Satisfiability in Gödel Logics

There is a visible asymmetry in logic research between the study of validity and the study of satisfiability in the sense of being true. Since Tarski`s seminal work, logics are usually identified with the set of their valid sentences. Therefore, the study of the properties of the set of tautologies of a logic are in the focus of attention. Not much attention, however, is paid to the issue of satisfiability. In particular, not much is known about the recursive enumerability of the set of unsatisfiable sentences of many important logics. This is not an issue in classical logic, where unsatisfiability is dual to validity. Indeed, testing the validity of a formula is equivalent to testing the unsatisfiability of its negation. However, this is not the case in many important non-classical logics, including Gödel logics, where the negation of a formula may be unsatisfiable, while the formula may still not be valid. The main goal of this project is to resolve this asymmetry for first-order Gödel logics, providing a uniform treatment of the validity and the satisfiability problem in as close analogy to classical logic as possible. (Gödel logics form one of the most important classes of logics between intuitionistic and classical logic.)

Satisability in Gödel Logics Abstract for publicity There is a visible asymmetry in logic research between the study of validity and the study of satisability in the sense of being true. Since Tarski`s seminal work, logics are usually identied with the set of their valid sentences. Therefore, the study of the properties of the set of tautologies of a logic are in the focus of attention. Not much attention, however, is paid to the issue of satisability. In particular, not much is known about the recursive enumerability of the set of unsatisable sentences of many important logics. This is not an issue in classical logic, where unsatisability is dual to validity. Indeed, testing the validity of a formula is equivalent to testing the unsatisability of its negation . However, this is not the case in many important non-classical logics, including Gödel logics, where the negation of a formula may be unsatisable, while the formula may still not be valid. The main result of this project is the resolution of this asymmetry for rst-order Gödel logics, providing a uniform treatment of the validity and the satisability problem in as close analogy to classical logic as possible. 1