*4.3. Figure III*

Figure III is different than previous figures. Firstly, on this figure, we can prove nontrivial syllogisms. Non-trivial syllogisms are syllogisms, as we said in the introduction, which contain intermediate quantifiers in both premises. This group of syllogisms is specific in that valid syllogisms work with a *common presupposition*. While in the previous figures, it was enough to always assume the presupposition and thus the non-emptiness of the fuzzy set for one fuzzy set, for non-trivial syllogisms it is necessary to assume the non-emptiness of the fuzzy set in the antecedent in both premises. This assumption is represented by the formula below.

$$(\exists x)((B|z) \, x \, \mathsf{dec}(B|z') \, x). \tag{27}$$

We denote the Formula (27) as a common presupposition of existential import of two fuzzy intermediate quantifiers (*Q*<sup>∀</sup>*Ev <sup>x</sup>*)(*<sup>B</sup>*, *A*) and (*Q*<sup>∀</sup>*Ev <sup>x</sup>*)(*<sup>B</sup>*,<sup>¬</sup>*<sup>A</sup>*).

We can order strongly valid syllogisms by monotonicity into triangles. As we can see in Theorems 24 and 26–28, these triangles are oriented differently than the triangles in Figure I and Figure II. In the proofs of Theorems 24 and 26–28, we strengthen the second premise to obtain strongly valid syllogisms in the columns. To obtain the strong validity the syllogisms in the first row, we strengthen the first premise.

In Theorem 29, we can order the strongly valid syllogisms into columns by monotonicity. In this Theorem, we use monotonicity to weaken the conclusion.
