A syllogism is an argument with two premises and one conclusion, built from three terms. The subject of the conclusion is the minor term, the predicate of the conclusion is the major term, and the term that appears in both premises but not in the conclusion is the middle term. The whole test is whether the middle term genuinely connects the other two.
Worked example: All pens are books. No book is a chair. Pens sit entirely inside books, and books share nothing with chairs, so pens share nothing with chairs. Conclusion "No pen is a chair" follows. Conclusion "Some chairs are pens" contradicts it and does not follow. Draw the circles first, read the conclusion second — that habit alone prevents most errors.