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Logic, Argument and Proof or Anything is possible if you don’t know what your talking about. He who will not reason is a bigot, he who cannot is a fool, and he who dares not is a slave. - Sir William Drummond, Academical Questions Two common types of reasoning. Trials I remember my favorite all time Far Side cartoon from Gary Larson. A scientist, replete with beard, long white lab coat, is standing on a tall, rickety ladder holding a cat, who has a rather sick, disgruntled look on its face. Another scientist, long beard, white coat, is standing by a chalkboard, and a vertical white line had been drawn down the middle of it. On the left side of the line, the words on its feet are underlined, and there are about 50 little tick marks sketched beneath the heading. On the right side of the line on its head are underlined, with no marks beneath these words. The caption reads “Clear for trial 51. Ready, set … ” Patterns Remember the child’s rhyme 2, 4, 6, 8, who do we appreciate? Why didn’t anyone ever stop and asked us what the next number in the pattern would be? A scientist freezes water 50 times in a row at 0 Celsius, then concludes water freezes at 0 Celsius. A forensic pathologist follows a pattern that temperature is lost from a dead body at a certain rate, from this pattern, the pathologist takes the temperature of the body, then predicts the time of death. This type of reasoning is called inductive reasoning because we base our conclusion on patterns we observe, going from repeated trials to form a conclusion. It is the type of reasoning you saw when you were in science class and you drew conclusions from experiments that you may have repeated over and over again. The words we use is that we are going from the specific to the general – from the observed pattern to then form the conclusion. Danger, Wil Robsinson, Danger – Every time you sit on the new couch, your allergies flare up. This observed pattern results in your decision to return the couch. You purchase a new couch. Once delivered, you sit down on it and your allergies flare up again. It takes you months to realize that your allergies are always worse at night, when you are routinely sitting on the couch watching TV. The point is that inductive reasoning it not enough. The common sense that we aptly call logic must come into play somewhere. And so, with inductive reasoning, while we base our conclusion on observed patterns, there is no guarantee the conclusion we draw is correct. A second form of reasoning reverses the previous thought process, going from the general to a specific, is called deductive reasoning. Here, we rely on general truths to imply a conclusion. For example, if I was to tell you no one who wins the lottery is unlucky, and then tell you my uncle once won the lottery, then you could conclude that my uncle is not unlucky. But, again, be careful deducing false conclusions. If the logic is sound, a false conclusion may still arrive if an assumption is false or a word has duplicitous meanings. All boys love comic books, you have a son, therefore your son must love comic books. I have two sons, one loves comic books, one doesn’t. So, where did the logic go wrong? It didn’t, because we assumed the assumptions were correct. The problem is one assumption was not correct. Not all boys love comics books. God is love, love is blind, Ray Charles is blind, therefore Ray Charles is God. The word blind here has dual meanings, thus as much as I revere Ray Charles, I do not subscribe to the notion that he is god. Legal contracts use sound logic all the time to bind us, often blind us. Politicians are known to use unsound logic to mislead us. Advertisers advertise the benefits of their product, leading us to infer that we need the product. This is pure delusion on our part. We need to develop a discriminating eye. What do I mean? Well, for example, just because a drug is known to decrease short term memory loss in Alztimer’s patients does not mean that not taking the drug will increase the rate of memory loss either. Yet, clever advertisements may have one inferring just that, especially if the person who is listening may be elderly, already afraid and thus prone to jump to conclusions. People who love sausage and the law should not watch either being made. Case in point. Pat Robertson, a one time presidential candidate, once said that he couldn’t prove that there weren’t Soviet missiles in Cuba, therefore, there might be. Years later, both John Kerry and George W. Bush said there was evidence of weapons of mass destruction in Iraq. But, was this evidence clear or did the government decide to enter into a war with Iraq with the logic that since we could not prove that Saddam did not own weapons of mass destruction, then the weapons might exist? Though correct, this sloppy logic is misleading. It is the same tactic defense attorney’s employ when confronting a jury. “And despite the fact that the defendant was known to give in to jealous rages, he had a visible cut the day after the murder, his blood was found at the scene, the victim’s blood was in the his car, the defendant’s glove was found behind the victim’s home, the Bruno Maglia foot print from the rarest of shoes matched the defendant’s shoes and was found at the scene, you can’t eliminate the possibility that someone else actually committed the murder, can you?” Come on. It will be our goal will be to avoid sloppy logic, be it inductive or deductive. Let’s now explore these two types of reasoning: Inductive reasoning is analyzing specific cases, observing a given pattern, and then forming a conclusion. We begin with number patterns because in determining these patterns you are training your mind to see the critical repetitions. Once this window is open, the sky is the limit and we can then migrate to other types of patterns. For a pattern to be established numerically, at least three numbers are needed, because for the pattern to be observed, you must see the pattern you found repeated at least once. Observing numerical patterns is not a trick of the mind, and it is not a case that either you see it or you don’t. You can usually see it, but just like anything else, you must know what you are looking for and how to articulate it. This means you need to find a way to represent the pattern mathematically. One type of representation is with an algebraic formula. Let’s begin with a gentle conversation. I ask you to find the next number in a sequence, and I give you 2, 5, 8, 11, … , most of you would say the next number is ‘14’. And if I gave you 302, 305, 308, 311, … and requested the next number, again, most of you would say ‘314’. If I ask you how do you know, you would probably tell me that we are adding 3 to each number in the sequence. But, if I give you 302, 305, 308, 311 and asked you for the 101st term down the line, most students eyes would narrow and say something like, “First off, this exercise is pointless and secondly, I can tell you the answer, but it is a needlessly tedious and would take me nearly an hour.” (I know you would, I have had this conversation a thousand times, literally.) And I would gently respond, “Why would it take you nearly an hour?” And you’d answer (is ‘you’d’ a word?) “because I would have to add three 101 times to 302.” And I would say, “so the pattern starts with 302 and it looks like you are telling me you will use repeated addition of 3 for 101 times. So, this could be done quick, right, because repeated addition is the same as what?” And you would in turn say “Multiplication. Yes. I see,” And you’d write 302 3 3 ... 3 302 3(101) 605 . 