into labelled rules. We show that these methods are closely related and we extend them to capture larger classes of intermediate logics.

On Deriving Nested Calculi for Intuitionistic Logics from Semantic Systems

“…, n} in R; † 2 states that v does not occur in the conclusion; † 3 stipulates that there exists a path of relational atoms (not necessarily directed) from v to w occurring in R; † 4 states that a does not occur in the conclusion. 6 consisting of treelike sequents. Note that if a derivation ends with a sequent of the form ⇒ w : A, then the derivation must necessarily contain a bottom treelike fragment since G(⇒ w : A) is a tree.…”

Intermediate Logics Admitting a Structural Hypersequent Calculus

“…Thus without loss of generality we may assume that q is of this form. In fact we may assume that the variables occurring in the terms [26]) Let P 0 = N 0 be a (countable) set of propositional variables and define sets of formulas P n , N n in the language of intuitionistic logic by the following grammar…”

“…7.2] and [40]. 3 However, by moving to the framework of hypersequent calculi [3,45,47] it is possible to construct cut-free hypersequent calculi for many well-known intermediate logics, see, e.g., [4,20,21,26]. Hypersequents are nothing but finite multisets of sequents.…”

Compositional Meaning in Logic

“…Overall, the impact and relevance of such a line of research is to be measured directly by its foundational character with respect to a better and deeper understanding of meaning in logics modeling complex phenomena and, of necessity, suitable general forms of compositional reasoning. Some highlights of the GeTFun project, covering topics such as truth-values, valuation semantics, bivalence, proof-systems and focusing, controlled nondeterminism, distance-based reasoning, information sources, paraconsistency, labeled and hybrid logic, inter alia, can be found in [2,10,12,20,19,21,24,25,29,32,45,44,55,56,58,74,50,7,3,27,26,59,11,47,57,43,49,60,54,1,66].…”