First-Order Logic

Abstract

In this chapter, we introduce a first-order logic. This logic differs in some respects from what is introduced in most logic text books as classical first-order logic. The reason for the differences is that our logic has been designed in such a way that it is convenient for talking about Java programs. In particular our logic includes a type system with subtyping, a feature not found in most presentations of first-order logic.

Not only the logic itself, but also our presentation of it is different from what is found in a logic textbook: We concentrate on the definition of the language and its meaning, motivating most aspects from the intended application, namely the description of Java programs. We give many examples of the concepts introduced, to clarify what the definitions mean. In contrast to an ordinary logic text, we hardly state any theorems about the logic (with the notable exception of Section 2.6), and we prove none of them. The intention of this book is not to teach the reader how to reason about logics, but rather how to use one particular logic for a particular purpose.

The reader interested in the theoretical background of first-order logic in the context of automated deduction might want to read the book of Fitting [1996], or that of Goubault-Larrecq and Mackie [1997]. There are a number of textbooks covering first-order logic in general: by Ben-Ari [2003], Enderton [2000], Huth and Ryan [2004], Nerode and Shore [1979], or for the mathematically oriented reader Ebbinghaus et al. [1984]. To the best of our knowledge the only textbooks covering many-sorted logic, but not allowing subsorts, are those by Manzano [1996] and Gallier [1986]. For the technical details of the particular logic described in this chapter, see [Giese, 2005].