$Π_{2}$-Rule Systems and Inductive Classes of Gödel Algebras
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866917836750848000 |
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| author | Almeida, Rodrigo Nicolau |
| author_facet | Almeida, Rodrigo Nicolau |
| contents | In this paper we present a general theory of $Π_{2}$-rules for systems of intuitionistic and modal logic. We introduce the notions of $Π_{2}$-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 $Π_{2}$-rule systems extending $\mathsf{LC}=\mathsf{IPC}+(p\rightarrow q)\vee (q\rightarrow p)$, 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 $\mathsf{LC}$: (1) we present a full classification of those inductive classes which are inductively complete, i.e., where all $Π_{2}$-rules which are admissible are derivable, and (2) show that the problem of admissibility of $Π_{2}$-rules over $\mathsf{LC}$ is decidable. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2311_07189 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | $Π_{2}$-Rule Systems and Inductive Classes of Gödel Algebras Almeida, Rodrigo Nicolau Logic Logic in Computer Science In this paper we present a general theory of $Π_{2}$-rules for systems of intuitionistic and modal logic. We introduce the notions of $Π_{2}$-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 $Π_{2}$-rule systems extending $\mathsf{LC}=\mathsf{IPC}+(p\rightarrow q)\vee (q\rightarrow p)$, 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 $\mathsf{LC}$: (1) we present a full classification of those inductive classes which are inductively complete, i.e., where all $Π_{2}$-rules which are admissible are derivable, and (2) show that the problem of admissibility of $Π_{2}$-rules over $\mathsf{LC}$ is decidable. |
| title | $Π_{2}$-Rule Systems and Inductive Classes of Gödel Algebras |
| topic | Logic Logic in Computer Science |
| url | https://arxiv.org/abs/2311.07189 |