Chapter 21 - Encoding Two-Valued Nonclassical Logics in Classical Logic

Abstract

Classical logic has been the principal logic used in applications for many years. One of its main application areas has been mathematics, simply because it is ideally suited for the formalization of mathematics. Its model theory and proof theory have had an enormous impact on the foundations of mathematics. Many logicians continue to hold the view that logic has four subareas: model theory, proof theory, recursion theory, and set theory. In all these four areas, classical logic reigns supreme. The chapter concerns with the traditional monotone two-valued nonclassical logics, and modal logic is the most prominent example of these logics. Despite this restriction, we are still faced with an overwhelming number of logics. Encoding these logics into other logics gives us a powerful tool for understanding them from both a logical, algorithmic, and computational point of view. The chapter first explains the main topics being discussed in the context of nonclassical logics; this will enable one to explain what this chapter is about, and what it is not about. The two most important properties of a logic are its expressive power and the (decidability of) the reasoning tasks that can be performed in the logic, in particular theoremhood. The chapter deals with expressive power. The expressive power of a logic may be analyzed from a number of angles. First of all, how do we express things in a given logic, or put differently, what is the syntax of the logic? This may seem a trivial point, but a poor syntax may be an important barrier to using a given logic. Second, what do we say in a given logic? That is, what is the meaning of the well formed formulae (wffs)? This is a non-trivial question, because it is application dependent and because it requires a thorough understanding of the intended application.