Logic and Argument 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 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 … ”

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 Celsius, then concludes water freezes at 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, 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, 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. 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 .

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.

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,

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, 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 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 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. Take the sequence 2, 6, 18, 54, … . In our notation,

So, the common ratio, r, is 3.

So, to find the , we have. More generally, we write for a first term , 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 10th term down the line?

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 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:

We use these multipliers as coefficients multiplied to the other terms in the pattern. For instance, . So, what if we wanted to find the 11th term? Since each term of the sequence is defined by the formula , we have . 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.

So, what is the eleventh term for this sequence? We could write

.

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 . If none of these tools help you, you are then challenged to find another pattern.

Exercise Set

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