Monadic Gödel logics

Gödel logics are one of the most important examples of manyvalued logics. They are both intermediate logics (logics between classical and intuitionistic logic) and prominent examples for fuzzy logics. The concept of fuzzy properties has been propagated by Lotfi Zadeh to represent inferences involving vague information, it is now the basis of various applications. Although this project addresses mathematical problems it is intended to incorporate the results into the most important expert system for medical diagnosis at the Vienna General Hospital. Fuzzy logics in general allow sentences to be assigned not only 1 for true and 0 for false, but also any number between 0 and 1 to represent relative falsity or truth. In Gödel logics, the meaning of these truth values only depends on their order. Kurt Gödel (19061978), one of the most famous logicians of modern age, described this group of logics to prove that infinitely many logics between intuitionistic and classic logic exist. This project deals in particular with the characterization of the monadic fragments of Gödel logics. These fragments are formed by restricting the predicate symbols to unary ones and constitute the most important fragment in the language of predicate logic. We aim to determine the decidability of the validity and of the satisfiability of sentences in this fragment and in the related onevariable and twovariable fragments. We also want to investigate the closely linked problem whether the validity of sentences in the monadic fragment of ukasiewicz logic is decidable. This is the most important among the open mathematical problems in the field of fuzzy logic.

The monadic fragment is one of the most important fragments of first-order logics. It formalizes e.g. the syllogisms of Aristoteles and most of rule-based systems. This fragment is decidable in classical logic. There is a lot of research going on with respect to this fragment due to its importance to publications.In this project we investigated the monadic fragments of Gödel logics.Gödel logics form an important class of intermediate logics. They are connected to Fuzzy reasoning. They are defined by closed subsets of [0; 1], which include 0 and 1.The project distinguished the problem of validity from the problem of 1-satisfiability. Validity is not dual to 1-satisfiability when addressing non classical logics. The project obtains a full classification of the decidability status of monadic first-order logics:1. The monadic fragment of all finitely-valued first-order Gödel logics is decidable with respect to validity and 1-satisfiability.2. The monadic fragment of all infinitely-valued Gödel logics is undecidable with respect to validity.3. The monadic fragment of infinitely-valued Gödel logics is decidable with respect to 1-satisfiability if and only if 0 is isolated in the set of truth values.