The New Oxford American Dictionary says this about the usage of conviction verses persuasion:

Although it is common to see convince and persuade used interchangeably, there are distinctions in meaning that careful writers and speakers try to preserve. Convince derives from a Latin word meaning ‘conquer, overcome.’ Persuade derives from a Latin word meaning ‘advise, make appealing, sweeten.’

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Proof convinces. Proof produces agreement through intellectual coercion. There are kinds of truth that are subject to proof, and these are objective truths. With a valid proof, to accept the premises is to accept the conclusion. One cannot resist the argument if one is intellectually decent. To reject the argument is to betray civility. With proof agreement is necessary.

But not all truths fall inside the jurisdiction of proof.

In fact, the most important truths — the ideas most significant to us, which in fact lend significance to objective truths — are outside the reach of proof’s coercive powers. They reside in the realm of freedom, where only persuasion has sway.

Appeals persuade. Appeals invite agreement. Appeals address the other as free beings, who decide out of desire, not necessity.

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Some people covet their freedom, but lack faith that freedom is theirs regardless of what is proven to them.

They see truth as a threat to their autonomy. And so they choose intellectual indecency.

And indeed, there are many people who cannot imagine voluntary agreement, who look for proofs that support their deepest convictions. They wish to convince others that their most important truths are the only valid truth, and to make others convicts of their own beloved beliefs. They attempt to argue others into submission.

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The easiest things to prove matter least.

The things that matter most are impossible to prove.

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If we wish to come to agreement on the things that matter most to us, we will have to rethink our notions of truth, and especially how this truth is established. We will not have certainty where we most desire it. We will never compel the types of agreements we crave.

, but we must come voluntarily and we must learn to invite others to agreement. As long as we will only come to an agreement when we cannot escape agreement — that is where we’ve been argued into submission —

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Compulsion by proof is only one mode of persuasion.

The modes of persuasion available to a person affects every aspect of his social existence. It determines the types of appeals he makes and accepts.