Sir William Drummond, Academical Questions

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

Remember that all time favorite 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 … ”

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. In forming our conclusion we say that we are going from the specific to the general – from the observed pattern to then form the conclusion. Danger, Will Robinson, 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, sometime. 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 someone was to tell you no one who wins the lottery is unlucky, and then tell you their uncle once won the lottery, you could conclude that their 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 we revere Ray Charles, most of us 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 we mean?

Well, for example, just because a drug is known to decrease the rate of short-term memory loss in Alzheimer’s patients does not mean that not taking the drug will increase the rate of memory loss either. Yet, clever advertisers may have one inferring just that, especially if the person who is listening is elderly, conceivably with impaired judgment, already afraid and thus prone to jump to conclusions. Given, the drug may improve the tasks involving memory in a large proportion of the Alzheimer's patients, such as remembering dates. And perhaps those taking the drug will never experience improved memory, but will find their memory loss did not increase either. But, the absence of the drug will not increase the rate of memory loss in the roughly 4 million Americans who have been diagnosed with Alzheimer's, a progressive disease for which there is no known cure.

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.

7 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 Iraq 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 Magli 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 skills:

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. We ask you to find the next number in a sequence, and give you 2, 5, 8, 11, … , most of you would say the next number is ‘14’. And if we gave you 302, 305, 308, 311, … and requested the next number, again, most of you would say ‘314’. If we asked how do you know, you would probably say that we are adding 3 to each number in the sequence. But, if given 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.” And our gentle response, “Why would it take you nearly an hour?” And you’d answer (is ‘you’d’ a word?) “because I would have to add three like 101 times to 302.” And we 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 1 4 2 4 3 . 101 times So, now, if we asked you for the 466th or even the 1,196th term and since you are able to articulate the pattern, you could derive an answer in a matter of moments. 302+ 3 + 3 + ... + 3 = 302 + 3(466) = 1700 1 4 2 4 3 466 times 302+ 3 + 3 + ... + 3 = 302 + 3(1,196) = 3890 1 4 2 4 3 1,196 times

8 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 1 4 2 4 3 101 times 605= 302 + 3(101) a= a + d n n 0   starting number common difference the nth term

or an = a0 + dn or y = b + mx

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, or difference, d, replaces m.

When first recognizing how to handle a pattern like this, many students feel 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. They looked at patterns and for that matter, life differently. Why? Because they are able to find patterns everywhere. Let’s use this pattern recognizing skill and colorize our world. Suppose you purchase a Condo and wanted to advertise it on Rentals Unlimited’s Internet sight. Their brochure advertises 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 want to rent your condo (which is gorgeous because it is situated near a travel resort.) It is Memorial Day. You purchased the Internet Site advertisement 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 renters 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 of your arithmetic sequence is 1700 and 13 percent of 1700 is 221. So, over the next year through Labor Day, you may expect 221 renters 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 renters in the next three years, through Labor Day. Now, you begin long term plans.

Let’s inductively 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,

9 right? Successive doubling. Another sequence: 2, 6, 18, 54, quick, what’s the next number? 54(3) = 72. Successive tripling. And yet another sequence: 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 10 th 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 was the multiplier. 2, 6, 18, 54, has a common multiplier, r, of 3. We need to mimic the pattern formation as we did earlier. We identify the first term (as we did with the arithmetic sequence) and then find the common multiplier (instead of common difference) and multiply it (instead of adding it) to the first term.

To find the common multiplier, we examine the ratio of consecutive elements from the sequence.

a0  2

a1 6 a1  6   3 a0 2

a2 18 Take the sequence 2, 6, 18, 54, … . In our notation, a2  18  3 a1 6

a3 54 a3  54  3 a2 18

a10  ? So, the common ratio, r, is 3. 2(3)(3)(3)...(3)= 2(3)10 = 2(59049) = 118098 So, to find the a10 , we have 1 442 4 43 . More generally, we 10 times n write an = a0 r for a first term a0 , a common ratio of r, and where n is the number of times r is multiplied to the first term.

