Now I claim that in systems where there is no distinction between predicate and sentence negation, we have ‘direct reference’. This is easily shown. Direct reference in a singular sentence is when the sentence is meaningless when the singular subject fails to refer. Assume that ‘a is F’ is not meaningless. If it is true, then there is a referent for ‘a’. If it is not true, the sentential negation ‘It is not the case that a is F’ is true. If sentential negation is equivalent to predicate negation, it follows that ‘a is non-F’ is true, and so a exists, and so, there is a referent for ‘a’. But (by excluded middle) either ‘a is F’ is true, or its contradictory (the sentential negation) is true. In either case, ‘a’ has a referent. Thus if ‘a is F’ is not meaningless, ‘a’ has a referent. Conversely if ‘a’ does not have a referent, ‘a is F’ is meaningless. But that is Direct Reference, as I have defined it.
This reasoning strikes me as correct. The notion of an atomic sentence is foundational for modern predicate logic (MPL). For such sentences there is no distinction between predicate and sentence negation. And given Edward's definition of 'direct reference,' I am persuaded that MPL entails direct refence for the subject terms of atomic sentences.
Note that I am substituting 'atomic sentence' for Edward's 'singular sentence.' Every atomic sentence (whether monadic or relational) is singular, but not every grammatically singular sentence is atomic. Or at least that is not obviously the case. Thus it is far from obvious that 'Peter smokes,' which is grammatically singular, is logically atomic. If one holds, with Russell, that grammatically proper names are definite descriptions in disguise, then the grammatically singular 'Peter smokes' will have an analysis that is existentially general.
Correct me if I'm wrong, but the subject term of an atomic sentence is what Russell called a logically proper name and distinguished from a grammatically proper name. So what Edward has shown is that Direct Reference holds for logically proper names. But this does not show that Direct Reference holds for ordinary names, grammatically proper names, such as 'Edward' in 'Edward is English' or 'Peter' in 'Peter smokes.'
Direct Reference for grammatically proper names, whether nonvacuous or vacuous (e.g. 'Vulcan') is false. But as far as I can see MPL is not committed to Direct Reference for such names. So while I am persuaded by Edward's reasoning above, I am not sure what its relevance is.