Propositional logic, First order logic and Higher order logic

Propositional logic

In Propositional logic symbols represent Facts (Propositions).
for example:

W - the person is a woman.

Syntactic elements

AND, OR, IMPLIES, EQUIVALENCE, NOT

First order logic (First order predicate calculus)

First-order logic represents the world in terms of objects and predicates. It uses the relations between objects and their properties. It also uses quantifiers and connectives.

syntactic examples

Objects - door, house, dog, ...
Relation - employer, sister, inside, ...
Properties - black, free, empty, hot, ...
Functions - first in, father of, friend of, ...

Syntactic elements

terms - objects
Quantifiers - UNIVERSAL (for all),  and EXISTENTIAL (there exists)
EQUALITY symbol (=)

Syntax

In propositional logic every expression is a sentence and every sentence expresses a fact. In addition to sentences First order logic also has terms that represent objects.

Higher-order logic

First order logic is so called because it works with the first order objects of the world. Higher-order logic enables us to quantify not only objects, but relations as well. What that can mean in practice is that two objects can be thought of as equal if and only if all of their properties are equivalent . We could say that two functions are equal when their arguments have the same value.

I'm not too sure of the value of HOL and it seems a bit contentious.