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 a similar but 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 previous number and if the result is the same when you divided successive pairs of numbers from the sequence, you have the common multiplier. Let’s try our inhospitable numbers. 8.2944/6.912 = 1.2 9.95328/8.2944 = 1.2 So, the common multiplier is 1.2 The 10 th n 10 term down the line? a10= a 0 r =6.912(1.2) = 42.79728215

Where would you see such a pattern in life, say, the 10th term of a geometric sequence? 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 for the next decade. We are taking 6.912 and multiplying it by 1.2 ten successive times. So, in ten

10 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 value it was originally worth.

Most of us do not have an extra 7 million dollars sitting around to plunk down on a blossoming minor league team, but if we are planning on purchasing a home, most of us will save money for a down payment. This brings us to the difference between true equity and the realistic equity. In the last problem from the Finance Chapter in this text, we discussed the equity that was saved five years into the payment plan for a $120,000 home. After paying monthly payments for 5 years, we found we still owed $89,569.61. So, the true equity in that problem was calculated to be $120,000 - $89,569.61or $30,430.39. But, if we factor in that most homes do appreciate in value over a five-year span, and moreover, if we then take that appreciation to be approximated at three percent per year, we have a geometric sequence. Our house, originally valued at $120,000 is now worth $120,000(1.03)5 = $139,112.89. So, our realistic equity could be considered $139,112.89 - $89,569.61 or $49,543.28.

There are two more patterns that pop up frequently. The first pattern is as predictable as a weather forecast on a summer day in Arizona. The signs of the terms oscillate between positive and negative. What causes this oscillation? A negative one is multiplied to the terms, over and over again. This common multiplier (-1) acts to oscillate the signs back and forth either by forcing the first to be positive or to be negative. Then the oscillation of signs for each subsequent term takes off from that point. We represent the common multiplier either of two ways: (- 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. For instance, -5,3, - 1,... = - 1鬃 5,1 3, - 1 � 1,...,( - 1)n+1 (5 2n )... . So, what if we wanted to find the 11th term? Since each term of the sequence is defined by the formula (- 1)n +1 (5 - 2n ) , we have 10+ 1 a10 =( - 1) (5 - 2(10)) = ( - 1)( - 15) = 15 . Does this check? Let’s list the terms of the sequence. Recalling the first term starts at n = 0, we have the following: -5, 3, -1, -1, 3, -5, 7, -9, 11, -13, 15. It checks.

The second pattern 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 , , , ..., ,... What is the eleventh term for this sequence? We could write 5 10 15 5(3)n 16- 2*10 - 4 a10 = = . 5(3)10 295,245

11 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)nor ( - 1) n +1 . If none of these tools help you, you are then challenged to find another pattern.

Exercise Set

What’s the next number? What is the 10th 2 5 8 8.  , , ,... term? Explain what lead you to the 3 7 11 conclusion. 9. 2, 42, 82, … 1. 2, 5, 8, … 10. 2, 125, 248, … 2. 30,120,480,... 11. 2, 10, 50, … 3. 20.0, 5.0, 1.25... 12. 2, - 1, 0.5, - 0.25… 4. -10, 20, - 30, 40, ... 13. 30.25, 166.375, 915.0625, ... 5. 15,- 150,1500... 14. 1, 4, 9, 16, … 2 5 8 6. , , ,... 15. 3, 6, 11, 18, … 7 7 7 0.008,- 0.0016, 0.00032... 2 5 8 16. 7. , , ,... 3 7 11 190 210 230 17. , , , ... 205 200 195

12 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

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

13 TV Some reality TV shows are scripted. x Survivor is a reality TV show Therefore, Survivor is scripted. SC

Answer: not valid SC

______

All fish swim. I swim. S x F Therefore, I’m a fish.

Answer: not valid

______

All birds fly. F A penguin is a bird. B Therefore, penguins fly. x

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 argument, where the conclusion follows from the premises, yet the conclusion is false because one of the premises is not true.

Exercise Set

For Problems 1-15, determine if the argument valid or not valid. Justify with an 2. Euler Diagram. All MTV videos are under twelve minutes. The Lost Seagulls video is not under twelve 1. minutes. All left handed batters have a hitter’s Therefore, the Lost Seagulls videos does not advantage. appear on MTV. Barry Bonds is left handed. Therefore, Barry Bonds has an advantage hitting.

