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Angell's logic of analytic containment AC has been shown to be characterized by a 9valued matrix NC by Ferguson, and by a 16valued matrix by Fine. We show that the former is the image of a surjective homomorphism from the latter, i.e., an epimorphic image. The epimorphism was found with the help of MUltlog, which also provides a tableau calculus for NC extended by quantifiers that generalize conjunction and disjunction. 

In this paper we give an analytic tableau calculus P L 1 6 for a functionally complete extension of Shramko and Wansing’s logic. The calculus is based on signed formulas and a single set of tableau rules is involved in axiomatising each of the four entailment relations ⊧ t, ⊧ f, ⊧ i, and ⊧ under consideration—the differences only residing in initial assignments of signs to formulas. Proving that two sets of formulas are in one of the first three entailment (...) 

The trilattice SIXTEEN₃ is a natural generalization of the wellknown bilattice FOUR₂. Cutfree, sound and complete sequent calculi for truth entailment and falsity entailment in SIXTEEN₃, are presented. 

In this paper we consider the theory of predicate logics in which the principle of Bivalence or the principle of NonContradiction or both fail. Such logics are partial or paraconsistent or both. We consider sequent calculi for these logics and prove Model Existence. For L4, the most general logic under consideration, we also prove a version of the CraigLyndon Interpolation Theorem. The paper shows that many techniques used for classical predicate logic generalise to partial and paraconsistent logics once the right (...) 

In this paper we show that, in Gentzen systems, there is a close relation between two of the main characters in algebraic logic and proof theory respectively: protoalgebraicity and the cut rule. We give certain conditions under which a Gentzen system is protoalgebraic if and only if it possesses the cut rule. To obtain this equivalence, we limit our discussion to what we call regular sequent calculi, which are those comprising some of the structural rules and some logical rules, in (...) 

When considering msequents, it is always possible to obtain an msequent calculus VL for every mvalued logic (defined from an arbitrary finite algebra L of cardinality m) following for instance the works of the Vienna Group for Multiplevalued Logics. The Gentzen relations associated with the calculi VL are always finitely equivalential but might not be algebraizable. In this paper we associate an algebraizable 2Gentzen relation with every sequent calculus VL in a uniform way, provided the original algebra L has a (...) 

This volume portrays the Polish or LvovWarsaw School, one of the most influential schools in analytic philosophy, which, as discussed in the thorough introduction, presented an alternative working picture of the unity of science. 

In [5], Béziau provides a means by which Gentzen’s sequent calculus can be combined with the general semantic theory of bivaluations. In doing so, according to Béziau, it is possible to construe the abstract “core” of logics in general, where logical syntax and semantics are “two sides of the same coin”. The central suggestion there is that, by way of a modification of the notion of maximal consistency, it is possible to prove the soundness and completeness for any normal logic. (...) 

This paper develops and motivates a paraconsistent approach to semantic paradox from within a modest inferentialist framework. I begin from the bilateralist theory developed by Greg Restall, which uses constraints on assertions and denials to motivate a multipleconclusion sequent calculus for classical logic, and, via which, classical semantics can be determined. I then use the addition of a transparent truthpredicate to motivate an intermediate speechact. On this approach, a liarlike sentence should be “weakly asserted”, involving a commitment to the sentence (...) 

The aim of this paper is to emphasize the fact that for all finitelymanyvalued logics there is a completely systematic relation between sequent calculi and tableau systems. More importantly, we show that for both of these systems there are al ways two dual proof sytems (not just only two ways to interpret the calculi). This phenomenon may easily escape one’s attention since in the classical (twovalued) case the two systems coincide. (In twovalued logic the assignment of a truth value and (...) 

A uniform construction for sequent calculi for finitevalued firstorder logics with distribution quantifiers is exhibited. Completeness, cutelimination and midsequent theorems are established. As an application, an analog of Herbrand’s theorem for the fourvalued knowledgerepresentation logic of Belnap and Ginsberg is presented. It is indicated how this theorem can be used for reasoning about knowledge bases with incomplete and inconsistent information. 