Objectives
Students will be able to:
- Identify propositions
- Compose and interpret negations of a statement
- Use logical connectors (and/or) and conditional statement (if,then)
- Use truth tables to find truth values of basic and complex statements
Section 1.1 The Language and Rules of Logic
Subsection 1.1.1 Logic
Logic is the study of reasoning. Our goal in this chapter is to examine arguments to determine their validity and soundness. In this section we will look at propositions and logical connectors that are the building blocks of arguments. We will also use truth tables to help us examine complex statements.
Subsection 1.1.2 Propositions
A proposition is a complete sentence that is either true or false. Opinions can be propositions, but questions or phrases cannot.
Example 1.1.2.
Which of the following are propositions?
- I am reading a math book.
- Math is fun!
- Do you like turtles?
- My cat
Solution.
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.
Subsection 1.1.3 Negation (not)
One way to change a proposition is to use its negation, or opposite meaning. We often use the word “not” to negate a statement.
Example 1.1.3.
Write the negation of the following propositions.
- I am reading a math book.
- Math is fun!
- The sky is not green.
- Cars have wheels.
Solution.
- Negation: I am not reading a math book.
- Negation: Math is not fun!
- Negation: The sky is green (or not not green).
- Negation: Cars do not have wheels.
Subsection 1.1.4 Negation of All and None
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.
Example 1.1.4.
Write two different negations of each statement.
- All college students take psychology.
- Dogs are never brown.
Solution.
- Negations: Not all college students take psychology, or, Some college students don’t take psychology.
- Negations: There are brown dogs, or, Some dogs are brown.
Subsection 1.1.5 Multiple Negations
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.
1
ygdp.yale.edu/phenomena/negative-concordIn the media and in ballot measures we often see multiple negations and it can be confusing to figure out what a statement means.
Example 1.1.5.
Read the statement and determine the outcome of a yes vote.
“Vote for this measure to repeal the ban on plastic bags.”
Solution.
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.
Example 1.1.6.
Read the statement and determine the outcome on mandatory minimum sentencing.
“The bill that overturned the ban on mandatory minimum sentencing was vetoed.”
Solution.
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.
Subsection 1.1.6 Logical Connectors: and and or
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).
Example 1.1.7.
Determine whether each compound statement is true or false.
- Six is an even number and Salem is the capital of Oregon.
- Six is an even number or Salem is the capital of Oregon.
- Guitars have strings and cats do not have whiskers.
- Guitars have strings or cats do not have whiskers.
- Whales are not mammals and spiders have ten legs.
- Whales are not mammals or spiders have ten legs.
Solution.
- True. Six is even and Salem is the capital of Oregon. Since both parts are true, the “and” statement is true.
- True. Since both parts are true, the “or” statement is true.
- False. Guitars have strings but cats do have whiskers. Both statements must be true for an “and” statement to be true.
- True. Since the first part is true, the “or” statement is true.
- False. Whales are mammals and spiders have eight legs, so both parts are false, making the “and” statement false.
- False. Neither of the parts are true, so the “or” statement is false.
Subsection 1.1.7 Exclusive vs. Inclusive Or
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.
Example 1.1.8.
Determine whether each or statement is inclusive or exclusive.
- Would you like a chicken or vegan meal?
- We want to hire someone who speaks Spanish or Chinese.
- Are you going to wear sandals or tennis shoes?
- Are you going to visit Thailand or Vietnam on your trip?
Solution.
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.
Subsection 1.1.8 Conditional Statements (if, then)
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 statement “If it is raining,” may be true or false for any given day. If the condition is true, then we will follow the course of action and go to the mall. If the condition is false, though, we haven’t said anything about what we will or won’t do.
Subsection 1.1.9 Basic Truth Tables
In logic we can use a truth table to analyze a complex statement by summarizing all the possibilities and their truth values (true or false). To do this, we break the statement down to its smallest elements, the propositions. Then we can see the outcome of the complex statement for all possible combinations of true and false for the propositions.
