224 Logic as a Tool I Direct proofs With this proof strategy we assume that all premises are true, and then try to deduce the desired conclusion by applying a sequence of correct – logical or substantially mathematical – inference steps Schematically, a direct proof takes the following form Assume P1 , , Pn (a sequence of valid inferences) Conclude C Typical patterns of direct proofs are derivations in Natural Deduction without using the rule of Reductio ad Absurdum Example 181 As an example, we provide a direct proof of the statement if n is an odd integer, then n2 is an odd integer Proof Suppose n is an odd integer Then there is an integer k such that n = 2k + Therefore, n2 = (2k + 1)(2k + 1) = 4k + 4k + = 2(2k + 2k ) + Therefore, n2 is odd That proves the statement In this proof only the first and last steps are logical, roughly corresponding to the derivation of A → B by assuming A and deducing B II Indirect proofs Indirect proofs are also known as proofs by assumption of the contrary or proofs by contradiction Typical patterns of indirect proofs are derivations in Semantic Tableaux, and also those derivations in Natural Deduction that use an application of the rule of Reductio ad Absurdum The idea of the indirect proof strategy is to assume that all premises are true while the conclusion is false (i.e., the negation of the conclusion is true), and try to reach a contradiction based on these assumptions, again by applying only valid inferences A contradiction is typically obtained by deducing a statement known to be false (e.g., deducing that + = 3) or by deducing a statement and its negation We can often reach a contradiction by deducing the negation of some of the premises (see the example below) which have been assumed to be true The rationale behind the proof by contradiction is clear: if all our assumptions are true and we only apply valid inferences, then all our conclusions must also be true By deducing a false conclusion we therefore show that, given that all original assumptions are true, the additional one that we have made (i.e., the negation of the conclusion) must have been wrong, that is, that the conclusion must be true