*4.2. Figure II*

The structures of the syllogisms of Figure II are similar to the structures of Figure I. We use monotonicity to weaken the conclusion in the proofs of Theorems 16–19. As we can see in these Theorems that we ordered the strongly valid syllogisms into triangles by monotonicity.

In Theorem 21, we ordered the syllogisms into columns by monotonicity. In the proof, we used monotonicity to strengthen the first premise.

In Theorem 22, we can find eight strongly valid syllogisms. We showed the proof of syllogism **(\*A)Ai-II**, in which we can see that its presupposition is a formula (<sup>∃</sup>*x*)(¬*Mx*), but the middle formula in this syllogism is (*Mx*). This is a consequence of the property of

contraposition (Lemma A1(h)). The formula representing the presupposition is related to the assumption that all formulas are not empty.
