Title: Subminimal Negation
Abstract:
Minimal Logic is considered in the original axiomatization (by Johansson 1937 and even Kolmogorov 1925) over the positive part of intuitionistic logic $\mathsf{IPC}$ with $\neg$ instead of $\bot$ with the axiom $(p\to q)\to ((p\to\neg q)\to\neg p)$. To investigate the different aspects of negation weaker logics are considered. Two important ones are the logic $\mathsf{CoPC}$ of contraposition, $(p\to q)\to(n\eg q\to\neg p)$ and the logic $\mathsf{NEF}$ of negative ex falso, $p\to (\neg p\to\neg q)$. For the semantics, intuitionistic Kripke models are equipped with a function from upsets to upsets to model negation. In this manner one still gets very well behaved models. Among other things, completeness and the finite model property are proved for a number of logics among which $\mathsf{CoPC}$ and the slightly weaker $\mathsf{NEF}$. This work is in cooperation with Ana Lucia Vargas and Almudena Colacito. |