How does predicate logic express existence, identity, predication, and class inclusion?

In predicate logic, the natural-language verb "to be" is treated as multiply ambiguous, with specific symbolic forms used to express existence, identity, predication, and class inclusion.1

Logical Expressions

• Predication: This involves attributing a property to a subject. For a subject t and a predicate B, it is expressed as $B(t)$ (e.g., "Tarzan is blond").1
• Identity: This expresses that two terms refer to the same object. For terms c and s, it is written as $c=s$ (e.g., "Clark Kent is Superman"). In higher-order systems, identity can also be defined as $x=y$ if and only if every property possessed by x is also possessed by y ($(\forall \varphi )(\varphi x\equiv \varphi y)$).12
• Existence: This asserts that an object exists. It is expressed using the existential quantifier $(\exists x)$, such as $(\exists x)(x=z)$, meaning "There is an x such that x is identical to z".1
• Class Inclusion: This expresses that one class is contained within another. For classes W and M, it is written as $(\forall x)(W(x)\supset M(x))$, meaning "For all x, if x is a whale, then x is a mammal".1