Goguen's contributions to fuzzy logic in retrospect

ABSTRACT

In the very early stage of the development of fuzzy logic, Joseph Goguen published profound work with lasting influence. Fifty years later, we provide an assessment of his contributions to fuzzy logic and thus pay tribute to Goguen, a former long-term member of the editorial board of this journal. We cover both Goguen's technical results, the research directions inspired by him, which have later been pursued, as well as his suggestions, which as yet remained explored to a lesser extent or practically unexplored.

KEYWORDS:

• Fuzzy logic
• fuzzy set

Disclosure statement

No potential conflict of interest was reported by the authors.

ORCID

R. Belohlavek  http://orcid.org/0000-0003-4924-3233

Notes

1. These two papers are among Goguen's three most cited papers: The most cited is Goguen (1967) with 2675 citations on Google Scholar and 1562 citations on Scopus; second is “Security policies and security models” (Proc. IEEE Comp. Soc. Symposium on Research in Security and Privacy, Oakland, CA, 1982, pp. 11–20) by Goguen and J. Meseguer with GS 2492 and Scopus 1051, third is Goguen (1968–9) with GS 1263 and Scopus 615. These papers also belong to the most cited papers on fuzzy logic.

2. Interestingly, Goguen seems not to have been aware of the existing work on propositional and predicate Łukasiewicz logic. He refers only to intuitionistic logic when considering algebraic structures of fuzzy logic and omits completely the then already vast number of results and the existing books on many-valued logics. As a result, he reinvents some then already existing definitions for predicate many-valued logics, such as how to model semantics of the existential and universal quantifiers in a many-valued setting.

3. He denotes formulas by P, Q, etc., implication by ⇒, and the truth value of P by $\left[P\right]$. We denote formulas by $\varphi ,\psi ,\dots$, logical connectives by $\to$ (implication), ⊗ (conjunction), …, their truth functions by ${\to }^{\cdot }$, ${\otimes }^{\cdot },\dots$, and the truth value of formula ϕ by $\parallel \varphi \parallel$.

4. The term “sorites paradox” sometimes refers to similar kinds of paradoxes involving vague predicates, e.g. the falakros paradox, also called the paradox of the bald man: A man with one hair is bald; if a man with n hairs is bald then a man with n + 1 hairs is bald; hence every man is bald.

5. However, there also exists a number of other, very different resolutions which have been proposed; see e.g. Rescher (2001) and Sainsbury (2009)

6. Completeness of Łukasiewicz logic was proved by Rose and Rosser in 1958; completeness of Gödel logic was proved by M. Dummett in 1959; see Bělohlávek, Dauben, and Klir (2017).

7. Using general partially ordered sets as sets of truth degrees was also mentioned by Zadeh (1965).

8. Goguen also presented additional reasons. First, one needs infima and suprema to define intersections and unions of fuzzy sets. Second, if $a_j$s represent performances of alternatives, then the performance of any alternative is at least the infimum of $a_j$s and at most the supremum of $a_j$s.

9. Most fuzzy logics assume commutativity as a natural property.

10. More precisely, the following theorem follows from results of classical model theory: ${\mathbf{L}}^U$ satisfies all Horn formulas satisfied by $\mathbf{L}$. This result was observed in Schwartz (1972).

11. Black suggested that $i(B,A)$ be based on the ratio $|\{u\mid B(u)<A(u\left)\right\}|/|\{u\mid B(u)>A(u\left)\right\}|$.

12. For instance, the fact that taking sets as objects and taking fuzzy relations between sets as morphisms forms a category means that composition of fuzzy relations is associative and has a neutral element.

13. Goguen was probably not aware that his definition of composition of morphisms is essentially a definition of the extension principle for functions with two arguments. He established some properties for it. The principle was then available for unary mappings in Zadeh (1965) with no further results. This fact seems unknown.

14. If L is a complete residuated lattice, $\mathbf{S}\mathbf{e}\mathbf{t}\left(L\right)$ is essentially a subcategory of $\mathbf{S}\left(L\right)$.

15. An ordinary topological space $⟨X,\tau ⟩$ is compact if every cover of X contains a finite subset which is still a cover of X. A cover of X a collection μ of open sets, i.e. $\mu \subseteq \tau$, whose union is X, i.e. $\bigcup _{A\in \mu }A=X$.

16. Tychonoff's theorem has been described as “probably the most important single theorem of general topology”; see Kelley (1955). The theorem says that a product of compact topological spaces is compact.

Additional information

Funding

This work was supported by the grant IGA_PrF_2019_034 of Palacky University Olomouc.

Notes on contributors

R. Belohlavek

Radim Belohlavek received Ph.D. degree in computer science from the Technical University of Ostrava, Czech Republic, in 1998, Ph.D. degree in mathematics from Palacky University, Olomouc, Czech Republic, in 2001, and D.Sc. degree in informatics and cybernetics from the Academy of Sciences of the Czech Republic in 2008. He is professor of computer science at Palacky University. Dr. Belohlavek's academic interests are in discrete mathematics, logic, uncertainty and information, and data analysis. He published two books, Fuzzy Relational Systems: Foundations and Principles (Kluwer, 2002) and Fuzzy Equational Logic (Springer, 2005, with Vilem Vychodil) and over 150 papers in conference proceedings and journals. Dr. Belohlavek is a Senior Member of IEEE (Institute of Electrical and Electronics Engineers), and a Member of ACM (Association for Computing Machinery) and AMS (American Mathematical Society), and is a member of editorial boards of several international journals.