101 times So, now, if I asked you for the 466th or even the 1,196th term and since you could articulate the pattern, then you could give me the answer if a matter of moments. 302 3 3 ... 3 302 3(466) 1700 466 times 302 3 3 ... 3 302 3(1,196) 3890 1,196 times So, the pattern is the first term, b, and then we would add the common difference, m, x times, where x is the term we are searching for. Doesn’t this look a little familiar, 302 3 3 ... 3 302 3(101) 605 101 times 605 302 3(101) an a0 d n starting number common difference term we are searching for or an a0 dn or y mx b This type of sequence is called arithmetic because we are adding the same number, d, to each term to find the next term done the line. This repeated addition is a nothing more than a linear relationship. We are dealing with integers {1, 2, 3, … } as oppose to real numbers. To show a distinction, we tend to use n instead of x. The common difference is the constant rate of change, d replaces m. When I first saw how to handle a pattern like this, I felt like Dorothy in the Wizard of Oz, where she opens the door of the black-n-white tornado toppled house and see her first glimpse of color. I looked at patterns and for that matter, life differently. Why? Because I was able to find patterns everywhere. Let’s use this pattern and colorize your world. Suppose you purchase a Condo and wanted to advertise it on Rentals Unlimited’s Internet sight. Their brochure advertise that initially there is always a large number of hits on your site the first week, with the same ol’ clients seeing a new condo and glancing at it. After the first week, the newness of your add will wear off its welcome, but still Rentals Unlimited advertises that each day another 3 people will access the sight. And, they tell you, you can reasonably expect 13 %, roughly 1/8, of the people who access the sight to rent your condo (which is gorgeous because it is situated near a travel resort.) It is Memorial Day. You purchased the Internet Site and the first week is over. You site tracker tells you 302 people have accessed your sight. You want to know roughly how many hits there will be on your site by the end of the summer so that you may estimate the numbers of rentals you can expect. There are 101 days between Memorial Day and Labor Day. So, according to our calculations, there will be 605 hits by Labor Day and so 0.13(605) = 78.65 or conservatively, 78 rentals for your condo. Now, this rental business is all new to you, so you ask your self, over the next year through Labor Day, how many rentals can I expect? So, 365+101 = 466. And the 466th term is 1700 and 13 percent of 1700 is 221. So, over the next year through Labor Day, you may expect 221 rentals to come from this site. How about over the next three years? 365 + 365 + 365 + 101 = 1196 the 1196th term is 3890. So, over the next three years, you may expect 13 percent of 3890, which is 505.7 or about 505 rentals in the next three years, through Labor Day. Now, you may plan long term. Let’s examine a second type of pattern. This pattern is not based on repeated addition, but rather repeated multiplication. It is often as obvious as the repeated addition, though. Let’s have another gentle conversation. 2, 4, 8, 16, quick, what’s the next number? 32, right? Successive doubling. 2, 6, 18, 54, quick, what’s the next number? 54(3) = 72. Successive tripling. 16, 8, 4, 2, quick what’s the next number? Successive halving, half of 2 is 1. But, again, the question looms, what is the 10th term down the line? Well, since we are hooking into patterns, let’s return to the last pattern we saw and examine the difference between what we did before and what we need to do now. In the first sequence, we are multiplying each term by 2, in the next sequence we were multiplying each term by 3, and in the last sequence, ½ . 2, 6, 18, 54, has a common multiplier, r, of 3. We need to mimic the pattern formation we did earlier. We find the first term, the common multiplier instead of common difference and multiply it instead of adding it. Arithmetic : 302 3 3 ... 3 302 3(101) 605 101 times b mx y Geometric : 2 (3)(3)(3)...(3) 2(3)10 2(59049) 118098 10 times an a0 n term we are searching for r 1st term common ration an a0 r n If we knew the knew we had a geometric sequence, but we did not know the common multiplier, how could we find it? Suppose we had inhospitable numbers: 6.912, 8.2944, 9.95328 …. We can check pretty quickly to realize that we are not adding the same number to each term, but are we multiplying the same number to each term? How do we check? Let’s return to an easier sequence that is geometric, like 2, 6, 18, 54, … Here we are multiplying each number by three. Patterns. Where is the three coming from? 6/2 = 3. 18/6 = 3. 54/18 = 3. Divide a number by the prior number and if the result is the same if you were to divide the successive number by the preceding number, you have the common multiplier. Let’s try our numbers. 8.2944/6.912 = 1.2 9.95328/8.2944 = 1.2 So, the common multiplier is 1.2 The 10th term down the line? a10 a0 r n 6.912(1.2)10 42.79728215 Where would you see such a pattern in life? Suppose a minor league sports franchise is currently worth 6.912 million dollars. Because of renewed interest in the league, we can expect the team’s value to grow by 20 % growth annually. So, we are just taking 6.912 and multiplying it to 1.2 ten times. So, in ten years, you may expect the team’s worth to be an estimated 42.79728215 million dollars or $ 42,797,282.15 as compared to its current $ 6,912,000 worth. There are two more patterns that pop up frequently. The first is as predictable as a weather forecast on a summer day in Arizona. The signs of the terms oscillate between positive and negative. So, the common multipliers that act to oscillate the signs back and forth are: (1)n 1, 1,1,... if n starts at 0. (1)n 1 1,1, 1,... if n starts at 0. We use these multipliers as coefficients multiplied to the other terms in the pattern: 5,3, 2,... 1*5,1*3, 1* 2,... So, 5,3, 2,... (1)n 1 (5 2n) The 10th term? (1) n 1 (5 2n) (1)101 (5 2(10)) (1)( 15) 15 The second is where the factions are governed by different patterns in the numerator and the denominator. Thus, we have a similar separation in mind for the terms, here separating the pattern from the numerator and the denominator. 16 14 12 16 2n , , , ... 5 10 15 5(3) n 16 2n 16 2 *10 4 The 10th term? n 10 5(3) 5(3) 295, 245 First test for the arithmetic sequence. Usually a student will first subtract the absolute value of consecutive numbers and then subtract the next two absolute values of consecutive numbers and if the common difference is the same, they will use the formula for the arithmetic sequence. Then test for a geometric sequence. If not, they divide the absolute value of consecutive numbers and then divide the next two absolute values of consecutive numbers. If the two quotients are the same, they pursue the geometric sequence formula. Then, if the signs oscillate back and forth, multiply the formula by (1) n or (1) n 1 . If none of these tools help you, you are then challenged to find another pattern. Problem Set What’s the next number? What is the 10th term? Explain what lead you to the conclusion. 1. 2, 5, 8, … 2. 30,120, 480,... 3. 20.0, 5.0, 1.25... 2 5 8 4. 10, 20, 30, 40, ... 5. 15, 150,1500... 6. , , , ... 7 7 7 2 5 8 2 5 8 7. , , , ... 8. , , , ... 9. 2, 42, 82, … 3 7 11 3 7 11 10. 2, 125, 248, … 11. 2, 10, 50, … 12. 2, - 1, 0.5, - 0.25… 13. 30.25, 166.375, 915.0625, ... 14. 1, 4, 9, 16, … 15. 3, 6, 11, 18, … 15. 0.008, 0.0016, 0.00032... 190 210 230 16. , , , ... 