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Two-premise categorical logic: ‘‘All windows are doors’’ and ‘‘No door is wall’’ – determine whether the derived negatives about windows and walls follow.

Difficulty: Medium

Only conclusion I follows
Only conclusion II follows
Either I or II follows
Neither I nor II follows
Both I and II follow

Correct Answer: Both I and II follow

Explanation:

Premises: (1) All windows are doors (W ⊆ D). (2) No door is a wall (D ∩ Wall = ∅). Conclusions: (I) No window is wall. (II) No wall is door.

Concept/Approach
Use subset chaining and properties of ‘‘No’’ (E-type) propositions, which are convertible: ‘‘No A are B’’ ⇒ ‘‘No B are A.’’
Deriving Conclusion I
Since W ⊆ D and D shares no elements with Wall, W shares none with Wall. Hence ‘‘No window is wall’’ follows.
Deriving Conclusion II
From ‘‘No door is wall’’ (No D are Wall), by conversion of an E-proposition, ‘‘No wall is door’’ holds. Hence II follows.
Common pitfalls
Forgetting that E-statements are symmetric; both directions are valid.
Final Answer
Both I and II follow.

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Two-premise categorical logic: ‘‘All windows are doors’’ and ‘‘No door is wall’’ – determine whether the derived negatives about windows and walls follow.

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