Syllogism — Practice

Question 1

Statements: All pencils are erasers. All erasers are sharpeners. Conclusions: I. All pencils are sharpeners. II. Some sharpeners are pencils.
(a) Only I follows (b) Only II follows (c) Both I and II follow (d) Neither follows

Accepted Answer

(c) Both I and II follow.

Chain the subsets: pencils ⊆ erasers ⊆ sharpeners, so every pencil is a sharpener — I follows. Since pencils sit inside sharpeners and classes are non-empty, some sharpeners are pencils — II follows by valid conversion.

Question 2

Statements: All doctors are readers. Some readers are writers. Conclusions: I. Some doctors are writers. II. All readers are doctors.
(a) Only I follows (b) Only II follows (c) Both follow (d) Neither follows

Accepted Answer

(d) Neither follows.

I: the reader–writer overlap can be drawn entirely away from the doctors circle, so it is not forced. II: 'All doctors are readers' puts doctors inside readers — readers can have members outside doctors, so 'All readers are doctors' is the classic invalid conversion. Neither conclusion survives.

Question 3

Statements: No table is a chair. Some chairs are sofas. Conclusions: I. Some sofas are not tables. II. Some sofas are tables.
(a) Only I follows (b) Only II follows (c) Either I or II follows (d) Neither follows

Accepted Answer

(a) Only I follows.

The sofas that overlap chairs lie entirely outside tables (since no chair is a table), so those sofas are definitely not tables — I is forced. II can be drawn true (other sofas touching tables) or false, so it is not forced. This is not an either–or case because I follows on its own.

Question 4

Statements: All coins are notes. Some notes are cards. Conclusions: I. Some coins are cards. II. No coin is a card.
(a) Only I follows (b) Only II follows (c) Either I or II follows (d) Both follow

Accepted Answer

(c) Either I or II follows.

I: the note–card overlap may or may not touch the coins circle — not forced. II: coins and cards could overlap — not forced. Both fail individually, both involve the same two terms (coins, cards), and 'some are' / 'none is' is a complementary pair covering all possibilities. That is exactly the either–or trigger.

Question 5

Statements: Some cups are glasses. All glasses are jars. Conclusions: I. All cups being jars is a possibility. II. Some cups are jars.
(a) Only I follows (b) Only II follows (c) Both follow (d) Neither follows

Accepted Answer

(c) Both follow.

II first: the cup–glass overlap lies inside jars, so some cups are definitely jars — forced, II follows. I is a possibility conclusion: draw the cups circle entirely inside jars (legal — nothing forbids it), so 'all cups are jars' is possible in at least one diagram — I follows.

Question 6

Statements: No pen is a pencil. All pencils are crayons. Conclusions: I. Some crayons being pens is a possibility. II. All crayons being pens is a possibility.
(a) Only I follows (b) Only II follows (c) Both follow (d) Neither follows

Accepted Answer

(a) Only I follows.

I: the part of the crayons circle outside pencils is unconstrained, so it can legally be drawn overlapping pens — one supporting diagram exists, the possibility holds. II is definitively blocked: if all crayons were pens, then pencils (which sit inside crayons) would also be pens — but 'No pen is a pencil' forbids that. No legal diagram supports II, so it fails.

Question 7

Statements: All keys are rings. Some rings are chains. No chain is a lock. Conclusions: I. Some rings are not locks. II. Some keys are chains. III. No key is a lock.
(a) Only I follows (b) Only I and II follow (c) Only II and III follow (d) All follow

Accepted Answer

(a) Only I follows.

I: the rings that overlap chains lie outside locks (no chain is a lock), so those rings are not locks — forced. II: the ring–chain overlap can be drawn away from keys — not forced. III: keys are only constrained to sit inside rings; nothing stops keys from overlapping locks — not forced. Only I survives.

Question 8

Statements: Some fruits are not sweets. All sweets are treats. Conclusions: I. Some fruits are not treats. II. Some treats are sweets.
(a) Only I follows (b) Only II follows (c) Both follow (d) Neither follows

Accepted Answer

(b) Only II follows.

II: sweets sit inside treats and are non-empty, so some treats are sweets — valid conversion, follows. I is the trap: the fruits outside sweets could still be inside treats (treats is bigger than sweets), so 'some fruits are not treats' is not forced — draw fruits fully inside treats with part outside sweets and I dies.

Question 9

Statements: Some tiles are bricks. Some bricks are stones. Conclusions: I. All stones are tiles. II. Some stones are not tiles.
(a) Only I follows (b) Only II follows (c) Either I or II follows (d) Neither follows

Accepted Answer

(c) Either I or II follows.

I: nothing forces all stones inside tiles — fails alone. II: stones could be drawn entirely inside tiles — fails alone. Same two terms (stones, tiles); 'All are' and 'Some are not' form the second complementary pair — exactly one of them must be true in any diagram. Either–or applies.

Question 10

Statements: All papers are files. No file is a folder. Some folders are racks. Conclusions: I. No paper is a folder. II. Some racks are not files. III. Some racks being papers is a possibility.
(a) Only I follows (b) Only I and II follow (c) Only II and III follow (d) All three follow

Accepted Answer

(d) All three follow.

I: papers ⊆ files and files are disjoint from folders, so papers are disjoint from folders — forced. II: the racks that overlap folders lie outside files (no file is a folder), so those racks are not files — forced. III is a possibility conclusion: rack members outside the folder overlap are unconstrained and can legally be drawn inside papers, so at least one supporting diagram exists — the possibility follows.

Practice — Syllogism

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