True or false?

The truth table for a simple true-false formula is as follows;

a AND NOT b IMPLIES a EXCLUSIVE OR b This formula governs the table values.

Truth table:
Value, a, b

False, T, T If a is true and b is true, the statement is contradictory and false.
Not both a and b can be true.
True, T, F If a is true and b is false, the statement (by the rule) is true by agreement.
True, F, T If a is false and b is true, the statement is true.
False, F, F If a is false and b is false, the statement is false.

The key operator is => IMPLIES. (a AND NOT b IMPLIES a EXCLUSIVE OR b)

The implication of the formula is agreement or disagreement. The statements must agree to disagree to be true.