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.
We present a sequent calculus for extensional mereology. It extends the classical first-order sequent calculus with identity by rules of inference corresponding to wellknown mereological axioms. Structural rules, including cut, are admissible.The aim of this paper is to investigate mereology [12,17] from the perspective of proof theory [10,16]. There are, we claim, mutual advantages. On the one hand, proof-theoretic methods will serve mereology by providing analytic calculi for various mereological theories that have been so far treated only axiomatically. We claim that analyticity in mereology is an important issue which only few have addressed so far. 1 Although in proof theory there is little consensus on how analyticity should be defined, it is generally accepted that in Gentzen's sequent calculus a proof is analytic when it contains no application of the cut rule. As a matter of fact, many of the standard derivations of mereological theorems, if rearranged in sequent calculi, are not analytic, in the sense cut is applied all the time. The question arises as to whether such cuts can be actually dispensed with, so that the theorem under consideration can have an analytic derivation. This paper is primarily concerned with answering positively to this question for extensional mereology.On the other hand, mereology is an interesting case study for the prooftheorist as it challenges the existing proof-theoretic techniques. While much of standard extensional mereology can be adequately dealt with following the general methodology of [11], it turns out that an important notion such as mereological fusion requires the method to be generalized. Thus, we first show how to formulate extensional mereology (without fusions) using inference rules that, once added to a base system for predicate logic (with identity), generate a system that still satisfies all the structural properties usually required to well-behaved sequent calculi. Secondly, we show how to extend 1 An exception is the recent work by Tennant [15] where a system of natural deduction with parallelized elimination rules for Lewesian mereology is developed. It will be discussed at some length in the concluding section. the methodology to cover also the case of binary fusions, using the so-called systems of rules recently proposed in [8]. Finally, we take into account unrestricted fusions as well.The paper is organized as follows. In Sect. 1 we briefly describe the general problem and explain the method we shall use. In Sect. 2 we introduce the sequent calculus for extensional mereology. In Sect. 3 we prove that such calculus is analytic. Then, in Sect. 4 we show that the extensionality principles can be systematically derived as theorems in our calculus by applying an effective proof-search procedure. Sections 5 and 6 are devoted to a prooftheoretic treatment of fusion. We leave Sect. 7 to a comparison with the existing literature and future research.
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