Introduction

What you’ll learn to do: Create truth tables to interpret statements and conditionals

In the previous section, we discussed fallacies (illogical arguments).  In this section, we focus on constructing logical statements and creating proper arguments.

Logic is a systematic way of thinking that allows us to deduce new information from old information and to parse the meanings of sentences. You use logic informally in everyday life and certainly also in doing mathematics. For example, suppose you are working with a certain circle, call it “Circle X,” and you have available the following two pieces of information.

1. Circle X has radius equal to 3.
2. If any circle has radius [latex]r[/latex], then its area is [latex]\pi{r}^{2}[/latex] square units.

You have no trouble putting these two facts together to get:

3. Circle X has area [latex]9\pi[/latex] square units.

You are using logic to combine existing information to produce new information. Since a major objective in mathematics is to deduce new information, logic must play a fundamental role. This chapter is intended to give you a sufficient mastery of logic.

Logical Statements

Truth Tables

A truth table is a table showing what the resulting truth value of a complex statement is for all the possible truth values for the simple statements.  We can visualize the relationship between a statement and its negation with a truth table as shown below:

This first row of this table illustrates that when [latex]P[/latex]  is true,  “not [latex]P[/latex]” is false.  The second row shows that when “not [latex]{P}[/latex]” is true, [latex]P[/latex] is false.

Often, we connect two or more statements together to form a new statement.  For example, when searching the internet, we use terms like “and” and “or” to help us find specific web pages that fit in the categories we are interested in.   When we search for “mystery and comedy” we are looking for a book that satisfies both of these conditions at the same time.  If we were to search for “mystery or comedy” we are looking for a book that is a mystery, a comedy, or both.

Notice here that or is not exclusive. This is a difference between the logic use of the word and common everyday use. When your significant other asks “do you want to go to the park or the movies?” they are usually proposing an exclusive choice – one option or the other, but not both. In mathematical logic, the or is not exclusive – more like being asked at a restaurant “would you like fries or a drink with that?” Answering “both, please” is an acceptable answer.

To make truth tables for the conjunction “P and Q,”  we must analyze all possible combinations of truth values for the individual statements, P and Q.   Since each of these two statements have two possible truth values (i.e.: either True or False), then there are a total of [latex]2\cdot 2[/latex] cases to consider, namely:

1. P is true and Q is true.
2. P is true and Q is false.
3. P is false and Q is true.
4. P is false and Q is false.

We analyze each of these four possibilities in the table below.

Similarly, there are four possible cases to consider for the disjunction “P or Q,”  as shown in the table below.

In most internet search engines, it is not necessary to include the word and; the search engine assumes that if you provide two keywords you are looking for both. In Google’s search, the keyword or has to be capitalized as OR, and a negative sign in front of a word is used to indicate not. Quotes around a phrase indicate that the entire phrase should be looked for. The search from the previous example on Google could be written:

Mexico university -“New Mexico”

Which Comes First?

Sometimes statements made in English can be ambiguous. For this reason, mathematical logic uses parentheses to show precedent, just like in algebraic order of operations.

The English phrase “Go to the store and buy me eggs and bagels or cereal” is ambiguous; it is not clear whether the requestors is asking for eggs always along with either bagels or cereal, or whether they’re asking for either the combination of eggs and bagels, or just cereal.

For this reason, using parentheses clarifies the intent:

Be aware that when a string of conditions is written without grouping symbols, it is often interpreted from the left to right, resulting in the latter interpretation.