 However, the theorems are inadequate vis-à-vis the intended meaning of “implication” and our actual modes of inference that the intended meaning tries to capture. So Lewis has in mind an intended meaning for the conditional connective \(\rightarrow\) or \(\supset\), and that is the meaning of the English word “implies”. The meaning of “implies” is that “of ordinary inference and proof” (1912: 531) according to which a proposition implies another proposition if the second can be logically deduced from the first. Given such an interpretation, (1) and (2) ought not to be theorems, thus propositional logic may be regarded as unsound vis-à-vis the reading of \(\rightarrow\) as logical implication. Consider (2) for example: from the sheer truth of a proposition \(p\) it does not (logically) follow that \(p\) follows logically from any proposition whatsoever. Additionally, given the intended, strict reading of \(\rightarrow\) as logical implication and the equivalence of \((\neg p\rightarrow q)\) and \((p\vee q)\), Lewis infers that disjunction too must be given a new intensional sense, according to which \((p\vee q)\) holds just in case if \(p\) were not the case it would have to be the case that \(q\).