Higher Order Predicate Logic:

In first order predicate logic, we are allowed to quantify over objects only. If we let  ourselves  to  quantify  over  predicate  or  function  symbols,  afterthat  we  have moved up to the more expressive higher order predicate logic. This means that we are representing meta-level information regarding our knowledge, such as "For all the functions we havespecified, they return the number 10 if the number 7 is the input given":

f, (f(7) = 10).

Multiple valued logics, where different truth value such as "unknown" is allowed. These have some o advantages of fuzzy logics, without essential worrying about probability.

Modal logics,  which cater for particular agents' beliefs regarding the world. For intended, one agent could trust that a certain statement is true, but another cannot. Modal logics help deal with statements that can be believed to be true to some, but not all agents.

Temporal logics, which make us able  to write sentences involving considerations of time, for an  example that a statement may become true sometime in the future.