Two statements, I and II, are followed by two conclusions, (a) and (b).…

Concept

In categorical logic, an existential statement says that at least one member belongs to a set. It does not identify that member beyond the stated property.

Two separate existential statements do not establish whether their witnesses are the same, whether the sets overlap, or whether any member exists outside their union. A conclusion follows only if it is true in every model satisfying the statements.

Application

1. Let R be the set of red flowers and B the set of blue flowers. Statement I gives R as non-empty, and Statement II gives B as non-empty.

2. To test conclusion (a), consider a model with one flower that is both red and blue and no other flowers. Both statements are true, but no flower lies outside R ∪ B, so conclusion (a) is not forced.

3. To test conclusion (b), consider two distinct flowers: one red-only flower and one blue-only flower. Both statements are true, but R ∩ B is empty, so conclusion (b) is not forced.

Cross-check and contrast

A valid conclusion must survive every model allowed by the statements. Each proposed conclusion has a countermodel, so neither is necessary.

• Only (a) follows: this requires an outside flower that the statements never guarantee.

• Only (b) follows: this requires a red-and-blue flower that the statements never guarantee.

• Both (a) and (b) follow: each of the two required existence claims can fail in a valid model.

• Neither (a) nor (b) follows: this is consistent with the countermodels for both conclusions.

Result

Therefore, neither (a) nor (b) logically follows.