Valid and Invalid ArgumentsStatement FormsArgumentsArgument FormsValidity of an Argument FormValidity of an ArgumentThe Form of an ArgumentExampleSlide 9Slide 10Slide 11Slide 12Slide 13Example: Invalid Argument Forms with True ConclusionsSlide 15Example: Valid Argument Forms with False ConclusionsSlide 17Modus PonensExamples of Modus PonensModus TollensExamples of Modus TollensOther Argument FormsSlide 23Slide 24FallaciesThe Fallacy of the ConverseSlide 27Fallacy of the InverseSlide 29Valid and Invalid ArgumentsLecture 4Section 1.3 Tue, Jan 23, 2007Statement FormsA statement is a sentence that is true or false.“If today is Wednesday, then Discrete Math meets today or I’m a baboon.”A statement form is the logical form of a statement, represented symbolically.p q rArgumentsAn 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 FormsAn 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 FormAn 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 ArgumentAn argument is valid if its argument form is valid, whether or not its premises are true.The Form of an ArgumentLet the premises be P1, …, Pn.Let the conclusion be C.The argument form is valid ifP1 Pn Cis a tautology.ExampleI will go fishing today.If the boss is in and I go fishing, then I will get fired.The boss is in.Therefore, I will get fired.Examplep = “I will go fishing today.”q = “The boss is in.”r = “I will get fired.”Argument form:pq p rq rExampleP1P2P3CP1 P2 P3 Cpq p rq rExampleP1P2P3CP1 P2 P3 Cpq p rq rT T TT T FT F TT F FF T TF T FF F TF F FExampleP1P2P3CP1 P2 P3 Cpq p rq rT T T TT F T FT T F TT T F FF T T TF T T FF T F TF T F FExampleP1P2P3CP1 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 ConclusionsAn argument form may be invalid even though its conclusion is true.If I go fishing, the boss will fire me.The boss fired me.Therefore, I went fishing.A true conclusion does not ensure that the argument form is valid.Example: Invalid Argument Forms with True ConclusionsAnother example.If 1 + 1 = 2, then pigs can fly.Pigs can fly.Therefore, 1 + 1 = 2.Example: Valid Argument Forms with False ConclusionsAn 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 ConclusionsAnother example.If 1 + 1 = 2, then pigs can fly.1 + 1 = 2.Therefore, pigs can fly.Modus PonensModus ponens is the argument formp qp qThis is also called a direct argument.Examples of Modus PonensIf it is Wednesday, then Discrete Math meets today. It is Wednesday. Therefore, Discrete Math meets today.If pigs can fly, then Discrete Math meets today. Pigs can fly. Therefore, Discrete Math meets today.Modus TollensModus tollens is the argument formp qq pThis is also called an indirect argument.It is equivalent to replacing p q with q p and then using modus ponens.Examples of Modus TollensIf it is Wednesday, then Discrete Math meets today. Discrete Math does not meet today. Therefore, it is not Wednesday.If pigs can fly, then Discrete Math meets today. Discrete Math does not meet today. Therefore, pigs cannot fly.Other Argument FormsFrom the specific to the generalp p qFrom the general to the specificp q pOther Argument FormsEliminationp qp qTransitivityp qq r p rOther Argument FormsDivision into Casesp qp rq r rFallaciesA fallacy is an invalid argument form.Two common fallaciesThe fallacy of the converse.The fallacy of the inverse.The Fallacy of the ConverseThe fallacy of the converse is the invalid argument formp qq pThis is also called the fallacy of affirming the consequent.ExampleIf it is Wednesday, then Discrete Math meets today. Discrete Math meets today. Therefore, it is Wednesday.Fallacy of the InverseThe fallacy of the inverse is the invalid argument formp qp qThis is also called the fallacy of denying the antecedent.ExampleIf pigs can fly, then Discrete Math meets today. Pigs cannot fly. Therefore, Discrete Math does not meet
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