Introduction to non-classical logics 2400-OG-EN-ICL

The course includes elements of modern symbolic logic, including introductory presentation of set theory, first-order classical logic, proof theory, the semantics of chosen systems and basic meta-logical results.
The study of various non-classical logics such as modal logic, many-valued logic, paraconsistent logic will be given.

Next to intuitions that led to a given system, the syntactical characterization will be proposed. From the proof-theoretical point of view, the syntactic characterization can have a form of various approaches, such as natural deduction, sequent calculus, tableau calculi, axiomatic systems, to mention only the most influential nowadays. For particular logics, the semantics will be also presented, aiming at the adequacy result for these logics. As an outcome, the completeness theorem for the selected logics will be formulated. Also, sketches of proofs of these theorems will be given. Considering non-classical logic, one can try to refer not only to the real world but also to some possible worlds. They can be interpreted in various ways: as ontological possibilities, moments of time, or states of affairs. In every case, specific interpretations can play the role of characterization of the semantics of a given system. So-called Kripke semantics can be used to express all these interpretations formally. In particular, this type of semantics can be applied to describe some classes of modal logics, such as normal and regular logics. On the other hand, examples of selected modal logics will be given together with the adequate formulation in terms of specific conditions imposed on Kripke structures (the so-called frames).
Taking into account that even in everyday situations, we can observe limitations of two-valued classical semantics, it is natural to try to extend the classical view. First, many-valued logics can be considered in such a context. They - in a sense - are similar to classical propositional logic. But they differ because they do not restrict the number of truth values to only two: they allow for a larger amount of truth degrees. In their case, the natural semantics is expressed with the help of logical matrices. Another way to obtain the non-classical approach is to change semantics so that some classical theses are no longer generally valid. One of the types of logics obtained in this way is the class of paraconsistent logics, for which - most often - Duns Scotus law is in a way suspended. Usually, the result is received by a reinterpretation of negation and also of the other connectives.