**5. Probabilistic and Fuzzy Logic in distinct sides of knowledge**

I think that the fuzzy interpretation, proposed by Kosko, of the wave-function |ψ|2 in Schrödinger's equation is very interesting. Indeed it seems to me that Schrödinger's equation regards the quantity and the quantum distribution of matter, and not the probability to find the particle in the region dV. However, in other fields of science it is not useful to try to reduce probabilistic logic to Fuzzy Logic or to treat the problems of probability with Fuzzy Logic. It is also wrong to reduce Fuzzy Logic to probabilistic logic. These two kinds of calculus have different fields of employment, different aims and give different informations about phenomena. An evidence is that probabilistic diagnosis and fuzzy diagnosis give different kinds of information about the health of the patient. In particular: probabilistic diagnosis drives in the choice of the possible diseases which could cause the symptoms, while fuzzy diagnosis gives the exact quantification of the strength of diseases. They are both useful in the study and in the cure of pathology but they do different tasks (cf. Licata, 2007). It is usual in literature to distinguish probabilistic logic from fuzzy logic, telling that the first is a way to formalize the "uncertainty" while the second is a method to treat "vagueness". In technical sense, uncertainty is the incompleteness of information, while vagueness is the absence of precise confines in the reference of *quantitative* adjectives, common names, etc. to objects of world (see §3). Nevertheless, some authors employed Fuzzy Logic to treat uncertainty (in the sense of incompleteness of information) and many theorists of probability think that probabilistic logic is a good way to treat vagueness. In general, it is clear that vagueness and uncertainty (in technical sense) can be theorized as two distinct areas of knowledge, studied by distinct methods. Given that uncertainty is understood as incompleteness of information, while vagueness regards an indefinite relationship between words and objects, it is possible to say that uncertainty and probabilistic logic fall in the area of "subjective knowledge", while vagueness and Fuzzy Logic fall in the area of "objective knowledge".

#### **6. Acknowledgments**

I thank Giuseppe Nicolaci and Marco Buzzoni for their irreplaceable help in the development of my research.

#### **7. References**

AA.VV., (1988). La Nuova Enciclopedia delle Scienze Garzanti, Garzanti editore, Milano.

Benzi, M. (1997). Il ragionamento incerto, Franco Angeli, Milano.


**Section 2** 

**Application to Management Problems** 