205 200 195 Deductive reasoning is analyzing the general truths and then drawing a specific conclusion. A common form of this reasoning is seen through syllogisms. Syllogisms are comprised of two parts, known truths called premises, followed by the conclusion drawn from these known truths. No one shorter than 5’ 4” can dunk. first premise All kindergartners are under 5’ 4” second premise Therefore, we deduce no kindergartner can dunk. conclusion We need to ask ourselves does the conclusion necessarily follow from the premises (known truths?) To answer this, we reword the question and ask our selves, “is the argument valid?”, which means “is the conclusion guaranteed from the premises?” And when we are dealing with general populations, one way we answer the question of the validity of an argument is to sketch out the populations as circles, inside a box that represents the rest of the universe, and we mark with an x where we want the concluding population to result. So, below, we draw a circle to show the population for the dunkers, labeling with a D, and draw another comparative circle to show where those under 5’ 4” would be in relation to the dunkers, and then we place an x to label where the kindergarteners would be located with respect to these populations. 5’4” x D Conclusion: the only place to draw the x is outside the dunkers circle, so no kindergartener can dunk and thus the argument is said to be valid. When generalizing about populations, we use words such as all, none or some. Standard Venn Diagrams are shown below. A A A B B B All A are B No A are B Some A are B TV x SC Some reality TV shows are scripted. Survivor is a reality TV show Therefore, Survivor is scripted. Answer: not valid __________________________________________________________ Sx F All fish swim. I swim. Therefore, I’m a fish. Answer: not valid __________________________________________________ F B x All birds fly. A penguin is a bird. Therefore, penguins fly. Answer: valid The problem here is that penguins don’t fly, their bones are to heavy, flippers too thin. Given the premises, the reasoning is logical, but the conclusion doesn’t make sense because one of the premises is false. This is an example of a valid arguments, where the conclusion follows from the premises, yet the conclusion is false because one of the premises is not true. Problem Sets For Problems 1-15, determine if the argument valid or not valid. Justify with an Euler Diagram. 1. All left handed batters have a hitter’s advantage. Barry Bonds is left handed. Therefore, Barry Bonds has an advantage hitting. 2. All MTV videos are under twelve minutes. The Lost Seagulls video is not under twelve minutes. Therefore, the Lost Seagulls videos does not appear on MTV. 3. Some students won’t get Pell grants. I am a student. Therefore, I won’t get a Pell grant. 4. All people smile. Coco, the gorilla who talks through sign language, is not a person. Therefore, Coco does not smile. 5. All cameras are expensive. Some cell phones are expensive. Therefore, some cell phones are cameras. 6. All cameras are expensive. Some cell phones are cameras. Therefore, some cell phones are expensive. 7. All beagles are dogs. Some 4 legged creatures are dogs. Therefore, some 4 legged creatures are beagles. 8. All beagles are dogs. Some 4 legged creatures are beagles. Therefore, some 4 legged creatures are dogs. 9. Spoiled milk doesn’t taste good. Some cheese is made from spoiled milk. Therefore, some cheese doesn’t taste good. 10. Spoiled milk doesn’t taste good. Some cheese is made from spoiled milk. Therefore, cheese doesn’t taste good. 11. All people smile. Gorillas are not people. Therefore, some gorillas smile. 12. All people smile. Some gorillas smile. Therefore, some gorillas are people. 13. Any animal that smiles is a person. A gorilla can smile. Therefore, a gorilla is a person. 14. Some gorillas smile. Some people smile. Therefore, some gorillas are people. 15. Some gorillas smile. All gorillas are not people. Therefore, all people don’t smile. For Problems 16-21. Writing exercise. Let’s reverse the thought process. We will give you the Venn Diagram, you give us the argument and tell us whether the argument is valid or invalid. Create you own and use your own creativity to do so. 16. 17. 18. x x x x x x 19. 20. 21. Symbolic Logic Everybody lies, but it doesn’t matter since nobody listens. Suppose I am in a contrary mood and tell you “all athletes are pampered.” Well, you take offense, maybe you are an athlete or have a friend, spouse or child that is one. So, what do you do? You say “that is not true?” Here is the key question, what did you just say when you said “that is not true?” What we are asking here is a common question in logic, we are trying to get at the core question, “what does it mean to negate a proposition?” So, first, we ask, what is a proposition? A statement or proposition is a senetence that is either true or false. In mathemtics, we call this a declarative sentences, because it decalres either truth or falsehood. In other words, it has a truth value, so we may in turn assign it a true or false. So, again, what’s a proposition? Statements like “Hillary Rodham Clinton is a senator.” This is true, at least at the time I am typing this. “I am a wallaby.” This is false, hopefully no discussion is needed here. “At Best Buy, all computer games are on sale this week” which is either true or false, depending on the week the statement is made. So, what’s not a proposition? Non-statements have no truth value, they are neither true or false. Can you think of sentences that are neither trueo ro false? “Hillary Rodham Clinton would make a great president.” Opinion. Opinions are never true or false, dispite how much you may wish to believe otherwise. “Wallabies are hideous.” Again, an opinion. “Is Best Buy having a sale on computer games this week?” A question searching for an answer, thus neither true or false. Consider the sentence “I am telling a lie.”. Is this a statement? If true, the speaker would be telling a lie, and so the sentence would not be true. But this is in the ture case. If false, the speaker would be telling the truth, but this is in the false case. This is called a paradox, or a self-contradiction. It is not a statement. Quantifiers Let’s return to someone having the audacity to say “all athletes are pampered.” If you reply “that is not true,” what are you really saying? No athlete is pampered because they must endure double duty, a rigorous training schedule as well as full time athletics. Or are you saying that while some atheltes are pampered, others are not. Well, let’s think this through. Usually, if you let the person finish the sentence, the polite thing to do in this society, the objector will say something like”that is not true. I am an ahtlete and I am not pampered.” And this is correct. The negation, saying something is not true, to that statement simply means that is it false that all athletes are pampered, so there must exist at least one athlete who is not pampered to render such a reply. Words such as all and some are called quantifiers. And in context, all means everyone, and some means there exists at least one. So, to negate these type of sentences, you ask your self what does it mean to be false. Some penguins fly. This is not true. Negation: This means there is not at least one penguin that flies. So, the proper negation: All penguins do not fly. Or more eloquently, No penguins fly. And this is technically correct, right, because in fact penguins can not fly. They wobble. No flight though. To negate a sentence invloving quantifiers, All p are q. Negation: Some p are not q. Some p are q. Negation: No p are q. Sometimes, we must take literary advantage to make the sentence flow. If we say. “numbers don’t lie (all p are not q),” we mean all numbers do not lie, and this data, we then conclude, must then reflect only what is correct numerically. But, if you reply, “well, sometimes the numbers can show bias,” this means you want to negate the sentence. So, there must exist a number that does lie or show bias, so “some numbers do lie (some p are not not q)” is a reasonable interpretation of the negation. Notice that we think of the statements in their positive form (q represents lie, not q represents don’t lie.). Problem Set: For the examples below, ask yourself what does it mean if you say the following sentence is not true. We have collected commonly heard expressions said by politicians over the years. This information is often misinformation. So, if your reaction is that this is not true, what are you really saying. For problems 1 – 13: Negate the following: 1. All drug companies spend more on advertising than research. 