Unit 1 - Basic Concepts in Logic PDF

argument arguments 2 3 argument premises, conclusion 4 Every logic book has at least one sensible example. The Power of Logic is a logic book. So, The Power of Logic contains at least one sensible example....

argument arguments 2 3 argument premises, conclusion 4 Every logic book has at least one sensible example. The Power of Logic is a logic book. So, The Power of Logic contains at least one sensible example. 5  statement NB 6 Deductive deductive inductive guarantee 7 probable 8  premises succeed connection premises conclusion. guaranteed 9 Sound Logical 10 11 12 Scenarios with different Valid Argument Invalid Argument truth values (Premises guaranteeing (Premises do not guarantee conclusion) conclusion) True premises If Harry loved Dumbledore, SOME Americans work in business. then Harry was sad when Donald Trump is an American. So, True conclusion Dumbledore died. So, Harry Donald Trump works in business. was sad when Dumbledore died. False premises All sharks are birds. All birds Every genius is a Mathematician. are politicians. So, all sharks Pokemon is a Mathematician. So, False conclusion are politicians. Pokemon is a genius. False premises All dogs are ants. All ants are Everything coloured is red. Steve mammals. So, all dogs are Harvey is a mortician. So, Steve True conclusion mammals. Harvey is hilarious. True premises All dogs are animals. All cats are animals. Hence, all dogs are cats. False conclusion Unknown All of the Cappadocians SOME hylidae are heterophoric. accepted perichoresis. Basil Maggie is heterophoric. So, Maggie truth value was a Cappadocians. So, Basil is a hylidae. accepted perichoresis. 13 14 15  Arguments Valid Arguments Invalid Arguments Valid Valid argument argument All invalid With all With at least arguments premises one false true are premises are unsound sound are unsound 16 17 linked 18 19 20 21  conditional statement antecedent consequent. EXAMPLE: IF Tom goes to school, THEN Tom gets an education. 22 it is raining the ground is wet 23  argument form argument form substitution instance 24  If A, then B. A. So, B. If A, then B. Not B. So, Not A. If A, then B. If B, then C. So, if A, then C. i) Either A or B. Not A. So, B. ii) Either A or B. Not B. So, A. Either A or B. If A, then C. If B, then D. So, either C or D. 25 26 27  Modus Ponens Example #1 If Tom lives in Kingston, then Tom lives in Jamaica. Tom lives in Kingston. ---------------------------------------------------------------- So, Tom lives in Jamaica. 28 1. Affirming the Consequent (AC) If A then B Conditional If A then B A Antecedent B Unlike MP, AC is an invalid ---------- ---------- argument form. Consequent So, A So, B Counter-example Example #1 If Tom lives in Kingston then Tom lives in Jamaica. (T) Tom lives in Jamaica (T) If Tom lives in Kingston, then ------------------------------------------------------ Tom lives in Jamaica. Tom lives in Kingston. (F) Tom lives in Kingston. ----------------------------------------- A common mistake in formal reasoning (such as Tom lives in Jamaica. AC) is a formal fallacy. Example #2 If John is lying, then you are innocent and John’s mother is guilty. John is lying. --------------------------------------------------------------------------------- You are innocent and John’s mother is guilty. Both arguments have the valid form modus ponens. So, both arguments are valid. 29  Modus Tollens (Denying the Consequent) Example #1 If Tom lives in Kingston, then Tom lives in Jamaica. It is not true that Tom lives in Jamaica. ----------------------------------------------------- It is not true that Tom lives in Kingston. 