Assume that the statement ‘All cows are herbivores’ is true and that cows…

Concept

A universal affirmative statement “All S are P” places the entire set S inside the set P. When S is known to have at least one member, this inclusion also guarantees that at least one S is P; this is the traditional immediate inference called subalternation.

Application

1. Let C be the set of cows and H be the set of herbivores.

2. “All cows are herbivores” gives C ⊆ H.

3. Because cows exist, choose any cow c. Then c ∈ C, and C ⊆ H gives c ∈ H.

4. Therefore at least one individual is both a cow and a herbivore: some cows are herbivores.

Contrast

• “No herbivores are cows” would require C and H to be disjoint, contrary to the stated inclusion and existence of cows.

• “Some herbivores are not cows” may be true in the real world, but C ⊆ H alone does not establish an H-member outside C.

• “All non-cows are non-herbivores” is the inverse of the given implication; an implication does not generally establish its inverse.

Cross-check

In a Venn diagram, the cow circle lies wholly inside the herbivore circle. Since the cow circle is non-empty, their overlap contains at least one member. Hence the immediate conclusion is “Some cows are herbivores.”