Chapter 7 Logic

7.1 Statements and Quantifiers

Learning Objectives

By the end of this section, you will be able to:

  • Identify logical statements
  • Represent statements in symbolic form
  • Negate statements in words
  • Negate statements symbolically
  • Translate negations between words and symbols
  • Express statements with quantifiers of all, some, and none
  • Negate statements containing quantifiers of all, some, and none

Have you ever built a club house, tree house, or fort with your friends? If so, you and your friends likely started by gathering some tools and supplies to work with, such as hammers, saws, screwdrivers, wood, nails, and screws. Hopefully, at least one member of your group had some knowledge of how to use the tools correctly and helped direct the construction project. After all, if your house isn’t built on a strong foundation, it will be weak and could possibly fall apart during the next big storm. This same foundation is important in logic.

In this section, we will begin with the parts that make up all logical arguments. The building block of any logical argument is a logical statement, or simply a statement. A logical statement has the form of a complete sentence, and it must make a claim that can be identified as being true or false.

When making arguments, sometimes people make false claims. When evaluating the strength or validity of a logical argument, you must also consider the truth values, or the identification of true or false, of all the statements used to support the argument. While a false statement is still considered a logical statement, a strong logical argument starts with true statements.

Identifying Logical Statements

An example of a logical statement with a false truth value is “All roses are red.” It is a logical statement because it has the form of a complete sentence and makes a claim that can be determined to be either true or false. It is a false statement because not all roses are red: some roses are red, but there are also roses that are pink, yellow, and white. Requests, questions, or directives may be complete sentences, but they are not logical statements because they cannot be determined to be true or false. For example, suppose someone said to you, “Please, sit down over there.” This request does not make a claim, and it cannot be identified as true or false; therefore, it is not a logical statement.

Example 1

Determine whether each of the following sentences represents a logical statement. If it is a logical statement, determine whether it is true or false.

a) Lil Wayne has won at least five Grammy awards.

This is a logical statement because it is a complete sentence that makes a claim that can be identified as being true or false. As of 2023, this statement is true: Lil Wayne won the following Grammys: Best Rap Album (2009), Best Rap Solo Performance (2009), Best Rap Song (2009), Best Rap Performance by a Duo or Group (2009), and Best Rap Performance (2017).

b) Please, eat your vegetables.

This is not a logical statement because, although it is a complete sentence, this request does not make a claim that can be identified as being either true or false.

c) All cats dislike dogs.

This is a logical statement because it is a complete sentence that makes a claim that can be identified as being true or false. This statement is false because some cats do like some dogs.

Exercise 1

Determine whether each of the following sentences represents a logical statement. If it is a logical statement, determine whether it is true or false.

a) The New Orleans Saints defeated the Indianapolis Colts in Super Bowl XLIV.

b) Taylor Swift released the song “You Belong with Me” in 2023.

c) Would you like some coffee or tea?

 

Solution

a) Logical statement, true

b) Logical statement, false

c) Not a logical statement; questions cannot be determined to be either true or false.

Representing Statements in Symbolic Form

When analyzing logical arguments that are made of multiple logical statements, symbolic form is used to reduce the amount of writing involved. Symbolic form also helps visualize the relationship between the statements in a more concise way in order to determine the strength or validity of an argument. Each logical statement is represented symbolically as a single lowercase letter, usually starting with the letter [latex]p[/latex]. To begin, you will practice how to write a single logical statement in symbolic form. This skill will become more useful as you work with compound statements in later sections.

Example 2

Write each of the following logical statements in symbolic form.

a) Barry Bonds holds the Major League Baseball record for total career home runs.

[latex]p[/latex]: Barry Bonds holds the Major League Baseball record for total career home runs.

The statement is labeled with a [latex]p[/latex]. Once the statement is labeled, use [latex]p[/latex] as a replacement for the full written statement to write and analyze the argument symbolically.

 

b) Some mammals live in the ocean.

[latex]q[/latex]: Some mammals live in the ocean.

The letter [latex]q[/latex] is used to distinguish this statement from the statement in Question 1, but any lowercase letter may be used.

 

c) The state bird of Louisiana is the brown pelican.

[latex]r[/latex]: The state bird of Louisiana is the brown pelican.