For example, let’s work with two propositions:
- R: You paid your rent this month.
- E: You paid your electric bill this month.
We will use these two propositions to demonstrate the truth tables for not, and, and or.
To set up a truth table, we list all the possible truth value combinations in a systematic way. The standard way of doing this is to make the first column half true, then half false, then cut the pattern in half with each succeeding column. For two propositions, the first two columns are shown to the right.
| R | E | R and E |
|---|---|---|
| T | T | |
| T | F | |
| F | T | |
| F | F |
The four possible combinations are
- Row 1: You have paid your rent and electric bill
- Row 2: You have paid your rent but not your electric bill
- Row 3: You have not paid your rent but you have paid your electric bill
- Row 4: You haven’t paid either your rent or electric bill (yet).
Once we fill in the starting columns, we add additional columns for the more complex statements. We can add as many columns as needed. Below are the basic truth tables for not, and, and or.
Basic Truth Tables.
Not: In the not R column, the truth value is the opposite of the value for R. For example, if R is true (you paid your rent) then not R (you did not pay your rent) is false.
| R | not R |
|---|---|
| T | F |
| F | T |
And: In the R and E column, you must have paid both your rent and electric bill. Otherwise R and E is false.
| R | E | R and E |
|---|---|---|
| T | T | T |
| T | F | F |
| F | T | F |
| F | F | F |
Or: In the R or E column, you must have paid either your rent or electric bill, or both (inclusive or). Otherwise R or E is false.
| R | E | R or E |
|---|---|---|
| T | T | T |
| T | F | T |
| F | T | T |
| F | F | F |
Subsection 1.1.10 Conditional Truth Tables
We talked about conditional statements (if, then statements), earlier. In logical arguments the first part (the “if” part) is usually a hypothesis and the second part (the “then” part) is a conclusion.
To understand the truth table 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 to social media, you’ll lose your job.” Under what conditions can you say that your friend was wrong?
There are four possible outcomes:
- You post the photo and lose your job
- You post the photo and don’t lose your job
- You don’t post the photo and lose your job
- You don’t post the photo and don’t lose your job
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.
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.
The four cases above correspond to the four rows of the truth table. For this truth table we will use P for “posting the photo,” and L for “losing your job.”
| P | L | If P, then L |
|---|---|---|
| T | T | T |
| T | F | F |
| F | T | T |
| F | F | T |
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.
We are using the words and, or, not and if then in this book, but if you look up other resources on truth tables you are likely to see these symbols.
Symbols used in other resources.
\(A \text{ and } B\) is written \(A \land B\)
\(A \text{ or } B\) is written \(A \lor B\)
\(\text{not } A\) may be written as ~\(A\) or \(\lnot A\)
\(\text{If } A\text{, then } B\) is written \(A\rightarrow B\)
Subsection 1.1.11 Truth Tables for Complex Statements
Truth tables really become useful when we analyze more complex statements. In this case we will have several columns. It helps to work from the inside out and create a column in the table for each intermediate statement.
Example 1.1.10.
Create a truth table for the statement A or not B
Solution.
When we create the truth table, we start with columns for the propositions, A and B. Then we add a column for not B because that is part of the final statement. Our last column is the final statement A or not B.
| A | B | not B | A or not B |
|---|---|---|---|
| T | T | ||
| T | F | ||
| F | T | ||
| F | F |
To complete the third column, not B, we take the opposite of the B column. Then to complete the fourth column, we only look at the A and the not B columns and compare them using or.
| A | B | not B | A or not B |
|---|---|---|---|
| T | T | F | T |
| T | F | T | T |
| F | T | F | F |
| F | F | T | T |
Subsection 1.1.12 Truth Tables with Three Propositions
To create a truth table with three propositions we need eight rows for all the possible combinations. We will first determine the columns we need to get to our final statement. Then we will fill in the first three columns using the same methodology as before. Start with half true, half false, then cut the pattern in half each time.
Example 1.1.11.
Create a truth table for the statement A and not (B or C)
Solution.
