According to Fred Sommers (The Logic of Natural Language, p. 166), ". . . one way of saying what an atomic sentence is is to say that it is the kind of sentence that contains only categorematic expressions." Earlier in the same book, Sommers says this:
In Frege, the distinction between subjects and predicates is not due to any difference of syncategorematic elements since the basic subject-predicate propositions are devoid of such elements. In Frege, the difference between subject and predicate is a primitive difference between two kinds of categorematic expressions. (p. 17)
Examples of categorematic (non-logical) expressions are 'Socrates' and 'mammal.' Examples of syncategorematic (logical) expressions are 'not,' 'every,' and 'and.' As 'syn' suggests, the latter expressions are not semantic stand-alones, but have their meaning only together with categorematic expressions. Sommers puts it this way: "Categorematic expressions apply to things and states of affairs; syncategorematic expressions do not." (164)
At first I found it perfectly obvious that atomic sentences have only categorematic elements, but now I have doubts. Consider the atomic sentence 'Al is fat.' It is symbolized thusly: Fa. 'F' is a predicate expression the reference (Bedeutung) of which is a Fregean concept (Begriff) while 'a' is a subject-expression or name the reference of which is a Fregean object (Gegenstand). Both expressions are categorematic or 'non-logical.' Neither is syncategorematic. And there are supposed to be no syncategorematic elements in the sentence: there is just 'F' and 'a.'
But wait a minute! What about the immediate juxtaposition of 'F' and 'a' in that order? That juxtaposition is not nothing. It conveys something. It conveys that the referent of 'a' falls under the referent of 'F'. It conveys that the object a instantiates the concept F. I suggest that the juxtaposition of the two signs is a syncategorematic element. If this is right, then it is false that atomic sentence lack all syncategorematic elements.
Of course, there is no special sign for the immediate juxtaposition of 'F' and 'a' in 'Fa.' So I grant that there is no syncategorematic element if such an element must have its own separate and isolable sign. But there is no need for a separate sign; the immediate juxtaposition does the trick. The syncategorematic element is precisely the juxtaposition.
Please note that if there were no syncategorematic element in 'Fa' there would not be any sentence at all. A sentence is not a list. The sentence 'Fa' is not the list 'F, a.' A (declarative) sentence expresses a thought (Gedanke) which is its sense (Sinn). And its has a reference (Bedeutung), namely a truth value (Wahrheitswert). No list of words (or of anything else) expresses a thought or has a truth value. So a sentence is not a list of its constituent words. A sentence depends on its constituent words, but it is more than them. It is their unity.
So I say there must be a syncategorematic element in 'Fa' if it is to be a sentence. There is need of a copulative element to tie together subject and predicate. It follows that, pace Sommers, it is false that atomic sentences are devoid of syntagorematic elements.
Note what I am NOT saying. I am not saying that the copulative element in a sentence must be a separate sign such as 'is.' There is no need for the copulative 'is.' In standard English we say 'The sea is blue' not 'The sea blue.' But in Turkish one can say Deniz mavi and it is correct and intelligible. My point is not that we need the copulative 'is' as a separate sign but that we need a copulative element which, though it does not refer to anything, yet ties together subject and predicate. There must be some feature of the atomic sentence that functions as the copulative element, if not immediate juxtaposition then something else such as a font difference or color difference.
At his point I will be reminded that Frege's concepts (Begriffe) are unsaturated (ungesaettigt). They are 'gappy' or incomplete unlike objects. The incompleteness of concepts is reflected in the incompleteness of predicate expressions. Thus '. . . is fat' has a gap in it, a gap fit to accept a name such as 'Al' which has no gap. We can thus say that for Frege the copula is imported into the predicate. It might be thought that the gappiness of concepts and predicate expressions obviates the need for a copulative element in the sentence and in the corresponding Thought (Gedanke) or proposition.
But this would be a mistake. For even if predicate expressions and concepts are unsaturated, there is still a difference between a list and a sentence. The unsaturatedness of a concept merely means that it combines with an object without the need of a tertium quid. (If there were a third thing, then Bradley's regress would be up and running.) But to express that a concept is in fact instantiated by an object requires more than a listing of a concept-word (Begriffswort) and a name. There is need of a syncategorical element in the sentence.
So I conclude that if there are any atomic sentences, then they cannot contain only categorematic expressions.