Classical Logic
Abstract
Western (deductive) logic originated in Greek antiquity. It found its first expression
in those works of the great philosopher Aristotle (384–322 BC) which
have come to be known as the Organon, i.e., ‘instrument’. Aristotle’s logic,
also known as syllogistics, was unsystematically concerned with patterns of
reasoning and argumentation. It remained in this rudimentary state relatively
unchanged and unchallenged until the second half of the nineteenth century.
At that time, logic underwent a period of unprecedented reform and modernization,
due in large part to the German mathematician Gottlob Frege (1848–
1925) It thus became more and more a
mathematical endeavor of studying the structure and peculiarities of artificial,
formal languages. In this new form, logic gave rise in the twentieth century to
disciplines such as theoretical informatics and programming languages, and
transformed our lives through computation, information processing, and the
Internet.
A formal language consists, in effect, of a particular alphabet and some
precise rules of forming, and transforming, strings over this alphabet. There
exist several types of formal languages analyzed in logic. Depending on their
structure, they are called first-order languages, second-order languages, and
so on. Today a logic is considered a theory of such a language and is, correspondingly,
referred to as a first-order logic, a second-order logic, and so on.
In this chapter, we shall outline a first-order logic as a paragon of deductive
logic. Its full name is: “Classical, first-order predicate logic with identity”.
What all these expressions mean exactly, will become clear below.
The logic we shall study first is termed classical logic because the idea to
create such an instrument, or ‘organon’, is rooted in Greek antiquity. Owing
to its origin, it is based on three time-honored Aristotelian doctrines. For these and several other reasons that we shall discuss later in
this chapter, it is also called a logic of the Aristotelian style, or an Aristotelian
logic for short.