35 Understanding Propositional Logic We now revisit the inductive definition of propositional formulae as special words in the alphabet of symbols used in propositional logic, by paying closer attention to the structure of the definition I emphasize the words is a propositional formula so we can see shortly how the definition transforms into an explicit definition and an induction principle Definition 18 The property of a word in AP L of being a propositional formula is defined inductively as follows Every Boolean constant (i.e., or ⊥) is a propositional formula Every propositional variable is a propositional formula If (the word) A is a propositional formula then (the word) ¬A is a propositional formula If each of (the words) A and B is a propositional formula then each of (the words) (A ∧ B ), (A ∨ B ), (A → B ), and (A ↔ B ) is a propositional formula The meaning of the inductive definition above can be expressed equivalently by the following explicit definition, which essentially repeats the definition above but replaces the phrase “is a propositional formula” with “is in (the set) FOR.” Definition 19 The set of propositional formulae FOR is the least set of words in the alphabet of propositional logic such that the following holds Every Boolean constant is in FOR Every propositional variable is in FOR If A is in FOR then ¬A is in FOR If each of A and B is in FOR then each of (A ∧ B ), (A ∨ B ), (A → B ), and (A ↔ B ) is in FOR This pattern of converting the inductive definition into an explicit definition is general and can be applied to each of the other inductive definitions presented here However, we have not yet proved that the definition of the set FOR given above is correct in the sense that the least (by inclusion) set described above even exists Yet, if it does exist, then it is clearly unique because of being the least set with the described properties We will prove the correctness later 1.4.1.4 The subgroup of a given group, generated by a set of elements I now provide a more algebraic example The reader not familiar with the notions of groups and generated subgroups can skip this safely Let G = G, ◦,−1 , e be a group and X be a subset of G The subgroup of G generated by X is the least subset [X ]G of G such that: e is in [X ]G Every element from X is in [X ]G