2 3 Valid and Invalid Arguments In mathematics and logic we de ne an argument as a series of statements followed by a conclusion De nition An sequence of statement forms All statements in an argument except for the nal one The are called is a sequence of statements and an The nal statement is called the is a symbol is read therefore and is normally placed before the conclusion To say that an argument form is statements are substituted for the statement variables in its premises if the resulting premises are all true then the conclusion is also true means that no matter what particular Example 1 Determine whether the following argument forms are valid or invalid 1 If we meet death then we say not today The phrase not today was not spoken Therefore we did not meet death Premise 1 Premise 2 Conclusion q p q p p q T T T F F T F F p q r Premise 1 p q r q Premise 2 Premise 3 Conclusion q p p q r 2 p q r p q q p r r q p T T T T T F T F T T F F F T T F T F F F T F F F 1 An argument form consisting of two premises and a conclusion is called a The rst and second premises are called the respectively which stands for method of a rming is an argument of the which stands for method of denying is an argument of the De nition and form form Example 2 Use modus tollens or modus ponens to provide conclusions to the following arguments 1 If a Lannister owes you a debt then it will be paid A Lannister owes you a debt This is an example of 2 If you see White Walkers then winter is coming Winter is not coming This is an example of If p then q p q If p then q q p 2 A two such forms modus tollens and modus ponens is a form of argument that is valid So far we have learned Rules of Inference Generalization Given a speci c statement we can generalize it The following argument forms are valid p p q q p q Example Sheldon Cooper is a theoretical physicist Therefore Sheldon Cooper is a theoretical physicist or a bus driver Specialization Given a condition on two statements we can apply it speci cally to one of the statements The following argument forms are valid Example Amy Farrah Fowler and Bernadette Rostenkowski are both biologists Therefore Bernadette Rostenkowski is a biologist Elimination Given a choice between two statements eliminating one of the statements leaves one possibility The following argument forms are valid p q p p q q p p q q p q p q Example Either Sheldon Cooper or Leonard Hofstadter is dating Amy Farrah Fowler Leonard Hofstadter is not dating Amy Farrah Fowler Therefore Sheldon Cooper is dating Amy Farrah Fowler 3 Transitivity The following argument form is valid p q q r p r Example If Sheldon wants to talk to Penny then he will knock on her door If Sheldon knocks on your door then he will knock on your door three times Therefore if Sheldon wants to talk to Penny then he will knock on her door three times Fallacies A Five common fallacies are is an error in reasoning that results in an invalid argument 1 2 3 4 5 Converse Error This type of error stems from the following awed logic Example If Barney Stinson is in MacLaren s then he is wearing a suit Barney Stinson is wearing a suit Therefore he is in MacLaren s p q q p 4 Inverse Error This type of error stems from the following awed logic p q p q Example If Barney Stinson is in MacLaren s then he is wearing a suit Barney Stinson is not in MacLaren s Therefore he is not wearing a suit Example 3 De nition Consider the examples in Example 1 and identify any possible errors in these arguments An argument is called true An argument that is not sound is called if and only if it is valid and all of its premises is Note that if you can show that by supposing the p is false leads logically to a contradiction then you can conclude that p is true This is called the The argument form for the contradiction rule is We will show this is a valid argument form p p c p c p p c p 5 Group Work Complete the following problems in groups of 2 or 3 1 A set of premises and a conclusion are given Use the valid argument forms to deduce the conclusion from premises giving a reason for each step a p q b q r c p s t d r e q u s f t 2 Use a truth table to determine whether the argument form is valid p q r q r p r 6