*1.2. Main Goals*

The main idea of this publication is to work with terms that also contain negated terms in the antecedent, and to study related valid fuzzy syllogisms. Typical examples of natural language expressions which are related to the graded Peterson's ([19]) cube are as follows:

> *Almost all students who do not like mathematics do not study technical fields*. *Most people who do not drink alcohol have healthy livers*.

A typical example of a syllogism with fuzzy intermediate quantifiers in both premises, which is called *non-trivial*, reads as follows:

> *P*1 : *Almost all people who do sports have healthy lungs. P*2 : *Almost all people who do sports do not have asthma. C* : *Some people who do not have asthma have healthy lungs.*

#### *1.3. Application of New Forms of Fuzzy Intermediate Quantifiers*

As mentioned above, there are several application areas where natural language expressions are used. We also gave, in the previous section, specific examples of natural language expressions that occur in both of the premises of syllogisms or are used to interpret natural data. Therefore, the idea is offered to first find and formally prove the validity of new forms of logical syllogisms, and further, to work on the use of these forms in the areas of fuzzy association analysis, language interpretation, linguistic summarization, the interpretation of time series, etc.

The paper is structured as follows: after the motivational introduction, the reader is acquainted with mathematical theory in the methods section. The third section contains important mathematical definitions of new forms of intermediate quantifiers. In this section, we follow with proofs of new forms of logical syllogisms. The results section is closed by concrete examples of valid forms of logical syllogisms. This section continues with a discussion section, in which we summarize the results achieved. We conclude this paper with a conclusion and statement of future directions.
