*4.4. Figure IV*

In Figure IV, we order strongly valid syllogisms only into columns by monotonicity. We can see that in Theorem 32, we use monotonicity to strengthen the second premise. We can also see that in Theorem 33, we use monotonicity to weaken the conclusion. Finally, we can see that in Theorem 35, we use monotonicity to strengthen the first premise.

#### **5. Conclusion and Future Work**

In the article, we followed up on previous results concerning the formal proof of fuzzy logic syllogisms in fuzzy natural logic. In the introduction to the article, we first set out the motivation for this, with various references to application areas that address the issue of fuzzy generalized quantifiers. We also introduced the reader to the main mathematical territories that shape natural fuzzy logic. The main results are contained in the third section, where we first presented the mathematical definitions of fuzzy intermediate quantifiers that form a graded Peterson's cube of opposition. We managed to formally prove, in the formal mathematical system, several new forms of logical syllogisms, the validity of which we semantically verified in the finite model. The main result is that all syntactically proven fuzzy syllogisms hold in every model.

We see further development of this article in two directions. We will first focus on extending the structure of valid fuzzy logical syllogisms by more premises. In the second part, we would like to mathematically propose Peterson's rules of distributivity, quality, and quantity for verifying the validity of logical syllogisms related to a graded Peterson's cube of opposition. The second main objective for the future is to program an algorithm based on these rules and verify the validity of new forms of fuzzy syllogisms automatically.

**Author Contributions:** Conceptualization: K.F. and P.M.; methodology, K.F. and P.M.; validation, K.F. and P.M.; formal analysis, K.F. and P.M.; investigation, K.F. and P.M.; resources, K.F. and P.M.; writing—original draft preparation, K.F. and P.M.; writing—review and editing, K.F. and P.M.; visualization, K.F. and P.M.; supervision, P.M.; project administration, P.M. All authors have read and agreed to the published version of the manuscript.

**Funding:** The work was supported by ERDF/ESF by the project "Centre for the development of Artificial Inteligence Methods for the Automotive Industry of the region", No. CZ.02.1.01/0.0/0.0/17- 049/0008414.

**Data Availability Statement:** Not applicable.

**Conflicts of Interest:** The authors declare no conflict of interest.
