part 1 Logic INTRODUCTION TO STRUCTURAL KINESIOLOGY CHAPTER Outline Foundations of Structural Kinesiology 2

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1 CONIRMING PAGES art CHAPER Logic INRODUCION O SRUCURAL KINESIOLOGY oundations of Structural Kinesiology Basic Biomechanical actors and Concets 57 Neuromuscular undamentals 4 Outline Statements and Quantifiers ruth ables yes of Statements 90 sob04590_ch0_ indd dai0079_ch0_00-0.indd Logical Arguments Euler Circles Summary 0/6/09 4:49:5 0/5/09 :46:7

2 CONIRMING PAGES art MAH IN he Art of Persuasion Everywhere you turn in modern society, somebody is try- be to examine the form of an argument and determine if ing to convince you of something. Vote for me! Buy my its conclusion follows logically from the statements in the roduct! Lease a car from our dealershi! Bailing out the argument. auto industry is a bad idea! You should go out with me this o that end, we ve written some claims below. Your job is weekend! Join our fraternity! Logic is sometimes defined to determine which of the arguments is valid, meaning that a as correct thinking or correct reasoning. Some eole refer conclusion can be logically drawn from a set of statements. to logic by a more casual name: common sense. Regardless he skills you learn in this will hel you to do so. INRODUCION O SRUCURAL KINESIOLOGY of what you call it, the ability to think logically is crucial for Where there s smoke, there s fire. all of us because our lives are inundated daily with advertise- Having a lot of money makes eole hay. My neighbor ments, contracts, roduct and service warranties, olitical debates, and news commentaries, to name just a few. Peole is a really hay guy, so he must have a lot of money. Every team in the SEC is good enough to lay in a often have roblems rocessing these things because of mis- bowl game. lorida State is not in the SEC, so they re interretation, misunderstanding, and faulty logic. not good enough to lay in a bowl game. c h a You t e rcan look u the truth or falseness of a fact on the Internet, but that won t hel you in analyzing whether a oundations of Structural Kinesiology certain claim is logically valid. he term common sense is cmisleading, h a t e r because evaluating logical arguments can be Neuromuscular undamentals 4 introa challenging and involved rocess. his duces the basic concets of formal symbolic logic and c h a Scriture ter is the word of God. I know this because it says so in the Bible. Basic Biomechanical actors and Concets 57 of mass destruction, we should go If Ira has weaons to war. It turns out that they don t have them, so we should not go to war. It will be a snowy day in Hawaii before ama Bay shows how to determine whether arguments are valid or makes it to the World Series. ama Bay layed in the invalid by using truth tables. One of our biggest goals will 008 World Series, so it must have snowed in Hawaii. or answers, see Math in the Art of Persuasion Revisited on Page 4 9 sob04590_ch0_ indd dai0079_ch0_00-0.indd9 0/6/09 4:50:07 0/5/09 :46:7

3 CONIRMING PAGES 9 art Chater Logic Section - Statements and Quantifiers LEARNING RNING OBJECIVES. Define and identify statements. As the world gets more comlex and we are bombarded with more and more information, it becomes more imortant than ever to be able to make sensible, objective evaluations of that information. One of the most effective tricks that advertisers, oliticians, and con artists use is to encourage emotions to enter into these evaluations. hey use carefully selected words and images that are designed to kee you from making decisions objectively. he field of symbolic logic was designed exactly for this reason. Symbolic logic uses letters to reresent statements and secial symbols to reresent words like and, or, and not. Use of this symbolic notation in lace of the statements themselves allows us to analytically evaluate the validity of the logic behind an argument without letting bias and emotion cloud our judgment. And these unbiased evaluations are the main goal of this. INRODUCION O SRUCURAL KINESIOLOGY. Define the logical connectives.. Write the negation of a statement. 4. Write statements symbolically. oundations of Statements cha t e r of sentences, including factual statein the English language there are many tyes ments, commands, oinions, and Biomechanical exclamations. In the actors objective study of Structural Kinesiology uestions, Basic logic, we will use only factual statements. and Concets 57 A statement is a4 declarative sentence that can be objectively determined to be true Neuromuscular undamentals or false, but not both. or examle, sentences like It is raining. he United States has sent a sace robe to Mars are statements because they are either true or false and you don t use an oinion to determine this. Whether a sentence is true or false doesn t matter in determining if it is a statement. Notice in the above examle the last statement is false, but it s still a statement. he following sentences, however, are not statements: Give me onion rings with my order. What oerating system are you running? Sweet! he guy sitting next to me is kind of goofy. he first is not a statement because it is a command. he second is not a statement because it is a uestion. he third is not a statement because it is an exclamation, and the fourth is not a statement because the word goofy is subjective; that is, it reuires an oinion. sob04590_ch0_ indd dai0079_ch0_00-0.indd9 0/6/09 4:50:07 0/5/09 :46:7

4 CONIRMING PAGES art EXAMPLE Section - Statements and Quantifiers 9 Recognizing Statements Decide which of the following are statements and which are not. (a) (b) (c) (d) (e) (f) Most scientists agree that global warming is a threat to the environment. Is that your lato? Man, that hurts! 8 6 his book is about database management. Everybody should watch reality shows. SOLUION Parts (a), (d), and (e) are statements because they can be judged as true or false in a nonsubjective manner. Part (b) is not a statement because it is a uestion. Part (c) is not a statement because it is an exclamation. Part (f ) is not a statement because it reuires an oinion. INRODUCION O SRUCURAL KINESIOLOGY ry his One Decide cide which of the following are statements and which are not.. Define and identify statements. oundations of Structural (a) Cool! (d) Cat can send text messages with (b) 8 5 her cell hone. Kinesiology is the host Basic actors (c) Ryan Seacrest of Biomechanical (e) When does the arty start? and Concets 57 American Idol. (f ) History is interesting. Neuromuscular undamentals 4 Simle and Comound Statements Statements can be classified as simle or comound. A simle statement contains only one idea. Each of these statements is an examle of a simle statement. hese cargo ants are khaki. My dorm room has three beds in it. Daytona Beach is in lorida. A statement such as I will take chemistry this semester, and I will get an A is called a comound statement since it consists of two simle statements. Comound statements are formed by joining two simle statements with what is called a connective. he basic connectives are and, or, if... then, and if and only if. Math Note In standard usage, the word then is often al omitted from a condition If of d statement; instea it snows, then I will go skiing, you d robably go just say, If it snows, I ll skiing. Each of the connectives has a formal name: and is called a conjunction, or is called a disjunction, if... then is called a conditional, and if and only if is called a biconditional. Here are some examles of comound statements using connectives. John studied for 5 hours, and he got an A. (conjunction) Luisa will run in a mini triathlon or she will lay in the camus tennis tournament. (disjunction) If I get 80% of the uestions on the LSA right, then I will get into law school. (conditional) We will win the game if and only if we score more oints than the other team. (biconditional) sob04590_ch0_ indd dai0079_ch0_00-0.indd9 0/6/09 4:50:08 0/5/09 :46:7

5 CONIRMING PAGES 94 art Chater Logic Classifying Statements as Simle or Comound EXAMPLE Math Note echnically, we ve given the names conjunction, disjunction, conditional, and biconditional to the now connectives, but from ole wh to on, we ll refer statements using these connectives by those names. or examle, we nd would call the comou d a le m Exa in statement on. cti jun dis Classify each statement as simle or comound. If it is comound, state the name of the connective used. (a) (b) (c) (d) Our school colors are red and white. If you register for Wii service, you will get days of free access. omorrow is the last day to register for classes. I will buy a hybrid or I will buy a motorcycle. SOLUION (a) (b) (c) (d) Don t let use of the word and fool you! his is a simle statement. his if... then statement is comound and uses a conditional connective. his is a simle statement. his is a comound statement, using a disjunction. INRODUCION O SRUCURAL KINESIOLOGY ry his One. Define the logical connectives. oundations of Classify ssify each statement as simle or o comound. If it is comound, state the name of the h connective ti used. d (a) he jacket is trendy and it is ractical. (b) his is an informative website on SDs. Structural I will go Basic Biomechanical (c) If itkinesiology does not rain, then windsurfi ng. (d) I will buy a flash drive or I willand buy aconcets zi drive. 57 (e) Yesterday was the deadline to withdraw from a class. actors Neuromuscular undamentals 4 Sidelight g A BRIE HISORY O LOO GI C he basic concets of logic can be attributed to Aristotle, who lived in the fourth century BCE. He used words, sentences, and deduction to rove arguments using techniues we will study in this. In addition, in 00 BCE. Euclid formalized geometry using deductive roofs. Both subjects were considered to be the inevitable truths of the universe revealed to rational eole. In the 9th century, eole began to reject the idea of inevitable truths and realized that a deductive system like Euclidean geometry is only true based on the original assumtions. When the original assumtions are changed, a new deductive system can be created. his is why there are different tyes of geometry. (See the Sidelight entitled NonEuclidean Geometry in Chater 0.) Eventually, several eole develoed the use of symbols rather than words and sentences in logic. One such erson was George Boole (85 864). Boole created the symbols used in this and develoed the theory of symbolic logic. He also used symbolic logic in mathematics. His manuscrit, entitled An Investigation into the Laws of hought, on Which Are ounded the Mathematical heories of Logic and Probabilities, was ublished when he was 9 in 854. Boole was a friend of Augustus De Morgan, who formulated De Morgan s laws, which we studied in Chater. Much earlier, Leonhard Euler (707 78) used circles to reresent logical statements and roofs. he idea was refined into Venn diagrams by John Venn (84 9). sob04590_ch0_ indd dai0079_ch0_00-0.indd94 0/6/09 4:50:08 0/5/09 :46:7

6 CONIRMING PAGES art Section - Statements and Quantifiers 95 Quantified Statements Math Note We ll worry later about determining whether statements involving tives uantifiers and connec, now or are true or false. and ng rni focus on lea. understanding the terms Quantified statements involve terms such as all, each, every, no, none, some, there exists, and at least one. he first five (all, each, every, no, none) are called universal uantifiers because they either include or exclude every element of the universal set. he latter three (some, there exists, at least one) are called existential uantifiers because they show the existence of something, but do not include the entire universal set. Here are some examles of uantified statements: Every student taking Math for Liberal Arts this semester will ass. Some eole who are Miami Hurricane fans are also Miami Dolhin fans. here is at least one rofessor in this school who does not have brown eyes. No Marlin fan is also a Yankee fan. INRODUCION O Negation SRUCURAL KINESIOLOGY he first and the fourth statements use universal uantifiers, and the second and third use existential uantifiers. Note that the statements using existential uantifiers are not all inclusive (or all exclusive) as the other two are. oundations of Neuromuscular he negation of a statement is a corresonding statement with the oosite truth value. or examle, for the statement My dorm room is blue, the negation is My dorm room is not blue. It s imortant to note that the truth values of these two are comletely oosite: one is true, and the other is false eriod. You can t negate My dorm room is blue by saying Myc hdorm is yellow, because it s comletely ossible a t eroom r that both statements are false. o make sure that you have a correct negation, check if one of the statements is true, the other must be false,actors and vice versa. he tyical Structuralthat Kinesiology Basic Biomechanical way to negate a simle statement is by adding the word not, as in these examles: and Concets 57 Statement undamentals 4 will win Saturday. Auburn I took a shower today. My car is clean. Math Note and he words each, every, ng, thi e all mean the sam in so what we say about all the to lies a n tio this sec, others as well. Likewise at and, some, there exists d ere sid con least one are to be the same and are ll. treated that way as we EXAMPLE Negation Auburn will not win Saturday. I did not take a shower today. My car is not clean. You have to be esecially careful when negating uantified statements. Consider the examle statement All dogs are fuzzy. It s not uite right to say that the negation is All dogs are not fuzzy, because if some dogs are fuzzy and others aren t, then both statements are false. All we need for the statement All dogs are fuzzy to be false is to find at least one dog that is not fuzzy, so the negation of the statement All dogs are fuzzy is Some dogs are not fuzzy. (In this setting, we define the word some to mean at least one.) We can summarize the negation of uantified statements as follows: Statement Contains... All do Some do Some do not None do Negation Some do not, or not all do None do, or all do not All do Some do Writing Negations Write the negation of each of the following uantified statements. (a) (b) (c) (d) Every student taking Math for Liberal Arts this semester will ass. Some eole who are Miami Hurricane fans are also Miami Dolhin fans. here is at least one rofessor in this school who does not have brown eyes. No Marlin fan is also a Yankee fan. sob04590_ch0_ indd dai0079_ch0_00-0.indd95 0/6/09 4:50:09 0/5/09 :46:7

