knows_K ≥ has_K
knows_K ≥ valid
valid ≥ true
has_K ≥ true
true ≥ asserts_K
true ≥ lax
and taking the reflexive, transitive closure. Then all the connectives are defined "at true". The connectives [[ ]], [ ], < >, !, and { } are defined as round trips through knows, has, asserts, valid, and lax, departing from and returning to true.
Afterwards tagged along with rwh and ben pierce and one of pierce's students for dinner, much technical conversation ensued.