No cats are dogs. All dogs are animals. Therefore, no cats are what?

In a categorical syllogism, a valid conclusion must be licensed by both premises through the shared middle term. When one premise is a universal negative ('No A is B') and the other is a universal affirmative ('All B is C'), the negative premise blocks any new affirmative link, but it does not license a brand-new negative link between the two outer terms (A and C) either — the only negative statement the premises actually guarantee is the one already given in the negative premise itself.

1. Premise 1, 'No cats are dogs', places cats and dogs in two groups that do not overlap.

2. Premise 2, 'All dogs are animals', places the entire dogs group inside the animals group.

3. Because Premise 1 is negative, it only guarantees that cats and dogs do not overlap — it says nothing about how cats relate to animals, mammals, or birds; none of those relations is fixed by the two statements.

4. So the blank can only be filled by the relation Premise 1 already establishes, not by any new relation.

Venn-diagram check: draw the dogs circle entirely inside the animals circle, and the cats circle entirely outside the dogs circle. The cats circle can still be drawn overlapping the animals circle, or entirely outside it — both drawings satisfy the two premises equally, so the cats-animals relation is never fixed and cannot be the conclusion. Only the cats-dogs relation from Premise 1 survives as certain. Hence the missing term is 'dogs' — this restates Premise 1, it is not a new deduction.