2. No child will be left behind. 3. Some of our troops did not obey the cease fire. 4. None of the previous rules apply. 5. This generation of youngsters don’t apply themselves. 6. All surveys are biased. 7. No president has ever written a good tell all book. 8. Some movie reviewers just do not get it. 9. Some politicians are not honest about their tax declarations. 10. All music videos are detrimental to our youth. 11. Some ball players use steroids. 12. All steroid use hurts the game of baseball. 13. No documentaries are fair and unbiased. Quantifiers Symbolic representations of quantifiers. We mathematicians are lazy. We consider it cumbersome to write the same three letters, all, to represent every one and just as cumbersome to write the same four letters, some, to mean at least one. So, we use the symbol ,called the universal quantifier, to mean all and the symbol , called the existential quantifier, to mean some (taken to at least one.) So, do we really use this notation because we are lazy? Abraham Lincoln once said of slavery “You can fool some of the people all of the time, and all of the people some of the time, but you can not fool all of the people all of the time.” He was discussing the deceit inherent with the institution of slavery. This form of predicate logic is the combination of propositional logic (“and”, “or”, “if … then … “, implications) with quantifiers (all, some) and it is used in every court of law, in every eloquent debate and in many political speeches. It is ingrained in the fabric of our society and is used to make sweeping generalizations, some true, some designed for the most noble of reasons (Lincoln) and some for the most nefarious of reasons. Debates or speeches, designed fluently enough, can turn a person’s mind around, making one believe the most implausible of things. To see the writing on the wall, we need to be able to read the writing on the wall. Let’s learn to read, mathematically. How does it work? Let’s look at a relatively benign example, where no one’s feelings can get hurt. Suppose you and I are arguing about one of us being too needy. We are lovers arguing, friends that have become too close, a parent talking to a child. Pick your poison. Let’s begin with that sentence “I need you.” The conversation quickly degenerates. “You need everyone.” “You need no one.” The conversation gets quickly out of hand and we get philosophical. “We all need someone.” “Everyone needs everyone.” “Everyone needs someone.” “Some people need everyone.” “We are always needed by someone.” “None of us needs to be needed all of the time.” Ready to throw your hands into the air? Aaahhh. Ok, let’s quantify this argument. We will use predicate logic to break down what we are saying in these statements. N(x,y) means x needs y. So, x is the one who needs, and y is the one who is needed. So, let’s see. What does xy, N ( x, y) mean? Loosely speaking, everyone needs everyone. Is this true? Does everyone in the world need everybody else in the world? Daily, of course not. But, in a theoretical way? Maybe for the world to survive peacefully, you could rationalize another answer. Let’s try a few. We will write the symbolic form of the phrase and try and think through whether or not the phrase it true. xy, N ( x, y) Everyone need someone. True? I think so. No man is an island. xy, N ( x, y) There is someone who needs everyone. I have met many people like this. yx, N ( x, y) There is someone who everyone needs. I don’t know. A religious person may answer this a truism. xy, N ( x, y) There is a person who needs someone. Yup. Probably quite a few. yx, N ( x, y) There is someone who is needed by somebody. Yup. My mom. How about yx, N ( x, y) Everybody is needed by someone. I hope so. Such methodical symbolism teaches us not to be sloppy we our thoughts. Consider this rather distasteful bur real world example. “You have slept with every woman we know.” “This is not true,” which means only that there exists at least one woman you know who you did not sleep with. I do not know if this truly help the argument. Now, back to Lincoln’s words about slavery “You can fool some of the people all of the time, and all of the people some of the time, but you can not fool all of the people all of the time.” We are quantifying people, x, and time, y. F(x,y) means x people can be fooled y of the time. You can fool some of the people all of the time xy, F ( x, y) there exists a person who can be fooled all of the time. and all of the people some of the time, xy, F ( x, y) everyone can be fooled at least once but you can not fool all of the people all of the time. xy, F ( x, y) everyone can be fooled all of the time But,, we are saying this is not true. It would be nice if we had symbols for the words and and not . We will have these symbols and we will develop these next section. It is the basis for propositional logic. But, first, let’s try a problem set. Now, I know what you thinking, this is a math class, dag-nab-it. Where are the numbers. Answer? In this following problem set. Example One The integers are the positive and negative whole numbers, to include zero. In set notation, they are { … -3, -2, -1, 0, 1, 2, 3, … }. Write each of the following statements about the integers symbolically, then designate which are true and which are false and comment with either an example, a counter example or your reasoning. 1. There are numbers that differ by 0.0001 x, y x y 0.0001 True, many pairs of numbers would suffice, say 1.0001 and 1.00001 2. There is an integer such that it is equal to triple itself. x x 3x True for x = 0. No we go backwards. Write each of the following statements about the integers in succinct English, then designate which are true and which are false and comment with either an example, a counter example or your reasoning. 3. x, y x y 0 There exist two integers such that one is larger than the other. True. Many pairs would suffice, try 2 and 1. 4. x, y ( x y) There exists an integer such that it is larger than every integer. False. There is no largest integer. Problem Set For problems 1-3. the integers are the positive and negative whole numbers, to include zero. In set notation, they are { … -3, -2, -1, 0, 1, 2, 3, … }. Write each of the following statements about the integers symbolically, then designate which are true and which are false and comment with either an example, a counter example or your reasoning. 1. Every integer is a solution to x 2 5 x 4 0 . 2. There are integers that are solutions to x 2 5 x 4 0 . 3. Every integer is a solution to x 2 5 x 4 ( x 4)( x 1) . For problems 4-6: Now we go backwards. Write each of the following statements about the integers in succinct English, then designate which are true and which are false and comment with either an example, a counter example or your reasoning. 