What properties does fuzzy logic preserve?

Question

I found the following statement in chapter 11 of Introduction to Fuzzy Logic by Rajjan Shinghal, discussing inference in fuzzy logic.

Approximate reasoning is carried out with fuzzy implications. Inferences can be made by generalized modus ponens even when a given fuzzy set is not identical to the fuzzy set in the antecedent of the fuzzy implication. This is done by first obtaining the relation between the antecedent and the consequent. The relation is then applied to the given fuzzy set. The procedure can be extended to compound rules.

This got me thinking: logic will generally preserve some property of the premises. For example, classical logic is truth-preserving and constructive logic is usually interpreted as being justification-preserving. So which property is fuzzy logic preserving?

Answer 1

You can see :

• Merrie Bergmann, An Introduction to Many-Valued and Fuzzy Logic, Oxford UP (2008), page 176-on.

There are various versions of Fuzzy logic, but - in a nutshell - we have the counterparts of the usual defintions :

A tautology is a formula that always has the value $1$ and a contradiction is a formula that always has the value $0$.

We have that :

every Fuzzy tautology is a classical tautology, and every Fuzzy contradiction is a classical contradiction, and every argument that is valid in Fuzzy logic is also classically valid.

But the converses do not hold: not every classical tautology is a Fuzzy tautology, and so on. For example, the classical tautology $P \lor ¬P$ isn’t a Fuzzy tautology.

See page 224-on for an axiom system :

that is sound that is, every theorem is a tautology of Fuzzy logic, and also weakly complete : every tautology of Fuzzy logic is a theorem of the axiomatic system (i.e. a formula derived from no assumptions).

However, the system is not strongly complete, where strong completeness means that, given any semantic entailment of a formula $P$ from a set $\Gamma$ of formulas, there is a derivation of $P$ from those formulas in the axiomatic system.

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For moe details, we can see :

• Petr Hájek, Metamathematics of Fuzzy Logic, Springer (1998)

as well as :

• Franco Montagna (editor), Petr Hájek on Mathematical Fuzzy Logic, Springer (2015).