14 3. 12. Some students won’t get Pell grants. All people smile. I am a student. Some gorillas smile. Therefore, I won’t get a Pell grant. Therefore, some gorillas are people.

4. 13. All people smile. Any animal that smiles is a person. Coco, the gorilla who talks through sign A gorilla can smile. language, is not a person. Therefore, a gorilla is a person. Therefore, Coco does not smile. 14. 5. Some gorillas smile. All cameras are expensive. Some people smile. Some cell phones are expensive. Therefore, some gorillas are people. Therefore, some cell phones are cameras. 15. 6. Some gorillas smile. All cameras are expensive. All gorillas are not people. Some cell phones are cameras. Therefore, all people don’t smile. Therefore, some cell phones are expensive. For Problems 16-20. Writing exercise. 7. Let’s reverse the thought process. We will All beagles are dogs. give you the Venn Diagram, you give us the Some 4 legged creatures are dogs. argument and tell us whether the argument Therefore, some 4 legged creatures are is valid or invalid. Create you own and use beagles. your own creativity to do so.

8. 16. 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. 17. 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.

15 18. 20.

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 sentence, because it decalres either truth or falsehood. In other words, it has a truth value, so we may in turn assign it aa being 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 this is being typed. At another time, it’s truth is certainly easy to verify. “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 true nor false? “Hillary Rodham Clinton would make a great president.” Opinion. Opinions are never true or false, dispite how much you may wish to

16 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 true 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 all 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 our 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, in response 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, 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 these 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.).

Exercise Set:

For the examples below, ask yourself what saying. For problems 1 – 13: Negate the does it mean if you say the following following: sentence is not true. We have collected 1. All drug companies spend more on commonly heard expressions said by advertising than research. politicians over the years. This information is often misinformation. So, if your reaction 2. No child will be left behind. is that this is not true, what are you really

17 3. Some of our troops did not obey the cease fire. 9. Some politicians are not honest about their tax declarations. 4. None of the previous rules apply. 10. All music videos are detrimental to our 5. This generation of youngsters don’t apply youth. themselves. 11. Some ball players use steroids. 6. All surveys are biased. 12. All anabolic steroid use hurts the game 7. No president has ever written a good tell of baseball. all book. 13. No documentaries are fair and unbiased. 8. Some movie reviewers just do not get it.

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 mean 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 a friend are arguing about one being too needy. 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, turning philosophical (and ugly). “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

18 needed. So, let’s see. What does "x " y, 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.

"x $ y, N ( x , y ) Every x needs a particular y. Everybody needs someone. True? I think so. No man is an island.

$x " y, N ( x , y ) One x needs every y. There is somebody who needs everyone. We all have met many people like this.

$y " x, N ( x , y ) There is a y needed by every x. There is someone who everybody needs. I don’t know. A religious person may answer this as a truism.

$x $ y, N ( x , y ) There is one x that needs one y. There is a person who needs someone. Yup. Probably quite a few.

$y $ x, N ( x , y ) There is a y needed by one x. There is someone who is needed by somebody. Yup. Your mom.

"y $ x, N ( x , y ) Every y is needed by one x. Everyone is needed by somebody. I hope so.

Such methodical symbolism teaches us not to be sloppy with our verbalization of 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 helps 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 $x " y, F ( x , y ) there exists a person who can be fooled all of the time. and all of the people some of the time, "x $ y, F ( x , y ) everyone can be fooled at least once but you can not fool all of the people all of the time. "x " y, F ( x , y ) everyone can be fooled all of the time

19 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 work through a problem set. Now, we know what you are thinking, this is a math class, dag-nab-it. Where are the numbers. Answer? In this following problem set.

Problem 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.