First let’s figure out the columns we will need. We have A, B, C, then we need the statement in the parentheses, (B or C). Then we need the negation of that column, not (B or C). Then we conclude with our final statement, A and not (B or C).
Here is the initial table:
| A | B | C | B or C | not (B or C) | A and not (B or C) |
|---|---|---|---|---|---|
| T | T | T | |||
| T | T | F | |||
| T | F | T | |||
| T | F | F | |||
| F | T | T | |||
| F | T | F | |||
| F | F | T | |||
| F | F | F |
Now we complete the columns one at a time. We use the B column and C column to complete B or C,. Then not (B or C) is the opposite of that column. For the final column we only need to look at the first and fifth columns, shaded in blue, with and. Here is the completed table.
| A | B | C | B or C | not (B or C) | A and not (B or C) |
|---|---|---|---|---|---|
| T | T | T | T | F | F |
| T | T | F | T | F | F |
| T | F | T | T | F | F |
| T | F | F | F | T | T |
| F | T | T | T | F | F |
| F | T | F | T | F | F |
| F | F | T | T | F | F |
| F | F | F | F | T | F |
For this statement A must be true and neither B or C can be true, so it is only true in the fourth row. For an example of this statement, let’s define these propositions in the context of professional baseball:
Let A = Anaheim wins, B = Baltimore wins, C = Cleveland wins.
Suppose that Anaheim will make the playoffs if: (1) Anaheim wins, and (2) neither Boston nor Cleveland wins. TFF is the only scenario in which Anaheim will make the playoffs.
Example 1.1.12.
Construct a truth table for the statement if m and not p, then r.
Solution.
First, it may help to add parentheses to help you clarify the order. Our statement could also be written, if (m and not p), then r. To build this table, we will build the statement in parentheses and then repeat the r column after it. It’s easier to read the conditional statement from left to right. Here are the columns for the table:
| m | p | r | not p | m and not p | r | If (m and not p), then r |
|---|---|---|---|---|---|---|
| T | T | T | ||||
| T | T | F | ||||
| T | F | T | ||||
| T | F | F | ||||
| F | T | T | ||||
| F | T | F | ||||
| F | F | T | ||||
| F | F | F |
For the fourth column, we take the opposite of p. Then we use the first and fourth columns to complete m and not p. With the r column repeated we can use columns five and six to complete our conditional statement. Here is the completed table:
| m | p | r | not p | m and not p | r | If (m and not p), then r |
|---|---|---|---|---|---|---|
| T | T | T | F | F | T | T |
| T | T | F | F | F | F | T |
| T | F | T | T | T | T | T |
| T | F | F | T | T | F | F |
| F | T | T | F | F | T | T |
| F | T | F | F | F | F | T |
| F | F | T | T | F | T | T |
| F | F | F | T | F | F | T |
When m is true, p is false, and r is false—the fourth row of the table—then the hypothesis m and not p will be true, but the conclusion is false, resulting in an invalid conditional statement; every other case gives a true result.
If you want a real-life situation that could be modeled by if m and not p, then r, consider this:
Let m = we order meatballs, p = we order pasta, and r = Ruba is happy.
The statement if m and not p, then r is, “if we order meatballs and don’t order pasta, then Ruba is happy”. If m is true (we order meatballs), p is false (we don’t order pasta), and r is false (Ruba is not happy), then the statement is false, because we satisfied the premise, but Ruba did not satisfy the conclusion.
In this section we have discussed propositions, logical connectors and truth tables. In the next section, we will look at set relationships before we analyze arguments.
Exercises 1.1.13 Exercises
1.
Which of the following are propositions?
- Pigs can fly.
- What?
- I don’t know.
- I like tofu.
2.
Which of the following are propositions?
- How far?
- Portland is not in Oregon.
- Portland Community College.
- It is raining.
3.
Write the negation of each proposition.
- I ride my bike to campus.
- Portland is not in Oregon.
4.
Write the negation of each proposition.
- You should see this movie.
- Lashonda is wearing blue.
