Common Logical Connectives

Understanding the following logical connectives, along with their properties, symbols, and names, will be key to applying the topics presented in this chapter. The chapter will discuss each connective introduced here in more detail.

The joining of two logical statements with the word “and” or “but” forms a compound statement called a conjunction. In logic, for a conjunction to be true, all the independent logical statements that make it up must be true. The symbol for a conjunction is [latex]\land .[/latex] Consider the compound statement “Derek Jeter played professional baseball for the New York Yankees, and he was a shortstop.” If [latex]p[/latex] represents the statement “Derrick Jeter played professional baseball for the New York Yankees” and if [latex]q[/latex] represents the statement “Derrick Jeter was a shortstop,” then the conjunction will be written symbolically as [latex]p \land q[/latex].

The joining of two logical statements with the word “or” forms a compound statement called a disjunction. Unless otherwise specified, a disjunction is an inclusive or statement, which means the compound statement formed by joining two independent clauses with the word or will be true if at least one of the clauses is true. Consider the compound statement “The office manager ordered cake for an employee’s birthday or they ordered ice cream.” This is a disjunction because it combines the independent clause “The office manager ordered cake for an employee’s birthday” with the independent clause “The office manager ordered ice cream” using the connective or. This disjunction is true if the office manager ordered only cake, only ice cream, or both cake and ice cream. Inclusive or means you can have one or the other or both!

Joining two logical statements with the word implies or using the phrase “if first statement, then second statement” is called a conditional or implication. The clause associated with the “if” statement is also called the hypothesis  or antecedent, while the clause following the “then” statement or the word implies is called the conclusion or consequent. The conditional statement is like a one-way contract or promise. The only time the conditional statement is false is if the hypothesis is true and the conclusion is false. Consider the following conditional statement: “If Pedro does his homework, then he can play video games.” The hypothesis/antecedent is the statement following the word if, which is “Pedro does/did his homework.” The conclusion/consequent is the statement following the word then, which is “Pedro can play his video games.”

Joining two logical statements with the connective phrase “if and only if” is called a biconditional. The connective phrase “if and only if” is represented by the symbol [latex]\leftrightarrow[/latex].  In the biconditional statement [latex]p \leftrightarrow q[/latex], [latex]p[/latex] is called the hypothesis or antecedent and [latex]q[/latex] is called the conclusion or consequent. For a biconditional statement to be true, the truth values of [latex]p[/latex] and [latex]q[/latex] must match. They must both be true or both be false.

The table below lists the most common connectives used in logic, along with their symbolic forms and the common names used to describe each connective.

[latex]\begin{array} {|c|c|c|} \hline \textbf{Connective}& \textbf{Symbol} & \textbf{Name} \\ \hline \text{and, but} & \land & \text{conjunction} \\ \hline \text{or} & \lor & \text{disjunction, inclusive or}\\ \hline \text{not} & \sim & \text{negation} \\ \hline \text{if..., then implies} & \rightarrow & \text{conditional, implication} \\ \hline \text{if and only if} & \leftrightarrow & \text{biconditional} \\ \hline \end{array}[/latex]

Note: These connectives, along with their names, symbols, and properties, will be used throughout this chapter and should be memorized.

Translating Compound Statements to Symbolic Form

To translate a compound statement into symbolic form, we take the following steps:

1. Identify and label all independent affirmative logical statements with a lowercase letter, such as [latex]p[/latex], [latex]q[/latex], or [latex]r[/latex].
2. Identify and label any negative logical statements with a lowercase letter preceded by the negation symbol, such as [latex]\sim{p}[/latex], [latex]\sim{q}[/latex], or [latex]\sim{r}[/latex].
3. Replace the connective words with the symbols that represent them, such as [latex]\land \text{, } \lor \text{, } \rightarrow \text{, or } \leftrightarrow[/latex].

Consider the previous example of your friend trying to get their driver’s license. Your friend passed the written test, but they did not pass the road test. Let [latex]p[/latex] represent the statement “My friend passed the written test.” And let [latex]\sim{q}[/latex] represent the statement “My friend did not pass the road test.” Because the connective but is logically equivalent to the word and, the symbol for but is the same as the symbol for and; replace but with the symbol [latex]\land[/latex]. The compound statement is symbolically written as [latex]p \land \sim{q}[/latex]. My friend passed the written test, but they did not pass the road test.

Note: When translating English statements into symbolic form, do not assume that every connective “and” means you are dealing with a compound statement. The statement “The stripes on the U.S. flag are red and white” is a simple statement. The word white doesn’t express a complete thought, so it is not an independent clause and does not get its own symbol. This statement should be represented by a single letter, such as [latex]s[/latex]: The stripes on the U.S. flag are red and white.

