# On the difference between traditional and deductive fuzzy logic

 Authors: Title: On the difference between traditional and deductive fuzzy logic Journal: Fuzzy Sets and Systems Volume 159 Number 10 Pages: 1153-1164 Year: 2008 Download from the publisher Preprint

#### Relation to the program of the Manifesto

The paper delimits the area of logic-based fuzzy mathematics more narrowly than an earlier paper From fuzzy logic to fuzzy mathematics: a methodological manifesto, which (following the optimism of Hájek's 1998 monograph) assumed that formal fuzzy logic can give foundations to all fuzzy mathematics. However, it turned out that traditional fuzzy mathematics actually deals with too many different phenomena and is in fact composed of several completely different parts. The field in which the logic-based approach is most fruitful is marked by a clear interpretation of membership degrees as degrees of truth (preserved under inference), while other areas of fuzzy mathematics work with a mixture of other notions of `degrees' (often not clarified enough). Naturally, logic-based methods apply less straightforwardly to such fields, even though formal fuzzy logic can sometimes help there as well. The paper therefore presents rather a more precise delimitation of the area of research than a retreat from the foundational program. -- LBehounek 16:35, 2 September 2008 (CEST)

#### The name of the class of logics

The name deductive fuzzy logic is in the paper applied both to a discipline and to a formally delimited class of logics (viz the intersection of the classes of Cintula's weakly implicative fuzzy logics and Ono's substructural logics as the logics of residuated lattices). As argued in a comment to the paper Fuzzy logics as the logics of chains, a better name for the class could perhaps be semilinear substructural logics (see the comments there). -- LBehounek 20:30, 2 September 2008 (CEST)

#### Truth degrees as upper intervals

In his talk at Logic, Algebra and Truth Degrees in Siena 2008, Josep Maria Font proposed the upper intervals $\displaystyle [\alpha,\to)=\{\beta\in L\mid\beta\ge\alpha\}$ to be called truth degrees, for all truth values $\displaystyle \alpha\in L$ . This proposal is quite consonant with my description of the "principle of persistence" in the present paper, and I regret that I did not get the idea of presenting the principle of persistence in this way, as it might have been more comprehensible for traditional fuzzy mathematicians than my expression "guaranteed thresholds" found in the paper. The rationale for Font's proposal is that so defined truth degrees, rather than truth values, are preserved by fully true implications (which was also one of the messages of my paper). -- LBehounek 16:09, 25 September 2008 (CEST)

#### Partial truth?

Although the paper uses the term partial truth frequently, it is not meant to engage in the philosophical dispute on the nature and bivalence of truth (the intended audience were researchers in traditional fuzzy logic rather than philosophers). The term was in fact used in the technical sense of "the (gradual) quality of propositions that is preserved under deductions in fuzzy logic". This quality is called here “partial truth” in analogy with the (bivalent) quality transmitted in deductions of classical logic, which is usually called truth. Whether we call the gradual quality "partial truth" or another name has no effect on the observations made in the article: the only important thesis is that similarly as classical logic operates salva veritatis, deductive fuzzy logics infer their conclusions salvo gradu - i.e., preserving the grades assigned to propositions, no matter whether they are interpreted as the degrees of truth, a measure of the underlying attributes,[1] costs,[2] or grades of any other kind. -- LBehounek 18:28, 5 November 2008 (CET)

## References

1. R. Keefe: Vagueness by numbers. Mind, 107:565–579, 1998.
2. L. Běhounek: Modeling costs of program runs in fuzzified propositional dynamic logic. In F. Hakl (ed.) Doktorandské dny ’08, pp. 11–18, Matfyzpress, Prague, 2008.