20 Exercise Set

For problems 1-10, rewrite the symbols as a x 2 5x  4(x  4)(x 1) . sentence in ordinary English. Use A(x,y) to mean x is afraid of y. Use 14. Every integer is a solution to M(x,y) to mean x owes money to y. 2 x  x(x  2)  2x . 1. "x $ y, A ( x , y ) 15. There are integers that are solutions to x 2  x 30 . 2. $x $ y, A ( x , y ) For problems 16-20: Now we go 3. $y $ x, A ( x , y ) backwards. Write each of the following statements about the integers in succinct "x $ y, A ( x , y ) 4. English, then designate which are true and which are false and comment with either an 5. $x " y, A ( x , y ) example, a counter example or your reasoning. 6. $y " x, A ( x , y ) 16.  x, y (x  y) 7. $x $ y, M ( x , y ) 17.  x, y (y 2  x) 8. "x $ y, M ( x , y ) 18.  x (x 2 0) 9. $y $ x, M ( x , y ) 19.  xy (x 2  x 2  y) 10."y $ x, M ( x , y ) 20. xy (x 2  x  y) For problems 11-15. the integers are the positive and negative whole numbers, to include zero. In set notation, they are For problems 21 to 23, let’s look at some { … -3, -2, -1, 0, 1, 2, 3, … }. Write quotations that have changed the hearts and each of the following statements about minds of mankind, advancing civilizations, the integers symbolically, then designate progress and even the course of love. which are true and which are false and Rewrite with quantifiers, and state whether comment with either an example, a or not you agree. counter example or your reasoning. 21. All religions must be tolerated … for … 11. Every integer is a solution to every man must get to heaven his own way. 2 – Frederick the Great x  5x  40 . 12. There are integers that are solutions to 22. Nobody’s perfect. Anonymous x 2  2x 150 . 23. Every man is an island un to himself. - 13. Every integer is a solution to Anonymous

21 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 many, 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.

I. Not P ~ P Negation Other words none, never

II. P and Q P Q Conjunction Other words, but, also, or a comma

III. P or Q P Q Disjunction Or

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

22 Other words, is equivalent to. This is considered the strongest form of a symbolic argument because it means the cause and effect of the statement goes both ways. That is P implies Q and Q implies P.

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儋 Q R The man lies, cheats but doesn’t steal. P儋 Q(~ R ) All men who lie, cheat. P Q The man lies, but doesn’t cheat or steal. P仝~ ( 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佼 Q ) R I won’t get an A if I don’t study or get enough sleep. ~ (P诋 Q ) (~ R ) The house will close if the money is wired or a gift is given. (Q诋 R ) P If you can smile and lie, you can cry and tell the truth. (P佼 Q ) ( R S )

Exercise Set

23 For problems 1-10, translate into symbols, 6. If Jack Rudy acted alone and died due to define each letter that you use. Try to write natural causes, I am a monkey’s uncle. P, Q , R or S in the affirmative. 7. If Medicare is cut, my parents will suffer 1. All democrats won’t support the tax and your parents will suffer. increase on the middle class. 8. No suspects were ever arrested solely 2. All gay couples can be legally married in because they were a celebrity. my state. 9. John Wilkes Booth was a phenomenal 3. If weapons of mass destruction were not actor, incredibly passionate but a lousy found, the war was not necessary. patriot.

4. No child will be left behind. 10. The CIA or the Mafia may have shot JFK or a lone gunman may have, but not the 5. We will never trust the government if the Cubans. country never resolves who killed JFK.

Truth Tables

P ~ P A truth table lists all possible combinations of true and false. Here, for a single T F statement P, P is either true or it is false. If the statement P is true, it’s negation is F T false. If the statement P is false, its negation is true.

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. That compound statement 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 our coming out party, people. We are not females, but we are teachers. 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, at least one of the two statements need to be true.

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 P Q T T T T F F F T F 24 F F F For a conjunction to be true, both statements must be true. Otherwise it is false.

P Q P Q T T T For a disjunction to be true, just one of the T F T statements need to be true. Otherwise it is false. 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. Handcuffed 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 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. We realize this may be in bad taste, but we will use the old lover’s promise anyway, “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.

P Q P Q In truth table terms, we have the following: If P holds true, I T T T promise Q will follow means if P is true and Q is true, the T F F conditional statement is true. And just as clearly, if P is true, F T T and Q is false, the promise has been broken, you won the F F T 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.

