Broadly speaking, a formal logic that employs technical symbols in the place of natural language.  Here is a conditional syllogism in natural language with its symbolic counterpart in modern propositional logic: 

ARGUMENT IN NATURAL LANGUAGE

If the weather is poor, then it is unpleasant.

But the weather is poor.

Therefore, it is unpleasant.

ARGUMENT IN SYMBOLIC FORM 

p ⸧ q 

p 

⸫ q 

Predicate, or first-order, logic refers to the branch of symbolic logic that uses mathematical quantification and relations as part of its symbolic language.  Although classical logic did not develop all of the aspects of modern propositional and predicate logic, it did develop, from the Middle Ages onward, a rigorous methodology of symbolization as well.