The classical syllogisms consist of propositions which use only simple (ie, one "variable") terms in the subject and predicate, so any argument containing propositions which use complex terms isn't a valid (classical) form, although it can still be handled within term logic by extended inference rules.
A complex term can be any Boolean expression. Being a term, it doesn't have a truth value, but the usual rules of simplification, combination etc, apply to such terms.
Boole was the first to show how such non-Aristotelian arguments could be formulated and "solved", but although his system was primarily a term logic, it was expressed in the language of mathematics (high school algebra). Towards the end of the 19th century John Neville Keynes (father of the famous economist), building on the work of Boole, Jevons, and Venn, developed a "syllogistic like" logical system which used the conventional Aristotelian propositional forms of A, E, I, O in which terms could be arbitrarily complex.
The set of inference rules needed to handle such arguments, together with many worked examples, can be found in appendix C (A Generalisation of Logical Processes in Their Application to Complex Propositions) of the final edition of Keynes' book Studies and Exercises in Formal Logic.
See also A.J. Baker's 1966 paper Non-empty complex terms, which contains a more succinct and rigorous account, as well as addressing issues of existential import.
An example of an argument which requires some of the extended inference rules is the following (taken from David Kelley's The Art of Reasoning, 1st ed.) :
- Some juveniles (J) who commit minor offenses (C) are put into prison (P).
- Any juvenile who is put into prison is exposed to all sorts of hardened criminals (E).
- A juvenile who is exposed to all sorts of hardened criminals will become bitter (B) and learn more techniques for committing crimes (T).
Any individual who learns more techniques for committing crimes is a menace to society (M), if he or she is bitter.
Therefore, some juveniles who commit minor offenses will be menaces to society.
Here's a proof of the conclusion :
1. Some JC are P premise
2. All JP are E premise
3. All E are BT premise
4. All BT are M premise
5. All JP are BT 2, 3, syllogism
6. All JP are M 4, 5, syllogism
7. All P are M or ~J 6, disjunctive transfer
8. Some JC are M or ~J 1, 7, syllogism
9. Some JC are J(M or ~J) 8, add superfluous conjunct
10. Some JC are JM 9, simplification of predicate
11. Some JC are M 10, drop superfluous conjunct (conclusion)