Use a truth table to determine that "division into cases" rule of inference is valid.
To determine the validity of the "division into cases" rule of inference using a truth table, we need to define the statements involved and construct a truth table that includes all possible combinations of truth values for these statements.
Let's denote the two statements involved as "P" and "Q." The "division into cases" rule of inference states that if we have a disjunction (P ∨ Q) and we want to deduce a conclusion, we can do so by considering two separate cases: one where P is true, and one where Q is true.
Here's how we can construct the truth table:
1. Begin by listing all possible combinations of truth values for P and Q. Since each statement can either be true (T) or false (F), we have four rows in total.
2. Next, determine the truth value of the disjunction (P ∨ Q) for each row. If either P or Q (or both) is true, then the disjunction is true; otherwise, it is false.
3. Finally, consider the conclusion we want to deduce based on the "division into cases" rule. We need to determine the truth value of this conclusion for each row by evaluating the disjunction (P ∨ Q) based on the cases outlined above.
By constructing the truth table using these steps, we can determine whether the rule is valid by checking if the conclusion follows logically from the truth values of the premises (disjunctions).
However, it's important to note that the "division into cases" rule is not usually evaluated using a truth table since it is not a traditional logical operator. Instead, it is considered a valid rule of inference because it follows from the general principles of logical reasoning.