21 Understanding Propositional Logic if B follows logically from A1 , , An we look for a truth assignment to the variables occurring in A1 , , An , B that renders all A1 , , An true and B false If we succeed, then we have proved that B does not follow logically from A1 , , An ; otherwise we want to prove that no such assignment is possible by showing that the assumption that it exists leads to a contradiction For example, let us check again that p → r, q → r |= (p ∨ q ) → r Suppose that for some assignment (p → r ) : T, (q → r ) : T and ((p ∨ q ) → r ) : F Then (p ∨ q ) : T and r : F , hence p : T or q : T Case 1: p : T Then (p → r) : F, that is, a contradiction Case 2: q : T Then (q → r) : F, again, a contradiction Thus, there is no assignment that falsifies the logical consequence above 1.2.2 Logically sound rules of propositional inference and logically correct propositional arguments We now apply the notion of logical consequence to define and check whether a given propositional argument is logically correct Let us first introduce some terminology Definition 12 A rule of propositional inference (inference rule, for short) is a scheme: P1 , , Pn C where P1 , , Pn , C are propositional formulae The formulae P1 , , Pn are called premises of the inference rule, and C is its conclusion An instance of an inference rule is obtained by uniform substitution of concrete propositions for the variables occurring in all formulae of the rule Every such instance is called a propositional inference, or a propositional argument based on that rule Definition 13 An inference rule is (logically) sound if its conclusion follows logically from the premises A propositional argument is logically correct if it is an instance of a logically sound inference rule Example 14 The following inference rule p, p → q q is sound, as we have already seen This rule is known as the Detachment rule or Modus Ponens, and is very important in the logical deductive systems called axiomatic systems which we will study in Chapter The inference Alexis is singing If Alexis is singing, then Alexis is happy Alexis is happy is therefore logically correct, being an instance of that inference rule