Syllogism — Categorical logic on 'door', 'dog', and 'cat' Statements: • No door is dog. • All the dogs are cats. Conclusions to evaluate: No door is cat. No cat is door. Some cats are dogs. All the cats are dogs.

Introduction / Context:
This is a classic syllogism problem involving universal negatives and affirmatives. The goal is to determine which conclusions logically follow from the given statements without adding external assumptions beyond standard exam conventions.

Given Data / Assumptions:

• Premise 1: 'No door is dog' (Door ∩ Dog = ∅).
• Premise 2: 'All dogs are cats' (Dog ⊆ Cat).
• Typical exam convention allows existential import for 'dogs' when inferring 'Some cats are dogs' from 'All dogs are cats' (i.e., dogs are assumed to exist).

Concept / Approach:
Use standard Venn logic. From Dog ⊆ Cat, every dog lies within the Cat circle. From No Door is Dog, the Door set is disjoint from Dog, but it might still overlap with the Cat region that is not Dog. Therefore, universal negatives about Door vs Cat do not necessarily follow.

Step-by-Step Solution:
From 'All dogs are cats': If at least one dog exists, then 'Some cats are dogs' is true → Conclusion (3) follows under common test assumptions.From 'No door is dog': We only know Door excludes the Dog subset. Cats outside Dog remain unrestricted regarding Door, so we cannot assert 'No door is cat' or its converse.There is no basis for 'All the cats are dogs' because Dog is a subset of Cat, not vice versa.

Verification / Alternative check:
Construct a model: Let Dog = {d1}, Cat = {d1, c2}, Door = {r1}. Then Premise 1 and 2 hold. Here, some cats are dogs (d1 ∈ Cat ∩ Dog) is true; but 'No door is cat' is not determined (r1 could be non-cat or cat); 'All cats are dogs' is false.

Why Other Options Are Wrong:
(1) and (2): Neither necessarily follows; Door could overlap Cat outside Dog.(4): Reverses the subset relation; not supported.Combinations including any of (1), (2), or (4) are therefore invalid.

Common Pitfalls:
Assuming that because doors are not dogs and dogs are cats, doors are not cats. This ignores the possibility of cats that are not dogs.

Final Answer:
Only (3)