**4. Discussion**

In the discussion section, we will comment on new forms of fuzzy syllogisms which we have proven in Section 3. In Section 3, we can see that we can order strongly valid syllogisms into triangles or columns by monotonicity. In the vertexes of these triangles are fuzzy syllogisms that consist of classical quantifiers or new forms of classical quantifiers. At the endpoints of the columns are also fuzzy syllogisms which contain classical quantifiers or new forms of classical quantifiers. Fuzzy syllogisms are proved by syntactic proofs, but we also use monotonicity to prove the strong validity of fuzzy syllogisms. We use monotonicity in three ways - to strengthen the first premise, to strengthen the second premise, or to weaken the conclusion. We also use Proposition 2 for the proofs.
