Let us say that a sentence is periphrastic if and only if there is a single word in that sentence such that we can remove the word and the result (i) is grammatical, and (ii) has the same truth value as the original sentence. For example:
 Roy murdered someone.
is periphrastic, since it is equivalent to:
 Roy murdered.
Thus, a sentence is not periphrastic if, for any word in the sentence, the result of removing the word is not grammatical, or the result of removing that word has a different truth value.
It should be noted that I am introducing “periphrastic” as a technical term here, and its use as defined above is different from (but connected to) its meaning in everyday English (further, the meaning here is significantly different from its technical meaning in grammar).
The notion of a sentence being periphrastic in this sense is simple, and at first glance we might think that we will always be able to determine whether a sentence is periphrastic merely by checking all of the sentences that can be obtained by removing one word. But like many other simple notions such as truth and knowability, it leads to puzzles – in this case, puzzles very similar to the truth-teller:
This sentence is true.
Consider the following sentence:
 I am periphrastic.
(We assume here that “I” is an informal way for a sentence to refer to itself).
Now,  is either periphrastic or not. But there seems to be no way to determine which it is.
If  is periphrastic, then there must be some word that we can remove from  such that the result is grammatical and true (since if  is periphrastic then, since that is what it says, it is true). The only way to remove a single word from  and obtain a grammatical result is:
 I am.
This sentence states that  exists, and is clearly true. Thus, the claim that  is periphrastic (and hence true) is completely consistent.
If  is not periphrastic, however, then the result of removing any word must either be ungrammatical or true (since  says that it is periphrastic, and thus in this case  is false). But, again, the only way to remove a single word from  and obtain a grammatical result is again , which is true. Thus, the claim that  is not periphrastic (and hence false) is completely consistent.
There would seem to be no other evidence that could settle the matter. Thus, even though it is obvious that  is either periphrastic or it is not, determining which seems impossible.
Interestingly, although most notions that allow for the construction of a truth-teller type puzzle also admit of a Liar-like paradoxical construction, I have failed to find an example of a paradoxical sentence that involves the idea of sentences being periphrastic. The obvious candidate to look at is:
 I am not periphrastic.
But there is nothing paradoxical about  – it is perfectly consistent to assume that  is false, and hence (contrary to what it says) periphrastic, since:
 I am not.
is a sentence obtained from  via the deletion of a single word that then has the same truth value as  – they are both false.
Perhaps my readers can do better. Is there a clearly paradoxical sentence that can be constructed using the notion of a sentence being periphrastic?
Featured image: Pieces Of The Puzzle by Hans. Public domain via Pixabay.