Truth tables are designed to show all the possible truth values that a statement, or a collection of statements, can have. It is necessary to find the truth values for a collection of statements because that is what an argument is–a collection of statements. But to construct a truth table for a single statement, it is necessary to set out an array of possible truth values for each and every simple statement that makes up the entire complex statement. And for arguments, the same procedure is followed.
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Note: I can’t write out statements in the proper symbolism, so I will use English words in place of the logical connectors. Remember, there are five of them.
Negation, symbolized by the squiggly line called the tilde, means ‘it is not the case that.’ I will use ‘not.’
Conjunction, symbolized by the dot, means ‘and.’ I will use ‘and.’
Disjunction, symbolized by the ‘v’ means ‘or.’ So I will use ‘or.’
Conditional, symbolized by the sideways horseshoe, means ‘if, then.’ I will use ‘if…. then,’
Biconditional. symbolized by the horizontal triple bar, means ‘if and only if.’ I will use ‘iff.’
Also, the way these operators behave is given by their definitions on pgs. 331 – 333.
Section 6.3 Questions
1. Look on page 342, the first page of the section. You are given a statement: If (A or not B) then B. Your job is to make a truth table for this statement. Find out how many horizontal rows of truth values is needed. Then work out the truth values of each simple component statement (i.e., the two As and the two Bs) on the rows below the full complex statement. Follow the procedure in the textbook, it is very clear.
2. Under the heading “Classifying Statements” define the three kinds of statements outlined there.
3. Under the heading “Comparing Statements” briefly describe the relations: logically equivalent, contradictory, consistent and inconsistent.
Section 6.4 Question (just one)
At the beginning of the section (pg. 351) you are given an argument in regular English. It goes like this.
1. If juvenile killers are as responsible for their crimes as adults are, then execution is a justifiable punishment.
2. Juvenile killers are not as responsible for their crimes as adults are.
3. Therefore, execution is not a justifiable punishment.
Your job is to symbolize the argument, then construct a truth table to find out is the argument is valid or not. Remember, validity means that it is impossible for the premises to be true and the conclusion false. (hint: this argument does what is called ‘denying the antecedent. Can you do that?)