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

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Pi2_Rule_Systems_and_Inductive_Classes_of_Godel_Algebras__Draft_.pdf - Submitted Version Download (478kB) |

## 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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