1. If there is nothing to show that the writer considers the doubt an unreasonable one, the word is always 'whether', which reminds us that there is a suppressed alternative:
   
   

   I doubt whether this is true (or not).
   Every one is at liberty to doubt whether ... (or not).

   
   
   To this part of the rule there is no exception.
   
   
2. If it is evident that the writer disapproves of the doubt, the words introducing it amount to an affirmation on his part that the thing doubted is undoubtedly true; the alternative is no longer offered; 'that' is therefore the word:
   
   

   I do not doubt that (i. e., I am sure that)...
   Who can doubt that...?

   
   
   This, however, is modified by 3.
   
   
3. The 'vivid' use of 'whether'. When the writer's point is rather the extravagance of the doubt than the truth of the thing doubted, 'whether' is often retained:
   
   

   It is as if a man should doubt whether he has a head on his shoulders.
   Can we imagine any man seriously doubting whether...?

   
   
   Here, according to 2., we ought to have 'that', since the writer evidently regards the doubt as absurd. But in the first sentence it is necessary for the force of the illustration that the deplorable condition of the doubter's mind should be vividly portrayed: accordingly, he is represented to us as actually handling the two alternatives. Similarly, in the second, we are invited to picture to ourselves, if we can, a hesitation so ludicrous in the writer's opinion. We shall illustrate this point further by a couple of sentences in which again the state of mind of the doubter, not the truth of the thing doubted, is clearly the point, but in which 'that' has been improperly substituted for the vivid 'whether':
   
   

   She found herself wondering at the breath she drew, doubting that another would follow.—Meredith.
   
   I am afraid that you will become so afraid of men's motives as to doubt that any one can be honest.—Trollope.

   
   
   The mistake commonly made is to use 'that' for 'whether' in violation of 1. 'Whether' is seldom used in place of 'that', and apparent violations of 2. often prove to be legitimate exceptions of the 'vivid' kind. Some of our examples may suggest that when the dependent clause is placed before the verb, 'that' appears because the writer had not decided what verb of doubt or denial to use. This is probably the true explanation of many incorrect thats, but is not a sufficient defence. It supplies, on the contrary, an additional reason for adhering to 'whether': the reader is either actually misled or at any rate kept in needless suspense as to what is going to be said, because the writer did not make up his mind at the right time how to say it. 'Whether' at the beginning at once proclaims an open question: after 'that' we expect (or ought to expect) 'I have no reason to doubt'.

There is nothing for it but to doubt such diseases exist.—H. G. Wells.

I do not think it would have pleased Mr. Thackeray; and to doubt that he would have wished to see it carried out determines my view of the matter.—Greenwood.

That the movement is as purely industrial as the leaders of the strike claim may be doubted.—Times.

And I must be allowed to doubt that there is any class who deliberately omit...—Times.

He may doubt that his policy will be any more popular in England a year or two hence than it is now.—Greenwood.

I doubt the correctness of the assertion... I doubt, I say, that Becky would have selected either of these young men.—Thackeray.

But that his army, if it retreats, will carry with it all its guns ... we are inclined to doubt.—Times.

It was generally doubted that France would permit the use of her port.—Times.