5.
Write a proposition that contains a double negative.
6.
Write a proposition that contains a triple negative.
7.
For each situation, decide whether the “or” is most likely exclusive or inclusive.
- An entrée at a restaurant includes soup or salad.
- You should bring an umbrella or a raincoat with you.
8.
For each situation, decide whether the “or” is most likely exclusive or inclusive.
- We can keep driving on I-5 or get on I-405 at the next exit.
- You should save this document on your computer or a flash drive.
9.
For each situation, decide whether the “or” is most likely exclusive or inclusive.
- I will wear a sweater or a jacket.
- My next vacation will be on the Oregon Coast or Mount Hood.
10.
For each situation, decide whether the “or” is most likely exclusive or inclusive.
- While in California I will go to the beach or Disneyland.
- The insurance agent offers car or boat insurance.
11.
Rewrite the statement in the conditional form if p, then q.
- Whenever it is sunny, I go swimming.
- I go see a movie on Fridays.
12.
Rewrite the statement in the conditional form if p, then q.
- I always carry an umbrella when it rains.
- On the weekend I like to hang out with friends.
13.
Translate each statement from symbolic notation into English sentences. Let A represent “Elvis is alive” and let K represent “Elvis is the King”.
- Not A
- A or K
- Not A and K
- If K, then not A
14.
Translate each statement from symbolic notation into English sentences. Let A represent “It rains in Oregon” and let B represent “I own an umbrella”.
- Not B
- A and not B
- If A, then B
- If not B, then A
15.
Translate each statement from English sentences into symbolic notation. Let A represent “I will protest” and let B represent “There is injustice.”
- There is injustice and I will protest.
- If there is injustice, then I will protest.
- I will protest if there is injustice.
- If there is not injustice, then I will not protest.
16.
Translate each statement from English sentences into symbolic notation. Let A represent “It’s time to eat” and let B represent “I am hungry.”
- It’s time to eat and I’m not hungry.
- It’s not time to eat.
- If it’s time to eat, then I’m hungry.
- If I’m not hungry then it’s not time to eat.
17.
Determine if the entire statement is true or false.
- An apple is a vegetable, or an apple is a fruit.
- Portland is not a city in Oregon.
18.
Determine if the entire statement is true or false.
- Fish can walk and birds can swim.
- If it is warm outside, then it is sunny.
Exercise Group.
Complete the truth table for each statement and write the meaning of each statement in the third column.
19.
Let A be: I live in Oregon.
Let B be: I go to Portland Community College
| A | B | A and B |
|---|---|---|
| T | ||
| T | ||
| F | ||
| F |
20.
Let A be: I am a psychology major
Let B be: I’m planning to transfer to Portland State
| A | B | A or B |
|---|---|---|
| T | ||
| T | ||
| F | ||
| F |
Exercise Group.
Complete the truth table for each statement.
21.
A and not B
| A | B | Not B | A and not B |
|---|---|---|---|
| T | |||
| T | |||
| F | |||
| F |
22.
Not (not A or B)
| A | B | Not A | Not A or B | Not (not A or B) |
|---|---|---|---|---|
| T | ||||
| T | ||||
| F | ||||
| F |
23.
Not (A and B and C)
| A | B | C | A and B and C | Not (A and B and C) |
|---|---|---|---|---|
| T | ||||
| T | ||||
| T | ||||
| T | ||||
| F | ||||
| F | ||||
| F | ||||
| F |
24.
Not A or (not B and C)
| A | B | C | Not A | Not B | Not B and C | Not A or (Not B and C) |
|---|---|---|---|---|---|---|
| T | ||||||
| T | ||||||
| T | ||||||
| T | ||||||
| F | ||||||
| F | ||||||
| F | ||||||
| F |
Exercise Group.
Create a complete truth table for each statement.
25.
Not(A and B) or C
26.
(A or B) and (A or C)
27.
If (A and B), then C
28.
If (A or B), then not C
29.
If (A and C), then not A
30.
If (B or C), then (A and B)