7 CONIRMING PAGES 96 art Chater Logic CAUION SOLUION (a) Some student taking Math for Liberal Arts this semester will not ass (or, not every student taking Math for Liberal Arts this semester will ass). (b) No eole who are Miami Hurricane fans are also Miami Dolhin fans. (c) All rofessors in this school have brown eyes. (d) Some Marlin fan is also a Yankee fan. Be esecially careful when negating statements. Remember that the negation of Every student will ass is not Every student will fail. ry his One INRODUCION O SRUCURAL KINESIOLOGY Write te the negation of each of the following uantified statements.. Write the negation of a statement. (a) (b) (c) (d) All cell hones have cameras. No woman can win the lottery. Some rofessors have PhDs. Some students in this class will not ass. oundations of Structural Kinesiology Symbolic Notation t e rot e Matc hha N Basic Biomechanical actors and Concets 57 Recall that one of our goals in this section is to write statements in symbolic form to hel us evaluate logical arguments objectively. Now we ll introduce the symbols and r Neuromuscular undamentals fou the of ee thr methods that will4 be used. he symbols for the connectives and, or, if... then, and if or, connectives in able - and only if are shown in able -. the order of the simle Simle statements in logic are usually denoted with lowercase letters like,, and tter: statements doesn t ma r. or examle, we could use to reresent the statement I get aid riday and to for examle, and reresent the statement I will go out this weekend. hen the conditional statement If e reresent the sam I get aid riday, then I will go out this weekend can be written in symbols as. comound statement. he symbol (tilde) reresents a negation. If still reresents I get aid riday, he same is true for the ) ion then reresents I do not get aid riday. connectives (disjunct. al) We often use arentheses in logical statements when more than one connective is and (bicondition the is ion involved in order to secify an order. (We ll deal with this in greater detail in the next he one excet ere wh ), ( al section.) or examle, there is a difference between the comound statements condition l. cia cru is er and ( ). he statement means to negate the statement first, then use the ord negation of in conjunction with the statement. or examle, if is the statement ido is a dog and is the statement Pumkin is a cat, then reads, ido is not a dog and Pumkin is a cat. he statement could also be written as ( ). ABLE - Symbols for the Connectives Connective Symbol Name and Conjunction or Disjunction if... then Conditional if and only if Biconditional sob04590_ch0_ indd dai0079_ch0_00-0.indd96 0/6/09 4:50:09 0/5/09 :46:7

8 CONIRMING PAGES art EXAMPLE 4 Section - Statements and Quantifiers 97 he statement ( ) means to negate the conjunction of the statement and the statement. Using the same statements for and as before, the statement ( ) is written, It is not the case that ido is a dog and Pumkin is a cat. he same reasoning alies when the negation is used with other connectives. or examle, means ( ). Examle 4 illustrates in greater detail how to write statements symbolically. Writing Statements Symbolically Let reresent the statement It is cloudy and reresent the statement I will go to the beach. Write each statement in symbols. (a) (b) (c) (d) I will not go to the beach. It is cloudy, and I will go to the beach. If it is cloudy, then I will not go to the beach. I will go to the beach if and only if it is not cloudy. (a) (b) (c) (d) his is the negation of statement, which we write as. his is the conjunction of and, written as. his is the conditional of and the negation of :. his is the biconditional of and not :. INRODUCION O SRUCURAL KINESIOLOGY SOLUION oundations of Structural Kinesiology Basic Biomechanical actors and Concets 57 ry his One 4 Let 4 reresent the statement I w will buy a Coke and reresent the statement Neuromuscular undamentals I will ill b buy some ocorn. Write W it each statement in symbols. (a) (b) (c) (d) I will buy a Coke, and I will buy some ocorn. I will not buy a Coke. If I buy some ocorn, then I will buy a Coke. I will not buy a Coke, and I will buy some ocorn. You robably noticed that some of the comound statements we ve written sound a little awkward. It isn t always necessary to reeat the subject and verb in a comound statement using and or or. or examle, the statement It is cold, and it is snowing can be written It is cold and snowing. he statement I will go to a movie, or I will go to a lay can be written I will go to a movie or a lay. Also the words but and although can be used in lace of and. or examle, the statement I will not buy a television set, and I will buy a CD layer can also be written as I will not buy a television set, but I will buy a CD layer. Statements written in symbols can also be written in words, as shown in Examle 5. EXAMPLE 5 ranslating Statements from Symbols to Words Write each statement in words. Let My dog is a golden retriever and My dog is fuzzy. (a) (b) (c) (d) (e) sob04590_ch0_ indd dai0079_ch0_00-0.indd97 0/6/09 4:50:09 0/5/09 :46:7

9 CONIRMING PAGES art 98 Chater Logic SOLUION (a) (b) (c) (d) (e) My dog is not a golden retriever. My dog is a golden retriever or my dog is fuzzy. If my dog is not a golden retriever, then my dog is fuzzy. My dog is fuzzy if and only if my dog is a golden retriever. My dog is fuzzy, and my dog is a golden retriever. ry his One 5 Write te each statement in words. Let L My friend is a football layer and M My ffriend i d iis smart. t INRODUCION O SRUCURAL KINESIOLOGY If this is your dog (which it s not, because it s mine), statement (e) describes it retty well. (b) (c) oundations of Structural Kinesiology Neuromuscular (b), (c), and (d ) are statements. undamentals 4 (a) (conjunction), (c) (conditional), and (d) (disjunction) are comound; (b) and (e) are simle. (a) (b) (c) (d) 4 (a) (e) Some cell hones don t have cameras. Some women can win the lottery. No rofessors have Ph.Ds. All students in this class will ass. (b) (c) Basic Biomechanical actors and Concets 57 Answers to ry his One (d) In this section, we defined the basic terms of symbolic logic and racticed writing statements using symbols. hese skills will be crucial in our objective study of logical arguments, so we re off to a good start. 4. Write statements symbolically. (a) 5 (a) My friend is not a football layer. (b) My friend is a football layer or my friend is smart. (c) If my friend is not a football layer, then my friend is smart. (d) My friend is smart if and only if my friend is a football layer. (e) My friend is smart and my friend is a football layer. (d) EXERCISE SE - Writing Questions. Define the term statement in your own words.. Exlain the difference between a simle and a comound statement.. Describe the terms and symbols used for the four connectives. 4. Exlain why the negation of All sring breaks are fun is not All sring breaks are not fun. sob04590_ch0_ indd dai0079_ch0_00-0.indd98 0/6/09 4:50:0 0/5/09 :46:7

10 CONIRMING PAGES art Section - Statements and Quantifiers 99 Real-World Alications or Exercises 5 4, state whether the sentence is a statement or not Please do not use your cell hone in class Nicki is a student in vet school. Who will win the student government residency? Neither Sam nor Mary arrives to the exam on time. You can carry a cell hone with you. Bill Gates is the creator of Microsoft. Go with the flow. Math is not hard. 4. Some eole who live in glass houses throw stones. 4. here is at least one erson in this class who won t ass. 4. Every hay dog wags its tail. 44. No men can join a sorority. 45. here exists a four-leaf clover. 46. Each erson who articiated in the study will get $ No one with green eyes wears glasses. 48. Everyone in the class was bored by the rofessor s lecture. 49. At least one of my friends has an iphone. 50. No one here gets out alive. INRODUCION O SRUCURAL KINESIOLOGY or Exercises 5 4, decide if each statement is simle or comound. 5. He goes to arties and hangs out at the coffee sho. 6. Sara got her hair highlighted. 7. Raj will buy an imac or a Dell comuter. 8. Euchre is fun if and only if you win. 9. ebruary is when Valentine s Day occurs. 0. Diane is a chemistry major.. ch a Ift eyou r win the Megabucks multistate lottery, then you will be rich.. He listened to ipod and he tyed a aer. oundations ofhisstructural Kinesiology and Alisha will both miss the sring break ch a Malcolm ter tri. Neuromuscular undamentals 4 or Exercises 5, identify each statement as a conjunction, disjunction, conditional, or biconditional Bob and om like stand-u comedians. Either he asses the test, or he fails the course. A number is even if and only if it is divisible by. Her nails are long, and they have rhinestones on them. I will go to the big game, or I will go to the library. If a number is divisible by, then it is an odd number. A triangle is euiangular if and only if three angles are congruent.. If your battery is dead, then you need to charge your hone overnight. or Exercises 8, write the negation of the statement he sky is blue. It is not true that your comuter has a virus. he dorm room is not large. he class is not full. It is not true that you will fail this class. He has large bices. or Exercises 9 50, identify the uantifier in the statement as either universal or existential. 9. All fish swim in water. 40. Everyone who asses algebra has studied. or Exercises 5 6, write the negation of the statements in Exercises or Exercises 6 7, write each statement in symbols. Let Sara is a olitical science major and let Jane is a uantum hysics major. Sara c6. ha t e r is a olitical science major, and Jane is a uantum hysics major. Basic actors 64. SaraBiomechanical is not a olitical science major. and 65. IfConcets Jane is not a57 uantum hysics major, then Sara is a olitical science major. 66. It is not true that Jane is a uantum hysics major or Sara is a olitical science major. 67. It is false that Jane is a uantum hysics major. 68. It is not true that Sara is a olitical science major. 69. Jane is a uantum hysics major, or Sara is not a olitical science major. 70. Jane is not a uantum hysics major, or Sara is a olitical science major. 7. Jane is a uantum hysics major if and only if Sara is a olitical science major. 7. If Sara is a olitical science major, then Jane is a uantum hysics major. or Exercises 7 8, write each statement in symbols. Let My dad is cool and My mom is cool. Let nerdy mean not cool My mom is not cool. Both my dad and my mom are nerdy. If my mom is cool, then my dad is cool. It is not true that my dad is cool. Either my mom is nerdy, or my dad is cool. It is not true that my mom is nerdy and my dad is cool. My mom is cool if and only if my dad is cool. Neither my mom nor my dad is cool. If my mom is nerdy, then my dad is cool. My dad is nerdy if and only if my mom is not cool. sob04590_ch0_ indd dai0079_ch0_00-0.indd99 0/6/09 4:50: 0/5/09 :46:7

11 CONIRMING PAGES 00 art Chater Logic or Exercises 8 9, write each statement in words. Let he lane is on time. Let he sky is clear ( ) ( ) or Exercises 9 0, write each statement in words. Let Mark lives on camus. Let rudy lives off camus ( ) ( ) ( ) Critical hinking 0. Exlain why the sentence All rules have excetions is not a statement. 04. Exlain why the sentence his statement is false is not a statement. INRODUCION O SRUCURAL KINESIOLOGY Section - ruth ables You can t believe everything you hear. Chances are you were taught this when you were younger, and it s retty good advice. In an ideal world, everyone would tell the truth all the time, but in the real world, it is extremely imortant to be able to searate c h a to t econvince r fact from fiction. When someone is trying you of some oint of view, the ability to logically evaluate the validity of an argument can be the difference between oundations of Structural Kinesiology Basic Biomechanical actors being informed and being deceived and maybe between you keeing and you being and Concets 57 searated from your hard-earned money! his section is all about deciding when a comound statement is or is not true, LEARNING RNING OBJECIVES based not on the4 situation itself, but simly on the structure of the statement and Neuromuscular undamentals the truth of the underlying comonents. We learned about logical connectives in. Construct truth Section -. In this section, we ll analyze these connectives using truth tables. A truth tables for negation, table is a diagram in table form that is used to show when a comound statement is disjunction, and true or false based on the truth values of the simle statements that make u the comconjunction. ound statement. his will allow us to analyze arguments objectively.. Construct truth tables for the conditional and biconditional.. Construct truth tables for comound statements. 4. Identify the hierarchy of logical connectives. Negation According to our definition of statement, a statement is either true or false, but never both. Consider the simle statement oday is uesday. If it is in fact uesday, then is true, and its negation ( ) oday is not uesday is false. If it s not uesday, then is false and is true. he truth table for the negation of looks like this. 5. Construct truth tables by using an alternative method. here are two ossible conditions for the statement true or false and the table tells us that in each case, the negation has the oosite truth value. Conjunction If we have a comound statement with two comonent statements and, there are four ossible combinations of truth values for these two statements: sob04590_ch0_ indd dai0079_ch0_00-0.indd00 0/6/09 4:50: 0/5/09 :46:7

