7 Understanding Propositional Logic • propositional variables p, q, r , possibly indexed, to denote unspecified propositions in the same way as we use algebraic variables to denote unspecified or unknown numbers; • the logical connectives that we already know; and • auxiliary symbols: parentheses (and) are used to indicate the order of application of logical connectives and make the formulae unambiguous Using these symbols we can construct propositional formulae in the same way in which we construct algebraic expressions from variables and arithmetic operations Here are a few examples of propositional formulae: , p, ¬⊥, ¬¬p, p ∨ ¬q, p1 ∧ ¬(p2 → (¬p1 ∧ ⊥)) There are infinitely many possible propositional formulae so we cannot list them all here However, there is a simple and elegant way to give a precise definition of propositional formulae, namely the so-called inductive definition (or recursive definition) It consists of the following clauses or formation rules: Every propositional constant or variable is a propositional formula If A is a propositional formula then ¬A is a propositional formula If A, B are propositional formulae then each of (A ∨ B ), (A ∧ B ), (A → B ), and (A ↔ B ) is a propositional formula We say that a propositional formula is any string of symbols that can be constructed by applying – in some order and possibly repeatedly – the rules above, and only objects that can be constructed in such a way are propositional formulae Note that the notion of propositional formula that we define above is used in its own definition; this is the idea of structural induction The definition works as follows: the first rule above gives us some initial stock of propositional formulae; as we keep applying the other rules, we construct more and more formulae and use them further in the definition Eventually, every propositional formula can be obtained in several (finitely many!) steps of applying these rules We can therefore think of the definition above as a construction manual prescribing how new objects (here, propositional formulae) can be built from already constructed objects I discuss inductive definitions in more detail in Section 1.4.5 From this point, I omit the unnecessary pairs of parentheses according to our earlier convention whenever that would not lead to syntactic ambiguity The formulae that are used in the process of the construction of a formula A are called subformulae of A The last propositional connective introduced in the construction of A is called the main connective of A and the formula(e) to which it is applied is/are the main subformula(e) of A I make all these more precise in what follows Example (Construction sequence, subformulae and main connectives) One construction sequence for the formula (p ∨ ¬(q ∧ ¬r )) → ¬¬r is p, q, r, ¬r, ¬¬r, q ∧ ¬r, ¬(q ∧ ¬r), p ∨ ¬(q ∧ ¬r ), (p ∨ ¬(q ∧ ¬r)) → ¬¬r