Lukasiewicz logic

The basic assumption upon which two-valued logic is based, that every proposition is either true or false, has been questioned since Aristotle. For example, propositions about future events are neither actually true, nor actually false, but potentially either. Consequently, their truth-value is undetermined, at least prior to the event. Propositions whose truth-value is problematic are not limited to future events. For example, propositions that imply physical or chemical measurements depend on the limitations of mea-surement. While several types of multivalued logic have been proposed, here we will discuss the infinite-valued logic, whose truth-values are represented by all the real numbers in the interval [0,1]. This is also called standard Lukasiewicz logic L. The primitives of this logic are defined as... 

It can be verified that the relations above reduce to their usual counterparts when applied to binary logic. The standard Lukasiewicz logic Li is isomorphic to fuzzy set theory based on the standard fuzzy operations in the same way the two-valued logic is isomorphic to the crisp set theory. The membership degree A(x) for x e X may be interpreted as the truth value of the proposition x is a member of the set A . The reciprocal is also valid. 

Lukasiewicz 1917-1920 Development of a well-founded three-valued logic... 

Now as we have noted earlier, fuzzy logic (FL) is an extension of MVL with truth values as fuzzy sets or more accurately as fuzzy restrictions on the truth. We will still call the truth space U defined on the interval [0, 1], but in FL a truth value will be a fuzzy subset t CU and Xr U- [0, 1]. We will still require rules to define the implication relation 1 as well as negation, conjunction C, disjunction D and equivalence. Zadeh suggested the use of the Lukasiewicz rules given above and they will be used in the rest of this book. Baldwin, Pilsworth and Guild [85, 86] have examined various alternative rules for implication. 

Finally, before discussing a longer example it is worth reminding ourselves Of the Lukasiewicz rules for the various logical relations. 

We can now attempt a modus ponens deduction using the Lukasiewicz implication rule given earlier, as our base logic. 

An exponent of the traditional multi-valued logical system is the technique introduced by Lukasiewicz in the 1920s, which is introduced, for example, in Lukasiewicz (1935 176). For the contribution of Lukasiewicz s work to temporal logic refer to Ohrstrom Hasle (1995 149). 

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