PUBLICATIONS DE L'INSTITUT MATHÉMATIQUE (BEOGRAD) (N.S.) Vol. 57(71) (dedicated to Djuro Kurepa), pp. 165178 (1995) 

Identity and permutationAleksandar KronFilozofski fakultet, Beograd, YugoslaviaAbstract: 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 AndersonBelnap conjecture). This property is equivalent to NOID (no identity!): if the axiomshema $(A\to A)$ 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 Gentzenstyle 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 axiomschema (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 AndersonBelnap conjecture is not equivalent to NOID. \par The system {\bf J}+ID is equivalent to {\bf RW}$_{\to}$ of relevance logic. Classification (MSC2000): 11A05 Full text of the article:
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