12 CONIRMING PAGES art Section - ruth ables Possibilities 0 Symbolic value of each. and are both true.. is true and is false.. is false and is true. 4. and are both false. So when setting u a truth table for a comound statement with two comonent statements, we ll need a row for each of the four ossibilities. Now we re ready to analyze conjunctions. Recall that a conjunction is a comound statement involving the word and. Suose a friend who s rone to exaggeration tells you, I bought a new comuter and a new ipod. his comound statement can be symbolically reresented by, where I bought a new comuter and I bought a new ipod. When would this conjunctive statement be true? If your friend actually had made both urchases, then of course the statement I bought a new comuter and a new ipod would be true. In terms of a truth table, that tells us that if and are both true, then the conjunction is true as well, as shown below. INRODUCION O SRUCURAL KINESIOLOGY oundations of On the other hand, suose your friend bought only a new comuter or only a new ipod, or maybe neither of those c h athings. t e r hen the statement I bought a new comuter and a new ipod would be false. In other words, if either or both of and are false, the comoundstatementbasic isbiomechanical false as well. With this information, we comlete Structuralthen Kinesiology actors the truth table for a basic conjunction: and Concets 57 Neuromuscular undamentals 4 Bought Bought Bought Bought comuter and ipod comuter, not ipod ipod, not comuter neither ruth Values for a Conjunction he conjunction is true only when both and are true. Disjunction Next, we ll look at truth tables for or statements. Suose your friend from the revious examle made the statement, I bought a new comuter or a new ipod (as oosed to and). If your friend actually did buy one or the other, then this statement would be true. And if he or she bought neither, then the statement would be false. So a artial truth table looks like this: Bought Bought Bought Bought comuter and ipod comuter, not ipod ipod, not comuter neither sob04590_ch0_ indd dai0079_ch0_00-0.indd0 0/6/09 4:50: 0/5/09 :46:7

13 CONIRMING PAGES 0 art Chater Logic Sidelight g Logical Gates and Comuter Design Logic is used in electrical engineering in designing circuits, which are the heart of comuters. he truth tables for and, or, and not are used for comuter gates. hese gates determine whether electricity flows through a circuit. When a switch is closed, the current has an uninterruted ath and will flow through the circuit. his is designated by a. When a switch is oen, the ath at the current is broken, and it will not flow. his is designated by a 0. he logical gates are illustrated here notice that they corresond exactly with our truth tables. his simle little structure is resonsible for the oeration of almost every comuter in the world at least until uantum comuters become a reality. If you re interested, do a Google search for uantum comuter to read about the future of comuting. AND Gate NO Gate OR Gate INRODUCION O SRUCURAL KINESIOLOGY But what if the erson actually bought both items? You might lean toward the statement I bought a new comuter or a new ipod being false. Believe it or not, it deends on what we mean by the word or. here are two interretations of that word, c h a tor. e r he inclusive or has the ossibility of known as the inclusive or and the exclusive both statements being true; but the exclusive or does not allow for this, that is, exactly oundations of Structural Kinesiology Basic Biomechanical actors one of the two simle statements must be true. In English when we use the word or, Concets 57 go to work or I will go to we tyically think of the exclusive or.and If I were to say, I will the beach, you would assume I am doing one or the other, but not both. In logic we generally use the4 inclusive or. When the inclusive or is used, the statement I will go Neuromuscular undamentals to work or I will go to the beach would be true if I went to both work and the beach. or the remainder of this we will assume the symbol reresents the inclusive or and will dro inclusive and just say or. he comleted truth table for the disjunction is Bought Bought Bought Bought comuter and ipod comuter, not ipod ipod, not comuter neither ruth Values for a Disjunction. Construct truth tables for negation, disjunction, and conjunction. he disjunction is true when either or or both are true. It is false only when both and are false. Conditional Statement A conditional statement, which is sometimes called an imlication, consists of two simle statements using the connective if... then. or examle, the statement If I bought a ticket, then I can go to the concert is a conditional statement. he first comonent, in this case I bought a ticket, is called the antecedent. he second comonent, in this case I can go to the concert, is called the conseuent. sob04590_ch0_ indd dai0079_ch0_00-0.indd0 0/6/09 4:50: 0/5/09 :46:7

14 CONIRMING PAGES art Section - ruth ables 0 Conditional statements are used commonly in every area of math, including logic. You might remember statements from high school algebra like If two lines are arallel, then they have the same sloe. Remember that we reresent the conditional statement If, then by using the symbol. o illustrate the truth table for the conditional statement, think about the following simle examle: If it is raining, then I will take an umbrella. We ll use to reresent It is raining and to reresent I will take an umbrella, then the conditional statement is. We ll break the truth table down into four cases. Case : It is raining and I do take an umbrella ( and are both true). Since I am doing what I said I would do in case of rain, the conditional statement is true. So the first line in the truth table is INRODUCION O SRUCURAL KINESIOLOGY Raining, take umbrella Case : It is raining and I do not take an umbrella ( is true and is false). Since I am not doing what I said I would do in case of rain, I m a liar and the conditional statement is false. So the second line in the truth table is Raining, do not take umbrella oundations of a take t e r an umbrella ( is false and is true). his recase : It is not raining andcihdo uires some thought. I never said in the original statement what I would do if it were Structuralnot Kinesiology no reason Basic Biomechanical actors raining, so there s to regard my original statement as false. Based on the and Concets 57 information given, we consider the original statement to be true, and the third line of the truth table is Neuromuscular undamentals 4 Not raining, take umbrella Case 4: It is not raining, and I do not take my umbrella ( and are both false). his is essentially the same as case I never said what I would do if it did not rain, so there s no reason to regard my statement as false based on what we know. So we consider the original statement to be true, and the last line of the truth table is Not raining, do not take umbrella or cases and 4, it might hel to think of it this way: unless we have definite roof that a statement is false, we will consider it to be true. ruth Values for a Conditional Statement he conditional statement is false only when the antecedent is true and the conseuent is false. sob04590_ch0_ indd dai0079_ch0_00-0.indd0 0/6/09 4:50: 0/5/09 :46:7

15 CONIRMING PAGES 04 art Chater Logic Biconditional Statement A biconditional statement is really two statements in a way; it s the conjunction of two conditional statements. or examle, the statement I will stay in and study riday if and only if I don t have any money is the same as If I don t have any money, then I will stay in and study riday and if I stay in and study riday, then I don t have any money. In symbols, we can write either or ( ) ( ). Since the biconditional is a conjunction, for it to be true, both of the statements and must be true. We will once again look at cases to build the truth table. Case : Both and are true. hen both and are true, and the conjunction ( ) ( ), which is also, is true as well. INRODUCION O SRUCURAL KINESIOLOGY Case : is true and is false. In this case, the imlication is false, so it doesn t even matter whether is true or false the conjunction has to be false. Basic Biomechanical and Neuromuscular undamentals 4 A technician who designs an automated irrigation system needs to decide whether the system should turn on if the water in the soil falls below a certain level or if and only if the water in the soil falls below a certain level. In the first instance, other inuts could also turn on the system. Case : is false and is true. his is case in reverse. he imlication is false, so the conjunction must be as well. c h a t e r oundations of Structural Kinesiology actors Concets 57 Case 4: is false and is false. According to the truth table for a conditional statement, both and are true in this case, so the conjunction is as well. his comletes the truth table. ruth Values for a Biconditional Statement. Construct truth tables for the conditional and biconditional. he biconditional statement is true when and have the same truth value and is false when they have oosite truth values. able - on the next age rovides a summary of the truth tables for the basic comound statements and the negation. he last thing you should do is to try and memorize these tables! If you understand how we built them, you can rebuild them on your own when you need them. ruth ables for Comound Statements Once we know truth values for the basic connectives, we can use truth tables to find the truth values for any logical statement. he key to the rocedure is to take it ste sob04590_ch0_ indd dai0079_ch0_00-0.indd04 0/6/09 4:50: 0/5/09 :46:7

16 CONIRMING PAGES art ABLE - Section - ruth ables 05 ruth ables for the Connectives and Negation Conjunction and Disjunction or Conditional if... then Biconditional if and only if Negation not INRODUCION O SRUCURAL KINESIOLOGY by ste, so that in every case, you re deciding on truth values based on one of the truth tables in able -. oundations of Structural Kinesiology c hexample ater Constructing a ruth able Basic Biomechanical actors and Concets 57 Neuromuscular undamentals 4a truth table for the statement. Construct SOLUION Ste Set u a table as shown. My leg isn t better, or I m taking a break is an examle of a statement that can be written as. he order in which you list the s and s doesn t matter as long as you cover all the ossible combinations. or consistency in this book, we ll always use the order for and for when these are the only two letters in the logical statement. Ste ind the truth values for by negating the values for, and ut them into a new column, column, marked. ruth values for are oosite those for. sob04590_ch0_ indd dai0079_ch0_00-0.indd05 0/6/09 4:50:4 0/5/09 :46:7

17 CONIRMING PAGES 06 art Chater Logic Ste ind the truth values for the disjunction. Use the and values for and in columns and, and use the disjunction truth table from earlier in the section. 4 he disjunction is true unless and are both false. he truth values for the statement are found in column 4. he statement is true unless is true and is false. INRODUCION O SRUCURAL ry hiskinesiology One Construct nstruct a truth table for the stat statement. oundations of When a statement contains arentheses, we find the truth values for the statements Structural Kinesiology in Examle Basic Biomechanical actors in arentheses first, as shown. his is similar to the order of oerations used in arithmetic and algebra. and Concets 57 Neuromuscular undamentals 4 EXAMPLE Constructing a ruth able Construct a truth table for the statement ( ). SOLUION Ste Set u the table as shown. It is not true that if it rains, then we can t go out is an examle of a statement that can be written as ( ). Ste ind the truth values for by negating the values for, and ut them into a new column. ruth values for are oosite those for. sob04590_ch0_ indd dai0079_ch0_00-0.indd06 0/6/09 4:50:5 0/5/09 :46:7

18 CONIRMING PAGES art Section - ruth ables 07 Ste ind the truth values for the imlication, using the values in columns and and the imlication truth table from earlier in the section. 4 he conditional is true unless is true and is false. Ste 4 ind the truth values for the negation ( ) by negating the values for in column 4. ( ) 4 5 INRODUCION O he negation has oosite SRUCURAL KINESIOLOGY values from column 4. he truth values for ( ) are in column 5. he statement is true only when and are both true. oundations of Structural Kinesiology ry his One Basic Biomechanical actors and Concets 57 Construct nstruct a truth table for the stat statement ( ). Neuromuscular undamentals 4 We can also construct truth tables for comound statements that involve three or more comonents. or a comound statement with three simle statements,, and r, there are eight ossible combinations of s and s to consider. he truth table is set u as shown in ste of Examle. EXAMPLE Constructing ruth able with hree Comonents Construct a truth table for the statement ( r). SOLUION Ste Set u the table as shown. I ll do my math assignment, or if I think of a good toic, then I ll start my English essay is an examle of a statement that can be written as ( r). r sob04590_ch0_ indd dai0079_ch0_00-0.indd07 0/6/09 4:50:6 0/5/09 :46:7

19 CONIRMING PAGES 08 art Chater Logic Again, the order of the s and s doesn t matter as long as all the ossible combinations are listed. Whenever there are three letters in the statement, we ll use the order shown above for consistency. Ste ind the truth value for the statement in arentheses, r. Use the values in columns and and the conditional truth table from earlier in the section. Put those values in a new column. r r he conditional is true unless is true and r is false. INRODUCION O SRUCURAL KINESIOLOGY 4 Ste ind the truth values for the disjunction ( r), using the values for from column and those for r from column 4. Use the truth table for disjunction from earlier in the section, c hand a tut e r the results in a new column. r r ( r) oundations of Structural Kinesiology Neuromuscular undamentals 4 5 Basic Biomechanical actors and Concets 57 he disjunction is true unless both and r are false. he truth values for the statement ( r) are found in column 5. he statement is true unless and r are false while is true. ry his One. Construct truth tables for comound statements. Construct nstruct a truth table for the stat statement ( ) r. In the method we ve demonstrated for constructing truth tables, we begin by setting u a table with all ossible combinations of s and s for the comonent letters from the statement. hen we build new columns, one at a time, by writing truth values for arts of the comound statement, using the basic truth tables we built earlier in the section. We have seen that when we construct truth tables, we find truth values for statements inside arentheses first. o avoid having to always use arentheses, a hierarchy of connectives has been agreed uon by those who study logic. sob04590_ch0_ indd dai0079_ch0_00-0.indd08 0/6/09 4:50:7 0/5/09 :46:7

