A sound and complete set of axioms and inference rules for syllogistic logic First, some syntax. We have an alphabet of six symbols: $\{A, E, I, O, X, {'}\}$. A variable is defined recursively as the symbol $X$ followed by zero or more $'$s. Now, let $V$ and $W$ be variables. A well-formed formula of syllogistic logic is one of these four formulas: $VAW, VEW, VIW, VOW$. We are going to read them as, respectively, All $V$'s are $W$'s, No $V$'s are $W$'s, Some $V$'s are $W$'s, and Some $V$'s are not $W$'s.
Now, to set up the semantics. I define a domain of discourse as the powerset of some set. Given a domain of discourse $D$, a valuation is a map from the set of variables $\{X, X', X'', X''', \dots\}$ to $D$. Given a valuation $f$, we assign a truth value to the formulas $VAW$, $VEW$, $VIW$, and $VOW$, as follows:

*

*The truth value of $VAW$ is $\top$ if $f(V) \subseteq f(W)$, and $\bot$ otherwise.

*The truth value of $VEW$ is $\top$ if $f(V) \cap f(W) = \emptyset$, and $\bot$ otherwise.

*The truth value of $VIW$ is $\top$ if $f(V) \cap f(W) \neq \emptyset$, and $\bot$ otherwise.

*The truth value of $VOW$ is $\top$ if $f(V) - f(W) \neq \emptyset$, and $\bot$ otherwise.

Finally, for the logical part. An argument is an ordered pair, the first component of which is a set of wffs, called premises, and the second component of which is a single wff, called the conclusion. An argument is said to be syllogistically valid iff there is no valuation on any domain that assigns the value $\top$ to all premises but $\bot$ to the conclusion.
Now, I have two questions. First, has there been anyone else in any paper or text that has formalized syllogistic logic the way I have done in this post? Second, has there been, in some paper or text, a sound and complete set of axioms and inference rules for syllogistic logic, along with the proof that it is indeed sound and complete? I would like some references for both questions.
 A: The classical syllogistic logic (along with many many variants) have indeed been studied from the point of view of mathematical logic. In addition to the references to Corcoran and Smiley given by Mauro in the comments, a completness theorem was proven earlier by Łukasiewicz and presented in his short book Aristotle's Syllogistic from the Standpoint of Modern Formal Logic.
The program of proving completeness theorems for variants of syllogistic logic lives on in the field of "natural logic" (which studies logical reasoning in systems based on natural language). One interesting phenomenon is that these "natural logics" are often decidable, and sometimes even computationally tractable, in sharp contrast to the more expressive but undecidable logics commonly used in mathematics.
Larry Moss and coauthors  have written a lot about this subject. A good survey is  Completeness theorems for syllogistic fragments (pdf). This paper contains proof systems and completeness theorem for several syllogistic logics, including the one with "all", "some" and "no" (in your notation, $A$, $E$, and $I$, but not $O$). The writing style is more modern (and more purely mathematical) than the earlier references listed above, so you may find it easier to read.
I've written to Larry to ask if he has a preferred reference for the completeness theorem for the exact logic you asked about, and I'll update this answer when he gets back to me.
