(1) Who wills the Good for the sake of a reward is double-minded.
The person who wills the Good for the sake of some other reward is not willing the Good as one thing, but willing two things. He does not will the Good in truth. Kierkegaard here, without mentioning him by name, appeals to Socrates in his arguments against sophists and orators:
In ancient times, this problem was also frequently an object of consideration. There were shameless teachers of impudence who thought it right to do wrong on a large scale and then to make it appear as if one willed the Good....But in ancient times there was also a simple sage, whose simplicity became a snare for the impudent ones' sophistry. He taught that in order really to be certain that it as the Good that man willed, one ought even to shun seeming good, presumably in order tha thte reward should not become tempting. For so different is the Good and the reward, when the reward is separately striven after, that the Good is the ennobling and sanctifying; the reward is the tempting. (pp. 69-70)
We aren't tempted unless there is some reward to be had; and if we will the Good in order to have that reward, then even though we are willing the Good, we are succombing to temptation. And we should be cautious even with regard to the possibility of succombing to temptation. Kierkegaard uses the example of lovers. A man who loves a woman because she is wealthy is not a true lover. When we deal with someone who loves a woman and thinks that her wealth is also a good thing, we would urge at least caution; the influence of the latter can make the former seem greater than it is. How do we know for sure when we have a true lover? When he loves the woman despite her wealth, in the sense that he could sincerely wish that she wasn't wealthy so that it would not cloud the issue.
It's important to be clear that, despite the language bordering on quietism (to which Kierkegaard is arguably very much inclined), Kierkegaard is not saying that everything that could be called a reward is ruled out. For instance, there can be a sense in which we might say that the Good carries a reward in itself, or that there is something rewarding about the Good itself; he himself talks about the reward that is added to the Good for all eternity "in the internal realm" as "homogeneous with the Good" (p. 74). The reward Kierkegaard has in mind here is that which is external to the Good, the kind that can come or go. It is this that allows the possibility of self-deception and therefore double-mindedness: we fool ourselves into thinking that we love the Good even though we really love the reward (like seeming good to others), just like the man can fool himself into thinking that he loves the woman even though he really loves the idea of being wealthy with her. The man can deceive the woman, too. But the Good is never deceived.
(2) Who wills the Good out of fear of punishment is double-minded.
The obverse side of willing the Good to get a reward is willing the Good to avoid a punishment; both indicate a reward-centered mind. Kierkegaard's argument is Socratic here, too:
He should fear to do wrong. But if he has done wrong, then he must, if he really wills one thing and sincerely wills the Good, desire to be punished, that hte punishment may heal him just as medicine heals the sick. If one who is sick fears the bitterness of the medicine, or fears "to let mhimself be cut and cauterized by teh physician," then what he really fears is--to get well.... (p. 79)
This argument is straight from Plato's Gorgias (although it is not explicitly mentioned), which is the source of the quotation; although the passages in the reward are not so telling, it was probably also the Gorgias that Kierkegaard primarily had in mind when he was discussing reward. Kierkegaard recognizes that it is a spiritual illness not to fear what you should fear; punishment, Platonically considered, is a medicine, and medicines are genuinely dangerous. But it is a worse spiritual illness to fear what you should not fear, such as medicine when you need it. Kierkegaard uses several analogies here. Fear of being poor can make a man of miser; it doesn't, however, give him the virtue of thrift. Fear of sexual disease can make a man a more moderate debauchee, but it never makes him chaste. Likewise, fear of punishment can make sinners hypocrites, but it can never make them pure of heart. Why? The problem is that they only will the Good for avoiding punishment. The miser's life makes a mockery of thrift, the moderated debauchee's life makes a mockery of chastity, and the sinner's attempt to avoid the hardship of seeming to be a sinner makes a mockery of the Good.
It is extraordinarily important, however, to be clear that here, as with reward, Kierkegaard is not saying that all things that might be called punishment are not to be feared; one could for instance call the loss of the Good a punishment. Indeed, he criticizes people who are squeamish about raising the possibility of an eternal punishment, because this, too, puts the Good under a condition and therefore makes it impossible to will it single-heartedly: such people who will the Good only under the condition of "if there is no eternal punishment" or "if I can escape eternal punishment" are as double-minded as anyone who focuses only on temporal punishment, and deceive themselves into thinking that they are willing the Good without fear of punishment. But if eternal punishment matters to how and what you would will where the Good is concerned, how can your willing of the Good not be something conditioned by fear of punishment? And regardless of the form, fear is an unsafe assistance to willing the Good: "Only one thing can help a man will the Good in truth: the Good itself" (p. 84). And who wills the Good in truth will accept, even hope for, whatever punishment the Good requires to waken him and lead him back to the Good when he strays.
(3) Who wills the Good so that it may be victorious through him is double-minded.
Far more subtle than the previous two is the person who wills the Good so as to be victorious, who takes the Good as a means for his own conquest or dominance. He wills to be the instrument of the Good, the chosen instrument, the one through whom the Good gets its victory. But the Good has eternal victory in and of itself. The reason this double-mindedness can enter into the picture is that the manifestation of this victory in time is very slow in coming. Over and over again people devoted to the Good die and it seems that they have managed to accomplish very little. Thus men charge around doing things, allegedly for the sake of the Good, as if the victory of the Good depended on it, and this, which is really the vice of impatience, they mistake for something more pure, namely, enthusiastic willing of the Good. The Good can only be willed in truth if it is willed patiently:
The Good puts on the slowness of time as a poor garment, and in keeping with this change of dress one who serves it must be clothed in the insignificant figure of the unprofitable servant. With the eye of his senses he is not permitted to see the Good in victory. Only with the eye of faith can he strive after its eternal victory. (p.103)
We can only will the Good when we no longer take it to be about us; when we will the Good not from self-assertion but simply because it alone is the Good itself. Once we, in our minds, split off the victory of the Good from the Good itself, we begin to deceive ourselves and become double-minded.
(4) Who wills the Good only up to a certain degree is double-minded.
This is a harsher sentence yet. The world is a busy place, with so many calls on our attention. Thus we rush around, distracted, and thereby enters in the danger of self-deception and therefore double-mindedness, because we can come to love the Good but only in a shallow way. When we hear of the Good, we may warm to it; we may be gripped in imagination and passion by talk of it. But we do not will it singleheartedly if we let the busy buzz of our lives give us excuses not to act on our love of it. Rather, we must recognize that there are no such excuses; such excuses are mere self-deceptions. As with the previous case, there is often an impatience that goes with this one, and we find ourselves with this double-mindedness when we try to reform, recognizing the medicine of punishment, accepting it, but, when we begin to be healed we simply step back out of the world and stop what we were doing. We are like patients who are continually sick because, though we go to the physician and trust the treatment, it is only up to the point; the doctor insists that we must be on the medicine for several more weeks or the sickness will come back with a resistance, but we feel fine, so we stop taking the medicine, and, sure enough, become sick again. And we do it over and over and over. Kierkegaard calls this the doubleness born of weakness, and says that it is the most common form of double-mindedness; we will the Good sincerely, but not with purity of heart.
And thus we come to the next stage of Purity of Heart Is to Will One Thing, namely, the answer to the question of what is required to have purity of heart.
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Soren Kierkegaard, Purity of Heart Is to Will One Thing, Steere, tr. Harper (1956).