Syllogism

A syllogism gives 2–3 statements (premises) and asks which conclusion(s) follow logically.

Statement types (the 4 standard forms):

• A (universal affirmative): "All A are B"
• E (universal negative): "No A is B"
• I (particular affirmative): "Some A are B"
• O (particular negative): "Some A are not B"

Venn diagram method — fast and reliable:

1. Draw circles for each category.
2. Place each premise into the diagram:
   • "All A are B" → A is fully inside B.
   • "No A is B" → A and B are non-overlapping.
   • "Some A are B" → A and B overlap (some shared region).
3. Check each conclusion against the diagram. A conclusion follows only if it is true in every arrangement allowed by the premises.

Key inference rules:

• "All A are B" implies "Some A are B" (true in standard logic syllogisms).
• "Some A are B" implies "Some B are A" (symmetry of overlap).
• "No A is B" implies "No B is A" (symmetric).
• "All A are B" does NOT imply "All B are A".
• "Some A are not B" does NOT imply "Some A are B" (different statements).

Either-or case: if a conclusion pair has a complementary form (like "Some X are Y" and "No X is Y") and at least one of them must be true given the premises, but neither must be true alone, then "either I or II follows".

Common conclusion-type pitfall: the question almost always offers a "Some X are Y" conclusion. If "All X are Y" is given, "Some X are Y" follows. If only "Some X are Y" is given, "All X are Y" does NOT follow.

Three-term syllogism:
"All apples are fruits. All fruits are healthy." → All apples are healthy (transitivity). Diagram check confirms.

Example 1:
Statements: All books are pens. All pens are erasers.
Conclusions: I. All books are erasers. II. Some erasers are books.
Venn: Books ⊂ Pens ⊂ Erasers. So all books are erasers (I follows). Therefore some erasers are books (II follows, since "All A are B" implies "Some B are A"). Answer: both I and II follow.

Example 2:
Statements: Some doctors are engineers. All engineers are graduates.
Conclusions: I. Some doctors are graduates. II. Some graduates are doctors.
Venn: Some doctors ∩ engineers (non-empty). Engineers ⊂ graduates. So the doctors who are engineers are also graduates. → Some doctors are graduates (I follows). By symmetry, some graduates are doctors (II follows).
Answer: both I and II follow.

Example 3:
Statements: All cats are mammals. No mammal is a bird.
Conclusions: I. No cat is a bird. II. Some birds are cats.
Venn: Cats ⊂ Mammals. Mammals ∩ Birds = ∅. So Cats ∩ Birds = ∅ as well. I follows. II contradicts. Answer: only I follows.

Example 4 — Either-or:
Statements: Some pens are red. Some pens are blue.
Conclusions: I. Some red are blue. II. No red is blue.
Venn: There may or may not be overlap between red and blue pens. Neither I nor II is necessarily true. But together they cover all cases (red and blue either overlap or don't). So Either I or II follows.

Example 5:
Statements: All mangoes are sweet. Some sweet things are bitter.
Conclusions: I. Some mangoes are bitter. II. Some bitter things are sweet.
Venn: Mangoes ⊂ Sweet. The "some bitter" portion of Sweet may or may not overlap with Mangoes. I does not follow. II is restatement of premise 2 (symmetry of "some"), so it follows. Answer: only II follows.

Speed tactic: write the categories in 1-letter abbreviations. Draw a quick rough Venn. If the conclusion can be imagined false with a valid diagram, it doesn't follow.