When multiple statements are present in later sections, you will want to be sure to use a different letter for each separate logical statement.

Exercise 2

Write each of the following logical statements in symbolic form.

a) The state flower of Louisiana is the magnolia.

b) Soccer is the most popular sport in the world.

c) All oranges are citrus fruits.

Solution

a) [latex]p[/latex]: The state flower of Louisiana is the magnolia.

b) [latex]q[/latex]: Soccer is the most popular sport in the world.

c) [latex]r[/latex]: All oranges are citrus fruits.

Negating Statements

Consider the false statement introduced earlier, “All roses are red.” If someone said to you, “All roses are red,” you might respond with “Some roses are not red.” You could then strengthen your argument by providing additional statements, such as “There are also white roses, yellow roses, and pink roses, to name a few.”

The negation of a logical statement has the opposite truth value of the original statement. If the original statement is false, its negation is true, and if the original statement is true, its negation is false. Most logical statements can be negated by simply adding or removing the word not. For example, consider the statement “Emma Stone has green eyes.” The negation of this statement would be “Emma Stone does not have green eyes.” The table below gives some other examples.

[latex]\begin{array} {|c|c|} \hline \textbf{Logical statement}& \textbf{Negation} \\ \hline \text{Gordon Ramsey is a chef.} & \text{Gordon Ramsey is not a chef.} \\ \hline \text{Tony the Tiger does not have spots.} & \text{Tony the Tiger has spots.}\\ \hline \end{array}[/latex]

The way you phrase your argument can impact its success. If someone presents you with a false statement, the ability to rebut that statement with its negation will provide you with the tools necessary to emphasize the correctness of your position.

Example 3

Write the negation of each logical statement in words.

a) Michael Phelps was an Olympic swimmer.

Add the word not to negate the affirmative statement: “Michael Phelps was not an Olympic swimmer.”

 

b) Tom is a cat.

Add the word not to negate the affirmative statement: “Tom is not a cat.”

 

c) Jerry is not a mouse.

Remove the word not to negate the negative statement: “Jerry is a mouse.”

Exercise 3

Write the negation of each logical statement in words.

a) Tim McGraw was not born in Louisiana.

b) Adele has a beautiful singing voice.

c) Leaves convert carbon dioxide to oxygen during the process of photosynthesis.

Solution

a) Tim McGraw was born in Louisiana.

b) Adele does not have a beautiful voice.

c) Leaves do not convert carbon dioxide to oxygen during the process of photosynthesis.

Negating Logical Statements Symbolically

The symbol for negation, or not, in logic is the tilde, [latex]\sim[/latex]. So, not [latex]p[/latex] is represented as [latex]\sim{p}[/latex]. To negate a statement symbolically, remove or add a tilde. The negation of not (not [latex]p[/latex]) is [latex]p.[/latex] Symbolically, this equation is [latex]\sim{(\sim{p})}=p[/latex].

Example 4

Write the negation of each logical statement symbolically.

a) [latex]p[/latex]: Michael Phelps was an Olympic swimmer.

To negate an affirmative logical statement symbolically, add a tilde: [latex]\sim{p}[/latex]

 

b) [latex]r[/latex]: Tom is not a cat.

Because the symbol for this statement is [latex]r[/latex], its negation is [latex]\sim{r}[/latex].

 

c) [latex]\sim{q}[/latex]: Jerry is not a mouse.

The symbol for this statement is [latex]\sim{q}[/latex], so to negate it, we simply remove the [latex]\sim[/latex] because [latex]\sim(\sim{q})=q[/latex]. The answer is [latex]q[/latex].

Exercise 4

Translating Negations between Words and Symbols

In order to analyze logical arguments, it is important to be able to translate between the symbolic and written forms of logical statements.

Examples

Given the statements

[latex]r[/latex]: Elmo is a red Muppet.

[latex]p[/latex]: Ketchup is not a vegetable.

a) Write the symbolic form of the statement “Elmo is not a red Muppet.”

“Elmo is not a red Muppet” is the negation of “Elmo is a red Muppet,” which is represented as [latex]r[/latex]. Thus, we would write “Elmo is not a red Muppet” symbolically as [latex]\sim{r}[/latex].

