Where gamma fails
The authors characterize a class of relevance logics in which Ackermann's rule \(\gamma\) is inadmissible. They make a case that one of these logics, called BN4, is best when you want four values: true, false, neither, both. Rule Gamma specifies: A, -A\(\vee B\vdash B\). After a brief discussion of the background of the ''Gamma Problem'', they present CWR. Roughly: CWR is the system R without A\&(A\(\to B)\to B\), (A\(\to.A\to B)\to (A\to B)\), (A\(\to -A)\to A\), but with axioms for classical (Boolean) negation \(\neg\) and a stricter (DeMorgan) negation: \(A\to f\) or -A. They present a matrix suitable for CWR4 and show that \(\gamma\) is inadmissible in CWR. BN4 is the set of formulas which this matrix characterizes. Their theorem states: \(\gamma\) fails for all subsystems of BN4 in which both \(A\to A\) and \(A\vee \neg A\) are theorem schemes.
scientific article; zbMATH DE number 3939328