## On fuzzy type theory.(English)Zbl 1068.03019

Fuzzy logic was originally invented to describe human reasoning. The original fuzzy logic covered only propositional formulas. Later, quantifiers were added to fuzzy logic. However, an important part of human reasoning is still missing from fuzzy logic: arguments that usually require a higher-order logic. For example, we can say that $$X$$ is a typical plant because it has all the properties that plants normally have. Such statements include quantifiers over all the properties; they are usually formalized in second-order logic. In the traditional (non-fuzzy) logic, one of the ways to describe higher-order logic is via a type theory. Type theory was originally designed to handle similar problems in set theory, where, in addition to sets of integers (i.e., in effect, properties of integers – properties of first order), we also have sets of sets of integers (i.e., properties satisfied by properties of first order). To formalize higher-order-type statements in fuzzy logic, the author proposes a new formalism that extends type theory to the multi-valued case of fuzzy logic. He also proves properties of this fuzzy type theory, e.g., its completeness. Interestingly, it turns out that these results are only possible within an appropriate formalization of fuzzy logic: e.g., it is essential to include a hedge “absolutely true” – defined as $$\Delta(a)=1$$ if $$a=1$$ and $$\Delta(a)=0$$ otherwise – into the set of basic fuzzy logic operations.

### MSC:

 03B52 Fuzzy logic; logic of vagueness

### Keywords:

fuzzy logic; type theory; higher-order logic

ETPS
Full Text:

### References:

  Andrews, P., A reduction of the axioms for the theory of propositional types, Fund. Math, 52, 345-350 (1963) · Zbl 0127.00701  Andrews, P., An Introduction to Mathematical Logic and Type Theory: To Truth Through Proof (2002), Kluwer: Kluwer Dordrecht · Zbl 1002.03002  Church, A., A formulation of the simple theory of types, J. Symbolic Logic, 5, 56-68 (1940) · JFM 66.1192.06  Esteva, F.; Godo, L., Monoidal t-norm based logictowards a logic for left-continuous t-norms, Fuzzy Sets and Systems, 124, 271-288 (2001) · Zbl 0994.03017  Gallin, D., Intensional and Higher-Order Modal Logic (With Applications to Montague Semantics) (1975), North-Holland: North-Holland Amsterdam · Zbl 0341.02014  Gottwald, S., A Treatise on Many-Valued Logics (2001), Research Studies Press Ltd: Research Studies Press Ltd Baldock, Herfordshire · Zbl 1048.03002  Hájek, P., Metamathematics of Fuzzy Logic (1998), Kluwer: Kluwer Dordrecht · Zbl 0937.03030  Henkin, L., Completeness in the theory of types, J. Symbolic Logic, 15, 81-91 (1950) · Zbl 0039.00801  Henkin, L., A theory of propositional types, Fund. Math, 52, 323-344 (1963) · Zbl 0127.00609  Hindley, J. R., Basic Simple Type Theory (1997), Cambridge University Press: Cambridge University Press Cambridge · Zbl 0906.03012  Höhle, U., Commutative residuated l-monoids, (Höhle, U.; Klement, E. P., Non-Classical Logics and Their Applications to Fuzzy Subsets. A Handbook of the Mathematical Foundations of Fuzzy Set Theory (1995), Kluwer: Kluwer Dordrecht), 53-106 · Zbl 0838.06012  Klawonn, F.; Kruse, R., Equality relations as a basis for fuzzy control, Fuzzy Sets and Systems, 54, 147-156 (1993) · Zbl 0785.93059  Martin-Löf, P., Intuitionistic Type Theory (1984), Bibliopolis: Bibliopolis Naples  Novák, V., Fuzzy Sets and Their Applications (1989), Adam Hilger: Adam Hilger Bristol · Zbl 0683.94018  Novák, V., Descriptions in full fuzzy type theory, Neural Network World, 13, 5, 559-569 (2003)  Novák, V.; Perfilieva, I.; Močkoř, J., Mathematical Principles of Fuzzy Logic (1999), Kluwer: Kluwer Boston, Dordrecht · Zbl 0940.03028  Perfilieva, I., Normal forms for fuzzy logic functions and their approximation ability, Fuzzy Sets and Systems, 124, 3, 371-384 (2001) · Zbl 0994.03019  Ranta, A., Type-Theoretical Grammar (1994), Clarendon Press: Clarendon Press Oxford · Zbl 0855.68073  Tichý, P., The Foundations of Frege’s Logic (1988), de Gruyter: de Gruyter Berlin · Zbl 0671.03001
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.