There might have been some confusion over the difference between a syllogism and an enthymeme. A syllogism is a formally structured logical appeal (using deductive reasoning) with a major premise, minor premise and a conclusion.
All snakes are reptiles (major premise)
Sally is a snake (minor premise)
Therefore, Sally is a reptile (conclusion)
The enthymeme is a sort of informal syllogism, a "rhetorical syllogism" in which the structure is technically incomplete (could I venture to say through "ellipsis"? Is this right?), but is nonetheless understood based on information we already know about the subject of the major premise.
Example: Some snakes are aggressive, therefore Sally could be aggressive.
We already know Sally is a snake from the previous example, so re-stating it in a minor premise would be redundant within the context of an enthymeme.