Question

,so Pis false. Thus values for P and Q that makes B false. Therefore B is not a R). We have seen in Example 2.24 Then tautology Exercise 21. Let A be the sentence [(P Q) n(Q → R)] (P that A is a tautology. Let B be the converse of A. Write out what B is in terms of P, Q, and R. show that B is not a tautology, by finding a combination of truth values for P, Q, and R that makes iB false. You should be able to do this without writing out a truth table. Exercise 22. Let A be the sentence (P Q) {[P => (Q → R)] (P = R). we saw in Exercise 17 that A is a tautology. Let B be the converse of A. Write out what B is in terms of P, Q, and R. Then show that B is not a tautology, by finding a combination of truth values for P, Q, and R that makes B false. You shoüld be able to do this without writing out a truth table Contradictions A contradiction is a sentence of the form QA-Q. Such a sentence is false whatever the truth value of Q may be. (Proof Either Q is true or Q is false. If Q is true, then -Q is false, so QA-Q is false. If Q is false, then QA-Q is false. Thus in either case, QA-Q is false.) How To Prove a Negative Sentence. usual way to prove a negative sentence -P is to assume P and to deduce a contradiction from is assumption. We shall now explain why this works. Let us begin by introducing a new logical symbol, th he symbol人may be read falsehood and should be thought of as standing for a false sentence. For has the instance, might stand for a contradiction same truth value as -P: Q Q.) As the following truth table shows, P Now if we assume P and deduce a contradiction Q A-Q, then by the method of conditional proof, the conditional sentence P be truc. (Q -Q) is true, so as the truth table (6) shows, the negative sentence「P must Exercise 23. Use the method of conditional proof to explain in words why the sentence is a tautology.18 (Do not use cases.) The way to prove a negative sentence, in this case -P, should also play a role in your proof. Be careful not to skip any steps. Be explicit about discharging assumptions. 18 By the way, this tautology is called modus tollenas. Actually, the full name for this which is latin for "the way to deny by denying." tautology is modhus tollendo tollens, Copyright Neil Falkner 2018

Exercise 23. I have the answer and it makes sense to me but I don't understand what the contradiction is here.

Answer 1

[(P implies Q) intersection ~ Q] implies ~p