5 Applications: Mathematical Proofs and Automated Reasoning Logical systems are used in practice to perform formal reasoning that goes beyond pure logic Most often they are used for mathematical reasoning, but they have also found numerous other applications in artificial intelligence, especially the field of knowledge representation and reasoning (e.g various description logics for ontologies), as well as in many areas of computer science including database theory, program analysis, and deductive verification In each of these areas the use of deductive systems is guided by the specific applications in mind, but the underlying methodology is the same: the purely logical engine of the deductive system, which consists of the general, logical axioms and inference rules is extended with specific, non-logical axioms and rules describing the particular subject area The logical and non-logical axioms and rules together constitute a formal theory in which the formal reasoning is performed by means of derivations from a set of assumptions in the chosen deductive system These assumptions are usually the relevant non-logical axioms, plus other specific ad hoc assumptions applying to the concrete domain or situation for which the reasoning is conducted Most typical mathematical theories describe classes of important relational structures such as sets, partial or linear orders, directed graphs, trees, equivalence relations, various geometric structures, etc., algebraic structures, such as lattices, Boolean algebras, groups, rings, fields, etc., or combined, for example ordered rings and fields Other important mathematical theories are intended to describe single structures of special interest, for instance: an axiomatic theory for the arithmetic of natural numbers, the most popular one being Peano Arithmetic; the subsystem of Presburger Arithmetic (involving only addition); or the first-order theory of the field of reals R or of the field of rational numbers Q, etc In this chapter I first discuss generally the logical structure, strategies and tactics of mathematical reasoning and proofs and then illustrate these with several examples in Logic as a Tool: A Guide to Formal Logical Reasoning, First Edition Valentin Goranko © 2016 John Wiley & Sons, Ltd Published 2016 by John Wiley & Sons, Ltd