In this chapter we’ll learn several different tools for decision making. We’ll look at logic, sets, Venn diagrams, percentages, rates and proportions. Then we’ll work with a problem solving process that can be applied to many types of situations.
Logic is the study of reasoning. Our goal in this section is to look at propositions, logical connectors and statements that are used in everyday life and examine their meaning.
The first and second items are propositions. The third one is a question and the fourth is a phrase, so they are not. We are not concerned right now about whether a statement is true or false. We will come back to that later when we look at compound statements.
Arguments are made of one or more propositions (called premises), along with a conclusion. Propositions may be negated, or combined with connectors like “and”, and “or”. Let’s take a closer look at how these negations and logical connectors are used to create more complex statements.
It is worth mentioning these qualifiers and how to negate them. If we were to make the statement, “All students read this book,” we could negate it by saying “Not all students read this book.” or “Some students don’t read this book.” Notice how this is very different from “No students read this book.”
If we were to negate, “No students read this book,” we could say, “A student is reading this book,” or even, “Some students do read this book.” It only takes one counterexample to negate an all or nothing statement and the phrases some do and some don’t are often used for this purpose. The same concept applies to similar words like everyone, nobody, always and never.
It is possible to use more than one negation in a statement. If you’ve ever said something like, “I can’t not go,” you are saying you will go. In fact, it’s often used for emphasis or a slightly different meaning, that you really must go. If someone says, “I don’t disagree,” they may be saying they don’t exactly agree but the person has a point. A double negative is similar to multiplying two negative numbers which gives a positive result. Using a third negation would then be equivalent to a single negation.
Note that in some instances a negation word like “no,” “nobody,” or “nothing” is used to emphasize rather than negate and this is called negative concord. For example, “I ain’t got no money,” is not a double negative but rather an emphasis of not having any money. This is common across many varieties of English and other languages. You can read more about negative concord at this site. In general, use your judgment and context cues to distinguish between a double negative and negative concord. We will use multiple negations but not negative concord in this book.
If you said that a yes vote would enable plastic bag usage, you are correct. The ban stopped plastic bag usage, so to repeal the ban would allow it again. This measure has a double negation and is also not very good for the environment.
In this case mandatory minimum sentencing would not be allowed. The ban would stop it, and the bill to overturn it was vetoed. This is an example of a triple negation.
When we use the word “and” between two propositions, it connects them to create a new statement that is also a proposition. For example, if you said “To finish this project, I need a screwdriver and a wrench,” then you are expressing the need for both tools. For an “and” statement to be true, the connected propositions must both be true. If even one proposition is false (for instance, you didn’t need a wrench) then the entire connected statement is false.
The word “or” between two propositions similarly connects the propositions to create a new statement. In this case, if you said “To finish this project, I need a screwdriver or a wrench,” then you are expressing the need for one of the tools (but probably not both). For an “or” statement to be true, at least one of the propositions must be true (or both could be true).
In English we often mean for or to be exclusive: one or the other, but not both. In math, however, or is usually inclusive: one or the other, or both. The thing we are including, or excluding is the “both” option.
The first or statement is a choice of one or the other, but not both, so it is exclusive. The second statement is inclusive because they could find a candidate who speaks both languages. The third statement is exclusive because you can’t wear both at the same time. The fourth statement is inclusive because you could visit both countries on your trip.
A conditional statement connects two propositions with if, then. An example of a conditional statement would be “If it is raining, then we’ll go to the mall.” The first part (the “if” part) is called the hypothesis and the second part (the “then” part) is called the conclusion.
The statement, “It is raining,” may be true or false for any given day. If the hypothesis is true, then we will follow the course of action and go to the mall. If the hypothesis is false, though, we haven’t said anything about what we will or won’t do.
To understand the truth values for a conditional statement it is helpful to look at an example. Let’s say a friend tells you, “If you post that photo on social media, you’ll lose your job.” Under what conditions can you say that your friend was wrong?
The only case where you can say your friend was wrong is the second case, in which you post the photo but still keep your job. This is the only time when a conditional statement is false.
Your friend didn’t say anything about what would happen if you didn’t post the photo, so you can’t say the last two statements are wrong. Even if you didn’t post the photo and lost your job anyway, your friend never said that you were guaranteed to keep your job if you didn’t post it.
In this case it can be useful to outline the possibilities and outcomes in a table. The four cases above correspond to the four rows of a truth table. If your class is using truth tables they are covered in a later section. For this table we will use P for “posting the photo,” and L for “losing your job.”
If the hypothesis (the “if” part) is false, we cannot say that the statement is a lie, so the result of the third and fourth rows is true. Notice that we are using a double negation in this explanation.
True. Pigs cannot fly (on their own) but birds can. Since the hypothesis is false, the statement is true regardless of whether the conclusion is true or false.
Some situations combine many “if, then”, “and” or “or” statements that may be inclusive or exclusive. Here’s an example where we need to evaluate a complex statement.
Determine which set(s) of qualifications meet the requirement: “To apply for this job, applicants must have a bachelor’s degree, an associate’s degree and 3 years of relevant experience, or a high school diploma or GED and 6 years of relevant experience.”
Lyssa has a bachelor’s degree and 2 years of relevant experience.
There are many combinations that work here. Lyssa, Ayan and Sylvia all meet the requirements or go beyond what is needed. Jordan does not have enough years of experience to go with a high school diploma for this job.
And and or statements can have their propositions reversed without affecting the truth value of the statement. If you need to buy bacon and eggs, that’s the same as buying eggs and bacon. The order does matter for a conditional statement, though. Sometimes during an argument, an if, then statement will be reversed or combined with not in different ways. There are names for the combinations and it is helpful to look at which statements are equivalent to each other. Here are the names for the different combinations of if, then and not statements.
Assume this original conditional statement is true: “If you take your dog to the park, they will be happy.” Write the converse, inverse and contrapositive of the original statement and determine which are equivalent to each other.
Converse: “If your dog is happy, then you took them to the park.” Notice we needed to change the tense to keep the order of events the same. If your dog is happy does that mean you took them to the park? Not necessarily, because there are other reasons your dog could be happy like getting a treat or taking a walk. So this is false.
Inverse: “If you don’t take your dog to the park, they will not be happy.” This is similar to the converse. Even if you don’t take your dog to the park, they could be unhappy for another reason. This is false. The inverse is logically equivalent to the converse.
Contrapositive: “If your dog is not happy, then you did not take them to the park.” Since the original statement is true, then this is also true. Taking your dog to the park would result in them being happy, so if they are not happy we can say that they did not go to the park. The contrapositive is logically equivalent to the original statement.
