Identity and permutation

Aleksandar Kron

Abstract: It is known that in the purely implicational fragment of the system {\bf TW}$_{\to}$ if both $(A\to B)$ and $ (B\to A)$ are theorems, then $A$ and $B$ are the same formula (the Anderson--Belnap conjecture). This property is equivalent to NOID (no identity!): if the axiom-shema $(A\to A)$ is omitted from {\bf TW}$_{\to}$ and the system {\bf TW}$_{\to}$-ID is obtained, then there is no theorem of the form $(A\to A)$. \par A Gentzen-style purely implicational system {\bf J} is here constructed such that NOID holds for {\bf J}. NOID is proved to be equivalent to NOE: there no theorem of {\bf J} of the form $((A\to A)\to B)\to B$, i.e., of the form of the characteristic axiom of the implicational system {\bf E}$_{\to}$ of entailment. \par If $ (p\to p)$ is adjoined to {\bf J} as an axiom-schema (ID), then there are theorems $(A\to B)$ and $(B\to A)$ such that $A$ and $B$ are distinct formulas, which shows that for {\bf J} the Anderson--Belnap conjecture is not equivalent to NOID. \par The system {\bf J}+ID is equivalent to {\bf RW}$_{\to}$ of relevance logic.

Classification (MSC2000): 11A05

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