Suppose your friend is trying to get a license to drive. In most places, they will need to pass some form of written test proving their knowledge of the laws and rules for driving safely. After passing the written test, your friend must also pass a road test to demonstrate that they can perform the physical task of driving safely within the rules of the law.
Consider the statement: "My friend passed the written test, but they did not pass the road test." This is an example of a compound statement , a statement formed by using a connective to join two independent clauses or logical statements. The statement, “My friend passed the written test,” is an independent clause because it is a complete thought or idea that can stand on its own. The second independent clause in this compound statement is, “My friend did not pass the road test.” The word "but" is the connective used to join these two statements together, forming a compound statement. So, did your friend acquire their driving license?.
This section introduces common logical connectives and their symbols, and allows you to practice translating compound statements between words and symbols. It also explores the order of operations, or dominance of connectives, when a single compound statement uses multiple 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 ∧ ∧ . Consider the compound statement, “Derek Jeter played professional baseball for the New York Yankees, and he was a shortstop.” If p p represents the statement, “Derrick Jeter played professional baseball for the New York Yankees,” and if q q represents the statement, “Derrick Jeter was a short stop,” then the conjunction will be written symbolically as p ∧ q . p ∧ q .
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 a least one of the clauses is true. Consider the compound statement, "The office manager ordered cake for 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 they ordered 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, ↔ . ↔ . In the biconditional statement, p ↔ q , p ↔ q , p p is called the hypothesis or antecedent and q q is called the conclusion or consequent. For a biconditional statement to be true, the truth values of p p and q q 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.
| Connective | Symbol | Name |
|---|---|---|
| and but | ∧ ∧ | conjunction |
| or | ∨ ∨ | disjunction, inclusive or |
| not | ~ | negation |
| if … … , then implies | → → | conditional, implication |
| if and only if | ↔ ↔ | biconditional |