20 CONIRMING PAGES art Section - ruth ables Math Note used When arentheses are the to emhasize order, statement r is he written as ( ) r. is r ent tem sta written as ( r). 09 Biconditional Conditional Conjunction, disjunction Negation When we find the truth value for a comound statement without arentheses, we find the truth value of a lower-order connective first. or examle, r is a conditional statement since the conditional ( ) is of a higher order than the disjunction ( ). If you were constructing a truth table for the statement, you would find the truth value for first. he statement r is a biconditional statement since the biconditional ( ) is of a higher order than the order of the conjunction ( ). When constructing a truth table for the statement, the truth value for the conjunction ( ) would be found first. he conjunction and disjunction are of the same order; the statement r cannot be identified unless arentheses are used. In this case, ( ) r is a disjunction and ( r) is a conjunction. INRODUCION O of Connectives EXAMPLE 4 Using the Hierarchy SRUCURAL KINESIOLOGY or each, identify the tye of statement using the hierarchy of connectives, and rewrite by using arentheses to indicate order. (a) oundations of StructuralSOLUION Kinesiology Neuromuscular (b) r (c) r (d) r Basic Biomechanical actors (a) he is higher than and the ;Concets the statement 57 is a disjunction and looks ( ) ( ) with arentheses. (b) he is higher than the or ; the statement is a conditional and looks undamentals 4 ( r) with arentheses. (c) he is higher than ; the statement is a biconditional and looks ( ) ( r) with arentheses. (d) he is higher than the ; the statement is a biconditional and looks ( ) r with arentheses. like like like like ry his One 4 4. Identify the hierarchy of logical connectives. or each, identify the tye of statement usi using the hierarchy of connectives, and rewrite by using arentheses to indicate order. (a) (b) r EXAMPLE 5 (c) (d) (e) r An Alication of ruth ables Use the truth value of each simle statement to determine the truth value of the comound statement. : O. J. Simson was convicted in California in 995. : O. J. Simson was convicted in Nevada in 008. r: O. J. Simson gets sent to rison. Statement: ( ) r sob04590_ch0_ indd dai0079_ch0_00-0.indd09 0/6/09 4:50:7 0/5/09 :46:7

21 CONIRMING PAGES 0 art Chater Logic SOLUION In robably the most ublicized trial of recent times, Simson was acuitted of murder in California in 995, so statement is false. In 008, however, Simson was convicted of robbery and kidnaing in Nevada, so statement is true. Statement r is also true, as Simson was sentenced in December 008. Now we ll analyze the comound statement. irst, the disjunction is true when either or is true, so in this case, is true. inally, the imlication ( ) r is true when both r and are true, which is the case here. So the comound statement ( ) r is true. ry his One 5 Using ng the simle statements in Examle 5, find the truth value of the comound statement ( ) r. INRODUCION O SRUCURAL KINESIOLOGY An Alternative Method for Constructing ruth ables In the next two examles, we will illustrate a second method for constructing truth tables so that you can make a comarison. he roblems are the same as Examles and. oundations of Structural Kinesiology Basic Biomechanical actors and Concets 57 c h a t e6 r Constructing a ruth able by Using an Alternative Method EXAMPLE Neuromuscular undamentals 4 Construct a truth table for ( ). SOLUION Ste Set u the table as shown. Ste Write the truth values for and underneath the resective letters in the statement as shown, and label the columns as and. Ste ind the negation of since it is inside the arentheses, and lace the truth values in column. Draw a line through the truth values in column since they will not be used again. ( ) ( ) ( ) sob04590_ch0_ indd dai0079_ch0_00-0.indd0 0/6/09 4:50:7 0/5/09 :46:7

22 CONIRMING PAGES art Section - ruth ables Ste 4 ind the truth values for the conditional ( ) by using the and values in columns and and the conditional truth table from earlier in the section. Place these values in column 4 and draw a line through the and values in columns and, as shown. ( ) 4 he conditional is true unless is true and ~ is false. INRODUCION O SRUCURAL KINESIOLOGY Ste 5 ind the negations of the truth values in column 4 (since the negation sign is outside the arentheses). It is not true that if it rains, then we can t go out, is an examle of a statement that can be written as ( ). ( ) 5 he negation has values oosite those in column 4. c4h a t e r e otstructural Math Nof oundations Kinesiology of (Basic Biomechanical actors he truth value ) is found in column 5. ortunately, these are the el It isn t necessary to lab same values we foundand in Examle. Concets 57 numbers h wit ns um col the h or to draw a line throug the in ues val Neuromuscular undamentals 4 the truth no columns when they are r, eve longer needed; how Construct nstruct a truth table for the statement ( ), using the alternative these two strategies can. method. ors err uce hel red ry his One 6 EXAMPLE 7 Constructing a ruth able by Using an Alternative Method Construct a truth table for the statement ( r). SOLUION Ste Set u the table as shown. r ( r) sob04590_ch0_ indd dai0079_ch0_00-0.indd 0/6/09 4:50:8 0/5/09 :46:7

23 CONIRMING PAGES art Chater Logic Ste Recoy the values of,, and r under their resective letters in the statement as shown. r ( r) INRODUCION O SRUCURAL KINESIOLOGY Ste Using the truth values in columns and and the truth table for the conditional ( ), find the values inside the arentheses for the conditional and lace them in column 4. r ( r) oundations of Structural Kinesiology Basic Biomechanical actors and Concets 57 he conditional is true unless is true and r is false. Neuromuscular undamentals 4 4 Ste 4 Comlete the truth table, using the truth values in columns and 4 and the table for the disjunction ( ), as shown. r ( 5 4 r) he disjunction is true unless and r are both false. he truth value for ( r) is found in column 5. hese are the same values we found in Examle. sob04590_ch0_ indd dai0079_ch0_00-0.indd 0/6/09 4:50:8 0/5/09 :46:7

24 CONIRMING PAGES art Section - ruth ables ry his One 7 Construct nstruct a truth table for the statement ( ( ) r using the alternative method. 5. Construct truth tables by using an alternative method. he best aroach to learning truth tables is to try each of the two methods and see which one is more comfortable for you. In any case, we have seen that truth tables are an effective way to organize truth values for statements, allowing us to determine the truth values of some very comlicated statements in a systematic way. Answers to ry his One INRODUCION O 4 SRUCURAL KINESIOLOGY (a) (b) (c) (d) (e) 5 ( ) c h a t er oundations of Structural Kinesiology c h a t er Disjunction; ( ) Conditional; ( ) r Biconditional; ( ) ( ) Conjunction; ( ) Biconditional; ( r) rue 6 cha t er ( ) Biomechanical actors Basic and Concets 57 Neuromuscular r undamentals r ( ) 4 r r ( ) r EXERCISE SE - Writing Exercises. Exlain the urose of a truth table.. Exlain the difference between the inclusive and exclusive disjunctions.. Exlain the difference between the conditional and biconditional statements. 4. Describe the hierarchy for the basic connectives. sob04590_ch0_ indd dai0079_ch0_00-0.indd 0/6/09 4:50:9 0/5/09 :46:7

25 CONIRMING PAGES 4 art Chater Logic Comutational Exercises or Exercises 5 4, construct a truth table for each ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) r (r ) ( ) (r ) ( r) ( ) ( r) ( ) r ( ) r ( r) ( ) ( r ) ( r) ( r) r ( ) ( ) ( r) ( r) ( r) ( ) ( ) r ( r) (r ) (r ) ( ) r Real-World Alications or Exercises 5 40, use the truth value of each simle statement to determine the truth value of the comound statement. Use the Internet if you need hel determining the truth value of a simle statement. 4. Write the comound statement in symbolic form, using conjunctions and the conditional. 4. Construct a truth table for the comound statement you wrote in Exercise If you take this roduct daily and don t cut your calorie intake by 0%, and then don t lose 0 ounds, is the claim made by the advertiser true or false? 45. If you take the roduct daily, don t cut your calorie intake by 0%, and do lose 0 ounds, is the claim true or false? c h adon t t e rtake the roduct daily, cut your calorie 46. If you intake by 0%, and do lose 0 ounds, is the claim truebasic or false?biomechanical actors INRODUCION O SRUCURAL KINESIOLOGY 5. : Jaan bombs Pearl Harbor. : he United States stays out of World War II. Statement: 6. : Barack Obama wins the Democratic nomination in 008. : Mitt Romney wins the Reublican nomination in 008. Statement: oundations of Structural Kinesiology 7. : NASA sends a manned sacecraft to the Moon. : NASA c h a tsends e r a manned sacecraft to Mars. Statement: Neuromuscular undamentals 4 8. : Michael Phels wins eight gold medals. : Michael Phels gets a large endorsement deal. Statement: 9. : Ale builds a ortable MP layer. : Ale stos making comuters. r: Microsoft releases the Vista oerating system. Statement: ( ) r 40. : Hurricane Katrina hits New Orleans. : New Orleans Suerdome is damaged. r: New Orleans Saints lay home games in 006 in Baton Rouge. Statement: ( ) r Exercises 4 46 are based on the comound statement below. A new weight loss sulement claims that if you take the roduct daily and cut your calorie intake by 0%, you will lose at least 0 ounds in the next 4 months. 4. his comound statement is made u of three simle statements. Identify them and assign a letter to each. and Concets 57 Exercises 47 5 are based on the comound statement below. he owner of a rofessional baseball team ublishes an oen letter to fans after another losing season. He claims that if attendance for the following season is over million, then he will add $0 million to the ayroll and the team will make the layoffs the following year. 47. his comound statement is made u of three simle statements. Identify them and assign a letter to each. 48. Write the comound statement in symbolic form, using conjunction and the conditional. 49. Construct a truth table for the comound statement you wrote in Exercise If attendance goes over million the next year and the owner raises ayroll by $0 million, but the team fails to make the layoffs, is the owner s claim true or false? 5. If attendance is less than million but the owner still raises the ayroll by $0 million and the team makes the layoffs, is the owner s claim true or false? 5. If attendance is over million, the owner doesn t raise the ayroll, but the team still makes the layoffs, is the owner s claim true or false? Critical hinking 5. Using the hierarchy for connectives, write the statement r by using arentheses to indicate the roer order. hen construct truth tables for ( ) r and ( r). Are the resulting truth values the same? Are you surrised? Why or why not? sob04590_ch0_ indd dai0079_ch0_00-0.indd4 0/6/09 4:50:9 0/5/09 :46:7

26 CONIRMING PAGES art Section - yes of Statements 54. he hierarchy of connectives doesn t distinguish between conjunctions and disjunctions. Does that matter? Construct truth tables for ( ) r and ( r) to hel you decide. 55. In 00, New York City Council was considering banning indoor smoking in bars and restaurants. Oonents of the ban claimed that it would have a negligible effect on indoor ollution, but a huge nega- 5 tive effect on the economic success of these businesses. Eventually, the ban was enacted, and a 004 study by the city deartment of health found that there was a sixfold decrease in indoor air ollution in bars and restaurants, but jobs, liuor licenses, and tax revenues all increased. Assign truth values to all the remises of the oonents claim; then write the claim as a comound statement and determine its validity. INRODUCION O SRUCURAL KINESIOLOGY Section - yes of Statements It s no secret that weight loss has become big business in the United States. It seems like almost every week, a new comany os into existence with the latest miracle ill to turn you into a suermodel. A tyical advertisement will say something like Use ofc hour ater LEARNING RNING OBJECIVES roduct may result in significant loss. hatsounds great, oundations Structuralweight Kinesiology Basic Biomechanical actors. Classify a of statement but think about what that stateand Concets 57 as a tautology, a ment really means. If use of the c h a t eself-contradiction, r or roduct may result in signifineither. Neuromuscular undamentals 4 loss, then it also may not result in weight loss at all! he statement cant weight. Identify logically could be translated as You will lose weight or you will not lose weight. Of euivalent statements. course, this statement is always true. In this section, we will study statements of this tye.. Write negations of comound statements. 4. Write the converse, inverse, and contraositive of a statement. autologies and Self-Contradictions In our study of truth tables in Section -, we saw that most comound statements are true in some cases and false in others. What we have not done is think about whether that s true for every comound statement. Some simle examles should be enough to convince you that this is most definitely not the case. Consider the simle statement I m going to Cancun for sring break this year. Its negation is I m not going to Cancun for sring break this year. Now think about these two comound statements: I m going to Cancun for sring break this year, or I m not going to Cancun for sring break this year. I m going to Cancun for sring break this year, and I m not going to Cancun for sring break this year. Hoefully, it s retty clear to you that the first statement is always true, while the second statement is always false (whether you go to Cancun or not). he first is an examle of a tautology, while the second is an examle of a self-contradiction. When a comound statement is always true, it is called a tautology. When a comound statement is always false, it is called a self-contradiction. sob04590_ch0_ indd dai0079_ch0_00-0.indd5 0/6/09 4:50:9 0/5/09 :46:7

