Final answer:
The two logical statements, 'If A, then B' and '(not A) or B', are equivalent as they are true when A and B are both true, or when A is false and B is true. They are both false only when A is true and B is false. This equivalence is due to the rules of conditional and disjunctive statements in logic.
Step-by-step explanation:
The logical statements in question, If A, then B and (not A) or B, are considered logically equivalent due to the principles of logic. The equivalence exists due to the truth values the statements hold under different circumstances.
True Circumstances
Both statements are true in the following scenarios:
- A is true and B is true: The conditional is satisfied (if A, then B), and the disjunction is also satisfied (A is not true, but B is).
- A is false and B is true: Even though A is false, the conditional holds since B is true, satisfying the 'then' part. In the disjunction, B’s truth makes the whole statement true.
False Circumstances
Both statements are false only when:
- A is true but B is false: This violates the conditional (if A, then not B), and for the disjunction, since A is true, the 'or B' portion cannot rescue the statement.
The reason these statements are identical lies in the way conditionals and logical disjunctions work. According to logic, an implication (if A, then B) is said to be false only when A is true and B is false. Similarly, the disjunction ((not A) or B) is false only under the same conditions, which makes them functionally the same in logical terms.