Extended Socratic Syllogisms? I'm not entirely sure where I might ask this, but there is a logic tag, so  I guess this fits the budget.
I am taking an introductory course on logic, mainly revolving around Syllogisms, or a logical argument structured with two premises (ordered by terms) preceding the conclusion. I find them fascinating, but are there extended forms of syllogisms? As in, is there a study in some sort of extended syllogism with three premises and a conclusions? Two premises and two conclusions? Or does the realm of syllogisms just end at "Syllogism?" Are there any books that cover this topic?
 A: For millennia, syllogistic logic as described by Aristotle was widely considered to be the fundament of all logic. It was only in the late 1800s with the work of Frege and his successors that it was recognized that the Aristotelian structure is not actually strong enough to represent many kinds of reasoning that are used routinely in mathematics.
What passes as mathematical logic since the early 1900s looks markedly different from the traditional set of syllogisms. There's still a concept of "from this and that we can conclude such-and-such", but calling them "syllogisms" has definitely fallen out of favor. Today we speak about "rules of inference" for these things, for example modus ponens: "From if $A$ then $B$ and $A$ we can conclude $B$".
(There are various rules of propositional reasoning that are still called "syllogisms" and fit into the framework of mathematical logic reasonably well. However, in a typical presentation most of these named rules will not figure explicitly; instead they are just examples of types of reasoning that can be derived from a set of more primitive axioms.) 
In various systems of mathematical logic one meets rules of inference with one, two or more premises. Rules with three or more premises are relatively uncommon, but mostly because it is often technically convenient to split them into a cascade of rules with fewer premises. There's not thought to be anything wrong with them as such.
On the other hand, rules with more than one conclusion are very rarely met. Instead of saying "from A and B we can conclude both of C and D", it is more convenient to have two rules stating "from A and B we can conclude C" and "from A and B we can conclude D". (There are logics where this isn't strictly true, such as linear logic, but still they are usually formalized using different tools than multi-conclusion rules).
Where in days long gone, students of philosophy had to learn by rote the names and properties of the umpteen different valid schemes for syllogism, modern mathematical logic instead is content with deriving them from a smaller set of more fundamental rules, as the need arises.

If you want to read more about these things, a good place to start might be Peter Smith's Teach Yourself Logic guide, which tries to be accessible to people coming to logic from the philosophy side (the author is a retired professor of philosophy) while still aiming straight for actual mathematical logic.
A: How about Lewis Caroll, in particular his book Symbolic Logic 
example 

What conclusion may be drawn from:
  (a)  No interesting poems are unpopular among people of real taste.
  (b)  No modern poetry is free from affectation.
  (c)  All your poems are on the subject of soap-bubbles.
  (d)  No affected poetry is popular among people of real taste.
  (e)  No ancient poem is on the subject of soap-bubbles.

http://www.math.hawaii.edu/~hile/math100/logice.htm 
http://en.wikipedia.org/wiki/Lewis_Carroll#Mathematical_work 
A: You have to take into account that already Aristotle discussed "chains" of predication, i.e. chains of syllogisms (and he considered also the infinite case); see :


*

*Aristotle, Posterior Analytics (editor Jonathan Barnes, 2nd ed, 1993), A21-22, page 29-on or here, page 52,


and the detailed discussion in :


*

*Jonathan Lear, Aristotle and Logical Theory (1980), page 17-on.

