Valid and Invalid ArgumentsArgumentsArgument FormsValidity of an Argument FormValidity of an ArgumentThe Form of an ArgumentExampleSlide 8Slide 9Example: Invalid Argument Forms with True ConclusionsExample: Valid Argument Forms with False ConclusionsSlide 12Modus PonensExamples of Modus PonensModus TollensExamples of Modus TollensOther Argument FormsSlide 18Slide 19FallaciesThe Fallacy of the ConverseSlide 22Fallacy of the InverseSlide 24Valid and Invalid ArgumentsLecture 3Section 1.3 Mon, Jan 23, 2006ArgumentsAn argument is a sequence of statements.The last statement is the conclusion.All the other statements are the premises.A mathematical proof is an argument.Argument FormsAn argument form is a sequence of statement forms.The last statement form is the conclusion.All the other statement forms are the premises.A mathematical proof follows an argument form.Validity of an Argument FormAn argument form is valid if its conclusion is true when its premises are true.Otherwise, the argument form is invalid.An invalid argument form is called a fallacy.Validity of an ArgumentAn argument is valid if its argument form is valid, whether or not its premises are true.The Form of an ArgumentLet the premises be P1, …, Pn.Let the conclusion be C.The argument form is valid ifP1    Pn  Cis a tautology.ExampleI will ride my bike today.If it is raining and I ride my bike, then I will get wet.It is raining.Therefore, I will get wet.Examplep = “I will ride my bike today.”q = “It is raining.”r = “I will get wet.”Argument form:pq  p  rq rExampleP1P2P3CP1  P2  P3  Cpq  p  rq rT T T T TT F T F TT T F T TT T F F TF T T T TF T T F TF T F T TF T F F TExample: Invalid Argument Forms with True ConclusionsAn argument form may be invalid even though its conclusion is true.If I eat my vegetables, I’ll be big and strong.I’m big and strong.Therefore, I ate my vegetables.A true conclusion does not ensure that the argument form is valid.Example: Valid Argument Forms with False ConclusionsAn argument form may be valid even though its conclusion is false.If I wait until the last minute to do my homework, then it will be a lot easier.I wait until the last minute to do my homework.Therefore, it will be a lot easier.A false conclusion does not mean that the argument form is invalid.Example: Valid Argument Forms with False ConclusionsAnother example.If 1 + 1 = 2, then pigs can fly.1 + 1 = 2.Therefore, pigs can fly.Modus PonensModus ponens is the argument formp  qp qThis is also called a direct argument.Examples of Modus PonensIf it is raining, then I am carrying my umbrella. It is raining. Therefore, I am carrying my umbrella.If pigs can fly, then I am carrying my umbrella. Pigs can fly. Therefore, I am carrying my umbrella.Modus TollensModus tollens is the argument formp  qq pThis is also called an indirect argument.It is equivalent to replacing p  q with q  p and then using modus ponens.Examples of Modus TollensIf it is raining, then I am carrying my umbrella. I am not carrying my umbrella. Therefore, it is not raining.If pigs can fly, then I am carrying my umbrella. I am not carrying my umbrella. Therefore, pigs cannot fly.Other Argument FormsFrom the specific to the generalp p  qFrom the general to the specificp  q pOther Argument FormsEliminationp  qp qTransitivityp  qq  r p  rOther Argument FormsDivision into Casesp  qp  rq  r rFallaciesA fallacy is an invalid argument form.Two common fallaciesThe fallacy of the converse.The fallacy of the inverse.The Fallacy of the ConverseThe fallacy of the converse is the invalid argument formp  qq pThis is also called the fallacy of affirming the consequent.ExampleIf it is raining, then I am carrying an umbrella. I am carrying an umbrella. Therefore, it is raining.Fallacy of the InverseThe fallacy of the inverse is the invalid argument formp  qp qThis is also called the fallacy of denying the antecedent.ExampleIf pigs can fly, then I am carrying an umbrella. Pigs cannot fly. Therefore, I am not carrying an