Agata Ciabattoni

From Axioms to Analytic Rules in Nonclassical Logics

We introduce a systematic procedure to transform large classes of (Hilbert) axioms into equivalent inference rules in sequent and hypersequent calculi. This allows for the automated generation of analytic calculi for a wide range of prepositional nonclassical logics including intermediate, fuzzy and substructural logics. Our work encompasses many existing results, allows for the definition of new calculi and contains a uniform semantic proof of cut-elimination for hypersequent calculi.

Hypersequent Calculi for Gödel Logics — a Survey

Hypersequent calculi arise by generalizing standard sequent calculi to refer to whole contexts of sequents instead of single sequents. We present a number of results using hypersequents to obtain a Gentzen-style characterization for the family of Gödel logics. We first describe analytic calculi for propositional finite and infinite-valued Gödel logics. We then show that the framework of hypersequents allows one to move straightforwardly from the propositional level to first-order as well as propositional quantification. A certain type of modalities, enhancing the expressive power of Gödel logic, is also considered.

Uniform Rules and Dialogue Games for Fuzzy Logics

We provide uniform and invertible logical rules in a framework of relational hypersequents for the three fundamental t-norm based fuzzy logics i.e., Łukasiewicz logic, Godel logic, and Product logic. Relational hypersequents generalize both hypersequents and sequents-of-relations. Such a framework can be interpreted via a particular class of dialogue games combined with bets, where the rules reflect possible moves in the game. The problem of determining the validity of atomic relational hypersequents is shown to be polynomial for each logic, allowing us to develop Co-NP calculi. We also present calculi with very simple initial relational hypersequents that vary only in the structural rules for the logics.

Algebraic proof theory for substructural logics: cut-elimination and completions

We carry out a unified investigation of two prominent topics in proof theory and order algebra: cut-elimination and completion, in the setting of substructural logics and residuated lattices. We introduce the substructural hierarchy – a new classification of logical axioms (algebraic equations) over full Lambek calculus FL, and show that a stronger form of cut-elimination for extensions of FL and the MacNeille completion for subvarieties of pointed residuated lattices coincide up to the level N2 in the hierarchy. Negative results, which indicate limitations of cut-elimination and the MacNeille completion, as well as of the expressive power of structural sequent calculus rules, are also provided. Our arguments interweave proof theory and algebra, leading to an integrated discipline which we call algebraic proof theory.

Towards a Semantic Characterization of Cut-elimination

We introduce necessary and sufficient conditions for a (single-conclusion) sequent calculus to admit (reductive) cut-elimination. Our conditions are formulated both syntactically and semantically.

On the (fuzzy) logical content of CADIAG-2

CADIAG-2 is a successful expert system assisting in the differential diagnosis in internal medicine. With its aid, conjectures about present diseases are derived from possibly vague information about a patients symptoms. In this paper we provide a mathematical formalisation of the inferential mechanism of CADIAG-2. A Gentzen-style calculus for the resulting logic is introduced and used to compare the systems behaviour with t-norm based fuzzy logics.

Finiteness in Infinite-Valued Σukasiewicz Logic

AbstractIn this paper we deepen Mundicis analysis on reducibility of the decision problem from infinite-valued Łukasiewicz logic

Expanding the realm of systematic proof theory

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