Assertion, denial and some cancellation rules in modal logic (Q1106191)

The author considers that principle that assertability conditions determine truth conditions. In any normal propositional modal logic with Ḻ to mean `it is assertable that', this principle can be formalized as the rule LA\(\leftrightarrow LB/A\leftrightarrow B\) (i.e. from the theoremhood of LA\(\leftrightarrow LB\) to infer \(A\leftrightarrow B)\). This rule is shown to trivialize any system containing either T or S4 in that it enables the proof of LA\(\leftrightarrow A\) for every wff. If however the rule is replaced by LA\(\leftrightarrow LB\), MA\(\leftrightarrow MB/A\leftrightarrow B\), then Sobocinski's K4 is obtained, which is \(S4+p\to (MLp\to Lp)+LMp\to MLp\).