As Arturo Magidin says, the basic kinds of formal logic studied today are propositional logic and first-order logic, and these can be used to model various aspects of syllogistic reasoning.
However, the actual study of syllogistic logic (also called traditional logic or term logic, the logic studied by Aristotle) is not very common in mathematics or mathematical logic today.
One reason for this is that there was a very complex set of conventions for when a statement in this logic is true, relating to which statements imply the existence of objects with the stated properties, and these conventions are not the ones that we use today.
For example, there was a common convention which can be called "universal affirmatives have existential import". In this convention, a statement "All S are P" is only true if (1) there is at least one thing that satisfies S and (2) every thing that satisfies S satisfies P. In particular, "All S are P" was taken to imply "Some S is P"; these two phrases are called "subalterns" of each other.
The relationship between the four claims "All S are P", "All S are not P", "Some S is P", and "Some S is not P" was laid out, in traditional logic, in the "square of opposition" (see e.g ).
In contemporary mathematics, we do not follow the same conventions, and some of the terminology from the square of opposition is completely lost. For example, no mathematical logic book I have ever seen uses the term "subaltern". And our modern convention is that "All S are P" does not imply the existence of any object satisfying S. Thus we accept that "All S are P" and "All S are not P" are both true if no S exists. In the square of opposition, those quoted statement are taken to be contradictory; if there are no S then "All S are P" is false and "All S are not P" is true.
Another reason that Aristotle's logic is not directly studied today is that it had no way to handle nested quantifiers. For example, the $\epsilon$-$\delta$ definition of a continuous function involves three levels of quantifiers. We want the logic we study to be able to handle that sort of statement.