27 CONIRMING PAGES 6 art Chater Logic CAUION Don t make the mistake of thinking that every statement is either a tautology or a selfcontradiction. We ve seen many examles of statements that are sometimes true and other times false. he samle statements above are simle enough that it s easy to tell that they are always true or always false based on common sense. But for more comlicated statements, we ll need to construct a truth table to decide if a statement is a tautology, a self-contradiction, or neither. Using a ruth able to Classify a Statement EXAMPLE Determine if each statement is a tautology, a self-contradiction, or neither. INRODUCION O SOLUION SRUCURAL KINESIOLOGY (a) ( ) (b) ( ) ( ) (c) ( ) (a) he truth table for statement (a) is oundations of Structural Kinesiology ( ) c ha t e r Basic Biomechanical actors and Concets 57 is always true, that is, a Since the truth table value consists of all s, the statement tautology. (b) he truth table for statement (b) is Let I am going to a Neuromuscular undamentals concert and I will wear black. ranslate each statement in Examle into a word statement using this choice of and. Can you redict which statements are tautologies, selfcontradictions, or neither? 4 ( ) ( ) Since the truth value consists of all s, the statement is always false, that is, a selfcontradiction. (c) he truth table for statement (c) is ( ) Since the statement can be true in some cases and false in others, it is neither a tautology nor a self-contradiction. ry his One. Classify a statement as a tautology, a selfcontradiction, or neither. Determine ermine if each statement is a ta tautology, a self-contradiction, or neither. (a) ( ) ( ) (b) ( ) (c) ( ) sob04590_ch0_ indd dai0079_ch0_00-0.indd6 0/6/09 4:50:0 0/5/09 :46:7

28 CONIRMING PAGES art Section - yes of Statements 7 Logically Euivalent Statements B A and B A he statements If the red dial is set to A, then use only seaker A and he red dial is not set to seaker A, or only seaker A is used can be modeled with the logic statements and. Next, consider the two logical statements and. he truth table for the two statements is Notice that the truth values for both statements are identical, that is,. When this occurs, the statements are said to be logically euivalent; that is, both comositions of the same simle statements have the same meaning. or examle, the statement If it snows, I will go skiing is logically euivalent to saying It is not snowing or I will go skiing. ormally defined, INRODUCION O SRUCURAL KINESIOLOGY EXAMPLE wo comound statements are logically euivalent if and only if they have the same truth table values. he symbol for logically euivalent statements is. Identifying Logically Euivalent Statements Determine if the two statements and are logically euivalent. oundationsb of Structural Kinesiology c h aa ter A and B SOLUION he truth table for the statements is Neuromuscular undamentals 4 In the red dial examle, would be If seaker A is not the only one used, then the red dial is not set to A. Basic Biomechanical actors and Concets 57 Since both statements have the same truth values, they are logically euivalent. ry his One. Identify logically euivalent statements. Determine ermine which two statements aare logically euivalent. (a) ( ) (b) (c) De Morgan s laws for logic give us an examle of euivalent statements. De Morgan s Laws for Logic or any statements and, ( ) and ( ) sob04590_ch0_ indd dai0079_ch0_00-0.indd7 0/6/09 4:50: 0/5/09 :46:7

29 CONIRMING PAGES 8 art Chater Logic Notice the similarities between De Morgan s laws for sets and De Morgan s laws for logic. De Morgan s laws can be roved by constructing truth tables; the roofs will be left to you in the exercises. De Morgan s laws are most often used to write the negation of conjunctions and disjunctions. or examle, the negation of the statement I will go to work or I will go to the beach is I will not go to work and I will not go to the beach. Notice that when you negate a conjunction, it becomes a disjunction; and when you negate a disjunction, it becomes a conjunction that is, the and becomes an or, and the or becomes an and. Using De Morgan s Laws to Write Negations EXAMPLE INRODUCION O SRUCURAL KINESIOLOGY Write the negations of the following statements, using De Morgan s laws. (a) (b) (c) (d) Studying is necessary and I am a hard worker. It is not easy or I am lazy. I will ass this test or I will dro this class. She is angry or she s my friend, and she is cool. SOLUION oundations of (a) Studying is not necessary or I am not a hard worker. (b) It is easy and I am not lazy. h adro t e rthis class. (c) I will not ass this test and I will cnot (d) She iskinesiology not angry and she friend,biomechanical or she is not cool. Structural is not my Basic actors and Concets 57 Neuromuscular undamentals 4One ry his Write te the negations of the followin following statements, using De Morgan s laws. (a) (b) (c) (d) I will study for this class or I will fail. I will go to the dance club and the restaurant. It is not silly or I have no sense of humor. he movie is a comedy or a thriller, and it is awesome. Earlier in this section, we saw that the two statements and are logically euivalent. Now that we know De Morgan s laws, we can use this fact to find the negation of the conditional statement. ( ) ( ) ( ) Note: ( ) his can be checked by using a truth table as shown. ( ) So the negation of is. sob04590_ch0_ indd dai0079_ch0_00-0.indd8 0/6/09 4:50: 0/5/09 :46:7

30 CONIRMING PAGES art EXAMPLE 4 Section - yes of Statements 9 or examle, if you say, It is false that if it is sunny, then I will go swimming, this is euivalent to the statement It s sunny and I will not go swimming. Writing the Negation of a Conditional Statement Write the negation of the statement If I have a comuter, then I will use the Internet. SOLUION Let I have a comuter and I will use the Internet. he statement can be negated as. his translates to I have a comuter and I will not use the Internet. INRODUCION O his One 4 SRUCURALryKINESIOLOGY. Write negations of comound statements. Write te the negation of the statemen statement If the video is oular, then it can be found on Youube. Y b c h a t eofr the basic comound statements. able - summarizes the negations oundations of Structural Kinesiology ABLE - Basic Biomechanical actors and Concets 57 Negation of Comound Statements Statement St t t N Negation ti E Euivalent i l tn Negation ti ( ) ( ) ( ) Neuromuscular undamentals 4 Variations of the Conditional Statement rom the conditional statement, three other related statements can be formed: the converse, the inverse, and the contraositive. hey are shown here symbolically. Statement Converse Inverse Contraositive Using the statement If essa is a chocolate Lab, then essa is brown as our original conditional statement, we find the related statements are as follows: Converse: If essa is brown, then essa is a chocolate Lab. Inverse: If essa is not a chocolate Lab, then essa is not brown. Contraositive: If essa is not brown, then essa is not a chocolate Lab. Notice that the original statement is true, but of the three related statements, only the contraositive is also true. his is an imortant observation one that we ll elaborate on shortly. sob04590_ch0_ indd dai0079_ch0_00-0.indd9 0/6/09 4:50: 0/5/09 :46:7

31 CONIRMING PAGES 0 art Chater Logic Writing the Converse, Inverse, and Contraositive EXAMPLE 5 Write the converse, the inverse, and the contraositive for the statement If you earned a bachelor s degree, then you got a high-aying job. SOLUION It s helful to write the original imlication in symbols:, where You earned a bachelor s degree and You got a high-aying job. Converse:. If you got a high-aying job, then you earned a bachelor s degree. Inverse:. If you did not earn a bachelor s degree, then you did not get a high-aying job. Contraositive:. If you did not get a high-aying job, then you did not earn a bachelor s degree. INRODUCION O SRUCURAL ry hiskinesiology One 5 Write te the converse, the inverse, an and the contraositive for the statement If you do well ll iin math th classes, l th then you are intelligent. oundations of h Note Mat ic Many eole using log he relationshis between the variations of the conditional statements can be Structural Kinesiology Basic Biomechanical actors determined by looking at the truth tables for each of the statements. and Concets 57 life assume that if in realneuromuscular undamentals 4 a statement is true, the ly converse is automatical ing true. Consider the follow son er statement: If a,000 earns more than $00 Since the original statement ( ) and the contraositive statement ( ) have buy er year, that erson can the same truth values, they are euivalent. Also note that the converse ( ) and the is se ver a Corvette. he con inverse ( ) have the same truth values, so they are euivalent as well. inally, can son stated as If a er t tha n notice that the original statement is not euivalent to the converse or the inverse since the, tte buy a Corve n tha re mo the truth values of the converse and inverse differ from those of the original statement. ns ear erson is Since the conditional statement is used so often in logic as well as mathe$00,000 er year. h th matics, a more detailed analysis is helful. Recall that the conditional statement may be far from the tru e to hav y ma son since a er is also called an imlication and consists of two simle statements; the first is the antents, make very large ayme cedent and the second is the conseuent. or examle, the statement If I jum into ee live in a tent, or work thr the ool, then I will get wet consists of the antecedent, I jum into the ool, and the jobs in order to afford the conseuent, I will get wet, connected by the if... then connective. exensive car. he conditional can also be stated in these other ways: imlies if only if is sufficient for is necessary for All are In four of these six forms, the antecedent comes first, but for if and is necessary for, the conseuent comes first. So identifying the antecedent and conseuent is imortant. sob04590_ch0_ indd dai0079_ch0_00-0.indd0 0/6/09 4:50: 0/5/09 :46:7

32 CONIRMING PAGES art Section - yes of Statements or examle, think about the statement If you drink and drive, you get arrested. Writing it in the different ossible forms, we get: Drinking and driving imlies you get arrested. You get arrested if you drink and drive. You drink and drive only if you get arrested. Drinking and driving is sufficient for getting arrested. Getting arrested is necessary for drinking and driving. All those who drink and drive get arrested. Of course, these all say the same thing. o illustrate the imortance of getting the antecedent and conseuent in the correct order, consider the if form, in this case You get arrested if you drink and drive. If we don t start with the conseuent, we get You drink and drive if you get arrested. his is comletely false there are any number of things you could get arrested for other than drinking and driving. INRODUCION O SRUCURAL KINESIOLOGY of a Conditional Statement EXAMPLE 6 Writing Variations Write each statement in symbols. Let A erson is over 6 6 and A erson is tall. oundations of (a) If a erson is over 6 6, then the erson is tall. c hbeing a t eover r 6 6. (b) Being tall is necessary for A erson is over if the Biomechanical erson is tall. Structural(c)Kinesiology 6 6 only Basic (d) Being 6 6 is sufficient for being tall. and Concets 57 (e) A erson is tall if the erson is over 6 6. actors Neuromuscular undamentals 4 SOLUION (a) (b) (c) (d) (e) If, then ; is necessary for ; only if ; is sufficient for ; if ; Actually, these statements all say the same thing! ry his One 6 4. Write the converse, inverse, and contraositive of a statement. Write te each statement in symbols. Let A student comes to class every day and A student t d t gets t a good d grade. (a) (b) (c) (d) A student gets a good grade if a student comes to class every day. Coming to class every day is necessary for getting a good grade. A student gets a good grade only if a student comes to class every day. Coming to class every day is sufficient for getting a good grade. In this section, we saw that some statements are always true (tautologies) and others are always false (self-contradictions). We also defined what it means for two statements to be logically euivalent they have the same truth values. Now we re ready to analyze logical arguments to determine if they re legitimate or not. sob04590_ch0_ indd dai0079_ch0_00-0.indd 0/6/09 4:50:4 0/5/09 :46:7

