Applications of multi-dimensional propositional logics

Ingolf Max (Leipzig) gives a talk at the MCMP Colloquium titled "Applications of multi-dimensional propositional logics". Abstract: Multi-dimensional propositional logics are formal systems which we get if we extend the language of classical propositional logic by ordered n-tuples of classical formulas and suitable operators having multi-dimensional expressions as their arguments. There are several kinds of motivation to deal with multi-dimensional logics in general and with multi-dimensional propositional logics in particular. One of them is connected with the program to reconstruct systems of non-classical logics within such a syntactically extended classical framework. Another kind of motivation is the possibility to show new basic aspects of formal systems which are of some important philosophical interest. Furthermore, it is possible to use the expressive power of such systems to translate expressions of natural languages (and, e.g., structured elements of music) into more complex formal ones. The familiar one-dimensional classical language plays then the role of a theoretical language. I will sketch the general form of multi-dimensional propositional systems with a fixed dimension n. It is possible to define several notions of validity (inconsistency) for ordered n-tuples of classical formulas using only the classical vocabulary. With respect to a more formal application of our logics it will be shown how finite many-valued logics can be equivalently reconstructed. But it is also interesting to see under which restrictions on our languages we get these formal results. With respect to a more philosophical application it will be demonstrated that the distinctions atomic–molecular, atomic–complex depend on the underlying logic, the choice of the logical complexity of basic expressions. With respect to empirical applications and leaving the strong analogy to finite many-valued logics we can include other well-defined parts of the whole language.