http://hdl.handle.net/11089/48181 Mon, 06 Apr 2026 20:35:04 GMT 2026-04-06T20:35:04Z https://dspace.uni.lodz.pl:443/bitstream/id/d6ba90f5-cbdf-4ee4-a943-6b1d6eb25317/ http://hdl.handle.net/11089/48181 http://hdl.handle.net/11089/48186 Harmony and Normalisation in Bilateral Logic del Valle-Inclan, Pedro In a recent paper del Valle-Inclan and Schlöder argue that bilateral calculi call for their own notion of proof-theoretic harmony, distinct from the usual (or ‘unilateral’) ones. They then put forward a specifically bilateral criterion of harmony, and present a harmonious bilateral calculus for classical logic.In this paper, I show how del Valle-Inclan and Schlöder’s criterion of harmony suggests a notion of normal form for bilateral systems, and prove normalisation for two (harmonious) bilateral calculi for classical logic, HB1 and HB2. The resulting normal derivations have the usual desirable features, like the separation and subformula properties. HB1-normal form turns out to be strictly stronger that the notion of normal form proposed by Nils Kürbis, and HB2-normal form is neither stronger nor weaker than a similar proposal by Marcello D’Agostino, Dov Gabbay, and Sanjay Modgyl. Tue, 18 Jul 2023 00:00:00 GMT http://hdl.handle.net/11089/48186 2023-07-18T00:00:00Z http://hdl.handle.net/11089/48185 Bilateral Rules as Complex Rules Ceragioli, Leonardo Proof-theoretic semantics is an inferentialist theory of meaning originally developed in a unilateral framework. Its extension to bilateral systems opens both opportunities and problems. The problems are caused especially by Coordination Principles (a kind of rule that is not present in unilateral systems) and mismatches between rules for assertion and rules for rejection. In this paper, a solution is proposed for two major issues: the availability of a reduction procedure for tonk and the existence of harmonious rules for the paradoxical zero-ary connective \(\bullet\). The solution is based on a reinterpretation of bilateral rules as complex rules, that is, rules that introduce or eliminate connectives in a subordinate position. Looking at bilateral rules from this perspective, the problems faced by bilateralism can be seen as special cases of general problems of complex systems, which have been already analyzed in the literature. In the end, a comparison with other proposed solutions underlines the need for further investigation in order to complete the picture of bilateral proof-theoretic semantics. Tue, 18 Jul 2023 00:00:00 GMT http://hdl.handle.net/11089/48185 2023-07-18T00:00:00Z http://hdl.handle.net/11089/48184 Supposition: A Problem for Bilateralism Kürbis, Nils In bilateral logic formulas are signed by + and –, indicating the speech acts assertion and denial. I argue that making an assumption is also speech act. Speech acts cannot be embedded within other speech acts. Hence we cannot make sense of the notion of making an assumption in bilateral logic. Attempts to solve this problem are considered and rejected. Tue, 18 Jul 2023 00:00:00 GMT http://hdl.handle.net/11089/48184 2023-07-18T00:00:00Z http://hdl.handle.net/11089/48183 Fractional-Valued Modal Logic and Soft Bilateralism Piazza, Mario; Pulcini, Gabriele; Tesi, Matteo In a recent paper, under the auspices of an unorthodox variety of bilateralism, we introduced a new kind of proof-theoretic semantics for the base modal logic \(\mathbf{K}\), whose values lie in the closed interval \([0,1]\) of rational numbers [14]. In this paper, after clarifying our conception of bilateralism – dubbed “soft bilateralism” – we generalize the fractional method to encompass extensions and weakenings of \(\mathbf{K}\). Specifically, we introduce well-behaved hypersequent calculi for the deontic logic \(\mathbf{D}\) and the non-normal modal logics \(\mathbf{E}\) and \(\mathbf{M}\) and thoroughly investigate their structural properties. Wed, 16 Aug 2023 00:00:00 GMT http://hdl.handle.net/11089/48183 2023-08-16T00:00:00Z