4. x, y ( x y) 5. x, y ( y 2 x) 6. x ( x 2 0) For problems 7 and 8, let’s look at some quotations that have changed the hearts and minds of mankind, advancing civilizations, progress and even the course of love. Rewrite with quantifiers, and state whether or not you agree. 7. All religions must be tolerated … for … every man must get to heaven his own way. – Frederick the Great 8. Nobody’s perfect. - Anonymous 9. Every man is an island un to himself. - Anonymous Connectives Propositions (statements) are connected with words to form compound sentences. Words such as not, and, or if … then … . Consider the statements: P: I am a female. Q: I am a teacher. Normally, it is easy to determine if a statement is true or false. Certainly, for me, P is false and Q is true. But, how about if we connect the statements with words or phrases? Again, our goal is to find the truth value of a sentence comprised of statements that are connected together some how. The way they are connected is with small words or phrases called connectives. P: I’m OK Q: It still bothers me. P Q means I am OK and it still bothers me. P Q means I am OK or it still bothers me. But, how about the following? What connective would you use to say: I am OK, but it still bothers me. I am OK, also it still bothers me. I am OK, it still bothers me. They are all compound statements with the word “and”. They all mean “I am OK and it still bothers me,” only you are emphasizing a different emotion with each sentence. So: I. Not P ~P Negation Other words none, never II. P and Q PQ Conjunction Other words, but, also, or a comma III. P or Q PQ Disjunction IV. If P then Q P Q Implication or Conditional Other words, all or no. Also, the following statements have the same meaning: “If P then Q”, “P implies Q”, “Q if P”, “P is sufficient for Q”, “Only if Q is P” Note: If we said “no penguins fly”, this is interpreted to mean “if it is a penguin, then it does not fly.” So, “no penguins fly” is the conditional statement P Q , where P is the statement “if it is a penguin” and Q is the statement “then it flies”. V. P if and only if Q P Q Equivalence or Biconditional Express in symbolic form. P: She was acquitted Q: The trial was fair She was acquitted and the trial was fair. P Q If she wasn’t acquitted, then the trial wasn’t fair. (~ P) (~ Q) It is not the case that she was acquitted or the trial was fair ~ ( P Q ) She wasn’t acquitted and the trial was fair. ~ P Q Express in symbolic form again, and we will take some liberty with our words. P: the man lies Q: the man cheats R: the man steals The man lies, cheats and steals. P Q R The man lies, cheats but doesn’t steal. P Q (~ R) All men who lie, cheat. P Q The man lies, but doesn’t cheat or steal. P ~ (Q R) Translate into symbols, define each letter that you use. Try to write P and Q in the affirmative. If you can hear, you can listen. P Q All penguins wobble when they walk. P Q All people don’t smile. P ~ Q No politicians are honest. P (~ Q) No documentaries are completely accurate and unbiased. P ~ (Q R) If I study and get enough sleep, I’ll get an A. ( P Q) R I won’t get an A if I don’t study or get enough sleep. ~ ( P Q) (~ R) The house will close if the money is wired or a gift is given. (Q R) P If you can smile and lie, you can cry and tell the truth. ( P Q) ( R S ) Problem Set: For problems 1-10, translate into symbols, define each letter that you use. Try to write P, Q , R or S in the affirmative. 1. All democrats won’t support the tax increase on the middle class. 2. All gay couples can be legally married in my state. 3. If weapons of mass destruction were not found, the war was not necessary. 4. No child will be left behind. 5. We will never trust the government if the country never resolves who killed JFK. 6. If Jack Rudy acted alone and died due to natural causes, I am a monkey’s uncle. 7. If Medicare is cut, my parents will suffer and your parents will suffer. 8. No suspects were ever arrested solely because they were a celebrity. 9. John Wilkes Booth was a phenomenal actor, incredibly passionate but a lousy patriot. 10. The CIA or the Mafia may have shot JFK or a lone gunman may have, but not the Cubans. Truth Tables As mentioned earlier, when you consider the statements: P: I am a female. Q: I am a teacher. Normally, it is easy to determine if a statement is true or false. For me, P Q states I am a female and I am a teacher, while P Q states I am a female or I am a teacher. Are the statements true or false? Let’s start with the first statement, “I am a female and I am a teacher. True or False? This is not my coming out party, people. I am not a female, but I am a teacher. P Q is false because for an ‘and’ statement to be true, both propositions must be true, while P Q is true because for an ‘or’ statement to be true, just one of the two statements need to be true. A truth table lists all possible combinations of true and false. P ~P T F F T When you have 2 statements to compare, there are four combinations of possibilities, right? Think back to the multiplication principle, there are 2x2 number of ways to order two quantities. They are: (T,T), (T,F), (F,T), and (F,F). P Q PQ T T T T F F F T F F F F P Q PQ T T T T F T F T T F F F Conditional So, suppose you walking down a dark street late at night, and suddenly, without warning, you are accosted by the police. They handcuffed you, and shackled, you hear them say, “Just look at you. This robbery was huge. You couldn’t have done this yourself.” You look back at the officer, stunned. He looks you up and down and says with a smirk, “If you robbed the store, you had an accomplice.” Accused! But it was the smirk was the last straw and although you know you should say nothing, you reply, “That’s not true.” What just happened? We will come back to this scenario in a moment. We use the symbols P Q to represent the conditional statement if P then Q. Realizing this may be in bad taste, we will use the old lover’s promise “if I win the lottery, then I will marry you.” Here, P is “I win the lottery” and Q is “I will marry you.” P is called the hypothesis (premise) and Q is called the conclusion. An implication or conditional statement needs to be thought of as a promise. Let’s analyze this conditional statement (or implication). The if P, then Q means if P occurs, then I promise Q will follow. So, if I win the lottery, I promise to marry you. Clearly, if I win the lottery and don’t marry you, the promise is broken, so the conditional statement was false. And if I do not win the lottery, I am under no obligation to marry you, so whether or not we get married doesn’t affect the promise I made. In other words, the conditional statement is true by default. In truth table terms, If P holds true, I promise Q will follow means if P is true and Q is true, the conditional statement is true. And just as clearly, if P is true, and Q is false, the promise has been broken, you won the lottery, but did not marry. So, the conditional statement is false. If P is false, you are under no obligation to keep the promise, so the obligation you had to marry me is not their anymore. The statement is true because it can not be false anymore, it is true by default. P Q PQ T T T T F F F T T F F T Now let’s return to the earlier scenario. Suppose you walking down the street, you’re accosted by the police. Handcuffed, they say “Just look at you. This robbery was huge. You couldn’t have done this yourself.” Nothing wrong yet. But, when he smirked and said “If you robbed the store, you had an accomplice,” your mistake was to reply. Why? Not