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).
This is not a logical statement because