Final answer:
A pair of propositions that cannot both be true and cannot both be false is described as Contradictory, which means there is a logical impossibility for both statements to be true at the same time. The right answer is A. Contradictory.
Step-by-step explanation:
If a pair of propositions cannot both be true, but also cannot both be false, they are mutually exclusive. This means that the truth of one proposition excludes the truth of the other, and vice versa. The correct answer to the question about what it means when a pair of propositions cannot both be true and cannot both be false is Contradictory.
According to the law of noncontradiction, contradictory propositions cannot be true in the same sense at the same time. A contradiction is a logical impossibility, which indicates that at least one of the beliefs held must be incorrect. Moreover, if propositions are contradictory, this indicates that they are not logically consistent, which is necessary to ensure that a set of beliefs can be coherent.
To illustrate this, consider the propositions 'It is raining' and 'It is not raining'. These propositions are contradictory because they cannot both be true at the same time. The law of noncontradiction dictates that if 'It is raining' is true, 'It is not raining' must be false, and vice versa.