because you spoke without an attorney present, but because of what you said. You were sloppy in your thought process. “That’s not true,” for a conditional statement means the hypothesis is true, and the conclusion is false. So, what just happened? You confessed, that’s what. Precisely and inexplicably, you said “I robbed the store, with no accomplice.” Clearly, that’s not what you meant to do. Other misconceptions in every day life? How about this: If aspirin cures a head ache, does not taking aspirin cause a head ache? Of course not. But, a common misconception is the direction of causation in a conditional statement. In other words, just because taking aspirin cures the headache, the negating the hypothesis, not taking the aspirin, will not necessarily negate the conclusion, and the negation of curing a headache is causing the headache. First, let’s write this as a compound statement. If you take an aspirin, then your head ache will be cured. Which of these statements have the same meaning? If you don’t take an aspirin, then your head ache won’t be cured. If your headache is cured, then you took an aspirin. If your head ached is not cured, then you did not take aspirin. Let’s rewrite the statement in symbolic form. Conditional: P Q If you take an aspirin, then your head ache will be cured. Converse of P Q is Q P If your headache is cured, then you took an aspirin. Inverse of P Q is (~ P) (~ Q) If you don’t take an aspirin, then your head ache won’t be cured. Contrapositive of P Q is (~ Q) (~ P) If your head ached is not cured, then you did not take aspirin. Rules for the Conditional 1. The conditional statement is equivalent to the contrapositive. 2. The converse is equivalent to the inverse. Biconditional A biconditional statement is a two-sided conditional statement, that is, it is a conditional statement whose cause and effect reads both directions. P Q is read “P if and only if Q” or equivalently “P is necessary and sufficient for Q”. This means “if P then Q” and “if Q then P”. In other words, P Q is equivalent to ( P Q) (Q P) . Let’s examine a biconditional statement. Suppose you were living in a society where if you went to war, then the act of going to war would cause a recession. Suppose your society was also such that every time the society went into a recession, the government in charge thought the way to get out of the recession was to go to war. Let P be “We will go to war” and Q be “a recession will occur”. The biconditional statement P Q is read “we will go to war if and only if a recession will occur.” This in turn means P implies Q and Q implies P. In other words, “ if we go to war, then a recession will occur” and “if a recession occurs, then we will go to war.” Each will cause the other. The two individual statements are necessary for each to occur. This means one is necessary for the other to occur and if one occurs, it is sufficient enough to cause the other to occur. Necessary and sufficient. Anecdotally, does it make sense that studying more frequently and getting better grades may be though of as another example of P Q . If you get study more frequently, this will cause your grades to go up. If your grades start to go up, you may choose to study more frequently to continue this trend. Also, if you studied less frequently, would your grades go down. Or if your grades go down, would you be apt to study less frequently because you would give up? The truth table for a biconditional should reflect this two-sided cause and effect we are discussing. In other words, a biconditional statement is true both P and Q are true (loosely you should think ‘they both occurred’) or if P and Q are not true (loosely you should think ‘they both did not occur’). If one is true and the other is false (loosely you should think ‘if P occurred, it did not cause Q to occur’), then the compound statement is false. Below is the truth table for P Q . P Q PQ QP ( P Q) (Q P) PQ T T T T T T T F F T F F F T T F F F F F T T T T Common Negations Many of the common negations we see in compound statements are familiar to us, we use them in our everyday language. Do we really know what we are saying? Of course, usually we do. How about the old fashion double negative. It’s false that I wasn’t dating her. This means I was dating her. I am not taking Tom or Jim to the movie premiere means I am not taking Tom and I am not taking Jim to the movie premiere. But how are you with some of the common misconceptions we find in our every day speech. If you said that it’s not true that I am taking Jim and Tom to the movie premiere, this means either you are not taking Jim or you are not taking Tom, or both. Is this what you meant? Or suppose your confronting your girlfriend of infidelity. You say something like, “You went out, and I didn’t go. That’s why I am jealous. You probably went out with your old boyfriend, Russ.” She rolls her eyes and says sarcastically, “Right,” and for the moment you are relieved. But when she adds, “If I went out, I was with him” , exhales and finishes with “but honey, you know that’s not true,” and rolls her eyes a second time. You storm out of the house. Where you right or wrong? Well, the answer depends upon why you were so jealous. If it was because she went out with Russ, you were wrong to storm out. But, if you were jealous because she went out without you, and you were just using Russ as an empty name filler, than heck yes, you were right to storm out. Because when she negated the conditional, she is reciting the second line of a truth table. She is telling you the hypothesis was true and the conclusion was false. It is no different than the gentleman we saw earlier who confessed to the crime without the help of an accomplice. She said she went out, but just not with Russ. So, if you are the jealous type, you then wonder just who she went out with. And so you storm out. We are examining some common negations. To show two statements are equivalent, they must have the same truth values of every possibility of true and false for each statement. This means they must have the same true tables. Check of the truth tables for each of the following: ~ ( ~ p) p . This is an exercise for you. Example 1. Construct a truth table for ~ ( P Q) and then for ~ P ~ Q . Solution. We construct the columns in order, working inside the parentheses first. Next, just as we read in English, we construct the truth tables from left to right. For ~ ( P Q) , we first find the column for conjunction connective ( P Q) then we negate that column. P Q PQ ~ ( P Q) T T T F T F F T F T F T F F F T In this next example, we do not have parentheses in our compound statement. So, first find the column for ~ P , then we find the column for ~ Q . We then join the two statements with the disjunction connective to find ~ P ~ Q . P Q ~P ~Q ~ P ~ Q T T T F F F F T F F T F T T F T F T T T F F F F T T T T Note, ~ ( P Q) ~ P ~ Q . This is referred to as one of DeMorgan’s Laws. Example 2. Construct a truth table for ~ ( P Q) and ~ P ~ Q to show ~ ( P Q) ~ P ~ Q . Then construct truth tables for ~ ( P Q) and P ~ Q to show ~ ( P Q) P ~ Q . Solution. P Q PQ ~ ( P Q) ~ P ~ Q T T T F F F F T F T F F F T F T T F T F F F F F T T T T P Q PQ ~ ( P Q) P ~ Q T T T F T F F T F F T T T T F T T F F F F F F T F F F T The first two are so common, they have a name, they are called DeMorgan’s Laws: ~ ( P Q) ~ P ~ Q and ~ ( P Q) ~ P ~ Q The last, the negation of a conditional is often the most misused negation in our every day language: ~ ( P Q) P ~ Q Example 3. Construct a truth table for [( P Q ) ( R Q )] ( P R ). First we must order our thoughts and proceed to order the columns in the truth table. We work inside the parentheses first. From left to right, we construct the column for the conditional P Q , then for the disjunction R Q . Next we connect the two statements with the conjunction connective ( P Q ) ( R Q ), thus completing what the compound statement inside the brackets. We then find the conclusion to the conditional P R . Now, we have the hypothesis and the conclusion for the conditional that controls the compound statement. Thus, lastly we find the conditional [( P Q ) ( R Q )] ( P R ). 