**1. Introduction**

The article aims to follow up on the achieved results concerning the formal proofs of fuzzy Peterson syllogisms.

The theory of syllogistic reasoning was investigated by several authors as a generalization of classical Aristotelian syllogisms ([1–3]). Categorical syllogisms ([4,5]) consist of three main parts: *the major premise*, *the minor premise*, and *the conclusion*. With the introduction of the term "generalized quantifier", which was proposed by Mostowski in [6], the study of logical syllogisms expanded to a group of generalized logical syllogisms. Such syllogisms include forms that contain different types of generalized quantifiers. One special group is intermediate quantifiers. This topic started to be interesting for linguists and philosophers, who laid a question of how to explain expressions that represent generalized quantifiers.

Peterson, in his book [7], is interested in the group of intermediate quantifiers which lie between classical quantifiers. He first philosophically analyzed and explained the meaning of intermediate quantifiers in terms of their position in the generalized square of opposition. Furthermore, he continued the study of the group of generalized logical syllogisms. Peterson's enlargement was established on an idea to substitute a classical quantifier with an intermediate one in the classical four figures, which returned in 105 new valid intermediate syllogisms. A check of the validity of a group of logical syllogisms was conducted using Venn diagrams, which was carried out by several authors [7–9]. Below, we present an example of a non-trivial fuzzy intermediate syllogism as follows:

> *P*1 : *Almost all people do not have a plane. P*2 : *Most people have a phone. C*:*Somepeoplewhohave a phonedonothaveaplane.*

The work of authors who dealt with generalized quantifiers was followed by several authors with the advent of the definition of a fuzzy set. Several authors followed up this approach. They introduced several forms of logical syllogisms with *fuzzy* generalized quantifiers. In 1985, L. Zadeh semantically interpreted a special group of fuzzy syllogisms

**Citation:** Fiala, K.; Murinová, P. A Formal Analysis of Generalized Peterson's Syllogisms Related to Graded Peterson's Cube. *Mathematics* **2022**, *10*, 906. https://doi.org/ 10.3390/math10060906

Academic Editor: ManuelOjeda-Aciego

Received: 27 January 2022 Accepted: 7 March 2022 Published: 11 March 2022

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**Copyright:** © 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

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with fuzzy intermediate quantifiers in both premises as well as in the conclusion. Below, we present a very famous *multiplicative chaining syllogism* ([10]) as follows:

$$\begin{array}{l} \text{Q}\_1 \text{ Y is } M\\ \text{Q}\_2 \text{ M is X} \\ \hline (\geq \text{Q}\_1 \otimes \text{Q}\_2) \text{ Y are X.} \end{array}$$

As was explained by Zadeh, the expression ≥ *Q*1 ⊗ *Q*2 in the conclusion of Zadeh's syllogism can be read as "at least *Q*1 ⊗ *Q*2".

Zadeh's work was later extended and sophisticated by many authors. From the point of view of fuzzy quantifiers, which are represented as intervals in Didier Duboise's approach in ([11,12]), Zadeh's special syllogism is compared by M. Pereira-Fariña in [13]. In the publications, Dubois et al. work with quantifiers represented as crisp closed intervals (*more than a half* = [0.5, 1], *around five* = [4, 6]). A typical example of such interval fuzzy syllogism appears as follows:

> *P*1: [5%, 10%] students have a job. *P*2: [5%, 10%] students have a child. *C*: [0%, 10%] students have a child and have a job.

Recall that a global overview of fuzzy generalized quantifiers and the various mechanisms for defining these quantifiers can be found in [14].

M. Pereira-Fariña et al. follow, in the next publication [15], by interpreting logical syllogisms with more premises. In this publication, a group of authors suggested a *general inference schema for syllogistic reasoning*, which was proposed as the transformation of the syllogistic reasoning study into an equivalent optimization problem.

The above-mentioned publications study and verify the validity of fuzzy syllogisms, especially semantically. In [16], a mathematical definition of fuzzy intermediate quantifiers based on the theory of evaluative linguistic expressions was proposed. The motivation, fundamental assumptions, and formalization of this theory are described in detail in [17]. Later, in [18], we focused on the syntactic construction of proofs of all 105 basic fuzzy logical syllogisms that relate to the graded Peterson's square of opposition (see [19]). Typical examples of natural language expressions contained in Peterson's logical syllogisms are as follows:

> *Most children like computer games*. *Most cats like to sleep.*

#### *1.1. Application of Generalized Quantifiers*

Generalized quantifiers offer several types of applications in economics, medicine, heavy industry, biology, etc. Let us first mention applications related to a group of fuzzy intermediate quantifiers which are represented by natural language expressions. One of the interesting applications is time series prediction, which has its application mainly in economic fields. In [20], the author proposed an interpretation, forecasting, and linguistic characterization of time series. The result makes it possible to obtain information about the data using natural language, which is much more understandable to the average user. To illustrate the reader, we present an example of the linguistic interpretation of economic time series, using natural language, as follows:

#### *Most (many, few)* analyzed time series stagnated recently, but their future trend is slightly *increasing* [20].

Another area that is very closely related to this application is getting new information from natural language data. Here, it is offered to use the theory of syllogistic reasoning and to obtain new information from natural language data using valid forms of syllogisms. Time series were also analyzed by a group of authors in [21]. There are also publications of authors who are interested in the linguistic summarization of data.

In [22], the authors introduced methods for using the linguistic database to summarize natural data (see [23–26]). Later, these methods were improved in [27] and implemented by Kasprzyk and Zadrozny [ ˙ 28]. Another linguistic summarization of process data was proposed in [29].

Another area of application is the use of fuzzy association analysis and the use of association rules to interpret natural language data. An algorithm for the interpretation of biological data using fuzzy intermediate quantifiers was proposed in [30].

> *Most irises with both small-length sepals and petals have small-width petals.*
