Perfect Pavelka Logic

Zadeh's Fuzzy Logic was introduced as a mathematical tool for deduction working with different degrees of truth, instead of the two truth degrees of Classical Logic. The next step taken by Pavelka was to admit assumptions and theorems which are also satisfied only to some truth degree, as well as deduction rules transferring these degrees to the deduced formulas. In the semantics these possibly, but not necessarily, correspond to the degrees of satisfaction of formulas. A significant consequent progress towards this direction was achieved by Hajek's work on Rational Pavelka Logic; it forms formulas with graded truth as couples of a formula and a rational truth value. The success of the Rational Pavelka Logic follows from the properties of the standard MV-algebra, i.e. the real unit interval [0, 1] with the Lukasiewicz operations, which was used as its semantical interpretation. The main mathematical result is that the Rational Pavelka Logic is sound and complete in the sense that the syntactical provability degree and the semantical degree of satisfaction coincide. In this paper an alternative to Pavelka's approach is investigated. Instead of the standard MV-algebra, the truth values are taken from a perfect MV-algebra, in particular, Chang's MV-algebra. This is the opposite extreme among MV-algebras, it admits only infinitesimals besides the two classical truth degrees. We argue that such an example also has a motivation in deduction imitating human reasoning. As Chang's MV-algebra is not complete, the provability degree need not exist. Nevertheless, as the main result the following weak completeness theorem is proved: If the provability degree of a formula exists, it coincides with the degree of satisfaction. This result is obtained by adding to Pavelka's original approach a new rule of inference, Hajek's book-keeping axioms and an axiom valid in Chang's MV-algebra. An example that clarifies the differences between Classical Logic, Rational Pavelka Logic and Perfect Pavelka Logic is given, and some new fuzzy rules of inference are introduced. (C) 2014 Elsevier B.V. All rights reserved.