33 CONIRMING PAGES art Chater Logic Answers to ry his One (a) Neither (c) autology 4 he video is oular and it cannot be found on Youube. (a) and (c) 5 (a) I will not study for this class and I will not fail. (b) I will not go to the dance club or the restaurant. (c) It is silly and I have a sense of humor. (d) he movie is not a comedy and it is not a thriller, or it is not awesome. Converse: If you are intelligent, then you do well in math classes. Inverse: If you do not do well in math classes, then you are not intelligent. Contraositive: If you are not intelligent, then you do not do well in math classes. 6 (b) Self-contradiction (a) (b) (c) INRODUCION O EXERCISE SE - SRUCURAL KINESIOLOGY (d) Writing Questions. Exlain the difference between a tautology and a self-contradiction. h a tstatement er. Is cevery either a tautology or a selfcontradiction? Why or not? oundations ofwhy Structural Kinesiology. Describe how to find the converse, inverse, and contraositive c h a t e r of a conditional statement. Comutational Exercises Neuromuscular undamentals 4 or Exercises 7 6, determine which statements are tautologies, self-contradictions, or neither ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) or Exercises 7 6, determine if the two statements are logically euivalent statements, negations, or neither. 7. ; 8. ; 9. ( ); 0. ( ); 4. How can you decide if two statements are logically euivalent? a you t e r decide if one statement is the negation 5. Howc hcan of another? Basic Biomechanical actors 6. Is a statement always logically euivalent to its and Concets 57 converse? Exlain ; ( ) ( r); ( ) ( r) ( ); ( ) ( ) r; r ( ) ( ) r; ( r) ; ( ) ( ) or Exercises 7, write the converse, inverse, and contraositive of each Real-World Alications In Exercises 4, use De Morgan s laws to write the negation of the statement he concert is long or it is fun. he soda is sweet or it is not carbonated. It is not cold and I am soaked. I will walk in the Race for the Cure walkathon and I will be tired. I will go to the beach and I will not get sunburned. he coffee is a latte or an esresso. he student is a girl or the rofessor is not a man. I will go to college and I will get a degree. It is right or it is wrong. Our school colors are not blue or they are not green. sob04590_ch0_ indd dai0079_ch0_00-0.indd or Exercises 4 49, let I need to talk to my friend and I will send her a text message. Write each of the following in symbols (see Examle 6). 4. If I need to talk to my friend, I will send her a text message. 44. If I will not send her a text message, I do not need to talk to my friend. 45. Sending a text message is necessary for needing to talk to my friend. 46. I will send her a text message if I need to talk to my friend. 47. Needing to talk to my friend is sufficient for sending her a text message. 0/6/09 4:50:5 0/5/09 :46:7

34 CONIRMING PAGES art Section -4 Logical Arguments 48. I need to talk to my friend only if I will send her a text message. 49. I do not need to talk to my friend only if I will not send her a text message. 50. Are any of the statements in Exercises 4 49 logically euivalent? or Exercises 5 56, write the converse, inverse, and contraositive of the conditional statement. 5. If he graduated with a bachelor s degree in management information systems, then he will get a good job. 5. If she does not earn $5,000 this summer as a barista at the coffeehouse, then she cannot buy the green ord ocus. 5. If the American Idol finale is today, then I will host a arty in my dorm room. 54. If my cell hone will not charge, then I will relace the battery. 55. I will go to Nassau for sring break if I lose 0 ounds by March. 56. he olitician will go to jail if he gets caught taking kickbacks. Critical hinking INRODUCION O SRUCURAL KINESIOLOGY 57. In this section, we wrote the negation of by using a disjunction. See if you can write the negation of by using a conjunction. 58. ry to write the negation of the biconditional by using only conjunctions, disjunctions, and negations. 59. Can you think of a true conditional statement about someone you know so that the converse is true as well? How about so that the converse is false? 60. Can you think of a true conditional statement about c h a someone ter you know so that the inverse is true as well? How about so that the inverse is false? oundations of Structural Kinesiology 6. Prove the first of De Morgan s laws by using truth tables: ( ) 6. Prove the second of De Morgan s laws by using truth tables: ( ) Basic Biomechanical actors and Concets 57 Neuromuscular undamentals 4 Section -4 Logical Arguments LEARNING RNING OBJECIVES. Define valid argument and fallacy.. Use truth tables to determine the validity of an argument.. Identify common argument forms. 4. Determine the validity of arguments by using common argument forms. Common sense is a funny thing in our society: we all think we have it, and we also think that most other eole don t. his thing that we call common sense is really the ability to think logically, to evaluate an argument or situation and decide what is and is not reasonable. It doesn t take a lot of imagination to icture how valuable it is to be able to think logically. We re retty well rotected by arents for our first few years of life, but after that the main tool we have to guide us through the erils of life is our brain. he more effectively that brain can analyze and evaluate information, the more successful we re likely to be. he work we ve done in building the basics of symbolic logic in the first three sections of this has reared us for the real oint: analyzing logical arguments objectively. hat s the toic of this imortant section. Valid Arguments and allacies A logical argument is made u of two arts: a remise or remises and a conclusion based on those remises. We will call an argument valid if assuming the remises are true guarantees that the conclusion is true as well. An argument that is not valid is called invalid or a fallacy. Let s look at an examle. Premise : Premise : Conclusion: All students in this class will ass. Rachel is a student in this class. Rachel will ass this class. We can easily tell that if the two remises are true, then the conclusion is true, so this is an examle of a valid argument. sob04590_ch0_ indd dai0079_ch0_00-0.indd 0/6/09 4:50:5 0/5/09 :46:7

35 CONIRMING PAGES 4 art Chater Logic Sidelight LO O GIC AND HE AR O OREE NSI CS Many students find it troubling that an argument can be considered valid even if the conclusion is clearly false. But arguing in favor of something that you don t necessarily believe to be true isn t a new idea by any means lawyers do it all the time, and it s commonly racticed in the area of formal debate, a style of intellectual cometition that has its roots in ancient times. In formal debate (also known as forensics), seakers are given a toic and asked to argue one side of a related issue. Judges determine which seakers make the most effective arguments and declare the winners accordingly. One of the most interesting asects is that in many cases, the contestants don t know which side of the issue they will be arguing until right before the cometition begins. While that asect is intended to test the debater s flexibility and rearation, a major conseuence is that oinion, and sometimes truth, is taken out of the mix, and contestants and judges must focus on the validity of arguments. A variety of organizations sonsor national cometitions in formal debate for colleges. he largest is an annual INRODUCION O SRUCURAL KINESIOLOGY chamionshi organized by the National orensics Association. Students from well over 00 schools articiate in a wide variety of categories. he 008 team chamions were ennessee State University, Kansas State University, California State Long Beach, and Western Kentucky University. oundations of Structural Kinesiology oint tobasic Biomechanical actors It s very imortant at this understand the difference between a true statement and a conclusion to a valid argument. A statement 57 that is known to be false can and Concets still be a valid conclusion if it follows logically from the given remises. or examle, consider this argument: Neuromuscular undamentals 4 Los Angeles is in California or Mexico. Los Angeles is not in California. herefore, Los Angeles is in Mexico.. Define valid argument and fallacy. his is a valid argument: if the two remises are true, then the conclusion, Los Angeles is in Mexico, must be true as well. We know, however, that Los Angeles isn t actually in Mexico. hat s the tricky art. In determining whether an argument is valid, we will always assume that the remises are true. In this case, we re assuming that the remise Los Angeles is not in California is true, even though in fact it is not. We will discuss this asect of logical arguments in greater deth later in this section. ruth able Method One method for determining the validity of an argument is by using truth tables. We will use the following rocedure. Procedure for Determining the Validity of Arguments Ste Write the argument in symbols. Ste Write the argument as a conditional statement; use a conjunction between the remises and the imlication ( ) for the conclusion. (Note: he is the same as but will be used to designate an argument.) sob04590_ch0_ indd dai0079_ch0_00-0.indd4 0/6/09 4:50:5 0/5/09 :46:7

36 CONIRMING PAGES art EXAMPLE Section -4 Logical Arguments 5 Ste Set u and construct a truth table as follows: Symbols Premise Premise Conclusion Ste 4 If all truth values under are s (i.e., a tautology), then the argument is valid; otherwise, it is invalid. Determining the Validity of an Argument INRODUCION O SRUCURAL KINESIOLOGY Determine if the following argument is valid or invalid. If a figure has three sides, then it is a triangle. his figure is not a triangle. herefore, this figure does not have three sides. SOLUION Ste Write the argument in symbols. Let he figure has three sides, and let he figure is a triangle. ranslated into symbols: oundations of Structural Kinesiology Basic Biomechanical actors (Premise) (Premise) and Concets 57 Neuromuscular undamentals 4 (Conclusion) A line is used to searate the remises from the conclusion and the three triangular dots mean therefore. Ste Write the argument as an imlication by connecting the remises with a conjunction and imlying the conclusion as shown. Premise ( ) Premise Conclusion Ste Construct a truth table as shown. ( ) [( ) ] Ste 4 Determine the validity of the argument. Since all the values under the are true, the argument is valid. ry his One Determine ermine if the argument is valid or invalid invalid. I will run for student government or I will join the athletic boosters. I did not join the athletic boosters. herefore, I will run for student government. sob04590_ch0_ indd dai0079_ch0_00-0.indd5 0/6/09 4:50:6 0/5/09 :46:7

37 CONIRMING PAGES 6 art Chater Logic Determining the Validity of an Argument EXAMPLE Determine the validity of the following argument. If a rofessor is rich, then he will buy an exensive automobile. he rofessor bought an exensive automobile. herefore, the rofessor is rich. SOLUION Ste Write the argument in symbols. Let he rofessor is rich, and let he rofessor buys an exensive automobile. INRODUCION O SRUCURAL KINESIOLOGY Ste Write the argument as an imlication. ( ) Ste Construct a truth table for the argument. ( ) oundations of Structural Kinesiology [( ) ] Basic Biomechanical actors Ste 4 Determine the validity of the argument. his argument is invalid since it is not a and Concets tautology. (Remember, when the values are not all57 s, the argument is invalid.) In this case, it cannot be concluded that the rofessor is rich. Neuromuscular undamentals 4 ry his One Determine ermine the validity of the following argu argument. If John blows off work to go to the layoff game, he will lose his job. John lost his job. herefore, John blew off work and went to the layoff game. CAUION Remember that in symbolic logic, whether or not the conclusion is true is not imortant. he main concern is whether the conclusion follows from the remises. Consider the following two arguments.. Either 4 or If 5, then I assed the math uiz. I did not ass the uiz. 5 he truth tables on the next age show that the first argument is valid even though the conclusion is false, and the second argument is invalid even though the conclusion is true! sob04590_ch0_ indd dai0079_ch0_00-0.indd6 0/6/09 4:50:6 0/5/09 :46:7

38 CONIRMING PAGES art Section -4 Logical Arguments Let be the statement 4 and be the statement 5. hen the first argument is written as ( ) Let be the statement 5 and be the statement I assed the math uiz. he second argument is written as ruth table for argument ( ) [( ) ] ruth table for argument ( ) INRODUCION O SRUCURAL KINESIOLOGY 7 [( ) ] he validity of arguments that have three variables can also be determined by truth tables, as shown in Examle. c hexample ater h a Argument ter Determining the Validity ofcan oundations of Structural Kinesiology Basic Biomechanical actors and Concets 57 Determine the validity of the following argument. Neuromuscular r r4 undamentals SOLUION Ste Write the argument in symbols. his has been done already. Ste Write the argument as an imlication. Make a conjunction of all three remises and imly the conclusion: ( r) ( r) ( ) Ste Construct a truth table. When there are three remises, we will begin by finding the truth values for each remise and then work the conjunction from left to right as shown. r r r ( r) ( r) [( r) ( r) ] ( ) Since the truth value for is all s, the argument is valid. sob04590_ch0_ indd dai0079_ch0_00-0.indd7 0/6/09 4:50:6 0/5/09 :46:7

39 CONIRMING PAGES 8 art Chater Logic. Use truth tables to determine the validity of an argument. ry his One Determine ermine whether the following argument is valid or invalid. r Common Valid Argument orms We have seen that truth tables can be used to test an argument for validity. But some argument forms are common enough that they are recognized by secial names. When an argument fits one of these forms, we can decide if it is valid or not just by knowing the general form, rather than constructing a truth table. We ll start with a descrition of some commonly used valid arguments. INRODUCION O SRUCURAL KINESIOLOGY. Law of detachment (also known by Latin name modus onens): oundations of Examle: If our team wins Saturday, then they go to a bowl game. Structural Kinesiology Basic Biomechanical Our team won Saturday. and Concets 57 actors herefore, our team goes to a bowl game.. Law of contraosition (Latin name modus tollens): Neuromuscular undamentals 4 Examle: If I try hard, I ll get an A. I didn t get an A. herefore, I didn t try hard.. Law of syllogism, also known as law of transitivity: r r Examle: If I make an illegal U-turn, I ll get a ticket. If I get a ticket, I ll get oints on my driving record. herefore, if I make an illegal U-turn, I ll get oints on my driving record. 4. Law of disjunctive syllogism: sob04590_ch0_ indd dai0079_ch0_00-0.indd8 0/6/09 4:50:7 0/5/09 :46:7

