Predicate logic is a system of mathematical logic. It uses predicates to express the state of certain things. Predicates can be thought of as "incomplete propositions" with a placeholder for objects or subjects that must be inserted to obtain a valid proposition. Predicate logic is different from propositional logic because it has the concept of quantifiers. Quantifiers indicate how many instances can be inserted in place of the variable in order to obtain a true proposition. This means that predicate logic can also make statements about quantity.
The best-known quantifiers are the existence quantifier (∃) and the universal quantifier (∀) . The existence quantifier means that there is at least one mathematical object from the universe (or "thing") that matches the predicate or formula. The universal quantifier means that all possible mathematical objects of the universe of discourse match a certain predicate or formula.
The notation of predicate logic defines that quantifiers directly precede (and thus introduce) variable names which are followed by other quantifiers or mathematical expression which use the introduced variables then.
which means "There is at least one cat which is black and which likes (one or more) dogs."
which means "It is not true that every cat doesn't like any dogs."
which means "There does not exist any cat-dog."