1 2 P Q R PQ R Q 12 PR [1 2] ( P R) T T T T T T T T T T F T T T F F T F T F T F T T T F F F F F F T F T T T T T F F F T F T T T F F F F T T T T F F F F F T F F F T Solution: So, we write the answer TFTTFFFT. Problem Set For questions 1 through 6, use the following statements. P: Arizona is the smallest state. Q: Arizona is west of New York. R: Arizona is in the Southern Hemisphere. S: Phoenix is in Arizona. Find the truth value: 1. ~ ( P S ) 2. ~ (Q P) 3. S ~ Q 4. ~ P ~ R 5. R (~ P Q) 6. P ~ R For problems 7-17, construct a truth table for the symbols below. 7. P ~ Q 8. ~ ( P ~ Q) 9. ~ P ~ Q 10. ~ ( P Q) 11. ~ (Q ~ P) 12. P (~ Q R) 13. ( P Q) ~ R) 14. P (Q ~ R) 15. ( P R) (Q ~ R) 16. Q ( P R) 17. Q ~ ( P R) Arguments So, how do we know if the facts together cause the conclusion we draw? Well, this is the sole purpose of this section. In this section, we will draw valid conclusions from known premises to deduce correctly. This is exactly what id done in a court of law by some well dressed attorney, presenting a series of statements that seem to point to some logical conclusion that is to be derived from his or her banter. The victim’s blood was found in the defendant’s car, the unusual footprint matches the rare shoes the defendant owned, the defendant’s time is unaccounted for when the murder occurred, and the glove worn by the attacker is the glove owned by the defendent. If you believe all of these facts, you must draw the conclusion that the defendant is guilty. Each point the attorney makes is called the hypothesis and the conclusion the attorney wants derived is called, well, the conclusion. Slowing this whole process down, the attorney makes point (hypothesis) after point (hypothesis) after point (hypothesis). “The blood of the victim was in the accused’s car and the accused’s shoe print was found at the murder scene and the defendant’s time is unaccounted for and the glove worn by the attacker is a glove the accused owns and the glove fits the accused (ok, maybe not this one). My point is, each point the attorney is trying to drive home is said in the manner of “This and this and this”. The attorney joins each point (hypothesis) with an and connective. He or she then finishes with: if you accept these points (this and this and this – the hypotheses), then we must conclude yadda-yadda-yadda (the conclusion). [h1 h2 h3 ... hn ] c So, an argument is valid if the conclusion necessarily follows from the hypotheses. This means that from the hypotheses, how ever many there are, the conclusion is guaranteed. To see if the conclusion is guaranteed, to see if the argument is valid, we first set up the argument symbolically. We then examine the resulting truth tables. The process for testing if an argument is valid is to write each hypothesis in symbolic form, join them with “and” connectives and the whole compound sentence become the hypothesis of a new conditional statement. The conclusion of the conditional statement is the conclusion to the argument. Again, the truth table must reflect that the conclusion follows from the hypotheses. So, the final column in the truth table must be a tautology, an expression that is true for all possible values of each hypothesis. We need all True’s in the final column. Problem 1 Is the following a valid argument? Let’s go back to the couple beseeched with jealousy. “If Russ was at the party, you surely would not have been there. But, you were there, so, Russ must have not have been at the party.” First we write the hypotheses, trying to keep them in the affirmative: P: Russ was at the party. Q: You were at the party. So, in argument form: 1. P ~ Q 2. Q Therefore, ~ P P Q ~Q P ~ Q Q [( P ~ Q) Q] ~P [( P ~ Q) Q] (~ P) T T F F T F F T T F T T F F F T F T F T T T T T F F T T F F T T Thus, a tautology, thus the conclusion is guaranteed, and the argument is valid. Problem 2 The salesman says, “The car has less than 50,000 miles on it and the tires are new.” You crawl on your knees and see for yourself that the tires are not new. You get up, look at the salesman and say, “The car doesn’t have 50,000 miles on it either.” You storm off and there is no sale. Are you justified in your response. We are asking ourselves if the following a valid argument? “If the car has less than 50,000 miles on it and the tires are new, and it turns out that the tires are not new, then the car didn’t really have 50,000 miles on it.” First we write the hypotheses: P: The car has 50,000 miles on it. Q: The tires are new. So, in argument form: 1. P Q 2. ~ Q Therefore, ~ P P Q P Q ~Q [( P Q) ~ Q)] ~P [( P ~ Q) Q] (~ P) T T T F F F T T F F T F F T F T T F F T T F F T T T T T Thus, a tautology, thus the conclusion is guaranteed, and the argument is valid. You were justified. Problem 3 How about a more complicated argument. Here are two cops talking. “If he were innocent, then he would not worry, do you agree?” “Sure.” “If he fled flea the scene, then he would worry.” “Yeah, I agree.” “So, it follows that if he were innocent, he would not flea the scene.” Valid or not? Write the hypotheses, connect them with conjunctions and write a statement. All in symbolic form. Remember to try and write each statement in the affirmative. P: He was innocent. Q: He was worried. R: He fled the scene. The argument: 1. P ~ Q 2. R Q Therefore, P ~ R We must construct a truth table for the argument [( P ~ Q ) ( R Q )] ( P ~ R ) 1 2 P Q R ~Q P~Q RQ 12 ~R P ~ R [1 2] ( p ~ R) T T T F F T F F F T T T F F F T F T T T T F T T T F F F F T T F F T T T T T T T F T T F T T T F T T F T F F T T T T T T F F T T T F F F T T F F F T T T T T T T Answer: Valid Consider an actor who just received bad reviews from some cheesy TV talk show host. The actor rationalizes the situation, “If he trashed my performance, then he wasn’t trashing my lifestyle. If he trashed me personally, then he’ed be trashing my lifestyle. Therefore, if he trashed my performance, he wasn’t trashing me personally.” Does this argument hold water, is it valid? Should the actor feel better? Write the hypotheses and the conclusion in symbolic form. P: He trashed my performance. Q: He trashed my lifestyle. R: He trashed me personally The argument: 1. P ~ Q 2. R Q Therefore, P ~ R Answer: Valid Contradiction is an expression that is false for all possible values of its prepositional variables. Problem Set: For problems 1 – 16, is the argument valid, not valid or a contradiction. Show a truth table to justify your answer. 1. If a student qualifies for non-resident financial aid, then they won’t qualify for any other financial aid. The student qualified for another form of financial aid. Therefore, the student won’t qualify for non-resident financial aid. 2. If the government cuts Medicare, then the elderly won’t have enough money for prescription drugs. If new funds are channeled toward helping the elderly, then the elderly will have enough money for prescription drugs. Therefore, if the government cuts Medicare, new funds won’t be channeled toward helping the elderly 3. No gay couples can legally marry in our county. My partner and I are legally married. Therefore, my partner and I are not gay. 4. No penguins fly. I don’t fly. Therefore, I am a penguin. 