Syllogism

Draw circles. A conclusion follows only if it is true in every diagram you can legally draw from the statements — not merely in the first one that comes to mind.

1. All + All. All A are B, all B are C, so all A are C. This one is safe, and its reverse "some C are A" is safe too.
2. All + Some. These give possibility, not certainty. From "all A are B" and "some B are C" nothing definite about A and C follows, because the C part of B may lie entirely outside A.
3. Negatives. From "no A is B", "no B is A" also follows. From "some A are not B" you may conclude nothing about the rest.
4. The either–or case. When two conclusions are complementary — "some A are B" and "some A are not B" — and each is true in some diagram but neither in all, the answer is that either one or the other follows.

The single most common mistake is using real-world knowledge. If the statement says "All flowers are stones", you must accept it as true and reason from it. Marks are lost the moment you think about what is actually the case.

The second most common is illicit conversion: turning "All A are B" into "All B are A". "Some" converts freely in both directions; "all" never does.

How to check: after drawing your first diagram, deliberately try to draw a second one that still obeys every statement but breaks your conclusion. If you succeed, the conclusion does not follow.

A conclusion follows only if true in every possible diagram · all + all → all; all + some → possibility only · no A is B ⇒ no B is A · "some" converts both ways, "all" never does · complementary pair true in different diagrams → either–or · never use real-world knowledge

Statements: All pens are books. All books are red. Conclusions: (I) All pens are red. (II) Some red things are pens. Which follows? → Draw the pen circle inside the book circle, and the book circle inside the red circle. Every pen is then inside red, so (I) holds; and pens form a part of red, so (II) holds as well. Both follow. Sanity check: try to redraw the circles so that a pen escapes the red circle — it cannot be done without breaking one of the two statements, which is what makes both conclusions certain rather than merely possible.