When a person declares that something is reasonable, the common usage of the word indicates that they are often only expressing a subjective assent to an idea or proposal. The very fact that many people say that a given claim "seems reasonable" demonstrates that they are not aiming for genuine rationality, with its deductive precision that does not hinge on subjective perceptions. They are instead aiming for personal persuasion. The two concepts are not the same.
I have yet to actually see a situation where the word reasonable is used in a way that is not in reference to someone's subjective threshold of persuasion--something that is short of absolute logical certainty. A claim is rational if it is in accordance with the necessary laws of logic, but "rational" is not the exact equivalent of what many mean by "reasonable." To be persuaded means nothing if one settles for an argument that is not entirely verifiable via the exercise of reason.
Persuasion is not the goal of a genuine seeker of truth; in fact, persuasion is absolutely useless and irrelevant to the pursuit of truth in itself. Proof is what those who desire the truth pursue, and nothing else. Anyone who seeks a lesser thing only reveals that they are content to believe without knowledge.
It is not surprising that many people settle for reasonableness instead of rationality, as intelligence is a scarce treasure. In a world of intellectual apathy, mediocrity, and unoriginality, a genuine rationalist will tower above those who cling to anything other than the laws of logic themselves. Because of this, he or she will ironically be able to enjoy both persuasion and proof alike, since it is only logical proof that should have peruasive power.
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