The problem is if they believe what I said

Question

"The problem is whether they believe what I said." is a correct sentence. How about "The problem is if they believe what I said."? This sentence is less commonly used but still OK?

Answer 1

Your first sentence involves three ellipses. One of these is implicit in the whether construction:

... whether they believe what I said or not.

The other omissions are not implicit in the sentence; they must be inferred by your hearer or reader from the discourse context. For instance, your particular example might represent:

The problem [for us] is [to discover] whether they believe what I said.
The problem [for them] is [to decide] whether they believe what I said.

In this sort of whether clause^† it is often possible to substitute if for whether—when we are speaking of a question, for instance:

The question is whether they believe what I said (or not).
The question if if they believe what I said (or not).

At first glance, there is no reason that if could not be substituted in your question:

The problem for us is to discover if they believe what I said.

I think most NS would agree that that is a well-formed sentence. However, it appears that this sentence does not occur, or occurs only very rarely, with ellipsis of the term represented here by to discover. I have just gone through the first hundred Google hits on “the problem is if”, and not one has the form of your sentence.

Why? I have a guess, but it is no more than that.

I think it is because the problem is if has a competing use in which the if heads a conditional clause which defines the ‘problem’ as contingent.

I’ve just told them that it wasn’t me who smashed up their car but Joe. The problem is if they believe what I said they’re probably going to go beat Joe to a bloody pulp.

The sequence The problem is if leads hearers and readers to expect a sentence like that; so to avoid confusion, speakers and writers avoid it. That doesn’t mean it’s wrong or ungrammatical; just that there is good reason for you to avoid it, too.

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^† Some linguists distinguish this sort of Wh- clause from other sorts and call it an “embedded question”.