34 Logic as a Tool 1.4 Supplementary: Inductive definitions and structural induction and recursion In section 1.1 I defined propositional formulae using a special kind of definition, which refers to the very notion it is defining Such definitions are called inductive They are very common and important, especially in logic, because they are simple, elegant, and indispensable when an infinite set of structured objects is to be defined Moreover, properties of an object defined by inductive definitions can be proved by a uniform method, called structural induction, that resembles and extends the method of mathematical induction used to prove properties of natural numbers Here I present the basics of the general theory of inductive definitions and structural induction Part of this section, or even all of it, can be skipped, but the reader is recommended to read it through 1.4.1 Inductive definitions Let us begin with well-known cases: the inductive definition of words in an alphabet and then natural numbers as special words in a two-letter alphabet Note the pattern 1.4.1.1 The set of all finite words in an alphabet Consider a set A Intuitively, a (finite) word in A is any string of elements of A We formally define the set of (finite) words in the alphabet A inductively as follows The empty string is a word in A If w is a word in A and a ∈ A, then wa is word in A The idea of this definition is that words in A are those, and only those, objects that can be constructed following the two rules above 1.4.1.2 The set of natural numbers We consider the two-letter alphabet {0, S }, where 0, S are different symbols, and formally define natural numbers to be special words in that alphabet, as follows is a natural number If n is a natural number then Sn is a natural number The definition above defines the infinite set {0, S 0, SS 0, SSS 0, · · · } Hereafter we denote S · · · n times · · ·S by n and identify it with the (intuitive notion of) natural number n 1.4.1.3 The set of propositional formulae Let us denote the alphabet of symbols used in propositional logic AP L Note that it includes a possibly infinite set PVAR of propositional variables