  The most familiar logics in the modal family are constructed from a weak logic called \(\bK\) (after Saul Kripke). Under the narrow reading, modal logic concerns necessity and possibility. A variety of different systems may be developed for such logics using \(\bK\) as a foundation. The symbols of \(\bK\) include ‘\({\sim}\)’ for ‘not’, ‘\(\rightarrow\)’ for ‘if…then’, and ‘\(\Box\)’ for the modal operator ‘it is necessary that’. (The connectives ‘\(\amp\)’, ‘\(\vee\)’, and ‘\(\leftrightarrow\)’ may be defined from ‘\({\sim}\)’ and ‘\(\rightarrow\)’ as is done in propositional logic.) \(\bK\) results from adding the following to the principles of propositional logic. 