30 2. Modus Tollens (MT) Denying the Antecedent (DA) If A then B Conditional If A then B Unlike MT, DA Not B Negation of Consequent Not A is an invalid ---------- ------------ argument form. So, Not A Negation of Antecedent So, Not B Counter-example Valid argument, in the form of MT If Tom lives in Kings, then Tom lives in Ja. (T) Since the form is valid, this individual argument is valid by virtue of having the valid form. Tom does NOT live in Kingston. (T) ------------------------------------------------------- So, Tom does NOT live in Jamaica. (F) Example #1 If Tom lives in Kings, then Tom lives in Ja. A common mistake in formal reasoning Tom does NOT live in Jamaica. ---------------------------------------------------------------- (suFormal fallacy. Invalid So, Tom does NOT live in Kingston. Invalid argument, in the form of DA Example #2 If you are healthy or wealthy, then you are lucky. It is not true that you are healthy or wealthy. --------------------------------------------------------------------- You are not lucky. 1.If A, then B 2.If B, then C. 3.So, if A, then C. 32 3. Hypothetical Syllogism (HS) If A then B Conditional A conditional statement is also called a If B then C Conditional “hypothetical statement ”. ---------- A syllogism is a three-step argument, i.e., So, if A then C. Conditional an argument with two premises and one conclusion. Example #2 If Tom loves his dog, then Tom wants his dog to be happy. If Tom wants his dog to be happy, then Tom will walk his dog everyday. ------------------------------------------------------------------------------------------------ So, if Tom loves his dog, then Tom will walk his dog everyday. Example #3 If everybody has what they want and nobody has what they don’t want, then everybody is happy. If everybody is happy then I am happy. ------------------------------------------------------------------------------------------------------------------------------------------ If everybody has what they want and nobody has what they don’t want, then I am happy. Since both arguments have the valid form hypothetical syllogism, both arguments are valid. Disjunctive Syllogism can take two (2) forms: 1.Either A or B. 1. Either A or B. 2.Not A. 2. Not B. 3.So, B. 3. So, A 34 4. Disjunctive Syllogism (DS) Either A or B Either A or B Disjunction The statement “A or B” is called Not A Not B Negation of One Disjunct the “ disjunction ” of the simpler ---------- ---------- statements “A” and “B”. So, B So, A The Other Disjunct The simpler statements “A” and “B” (which make up the Example #1 disjunction) are called the “ Either Tom is Mary’s friend OR Tom is my friend. disjuncts ”. Tom is NOT Mary’s friend. ---------------------------------------------------------- Tom is my friend. Example #2 Either Mary is happily married as well as busy at work OR she is having an affair. Mary is NOT having an affair. ----------------------------------------------------------------------------------------------------------- Mary is happily married as well as busy at work. In its inclusive sense, the disjunction “A or B” means “A or B or both”. In its exclusive sense, the disjunction “A or B” means “A or B but not both”. The argument form, disjunctive syllogism, is valid for both senses of the disjunction. NOTE : In this unit, we use the term “or” in its inclusive sense, unless otherwise specified.  It combines both conditional and disjunctive statements 1.Either A or B. 2.If A, then C. 3.If B, then D. 4.So, either C or D. 36 5. Constructive Dilemma (CD) Either A or B Disjunction If A, then C. Conditional If B, then D Conditional Example #1 ---------- So, C or D Disjunction Either I take a taxi or I take a bus. If I take a taxi then I will be too early. If I take a bus then I will be too late. -------------------------------------------------- So, I will be too early or I will be too late. Example #2 Either You keep your promise or you don’t keep your promise. If you keep your promise then you will be happy. If you don’t keep your promise then you will be unhappy. ------------------------------------------------------------------------------ You will be happy or you will be unhappy.  counterexample the premises are true, and conclusion is false  well- known truths well-known falsehood PREM 1: If Jenny is a neurologist, then she is a doctor. PREM 1: If A then B. PREM 2: Jenny is NOT a doctor. PREM 2: Not B. CONCL: So, Jenny is a neurologist. CONCL: So, A. PREM 1: If Usain Bolt is less than 2 ft tall, then he is less than 3 ft tall. [True] PREM 2: Usain Bolt is NOT less than 3 feet tall. [True] CONCL: So, Usain Bolt is less than 2 feet tall. [False].