Translating Compound Statements in Symbolic Form with Parentheses into Words

When parentheses are written in a logical argument, they group a compound statement together just like when calculating numerical or algebraic expressions. Any statement in parentheses should be treated as a single component of the expression. If multiple parentheses are present, work with the innermost parentheses first.

Consider your friend’s struggles to get their license to drive. Let [latex]p[/latex] represent the statement “My friend passed the written test,” let [latex]q[/latex] represent the statement “My friend passed the road test,” and let [latex]r[/latex] represent the statement “My friend received a driver’s license.” The statement [latex](p \land q) \rightarrow r[/latex] can be translated into words as follows: the statement [latex]p \land q[/latex] is grouped together to form the hypothesis of the conditional statement and [latex]r[/latex] is the conclusion. The conditional statement has the form “if [latex]p \land q[/latex], then [latex]r[/latex].”

Therefore, the written form of this statement is “If my friend passed the written test and they passed the road test, then my friend received a driver’s license.”

Sometimes a compound statement within parentheses may need to be negated as a group. To accomplish this, add the phrase “it is not the case that” before the translation of the phrase in parentheses. For example, using [latex]p[/latex], [latex]q[/latex], and [latex]r[/latex] of your friend obtaining a license, let’s translate the statement [latex]\sim{(p \land q)} \rightarrow \sim{r}[/latex] into words. In this case, the hypothesis of the conditional statement is [latex]\sim{(p \land q)}[/latex], and the conclusion is [latex]\sim{r}[/latex]. To negate the hypothesis, add the phrase “it is not the case” before translating what is in parentheses. The translation of the hypothesis is the sentence “It is not the case that my friend passed the written test and they passed the road test,” and the translation of the conclusion is “My friend did not receive a driver’s license.” So a translation of the complete conditional statement, [latex]\sim{(p \land q)} \rightarrow \sim{r}[/latex], is “If it is not the case that my friend passed the written test and the road test, then my friend did not receive a driver’s license.”

Note: It is acceptable to interchange proper names with pronouns and remove repeated phrases to make the written statement more readable as long as the meaning of the logical statement is not changed.

The Dominance of Connectives

The order of operations for working with algebraic and arithmetic expressions provides a set of rules that allow consistent results. For example, if you were presented with the problem [latex]1+2 \times 3[/latex], and you were not familiar with the order of operations, you might assume that you calculate the problem from left to right. If you did so, you would add 1 and 3 to get 4 and then multiply this answer by 2 to get 8, resulting in an incorrect answer. Try inputting this expression into a scientific calculator. If you do, the calculator should return a value of 7, not 8.

Scientific Calculator

The order of operations for algebraic and arithmetic operations states that all multiplication must be applied prior to any addition. Parentheses are used to indicate which operation—addition or multiplication—should be done first. Adding parentheses can change and/or clarify the order. The parentheses in the expression [latex]1+2 \times 3[/latex] indicate that 3 should be multiplied by 2 to get 6, and then 1 should be added to 6 to get 7: [latex]1+2 \times 3 = 7[/latex].

As with algebraic expressions, there is a set of rules that must be applied to compound logical statements in order to evaluate them with consistent results. This set of rules is called the dominance of connectives. When evaluating compound logical statements, connectives are evaluated from least dominant to most dominant as follows:

1. Parentheses are the least dominant connective. So any expression inside parentheses must be evaluated first. Add as many parentheses as needed to any statement to specify the order to evaluate each connective.
2. Next, we evaluate negations.
3. Then, we evaluate conjunctions and disjunctions from left to right, because they have equal dominance.
4. After evaluating all conjunctions and disjunctions, we evaluate conditionals.
5. Lastly, we evaluate the most dominant connective, the biconditional. If a statement includes multiple connectives of equal dominance, then we will evaluate them from left to right.

Let’s revisit your friend’s struggles to get their driver’s license. Let [latex]p[/latex] represent the statement “My friend passed the written test,” let [latex]q[/latex] represent the statement “My friend passed the road test,” and let [latex]r[/latex] represent the statement “My friend received a driver’s license.” Let’s use the dominance of connectives to determine how the compound statement [latex]p \land \sim{q} \rightarrow r[/latex] should be evaluated.

Step 1: There are no parentheses, which are the least dominant of all connectives, so we can skip over that.

Step 2: Because negation is the next least dominant, we should evaluate [latex]\sim{q}[/latex] first. We could add parentheses to the statement to make it clearer which connective needs to be evaluated first: [latex]p \land \sim{q} \rightarrow r[/latex] is equivalent to [latex]p \land (\sim{q}) \rightarrow r[/latex].

Step 3: Next, we evaluate the conjunction, [latex]\land[/latex]. [latex]p \land \sim{q} \rightarrow r[/latex] is equivalent to [latex](p \land ((\sim{q})) \rightarrow r[/latex]

Step 4: Finally, we evaluate the conditional, [latex]\rightarrow[/latex], as this is the most dominant connective in this compound statement.