 

b) Translate the statement [latex]\sim{p}[/latex] into words.

Because [latex]p[/latex] is the symbol representing the statement “Ketchup is not a vegetable,” [latex]\sim{p}[/latex] is equivalent to the statement “Ketchup is a vegetable.”

Exercise 5

Expressing Statements with Quantifiers of All, Some, or None

quantifier is a term that expresses a numerical relationship between two sets or categories. For example, all squares are also rectangles, but only some rectangles are squares, and no squares are circles. In this example, all, some, and none are quantifiers. In a logical argument, the logical statements made to support the argument are called premises, and the judgment made based on the premises is called the conclusion. Logical arguments that begin with specific premises and attempt to draw more general conclusions are called inductive arguments.

Consider, for example, a parent walking with their three-year-old child. The child sees a cardinal fly by and points it out. As they continue on their walk, the child notices a robin landing on top of a tree and a duck flying across to land on a pond. The child recognizes that cardinals, robins, and ducks are all birds, then excitedly declares, “All birds fly!” The child has just made an inductive argument. They noticed that three different specific types of birds all fly, then synthesized this information to draw the more general conclusion that all birds can fly. In this case, the child’s conclusion is false.

The specific premises of the child’s argument can be paraphrased by the following statements:

  • Premise: Cardinals are birds that fly.
  • Premise: Robins are birds that fly.
  • Premise: Ducks are birds that fly.

The general conclusion is “All birds fly!”

All inductive arguments should include at least three specific premises to establish a pattern that supports the general conclusion. To counter the conclusion of an inductive argument, it is necessary to provide a counterexample. The parent can tell the child about penguins or emus to explain why that conclusion is false.

On the other hand, it is usually impossible to prove that an inductive argument is true. So inductive arguments are considered either strong or weak. Deciding whether an inductive argument is strong or weak is highly subjective and often determined by the background knowledge of the person making the judgment. Most hypotheses put forth by scientists using what is called the “scientific method” to conduct experiments are based on inductive reasoning.

In the following example, we will use quantifiers to express the conclusion of a few inductive arguments.

Example 6

For each series of premises, draw a logical conclusion to the argument that includes one of the following quantifiers: all, some, or none.

a) Squares and rectangles have four sides. A square is a parallelogram, and a rectangle is a parallelogram. What conclusion can be drawn from these premises?

The conclusion you would likely come to here is “Some four-sided figures are parallelograms.” However, it would be incorrect to say that all four-sided figures are parallelograms because there are some four-sided figures, such as trapezoids, that are not parallelograms. This is a false conclusion.

 

b) [latex]1+2=3[/latex], [latex]6+7=13[/latex], and [latex]23+24=47[/latex]. Of these, 1 and 2, 6 and 7, and 23 and 24 are consecutive integers; 3, 13, and 47 are odd numbers. What conclusion can be drawn from these premises?

From these premises, you may draw the conclusion “All sums of two consecutive counting numbers result in an odd number.” Most inductive arguments cannot be proven true, but several mathematical properties can be. If we let [latex]n[/latex] represent our first counting number, then [latex]n+1[/latex] would be the next counting number and [latex]n+(n+1)=2n+1[/latex]. Because [latex]2n[/latex] is divisible by two, it is an even number, and if you add 1 to any even number the result is always an odd number. Thus, the conclusion is true!

 

c) Sea urchins live in the ocean, and they do not breathe air. Sharks live in the ocean, and they do not breathe air. Eels live in the ocean, and they do not breathe air. What conclusion can be drawn from these premises?

Based on the premises provided, with no additional knowledge about whales or dolphins, you might conclude “No creatures that live in the ocean breathe air.” Even though this conclusion is false, it still follows from the known premises and thus is a logical conclusion based on the evidence presented. Alternatively, you could conclude “Some creatures that live in the ocean do not breathe air.” The quantifier you choose to write your conclusion with may vary from another person’s based on how persuasive the argument is. There may be multiple acceptable answers.