5. No penguins fly. A duck flies. Therefore, a duck is not a penguin. 6. If we continue with the deforestation of the Amazon Rainforest, then we will not have enough oxygen to breath. We have enough oxygen to breath. Therefore, we are not continuing to deforest the Amazon Rainforest. 7. If terrorism stopped tomorrow, then there would be no need for war. If we continue to increase the budget for the military, then we will have a need for war. Therefore, if terrorism stopped tomorrow, we won’t continue to increase the budget for the military. 8. Gas prices went up. If we looked for alternate fuel sources, gas prices would go down. Therefore, we looked for alternate fuel sources. 9. If the merchandise is defective, the company will suffer. If the company suffers, I will lose my job. I lost my job. Therefore, the merchandise was defective. 10. If Mozart were alive, he would hate contemporary music. If Mozart were alive, he would roll over in his grave. Therefore, Mozart hates contemporary music and he would roll over in his grave. 11. If Mozart were alive, he would hate contemporary music. If Mozart were alive, he would roll over in his grave. Mozart rolled over in his grave. Therefore, Mozart hates contemporary music. 12. No one who lives in extreme poverty is helped by society. I am helped by society. Therefore, I don’t live in extreme poverty. 13. No one who lives in extreme poverty is helped by society. I am not helped by society. Therefore, I live in extreme poverty. 14. No one who believes in religion can believe in evolution. I believe in evolution. Therefore, I do not believe in religion. 15. If a fetus has a soul, then abortion should not be legal. If you believe in woman’s choice, then abortion should be legal. If life begins at conception, then a fetus has a soul. Therefore, if a fetus has a soul, then you don’t believe in woman’s choice. Hint: Use 4 statements, 16 rows on the truth table, to determine the validity of the argument. 16. No inmate should face the death penalty. If someone plots to overthrow our government, then they should face the death penalty. If someone kills in self defense, then they should still be an inmate. Therefore, if someone kills in self defense, then they won’t plan to overthrow the government. Hint: Use 4 statements, 16 rows on the truth table, to determine the validity of the argument. Fallacy Mort Sahl once said of the 1980 presidential election that the people’s choice that year was not really Ronald Reagan because people weren’t so much voting for Reagan as they were casting a vote against Carter. He went on to say that had Reagan ran unopposed, he would have lost. A fallacy is a misleading notion or an erroneous belief. There are many types of fallacies, most of which are known by politicians, lawyers and the media. Often, they are simply a misuse of a logical connective, sometimes they are nothing more than a slight of hand of common sense. There is the false dilemma fallacy, where a limited number of options is given, when really there are more options. The lawyer who accuses “so you either loved him or you killed him.” The protester who screams “America, love it or leave it.” The politician who extorts “he is either for the war in Iraq or against it.” Black and white. No gray area. There is the fallacy of ignorance. If something is not false, then it must be true. Earlier we cited that Pat Robertson, a one time presidential candidate, once said that he couldn’t prove that there weren’t Soviet missiles in Cuba, therefore, there might be. Many people truly believe that since we can not prove aliens don’t exist, they do. There is the fallacy of the slippery slope. It is when we use the conditional statement incorrectly. If we pass laws prohibiting the use of automatic weapons, it won’t be long before laws will be passed taking away our right to bear arms. If we allow Jake Plumber to wear number 40 on his helmet commemorating Pat Tillman, who lost his life in Afghanistan, it won’t be long before another NFL player wears inappropriate head gear advertising a company. There is the fallacy of the statistical syllogism. This is when a general rule mitigates the exception. It is when a statement is usually true, but not always. You should never speed in a school zone, but what about if you are rushing to the hospital with an emergency? Liberals favored not going to war in Iraq, but John Kerry was for the war. There are the fallacies of the too broad or the too narrow. This is when we use the biconditional statements incorrectly. It is a newspaper if and only if it contains the daily news. Conservatives were for the war in Iraq. Squares have four sides. Well, Cnn.com contains the daily news and it is not a newspaper. John Kerry and Joe Lieberman were both liberals who were for the war in Iraq initially. And how many objects have four sides that are not squares? Then there is the fallacy of the non-transitive. It was said that in the 2004 democratic primaries, that if it was a two man race, Kerry could have beaten Edwards, but Edwards could have beaten Sharpton and Sharpton could have beaten Kerry. So, if Edwards could have beaten Sharpton who could have beaten Kerry, couldn’t Edwards have beaten Kerry? Anyway, Kerry was elected because it was not a two man race. And if we do have a two man race, as we do in a presidential election, shouldn’t we be extra careful which two men run… Problem Set State the type of fallacy. 1. If we allow Terrell Owens to sign a football after scoring a touchdown and hand it to their agent in the stands, it won’t be long before players bring their agents down on the field to join in the post touch down celebration. 2. You should never scream fire in a crowded theatre. 3. Either your for woman’s choice or you look at abortion as the murder of the unborn. 4. Liberals are those people who are high government spending. 5. You can’t prove to me that the house isn’t haunted, it might well be. Lastly, while fallacies are anomalies in intuition that are contrary to common sense, sound logical reasoning is the root of reliable intuition that serves as the basis for common sense. How does one’s intellect grow? Through gaining knowledge? Certainly. But, the growth of one’s intellect is based on so much more than just the rote acquisition of knowledge. So, how else does one’s intellect grow? Through the honing of intuition and the development of common sense. Through good decision making. Through understanding arguments presented. Through gaining knowledge of how to react to pointed questions. Through the recognition of patterns. Through developing the capability to think through a problem with an acute understanding of both the question as well as its implications. Through the ability to see reasonability of data, numbers or patterns that we are confronted with daily. When confronted by numbers, patterns or arguments, one’s intellect is what enables us to adequately combat any natural question that may arise. And isn’t developing this intellect nothing more than developing our the ability to think and reason mathematically. With a little mathematical literacy, we enable ourselves to function more proficiently in this numerate society we live. This is where the rainbow ends. Go back to your life. Commit yourself to excellence, regardless of your field of endeavor. Scrutinize all you see, critical of the information that besieges you each day, appreciative in the timeless beauty that surrounds you and mostly, confident in your ability to adapt and grow. Find your slice of heaven among the other 6.3 billion people on this third rock away from one of 100 billion stars in this galaxy and you will shine.