Spurred on by this, I’ve done some more research. I think I’ve found a definition of the sort they might have in mind in the appendix to John Lyons’ Chomsky (Harvester Press, 1977). It is a definition of ‘sentence’ for the purposes of formal syntax and it doesn’t depend on any hidden semantic terms. A sentence is a finite sequence, or string, of elements drawn from a finite vocabulary (p. 157), where a vocabulary is a non-empty set of word types. The same finite vocabulary underlies a vast and diverse class of possible strings formed from that vocabulary. Subsets of this class are the possible languages based on this vocabulary. The grammatically well-formed sentences of a language are all and only those sentences in that language (so an ungrammatical sentence isn’t a sentence of that language at all). The task of the formal syntactician is to write down the formal grammatical rules that generate the sentences of one of these languages. And assuming that natural languages (from particular idiolects or national languages) are formal languages, to set down what grammatical rules generate that natural language. Additionally, grammarians should seek out and locate those grammatical features which are universal (common to all human languages)
The existence of this style of definition of a sentence doesn’t affect the philosophical point I would make about the respective statuses of syntax and semantics (namely that semantics has ultimate priority) because the definition presupposes a lot of interpretative stage-setting before it is applicable.
Consider a formal analogy. We could call it ‘formal architecture’. We could offer a definition of a building formally analogous to the definition of sentence used in formal syntax. A ‘building’, on this view, would be a finite construction out of elements from a finite set of building materials (slabs, blocks, etc, … and bricks). Architectural styles (modernist, art-deco, Gothic, Classical, and so on) correspond in formal architecture to languages. They are classes of possible buildings. The formal architect’s goal would be to codify the rules that generate exactly the buildings of particular architectural styles and possibly identify universal structural features of any architectural style.
Now assuming the formal parallel holds, it is clear that the definitions of ‘sentence’ and ‘building’ in this formal sense fall a long way short of saying what a sentence or a building is. Because the formal details are parallel, the only things that distinguish sentences and buildings are not included in the definition. For all the definitions say, sentences are buildings, which means that neither sentences nor buildings (so-defined) are sentences or buildings as we know them but are rather more abstract things.
If we want to know the nature of what commonsense calls sentences, we need to appeal to something besides their formal structure. And I suggest this is where semantic and pragmatic considerations are relevant. A sentence is something that can be true or false, whereas a building cannot. A building can keep the rain off your head, but a sentence cannot. And so on. Attending to these differences is part of identifying the subject matter of linguistics. Formal syntax as opposed to formal architecture cannot even get started without appeal to semantic considerations.