PP-2023-06: Π2 -Rule Systems and Inductive Classes of Gödel Algebras

PP-2023-06: Almeida, Rodrigo N. (2023) Π2 -Rule Systems and Inductive Classes of Gödel Algebras. [Pre-print] (Submitted)

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Abstract

In this paper we present a general theory of Π2-rules for systems of intuitionistic and modal logic. We introduce the notions of Π2-rule system and of an Inductive Class, and provide model-theoretic and algebraic completeness theorems, which serve as our basic tools. As an illustration of the general theory, we analyse the structure of inductive classes of Gödel algebras, from a structure theoretic and logical point of view. We show that unlike other well-studied settings (such as logics, or single-conclusion rule systems), there are continuum many Π2-rule systems extending LC=IPC+(p→q)∨(q→p), and show how our methods allow easy proofs of the admissibility of the well-known Takeuti-Titani rule. Our final results concern general questions admissibility in LC: (1) we present a full classification of those inductive classes which are inductively complete, i.e., where all Π2-rules which are admissible are derivable, and (2) show that the problem of admissibility of Π2-rules over LC is decidable.

Item Type:	Pre-print
Report Nr:	PP-2023-06
Series Name:	Prepublication (PP) Series
Year:	2023
Uncontrolled Keywords:	algebraic logic, model theory, admissibility, godel algebra
Subjects:	Logic
Depositing User:	Rodrigo Nicolau Almeida
Date Deposited:	16 Nov 2023 20:05
Last Modified:	16 Nov 2023 20:05
URI:	https://eprints.illc.uva.nl/id/eprint/2284

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