Exercise 6

For each series of premises, draw a logical conclusion to the argument that includes one of the following quantifiers: all, some, or none.

a) [latex]1+2=3[/latex], [latex]5+6=11[/latex], and [latex]14+15=29[/latex]. Of these, 1 and 2 are consecutive integers, 5 and 6 are consecutive integers, and 14 and 15 are consecutive integers. Also, their sums, 3, 11, and 29, are all prime numbers. Prime numbers are positive integers greater than 1 that are only divisible by 1 and the number itself. What conclusion can you draw from these premises?

b) A robin is a bird that lays blue eggs. A chicken is a bird that typically lays white and brown eggs. An ostrich is a bird that lays exceptionally large eggs. If a bird lays eggs, then they do not give live birth to their young. What conclusion can you draw from these premises?

c) All parallelograms have four sides. All rectangles are parallelograms. All squares are rectangles. What additional conclusion can you make about squares from these premises?

Solution

a) The sum of some consecutive integers results in a prime number.

b) No birds give live birth to their young.

c) All squares are parallelograms and have four sides.

Negating Statements Containing Quantifiers

Recall that the negation of a statement will have the opposite truth value of the original statement. There are four basic forms that logical statements with quantifiers take on.

  • All [latex]A[/latex] are [latex]B[/latex].
  • Some [latex]A[/latex] are [latex]B[/latex].
  • No [latex]A[/latex] are [latex]B[/latex].
  • Some [latex]A[/latex] are not [latex]B[/latex].

The negation of logical statements that use the quantifiers allsome, or none is a little more complicated than just adding or removing the word not.

For example, consider the logical statement “All oranges are citrus fruits.” This statement expresses a subset relationship. The set of oranges is a subset of the set of citrus fruit. This means that there are no oranges that are outside the set of citrus fruit. The negation of this statement would have to break the subset relationship. To do this, you could say, “At least one orange is not a citrus fruit.” Or, more concisely, “Some oranges are not citrus fruits.” It is tempting to say “No oranges are citrus fruits,” but that would be incorrect. Such a statement would go beyond breaking the subset relationship, to stating that the two sets have nothing in common. The negation of “[latex]A[/latex] is a subset of [latex]B[/latex]” would be to state that “[latex]A[/latex] is not a subset of [latex]B[/latex],” as depicted by the Venn diagrams below.

Two charts show a large and small circle. The first chart titled Statement depicts all oranges are as citrus fruit shows a large circle labeled citrus fruit with a small circle labeled oranges inside it. An 'x' mark is indicated in the center of the small circle. The first chart represents U is equal to fruit. The second chart titled Negation depicts some oranges that are not citrus fruit shows a large circle labeled citrus fruit with a small circle labeled oranges located on the boundaries of the large circle. An 'x' mark is indicated in the small circle, outside the large circle. The second chart represents U is equal to fruit.
Figure 1. A statement and the negation of the statement

The statement “All oranges are citrus fruits” is true, so its negation, “Some oranges are not citrus fruits,” is false.

Now consider the statement “No apples are oranges.” This statement indicates that the set of apples is disjointed from the set of oranges. The negation must state that the two are not disjoint sets, or that the two sets have at least one member in common. Their intersection is not empty. The negation of the statement, “[latex]A[/latex] intersection [latex]B[/latex], is the empty set” is the statement that “[latex]A[/latex] intersection [latex]B[/latex] is not empty,” as depicted in the Venn diagram below.

Two charts show two circles. The first chart titled Statement depicts all no apples are oranges shows a pink-colored circle labeled apples and a green-colored circle labeled oranges. The first chart represents U is equal to fruit. The second chart titled Negation depicts some apples are oranges shows two circles in the form of a Venn diagram. The first pink-colored circle is labeled apples and the second green-colored circle is labeled oranges. The region found between the two circles is in orange. An 'x' mark is indicated in this orange-colored region. The second chart represents U is equal to fruit.
Figure 2. A statement and the negation of the statement

The negation of the true statement “No apples are oranges” is the false statement “Some apples are oranges.”

The following table summarizes the four different forms of logical statements involving quantifiers and the forms of their associated negations, as well as the meanings of the relationships between the two categories or sets [latex]A[/latex] and [latex]B[/latex].