40 CONIRMING PAGES art Section -4 Logical Arguments 9 Examle: You re either brilliant or insane. You re not brilliant. herefore, you re insane. Common allacies Math Note It s more common to mistakenly think that an id invalid argument is val y wa er oth rather than the ay uld sho you around, so secial attention to the common fallacies listed. Next, we will list some commonly used arguments that are invalid.. allacy of the converse: INRODUCION O SRUCURAL KINESIOLOGY Examle: If it s riday, then I will go to hay hour. I am at hay hour. herefore, it is riday. c hgo a to t ehay r his is not valid! You can hour other days, too. oundations of Structural Kinesiology Basic Biomechanical actors. allacy of the inverse: and Concets 57 Neuromuscular undamentals 4 Examle: If I exercise every day, then I will lose weight. I don t exercise every day. herefore, I won t lose weight. his is also not valid. You could still lose weight without exercising every day.. allacy of the inclusive or: Examle: I m going to take chemistry or hysics. I m taking chemistry. herefore, I m not taking hysics. Remember, we ve agreed that by or we mean one or the other, or both. So you could be taking both classes. You will be asked to rove that some of these are invalid by using truth tables in the exercises. sob04590_ch0_ indd dai0079_ch0_00-0.indd9 0/6/09 4:50:7 0/5/09 :46:7

41 CONIRMING PAGES 0 art Chater Logic Sidelight g CIRCULAR REASO O NING susect how he knows that Sue can be trusted, and he says, I can assure you of her honesty. Ultimately, the susect becomes even more susect because of circular reasoning: his argument boils down to I am honest because I am honest. While this examle might seem blatantly silly, you d be surrised how often eole try to get away with this fallacy. A Google search for the string circular reasoning brings u hundreds of arguments that are thought to be circular. Circular reasoning (sometimes called begging the uestion ) is a sneaky tye of fallacy in which the remises of an argument contain a claim that the conclusion is true, so naturally if the remises are true, so is the conclusion. But this doesn t constitute evidence that a conclusion is true. Consider the following examle: A susect in a criminal investigation tells the olice detective that his statements can be trusted because his friend Sue can vouch for him. he detective asks the INRODUCION O SRUCURAL KINESIOLOGY Recognizing Common Argument orms EXAMPLE 4 Determine whether the following arguments are valid, using the given forms of valid arguments and fallacies. (c) (a) (b) (d) r s s t r t SOLUION (a) his is the law of detachment, therefore a valid argument. oundations of Structural Basic Biomechanical actors (b) his fikinesiology ts the law of contraosition with the statement substituted in lace of, and Concets 57 so it is valid. (c) his fits the fallacy of the converse, using statement and rather than and, so it is an invalid Neuromuscular undamentals 4 argument. (d) his is the law of syllogism, with statements r, s, and t, so the argument is valid. ry his One 4. Identify common argument forms. Determine ermine whether the arguments are valid, val using the commonly used valid arguments and fallacies. (a) EXAMPLE 5 (b) r s (c) (d) ( ) r s r r ( ) Determining the Validity of an Argument by Using Common Argument orms Determine whether the following arguments are valid, using the given forms of valid arguments and fallacies. (a) If you like dogs, you will live to be 0. You like dogs. herefore, you will live to be 0. (b) If the modem is connected, then you can access the Web. he modem is not connected. herefore, you cannot access the Web. sob04590_ch0_ indd dai0079_ch0_00-0.indd0 0/6/09 4:50:8 0/5/09 :46:7

42 CONIRMING PAGES art Section -4 Logical Arguments (c) If you watch Big Brother, you watch reality shows. If you watch reality shows, you have time to kill. herefore, if you have time to kill, you watch Big Brother. (d) he movie Scream is a thriller or a comedy. he movie Scream is a thriller. herefore, the movie Scream is not a comedy. (e) My ipod is in my backack or it is at my friend s house. My ipod is not in my backack. herefore, my ipod is at my friend s house. SOLUION INRODUCION O SRUCURAL KINESIOLOGY oundations of (a) In symbolic form this argument is ( ). We can see that this is the law of detachment, so the argument is valid. (b) In symbolic form this argument is ( ). his is the fallacy of the inverse, so the argument is invalid. (c) In symbolic form this argument is ( ) ( r) (r ). We know by the law of transitivity that if ( ) ( r), then r. he given conclusion, r, is the converse of r, so is not euivalent to r (the valid conclusion). So the argument is invalid. (d) In symbolic form the argument is ( ). his is the fallacy of the in clusive or, so the argument is invalid. In symbolic form ( ). his is the law of disjunctive Structural(e)Kinesiology the argument Basic isbiomechanical actors syllogism, so the argument is valid. and Concets 57 Neuromuscular undamentals 4 ry his One 5 4. Determine the validity of arguments by using common argument forms. Determine ermine whether the following argument arguments are valid using the given forms of valid arguments and fallacies. (a) If Elliot is a freshman, he takes English. Elliot is a freshman. herefore, Elliot takes English. (b) If you work hard, you will be a success. You are not a success. herefore, you do not work hard. (c) Jon is chea or financially broke. He is not financially broke. herefore, he is chea. (d) If Jose asks me out, I will not study riday. I didn t study riday. herefore, Jose asked me out. here are two big lessons to be learned in this section. irst, sometimes an argument can aear to be legitimate suerficially, but if you study it carefully, you may find out that it s not. Second, the validity of an argument is not about whether the conclusion is true or false it s about whether the conclusion follows logically from the remises. sob04590_ch0_ indd dai0079_ch0_00-0.indd 0/6/09 4:50:9 0/5/09 :46:7

43 CONIRMING PAGES art Chater Logic Answers to ry his One Valid 4 (a) Valid (b) Invalid (c) Invalid Invalid 5 (a) Valid (b) Valid (c) Valid (d) Valid (d) Invalid Invalid INRODUCION O Writing Exercises SRUCURAL KINESIOLOGY EXERCISE SE -4. Describe the structure of an argument.. Is it ossible for an argument to be valid, yet have a false conclusion? Exlain your answer.. Is it ossible for an argument to be invalid, yet have a true conclusion? Exlain your answer. c h ayou t eare r setting u a truth table to determine 4. When the validity of an argument, what connective is used between the remises of an argument? What connective is used between the remises and the conclusion? 5. Describe what the law of syllogism says, in your own words. c h a why t e r the fallacy of the inclusive or is a 6. Describe fallacy. oundations of Structural Kinesiology Basic Biomechanical actors and Concets 57 Exercises Comutational Neuromuscular 4 or Exercises 7 6, using truthundamentals tables, determine whether each argument is valid r r 7. Write the law of detachment in symbols; then rove that it is a valid argument by using a truth table. 8. Write the law of contraosition in symbols; then rove that it is a valid argument by using a truth table. 9. Write the law of syllogism in symbols; then rove that it is a valid argument by using a truth table. 0. Write the law of disjunctive syllogism in symbols; then rove that it is a valid argument by using a truth table.. Write the fallacy of the converse in symbols; then rove that it is not a valid argument by using a truth table.. Write the fallacy of the inclusive or in symbols; then rove that it is not a valid argument by using a truth table. or Exercises, determine whether the following arguments are valid, using the given forms of valid arguments and fallacies r r. 7.. r r sob04590_ch0_ indd dai0079_ch0_00-0.indd 0/6/09 4:50:9 0/5/09 :46:7

44 CONIRMING PAGES art Section -4 Logical Arguments Real-World Alications or Exercises 4, identify,, and r if necessary. hen translate each argument to symbols and use a truth table to determine whether the argument is valid or invalid.. If it rains, then I will watch the Real World marathon. It did not rain. I did not watch the Real World marathon. 4. ed will get a Big Mac or a Whoer with cheese. ed did not get a Whoer with cheese. ed got a Big Mac. 5. If Julia uses monster.com to send out her resume, she will get an interview. Julia got an interview. 4. If you back u your hard drive, then you are rotected. Either you are rotected or you are daring. If you are daring, you won t back u your hard drive. or Exercises 4 50, write the argument in symbols; then decide whether the argument is valid by using the common forms of valid arguments and fallacies. 4. If the cheese doesn t melt, the nachos are ruined. he nachos are ruined. he cheese didn t melt. INRODUCION O SRUCURAL KINESIOLOGY Julia used monster.com to send out her resume. 6. If it snows, I can go snowboarding. It did not snow. I cannot go snowboarding. 7. I will go to the arty if and only if my ex-boyfriend is not going. c h a My t e rex-boyfriend is not going to the arty. I will go toofthe arty. oundations Structural Kinesiology 8. If you are suerstitious, then do not walk under a c h a ladder. ter If you do not walk under a ladder, then you are Neuromuscular undamentals 4 suerstitious. You are suerstitious and you do not walk under a ladder. 9. Either I did not study or I assed the exam. I did not study. I failed the exam. 40. I will run the marathon if and only if I can run 0 miles by Christmas. I can run 0 miles by Christmas or I will not run the marathon. If I ran the marathon, then I was able to run 0 miles by Christmas. 4. If you eat seven ieces of izza, then you will get sick. If you get sick, then you will not be able to go to the game. 44. I studied or I failed the class. I did not fail the class. I studied. 45. If I go to the student symosium on environmental issues, I will fall aslee. If the seaker is interesting, I will not fall aslee. If I go to the student symosium on environmental issues, the seaker will not be interesting. Ift it c46. ha e ris sunny, I will wear SP 50 sun block. It is not sunny. Basic Biomechanical actors I will not wear SP 50 sun block. and Concets If I get an A in this class, I will do a dance. I did a dance. I got an A in this class. 48. I will not wear a Seedo at the beach or I will be embarrassed. I am not embarrassed. I did not wear a Seedo at the beach. 49. If I see the movie at an IMAX theater, I will get dizzy. I see the movie at an IMAX theater. I get dizzy. 50. I will backack through Euroe if I get at least a.5 grade oint average. I do not get at least a.5 grade oint average. I will not backack through Euroe. If you eat seven ieces of izza, then you will not be able to go to the game. Critical hinking 5. Oscar Wilde once said, ew arents nowadays ay any regard to what their children say to them. he old-fashioned resect for the young is fast dying out. his statement can be translated to an argument, as shown next. sob04590_ch0_ indd dai0079_ch0_00-0.indd If arents resected their children, then arents would listen to them. Parents do not listen to their children. Parents do not resect their children. 0/6/09 4:50:9 0/5/09 :46:7

45 CONIRMING PAGES 4 art Chater Logic Using a truth table, determine whether the argument is valid or invalid. 5. Winston Churchill once said, If you have an imortant oint to make, don t try to be subtle or clever. Use a ile driver. Hit the oint once. hen come back and hit it again. hen a third time a tremendous wack! his statement can be translated to an argument as shown. If you have an imortant oint to make, then you should not be subtle or clever. You are not being subtle or clever. You will make your oint. Using a truth table, determine whether the argument is valid or invalid. 5. Write an argument matching the law of syllogism that involves something about your school. hen exlain why the conclusion of your argument is valid. 54. Write an examle of the fallacy of the inverse that involves something about your school. hen exlain why the conclusion of your argument is invalid. 55. Look u the literal translation of the Latin term modus onens on the Internet, and exlain how that alies to the law of detachment. 56. Look u the literal translation of the Latin term modus tollens on the Internet, and exlain how that alies to the law of contraosition. INRODUCION O SRUCURAL KINESIOLOGY Section -5 Euler Circles Abraham Lincoln once said, You can fool some of the eole all of the time, and all of the eole some of the time, but you cannot fool all of the eole all of the time. clincoln hater was a really smart guy he understood the ower of logical argumentsand the fact that Biomechanical actors oundations of Structural Kinesiology Basic cleverly crafted hrases could be an effective and Concets 57 tool in the art of ersuasion. What s interest LEARNING RNING OBJECIVES ing about this uote from our ersective is Neuromuscular undamentals the liberal use of 4 the uantifiers some and all.. Define syllogism. In this section, we will study a articular tye. Use Euler circles to of argument that uses these uantifiers, along determine the validity with no or none. A techniue develoed by of an argument. Leonhard Euler way back in the 700s is a useful method for analyzing these arguments and testing their validity. Euler circles are diagrams similar to Venn diagrams. We will use them to study arguments using four tyes of statements. he statement tyes are listed in able -4, and the Euler circle that illustrates each is shown in igure - on the next age. Each statement can be reresented by a secific diagram. he universal affirmative All A is B means that every member of set A is also a member of set B. or examle, ABLE -4 ye General orm Examle Universal affirmative All A is B All chickens have wings. Universal negative No A is B No horses have wings. Particular affirmative Some A is B Some horses are black. Particular negative Some A is not B Some horses are not black. sob04590_ch0_ indd dai0079_ch0_00-0.indd4 0/6/09 4:50:9 0/5/09 :46:7