25 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 of the conditional statement in every day life? Remember our earlier discussion on administering drugs for Alzheimer’s patients. Initially, we briefly discussed administering a drug that known to decrease the rate of short-term memory loss in Alzheimer’s patients. Let’s reconstruct the sentence in the form of a conditional statement. It would read “If drug X is administered to Alzheimer’s patients, then it will decrease the rate of short-term memory loss.” In symbols, this would be P Q , where P is “drug X is administered to Alzheimer’s patients” and Q is “the drug will decrease the rate of short-term memory loss.”

Also recall, a common misconception is the direction of the causation. In other words, the statement “If patient exhibits a decrease in short term memory loss, the patient must have been given drug X” does not necessarily follow from the conditional statement. Maybe the patient was administered drug Y instead.

A second misconception for the conditional statement is assuming that not P not Q. For Alzheimer’s patients, we discussed the implication that not taking the drug will increase the rate of memory loss. As mentioned earlier, this does not follow either.

Which of these statements have the same meaning? If you take drug X, then your rate of short term loss will be decreased. If your short term memory loss is decreased, then you must have taken drug X. If you do not take drug X, then your rate of short term loss will not decrease. If your short term memory loss is not decreased, then you must not have taken drug X.

Let’s rewrite the statement in symbolic form using the proper terminology. Conditional: P Q If you take drug X, then your rate of short term loss will be decreased. Converse of P Q is Q P If your short term memory loss is decreased, then you must have taken drug X. Inverse of P Q is (~P ) (~ Q ) If you do not take drug X, then your rate of short term loss will be not decrease. Contrapositive of P Q is (~Q ) (~ P ) If your short term memory loss is not decreased, then you must not have taken drug X.

26 To see which statements are equivalent to each other, we will examine the truth table for each of the four and where we have equivalent columns, the statements are in turn equivalent.

P Q P Q Q P (~P ) (~ Q ) (~Q ) (~ P ) T T T T T T F T F F T F T F F F T T F T T T F F F T T T F F T F F F T T F F T F T F T T T T T F

Note, as we created the truth tables, we found pairs of statements to be equivalent. From this pattern, we can create rules for the conditional statement.

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 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 when 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.

27 Below is the truth table for P  Q . P Q P Q Q P (P Q ) ( Q P ) P Q 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, Fred.” 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 Fred, you were wrong to storm out. But, if you were jealous because she went out without you, and you were just using Fred 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 Fred. So, if you are the jealous type, you then wonder just whom she went out with. And 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 or 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.

28 Problem One

Construct a truth table for ~ (P Q ) and then for ~P ~ Q .

Solution

P Q P Q ~ (P Q ) We construct the columns in order, working inside T T T F the parentheses first. Next, just as we read in T F F T English, we construct the truth tables from left to F T F T right. For ~ (P Q ) , we first find the column for F F F T conjunction connective (P Q ) then we negate that column.

P Q ~ P ~ Q ~P ~ Q In this next example, we do not have T T F F F F F parentheses in our compound statement. So, T F F T F T T first find the column for ~ P , then we find F T T F T T F the column for ~ Q . We then join the two F F T T T T T statements with the disjunction connective to find ~P ~ Q .

Note, ~ (P Q ) 黑~P ~ Q . This is referred to as one of DeMorgan’s Laws.

Problem Two

Construct a truth table for ~ (P Q ) and ~P ~ Q to show ~ (P诤 Q ) ~ P ~ Q . Then construct truth tables for ~ (P Q ) and P~ Q to show ~ (P Q ) P ~ Q .

Solution

P Q P Q ~ (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 P Q ~ (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

These two equivalent statements are known as DeMorgan’s Laws: ~ (P俸 Q ) ~ P ~ Q and ~ (P诤 Q ) ~ P ~ Q

29 The last equivalency, the negation of a conditional, as seen earlier, is often misused in our every day language: ~ (P Q ) P ~ Q

Back to our story regarding the shackled man who was accused of robbery, what should he have replied? When accused with, “if you robbed the store, you had an accomplice,” the man should be tempted to say, “that’s not true.” But, let’s examine the negation of “if you robbed the store, you had an accomplice” Let P be “you robbed the store” and Q be “you had an accomplice.” So, the negation looks like, symbolically, ~ (P Q ) P ~ Q . The shackled man should say P and not Q. Meaning, “I could have done it myself? Welcome to the world of math humor.

