 The consequential conditional in the logics of consequential implication investigated by Pizzi (1977, 1991, 1993, 1996, 2004, 2005, 2008, 2018) and Pizzi and Williamson (1997, 2005) validates Aristotle’s theses but fails to validate Boethius’ theses. It is thus connexive only in a weak sense, but since the consequential implication is a strict conditional that is required to satisfy some extra condition, also logics of consequential implication fit into the classificatory scheme provided by the semantical perspective. The following table is a slight extension of the summarizing overview from Omori and Wansing 2019 (with pointers to the relevant sections of the present entry), where the approaches above the double line adjust (support of) truth conditions (or add semantical machinery to standard truth conditions), whereas the approach below the double line tweaks (support of) falsity conditions: