Approximate Reasoning

In accordance with Hilbert’s approach, in this chapter we will show that it is possible to define the deduction operator of a fuzzy logic by a suitable extension of the notions of inference rule and proof. To justify such an extension we start from the famous paradox, the “Heap Paradox”, involving the vague predicate “is small”. A similar paradox is the “Bald Man Paradox”. An interesting treatment of such paradoxes by means of fuzzy logic can be found in Goguen [1968/69]. The paradox runs as follows: (a)If you add one stone to a small heap, it remains small.(b)A heap containing just one stone is small, therefore:(c)every heap is small.