Problem Three

Construct a truth table for [( P Q )  ( R Q )] ( P R ).

Solution

We have three statements. How many rows will be needed for the truth table? We are analyzing three statements, where each statement could be either true or false. This means we have two choices, T or F, paired with two more choices, T or F, (four choices) each paired with two more choices, T or F. So we have 2 x 2 x 2 = 8 choices altogether. Thus eight rows on our truth table.

First we must order our thoughts and proceed to order the columns in the truth table. We work inside the brackets 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 P Q R Q 1  2 P R [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.

30 Exercise Set

For questions 1 through 6, use the following For problems 7-17, construct a truth table statements. for the symbols below. P: Arizona is the smallest state. Q: Arizona is west of New York. 7. P ~ Q R: Arizona is in the Southern Hemisphere. S: Phoenix is in Arizona. 8. ~ (P ~ Q) Find the truth value: 9. ~ P ~ Q 1. ~ (P S ) 10. ~ (P  Q) 2. ~ (Q P ) 11. ~ (Q ~ P) 3. S~ Q 12. P  (~ Q  R) 4. ~P ~ R 13. (P仝 Q ) ~ R ) 5. R诋(~ P Q ) 14. P  (Q ~ R) 6. P~ 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 is done in a court of law by some well dressed attorneys, 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 defendant. 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.

31 Slowing this whole process down, the attorney makes point (hypothesis) after point (hypothesis) after point (hypothesis). “The blood of the victim was in the defendant’s car and the defendant’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 defendant owns and the glove fits the defendant (ok, maybe not this one). My point is, each point the attorney is trying to drive home is verbalized 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

An argument is valid if the conclusion necessarily follows from the hypotheses. This means that from the hypotheses, however 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. 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 to affirm validity of an argument.

Problem One

Is the following a valid argument? Let’s go back to the couple beseeched with jealousy. “If Fred was at the party, you surely would not have been there. But, you were there, so, Fred must have not have been at the party.”

Solution

First we write the hypotheses, trying to keep them in the affirmative: P: Fred was at the party. Q: She was at the party.

So, in argument form: 1. P~ Q 2. Q Therefore, ~ P

32 Treating the argument as a collection of hypotheses connected with “ands”, to form a single implication, we have the following truth table to test.

1 2 [1 2] c [1佼 2] (c ) 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 Two

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 probably doesn’t have 50,000 miles on it either.” You leave and there is no sale. Is your response justified?

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 less than 50,000 miles on it.”

Solution

We write the hypotheses: P: The car has less than 50,000 miles on it. Q: The tires are new.

So, converting to symbols, we have the following 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.

33 Problem Three

How about a more complicated argument. Here are two lawyers 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?

Solution

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 R Q 1  2 ~ 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 late night 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 mental analysis (argument) hold water, is it valid? How should the actor feel better?

34 The analysis of the argument will follow as it did a moment ago. We will write the hypotheses and the conclusion in symbolic form.

P: He trashed my performance. Q: He trashed my lifestyle. R: He trashed me personally

We then construct the argument: 1. P~ Q 2. R Q Therefore, P~ R

And low and behold, note the argument, logistically, is identical to the argument posed by the two lawyers in the prior problem, thus the argument the actor statement is more than a rationalization, it is valid. The actor’s performance was judged to be below expectations, and was not personally attacked. Of course, perhaps a better performance next time will circumvent the need for such mental anguish to follow. .

Contradiction is an expression that is false for all possible values of its prepositional variables.

