**2. Related Work**

Both Fuzzy Property Grammars (FPGr) and Fuzzy Natural Logic (FNL) can provide strategies to approach linguistic universals and the measurement of language complexity.

Regarding linguistic universality, we consider that the concept of universal can be better defined considering a continuous scale than adopting a discrete perspective. We disregard the fact that only those linguistic rules shared by all known languages—roughly 7000—can be regarded as linguistic universals. On the other hand, we define "universality" as a continuum using the FNL's gradient features of the theory of evaluative expressions. As a result, we shall continue to respect the two extreme points that already exist: 0 (nonuniversal) and 1 (full-universal). However, we create a spectrum in which we shall fit those linguistic rules known as "*quasi-universals*" between these two positions. On the other hand, FPGr and FNL make it possible to devise universal models, that is languageindependent models, that can be applied to all natural languages, and they use a fuzzygradient technique to describe linguistic universals in fuzzy terms [0, 1] rather than labeling universals with a confusing nomenclature. The times that a fuzzy universal is fulfilled or violated in a fuzzy grammar determines the fuzzy degree membership of a linguistic rule. To create a Fuzzy Universal Grammar, we will use the model of FPGr. Finally, FPGr is a cheap and reusable architecture to define linguistic phenomena and their variation and makes it possible to advance in the systematization of variation in languages.

Regarding the complexity of language, it can be captured in quantitative terms (absolute complexity), such as the more rules a grammar has, the more complex it is. Therefore, if we have a system that provides all the rules of a language, we could capture its degree of complexity under the architecture of FNL of the theory of evaluative expressions. It is also possible to measure complexity between languages with a FPGr. The more rules are shared between two languages, the less complexity will be found between those two languages. The fewer rules those languages share, the more complex they will be in relation to each other. However, this approach will have a higher cost, since it will demand checking how many rules of a targeted language are shared with respect to all the other languages. That is why we have disregarded such an approach, and we have implemented a Fuzzy Universal Property Grammar that will consider all the possible combinations of rules. Therefore, for every single set of languages, we will only need to check coincidences in our Fuzzy Universal Property Grammar. Thus, a value in terms of degree [0–1] will arise from the number of coincidences defining universality: as in what is the membership degree of a set of rules of a language with respect to a Fuzzy Universal Property Grammar. In this way, the notion of universality can help measure the relative complexity of a language, assuming that those languages that have a lot of specific rules are meant to be more complex. That is, the more universal a language is, the less complex it is; the less universal a language is, the more complex it is.

In the following, we present the models of Fuzzy Universal Property Grammar and Fuzzy Natural Logic as a strategy to define linguistic universality and language complexity as vague concepts.

#### *2.1. Fuzzy Property Grammars for Linguistic Universality*

Fuzzy Property Grammars (FPGr) [34–36] combine the formalism of Fuzzy Natural Logic [37–43] and linguistic constraints typically used in linguistics. The higher-order fuzzy logic as a formalism describes the grammar at a higher level (abstractly), enabling a mathematical formalization of the degrees of grammaticality. In comparison, linguistic constraints allow us to describe vague phenomena on a local-sentence level, characterizing the objects (constraints) as prototypical and borderline ones. Therefore, both assets are necessary to build an FPGr. There are three key concepts of an FPGr: linguistic constraint, universe of the linguistic domains and fuzzy grammar, and linguistic construction.
