Syllogism Basics

• How to read two (or more) statements ("All A are B", "Some B are C", "No B are C") and work out which conclusion must always be true.
• The difference between a conclusion that necessarily follows and one that is merely possible (a common trap in olympiad reasoning).
• Standard valid patterns: All-All chains, All-No chains, Some-All chains, and Some-No chains.

1. "All A are B" + "All B are C" ⊢ "All A are C" — a valid chain (every A is inside C).
2. "All A are B" + "No B are C" ⊢ "No A are C" — a valid chain (A and C never overlap).
3. "Some A are B" + "All B are C" ⊢ "Some A are C" — valid, since the A's that are B must also be C.
4. Trap pattern: "All A are B" + "All C are B" does not let you conclude anything fixed about A and C — both are just subsets of B, and could overlap fully, partly, or not at all. The correct answer here is "Conclusion does not necessarily follow."

Statements: "All cats are animals. All animals are living things." Which conclusion follows?

Step 1 — draw circles: Cats inside Animals, Animals inside Living things
Step 2 — so Cats must be inside Living things too
Answer: "All cats are living things." (valid — Barbara form)

• Assuming any two statements about the same three groups must combine into a definite conclusion — sometimes none does.
• Confusing "Some A are B" with "All A are B" — "some" only guarantees a part, not the whole group.
• Picking a conclusion that "sounds right" without checking it holds in every possible case (draw a circle diagram to be sure).

• "All roses are flowers. All flowers are plants." What follows?
• "All pens are stationery. All pencils are stationery." Does "All pens are pencils" follow? Why or why not?
• "Some doctors are teachers. All teachers are graduates." What follows?

Open the Practice tab for graded questions on Syllogism Basics.