Exercise Set

For problems 1 – 16, is the argument valid, not valid or a contradiction. Show a truth table to justify your answer. 2. No gay couples can legally marry in our A word of caution so that the student is not county. My partner and I are legally mislead in this exercise set. The point in married. Therefore, my partner and I are not this exercise set is not to overtly or gently gay. propagate a point of view. The point is much loftier. It is to illustrate the power of 3. No penguins fly. I don’t fly. Therefore, persuasion inherent in an argument. Recall I am a penguin. the entirety of this chapter. Even if an argument is valid, what interpretation should 4. No penguins fly. A duck flies. you give? Are the hypotheses true? Does Therefore, a duck is not a penguin. the conclusion imply what you think it implies? You must decide these points by 5. If we continue with the deforestation of yourself as you proceed through this the Amazon Rainforest, then we will not exercise set. have enough oxygen to breath. We have enough oxygen to breath. Therefore, we are 1. If a student qualifies for non-resident not continuing to deforest the Amazon financial aid, then they won’t qualify for any Rainforest. other financial aid. The student qualified for another form of financial aid. Therefore, the 6. If terrorism stopped tomorrow, then there student won’t qualify for non-resident would be no need for war. If we continue to financial aid. increase the budget for the military, then we

35 will have a need for war. Therefore, if 13. No one who believes in religion can terrorism stopped tomorrow, we won’t believe in evolution. I believe in evolution. continue to increase the budget for the Therefore, I do not believe in religion. military. 14. If the government cuts Medicare, then 7. Gas prices went up. If we looked for the elderly won’t have enough money for alternate fuel sources, gas prices would go prescription drugs. If new funds are down. Therefore, we looked for alternate channeled toward helping the elderly, then fuel sources. the elderly will have enough money for prescription drugs. Therefore, if the 8. If the merchandise is defective, the government cuts Medicare, new funds won’t company will suffer. If the company be channeled toward helping the elderly. suffers, I will lose my job. I lost my job. Therefore, the merchandise was defective. 15. If a fetus has a soul, then abortion should not be legal. If you believe in 9. If Mozart were alive, he would hate woman’s choice, then abortion should be contemporary music. If Mozart were alive, legal. If life begins at conception, then a he would roll over in his grave. Therefore, fetus has a soul. Therefore, if a fetus has a Mozart hates contemporary music and he soul, then you don’t believe in woman’s would roll over in his grave. choice. Hint: Use 4 statements, 16 rows on the truth table, to determine the validity of the argument. 10. If Mozart were alive, he would hate contemporary music. If Mozart were alive, 16. No inmate should face the death he would roll over in his grave. Mozart penalty. If someone plots to overthrow our rolled over in his grave. Therefore, Mozart government, then they should face the death hates contemporary music. penalty. If someone kills in self defense, then they should still be an inmate. 11. No one who lives in extreme poverty is Therefore, if someone kills in self defense, helped by society. I am helped by society. then they won’t plan to overthrow the Therefore, I don’t live in extreme poverty. government. Hint: Use 4 statements, 16 rows on the truth table, to determine the validity of the argument. 12. No one who lives in extreme poverty is helped by society. I am not helped by 17. Design your own contradictory society. Therefore, I live in extreme argument and build a truth table that is on poverty. the other end of the spectrum from the tautology. The final column needs to be all ‘false’.

36 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 fallacy of the false choices, where a limited number of options are given, and you know 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 the ignorant. 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 takes this premise one step further and wears inappropriate headgear to advertise a company. This out cry occurred in the Fall of 2004 and the NFL experienced the public’s chorus of disapproval.

There is the fallacy of the general rule. 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. News is in 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 are both liberals who were for the war in Iraq initially. And how many objects have four sides that are not squares?

Then there is exposed 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 and Sharpton 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 presidential elections from our life time, shouldn’t we be extra careful which two men run…

37 Exercise Set

State the type of fallacy. 6. “Mom, you can’t tell me I didn’t see a 1. If we allow Terrell Owens to sign a ghost”, the little boy said, “I saw it.” football after scoring a touchdown and hand it to their agent in the stands, it won’t be 7. Either you get a flue shot, or you’ll get long before players bring their agents down sick. on the field to join in the post touch down celebration. 8. The risk of executing the innocent precludes the use of the death penalty 2. You should never scream fire in a because when the state executes the crowded theatre. innocent, the very moral fiber of this country will be the next to go. 3. Either you are for woman’s choice or you look at abortion as the murder of the unborn. 9. Never yell fire in a crowded theatre.

4. Liberals are those people who are big on 10. Divorce causes children to withdraw, so government spending. my withdrawn child must be upset about my divorce. 5. You can’t prove to me that the house isn’t haunted, it might well be.