46 CONIRMING PAGES art Section -5 Euler Circles A B 5 B A (a) Universal affirmative All A is B A (b) Universal negative No A is B B A B INRODUCION O SRUCURAL KINESIOLOGY (c) Particular affirmative Some A is B (d) Particular negative Some A is not B igure - the statement All chickens have wings means that the set of all chickens is a subset of the set of animals that have wings. c h aa tise B r means that no member of set A is a member he universal negative No ses of set B. In other words, set A and set B are disjoint sets. or examle, No horses If I say that Some hor oundations of not Structuralhave Kinesiology Basic are black, you can wings meansthat the set of allbiomechanical horses and the set ofactors all animals with wings are ses om assume that S e hor and Concets 57 disjoint (nonintersecting). c h aare tnot e r black. If I say, he articular affirmative Some A is B means that there is at least one member Some horses are not of set A that is also a member of set B. or examle, the statement Some horses are e um ass Neuromuscular black, you cannot undamentals 4 black means that there is at least one horse that is a member of the set of black aniare ses that Some hor mals. he in igure -(c) means that there is at least one black horse. black. he articular negative Some A is not B means that there is at least one member of set A that is not a member of set B. or examle, the statement Some horses are not black means that there is at least one horse that does not belong to the set of black animals. he diagram for the articular negative is shown in igure -(d ). he is laced in circle A but not in circle B. he in this examle means that there exists at least one horse that is some color other than black. Many of the arguments we studied in Section -4 consisted of two remises and a conclusion. his tye of argument is called a syllogism. We will use Euler circles to test. Define syllogism. the validity of syllogisms involving the statement tyes in able -4. Here s a simle examle: Math Note Premise Premise All cats have four legs. Some cats are black. Conclusion herefore, some four-legged animals are black. Remember that we are not concerned with whether the conclusion is true or false, but only whether the conclusion logically follows from the remises. If yes, the argument is valid. If no, the argument is invalid. Euler Circle Method for esting the Validity of an Argument o determine whether an argument is valid, diagram both remises in the same figure. If the conclusion is shown in the figure, the argument is valid. sob04590_ch0_ indd dai0079_ch0_00-0.indd5 0/6/09 4:50:0 0/5/09 :46:7

47 CONIRMING PAGES 6 art Chater Logic Many times the remises can be diagrammed in several ways. If there is even one way in which the diagram contradicts the conclusion, the argument is invalid since the conclusion does not necessarily follow from the remises. Examles,, and show how to determine the validity of an argument by using Euler circles. Using Euler Circles to Determine the Validity of an Argument EXAMPLE Use Euler circles to determine whether the argument is valid. All cats have four legs. Some cats are black. INRODUCION O SOLUION SRUCURAL KINESIOLOGY herefore, some four-legged animals are black. he first remise, All cats have four legs, is the universal affirmative; the set of cats diagrammed as a subset of four-legged animals is shown. our-legged animals Cats oundations of Structural Kinesiology Neuromuscular Basic Biomechanical actors and Concets 57 he second remise, Some cats are black, is the articular affirmative and is shown by lacing an in the intersection of the cats circle and the black animals circle. he undamentals 4 diagram for this remise is drawn on the diagram of the first remise and can be done in two ways, as shown. our-legged animals Cats our-legged animals Black animals Cats Black animals he conclusion is that some four-legged animals are black, so the diagram for the conclusion must have an in the four-legged animals circle and in the black animals circle. Notice that both of the diagrams corresonding to the remises do have an in both circles, so the conclusion matches the remises and the argument is valid. Since there is no other way to diagram the remises, the conclusion is shown to be true without a doubt. ry his One Use Euler circles to determine whe whether the argument is valid. All college students buy textbooks. Some book dealers buy textbooks. herefore, some college students are book dealers. sob04590_ch0_ indd dai0079_ch0_00-0.indd6 0/6/09 4:50: 0/5/09 :46:7

48 CONIRMING PAGES art EXAMPLE Section -5 Euler Circles 7 It isn t necessary to use actual subjects such as cats, four-legged animals, etc. in syllogisms. Arguments can use letters to reresent the various sets, as shown in Examle. Using Euler Circles to Determine the Validity of an Argument Use Euler circles to determine whether the argument is valid or invalid. Some A is not B. All C is B. Some A is C. SOLUION INRODUCION O SRUCURAL KINESIOLOGY he first remise, Some A is not B, is diagrammed as shown. A B he second remise, All C cishb, by lacing circle C inside circle B. a tise diagrammed r his can be done in several ways, as shown. oundations of Structural Kinesiology A Neuromuscular undamentals 4 Basic Biomechanical actors and BConcets 57A B C A C B C he third diagram shows that the argument is invalid. It matches both remises, but there are no members of A that are also in C, so it contradicts the conclusion Some A is C. ry his One Use Euler circles to determine whe whether the argument is valid. Some A is B. Some A is not C. Some B is not C. Let s try one more secific examle. sob04590_ch0_ indd dai0079_ch0_00-0.indd7 0/6/09 4:50: 0/5/09 :46:7

49 CONIRMING PAGES 8 art Chater Logic Using Euler Circles to Determine the Validity of an Argument EXAMPLE Use Euler circles to determine whether the argument is valid. No criminal is admirable. Some athletes are not criminals. Some admirable eole are athletes. SOLUION Diagram the first remise, No criminal is admirable. INRODUCION O SRUCURAL KINESIOLOGY Criminals Admirable Peole We can add the second remise, Some athletes are not criminals, in at least two different ways: Criminals Admirablec h a t e r Criminals Peole Athletes oundations of Structural Kinesiology Admirable Peole Basic Biomechanical actors and Concets 57 Athletes Neuromuscular undamentals 4 In the first diagram, the conclusion aears to be valid: some athletes are admirable. But the second diagram doesn t suort that conclusion, so the argument is invalid. ry his One. Use Euler circles to determine the validity of an argument. Use Euler circles to determine whe whether the argument is valid. All dogs bark. No animals that bark are cats. No dogs are cats. We have now seen that for syllogisms that involve the uantifiers all, some, or none, Euler circles are an efficient way to determine the validity of the argument. We diagram both remises on the same figure, and if all ossible diagrams dislay the conclusion, then the conclusion must be valid. Answers to ry his One Invalid Invalid Valid sob04590_ch0_ indd dai0079_ch0_00-0.indd8 0/6/09 4:50: 0/5/09 :46:7

50 CONIRMING PAGES art Section -5 Euler Circles 9 EXERCISE SE -5 Writing Exercises. Name and give an examle of each of the four tyes of statements that can be diagrammed with Euler circles.. Exlain how to decide whether an argument is valid or invalid after drawing Euler circles.. What is a syllogism? 4. How do Euler circles differ from Venn diagrams? Comutational Exercises or Exercises 5 4, draw an Euler circle diagram for each statement. 8. All S is. No S is R. Some is R. INRODUCION O SRUCURAL KINESIOLOGY 5. All comuters are calculators. 6. No unicorns are real. 7. Some eole do not go to college. 8. Some CD burners are DVD burners. 9. No math courses are easy. 0. Some fad diets do not result in weight loss.. Some laws in the United States are laws in Mexico.. All members of Mensa are smart.. No cheeseburgers are low in fat. 4. Some oliticians are crooks. or Exercises 5 4, determine whether each argument is oundations valid or invalid. of Structural Kinesiology 5. All X is Y. Some Y is Z. Neuromuscular Some X is Z. 9. No M is N. No N is O. Some M is not O. 0. Some U is V. Some U is not W. No W is U. Some c. ha t e r A is not B. No A is C. Basic Biomechanical Some A is not C. actors and Concets 57. All P is Q. undamentals 4 6. Some A is not B. No B is C. Some A is not C. 7. Some P is Q. No Q is R. Some P is not R. All Q is R. All P is R.. No S is. No is R. No S is R. 4. Some M is N. Some N is O. Some M is O. Real-World Alications or Exercises 5 8, use Euler circles to determine if the argument is valid. 5. All hones are communication devices. Some communication devices are inexensive. Some hones are inexensive. 6. Some students are overachievers. No overachiever is lazy. Some students are not lazy. 7. Some animated movies are violent. No kids movies are violent. No kids movies are animated. 8. Some rotesters are angry. Some rotesters are not civil. Some civil eole are not angry. 9. Some math tutors are atient. No atient eole are demeaning. Some math tutors are not demeaning. 0. Some students are hard-working. Some hard-working eole are not successful. Some students are not successful.. Some movie stars are fake. No movie star is talented. No fake eole are talented. sob04590_ch0_ indd dai0079_ch0_00-0.indd9 0/6/09 4:50: 0/5/09 :46:7

51 CONIRMING PAGES 40 art Chater Logic. Some CEOs are women. Some women are tech-savvy. Some CEOs are not tech-savvy.. Some juices have antioxidants. Some fruits have antioxidants. No juices are fruits. 4. Some funny eole are sad. No serious eole are sad. No funny eole are serious. 5. Some women have highlighted hair. All women watch soa oeras. All eole who watch soa oeras are emotional. Some emotional eole watch soa oeras. 6. All students text message during class. Some students in class take notes. All students who take notes in class ass the test. Some students ass the test. 7. Some birds can talk. Some animals that can talk can also moo. All cows can moo. Some cows can talk. 8. All cars use gasoline. All things that use gasoline emit carbon dioxide. Some cars have four doors. Some things with four doors emit carbon dioxide. INRODUCION O SRUCURAL KINESIOLOGY Critical hinking or Exercises 9 4, write a conclusion so that the argument is valid. Use Euler circles. 9. All A is B. AllcBhis a C.t e r oundations of Structural Kinesiology 40. No M is P. AllcShisa M. ter 4. All calculators can add. No adding machines can make breakfast. 4. Some c heole a t e rare rejudiced. All eole have brains. Basic Biomechanical actors and Concets 57 Neuromuscular undamentals 4 sob04590_ch0_ indd dai0079_ch0_00-0.indd40 0/6/09 4:50: 0/5/09 :46:7

52 CONIRMING PAGES art Section - C H A P E R Summary Imortant erms Imortant Ideas Statement Simle statement Comound statement Connective Conjunction Disjunction Conditional Biconditional Negation ormal symbolic logic uses statements. A statement is a sentence that can be determined to be true or false but not both. A simle statement contains only one idea. A comound statement is formed by joining two or more simle statements with connectives. he four basic connectives are the conjunction (which uses the word and and the symbol ), the disjunction (which uses the word or and the symbol ), the conditional (which uses the words if... then and the symbol ), and the biconditional (which uses the words if and only if and the symbol ). he symbol for negation is. Statements are usually written using logical symbols and letters of the alhabet to reresent simle statements. ruth table A truth table can be used to determine when a comound statement is true or false. A truth table can be constructed for any logical statement. INRODUCION O SRUCURAL KINESIOLOGY - - oundations A statement that is always true is called a tautology. A statement that is autology always false is called a self-contradiction. wo statements that have the Self-contradiction same truth values are said to be logically euivalent. De Morgan s laws Logically euivalent c hnegation a t e r of a conjunction or disjunction. rom the are used to find the statements conditional statement, three other statements can be made: the converse, Converse of Structural Kinesiology Basic Biomechanical actors the inverse, and the contraositive. Inverse and Concets 57 Contraositive De Morgan s laws Neuromuscular undamentals MAH IN Argument Premise Conclusion ruth tables can be used to determine the validity of an argument. An argument consists of two or more statements called remises and a statement called the conclusion. An argument is valid if when the remises are true, the conclusion is true. Otherwise, the argument is invalid. Syllogism Euler circles Universal affirmative Universal negative Particular affirmative Particular negative A mathematician named Leonhard Euler develoed a method using circles to determine the validity of an argument that is articularly effective for syllogisms involving uantifiers. his method uses four tyes of statements: () the universal affirmative, () the universal negative, () the articular affirmative, and (4) the articular negative. Euler circles are similar to Venn diagrams. he Art of Persuasion REVISIED All the arguments on he first is invalid because there are sources of smoke the list are logically invalid other than fire. he second is invalid because it s ossible to excet the last one. Even be hay for some reason other than having lots of money. he though the claim made by the third is invalid because the initial statement says nothing about last argument is false it did not snow in Hawaii in 008 teams not in the SEC. he fourth is invalid because it is circu- this is so because the first remise is false. Remember, the lar reasoning (see Sidelight in Section -4). he fifth is invalid validity of an argument is based on whether it follows from because the first statement doesn t say that weaons of mass the remises, not on its actual truth. destruction are the only reason to go to war. 4 sob04590_ch0_ indd dai0079_ch0_00-0.indd4 0/6/09 4:50: 0/5/09 :46:7