[latex]\begin{array} {|l|l|} \hline \textbf{Logical statements with quantifiers}& \textbf{Negation of logical statements with quantifiers} \\ \hline \text{Form: All } A \text{ are } B. & \text{Form: Some } A \text{ are not } B. \\ \text{Means: } A \text{ is a subset of } B\text{, } A \subset B. & \text{Means: } A \text{ is not a subset of } B\text{, } A \not\subset B. \\ \text{All zebras have stripes. (True)} & \text{Some zebras do not have stripes. (False)} \\ \hline \text{Form: Some } A \text{ are } B. & \text{Form: No } A \text{ are } B. \\ \text{Means: } A \text{ intersection } B\text{ is not empty, } A \cap B \neq \emptyset . & \text{Means: } A \text{ intersection } B \text{ is empty, } A \cap B = \emptyset. \\ \text{Some fish are sharks. (True)} & \text{No fish are sharks. (False)} \\ \hline \text{Form: No } A \text{ are } B. & \text{Form: Some } A \text{ are } B. \\ \text{Means: } A \text{ intersection } B\text{ is empty, } A \cap B = \emptyset . & \text{Means: } A \text{ intersection } B \text{ is not empty, } A \cap B \neq \emptyset. \\ \text{No trees are evergreens. (False)} & \text{Some trees are evergreens. (True)} \\ \hline \text{Form: Some } A \text{ are not } B. & \text{Form: All } A \text{ are } B. \\ \text{Means: } A \text{ is not a subset of } B \text{, } A \not\subset B. & \text{Means: } A \text{ is a subset of } B\text{, } A \subset B. \\ \text{Some horses are not mustangs. (True)} & \text{All horses are mustangs. (False)} \\ \hline \end{array}[/latex]

As a reminder from the Sets chapter, “[latex]A \text{ is a subset of } B[/latex]” means that every member of set [latex]A[/latex] is also a member of set [latex]B[/latex]. The intersection of two sets [latex]A[/latex] and [latex]B[/latex] is the set of all elements they share in common. If [latex]A[/latex] intersection [latex]B[/latex] is the empty set, then sets [latex]A[/latex] and [latex]B[/latex] do not have any elements in common. The two sets do not overlap. They are disjoint.

Video for Extra Help

Example 7

Given the statements

[latex]p[/latex]: All leopards have spots.

[latex]r[/latex]: Some apples are red.

[latex]s[/latex]: No lemons are sweet.

Write each of the following symbolic statements in words.

a) [latex]\sim{p}[/latex]

The statement “All leopards have spots” is [latex]p[/latex] and has the form “All [latex]A[/latex] are [latex]B[/latex]. According to the table above, the negation will have the form “Some [latex]A[/latex] are not [latex]B[/latex].” The negation of [latex]p[/latex] is the statement “Some leopards do not have spots.”

 

b) [latex]\sim{r}[/latex]

The statement “Some apples are red” has the form “Some [latex]A[/latex] are [latex]B[/latex].” According to the table above, the negation will have the form “No [latex]A[/latex] are [latex]B[/latex],” indicating that [latex]A[/latex] and [latex]B[/latex] do not intersect. This results in the opposite truth value of the original statement, so the negation of “Some apples are red” is the statement “No apples are red.”

 

c) [latex]\sim{s}[/latex]

Because [latex]s[/latex] is the statement “No lemons are sweet,” [latex]s[/latex] is asserting that the set of lemons does not intersect with the set of sweet things. The negation of [latex]s[/latex], [latex]\sim{s}[/latex], must make the opposite claim. It must indicate that the set of lemons intersects with the set of sweet things. This means at least one lemon must be sweet. The statement “Some lemons are sweet” is [latex]\sim{s}[/latex]. The negation of the statement, “No [latex]A[/latex] are [latex]B[/latex],” is the statement “Some [latex]A[/latex] are [latex]B[/latex],” as indicated in the table above.

Exercise 7

Given the statements

[latex]\sim{p}[/latex]: Some apples are not sweet.

[latex]r[/latex]: No triangles are squares.

[latex]s[/latex]: Some vegetables are green.

Write each of the following symbolic statements in words.

a) [latex]p[/latex]

b) [latex]\sim{r}[/latex]

c) [latex]\sim{s}[/latex]

Solution

a) All apples are sweet.

b) Some triangles are squares.

c) No vegetables are green.

definition

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