Modern Mathematical Logic

Now that we have successfully worked through several important aspects of propositional logic, it is time to move on to a much more substantial and important logic: first-order logic. We gave a basic overview of the fundamental ideas in the introduction. Fundamentally, many areas of mathematics deal with mathematical structures consisting of special constants, relations, and functions, together with certain axioms that these structures obey. We want our logic to be able to handle different types of situations, so we allow ourselves to vary the number and types of these objects. For example, in group theory, we have a special identity element and a binary function corresponding to the group operation. If we want, we can also add in a unary function corresponding to the inverse operation. For ring theory, we have two constants for 0 and 1, along with two binary functions for addition and multiplication (and possibly a unary function for additive inverses). For partial orderings, we just have one binary relation. Any such choice gives rise to a language.