This is not the first time I have encountered this way of thinking that you apply here. It is as if this was pure mathematics, and we are all expected to have autism, or even to be meat-based mechanical devices. I know that mathematicians often attempt to speak in such a manner. So the speaker is permitted to utter no implications, and the hearer is permitted to make no inferences; it is as if we are all computers, but I long ago decided that this is simply not tenable. Perhaps we are limited to using set theory, or first order logic? But set theory was literally born bearing the burden of dealing with “uncountable” numbers, and first order logic cannot do all that higher order logic can.

We have seen the counting numbers expressed as an endless sequence of embedded sets (or even lambdas!). But with higher order logic, predicates can be elements of higher predicates. So if the number 7 is defined as a second order predicate, we can express that some predicate P bears seven member elements by simply writing =// 7P //=, where 7 is just a second order predicate. And then this “implies” that (using =// _E //= to indicate the “existential operator”) =// _E_e( P ) //= ; which is to say that there exists at least one e that is a member of (first order) P (that e being one of 7 in this example). But higher order logic suffers from a Cassandra syndrome, and seems to tell people things they don’t want to think about.

So I suspect there will be some “informal” implications here, by necessity. A ballot is a piece of paper (or at least the representation of such) upon which electors can vote for political candidates. Certain (sometimes untrustworthy) officials decide which candidates will be “on” this ballot, and hopefully electors can vote for write-in candidates that the officials did not place “on” the ballot. So let us consider two kinds of ballots, “one to ten” and “zero to nine.” They are:

=// Joe Blow: (1), (2), (3), (4), (5), (6), (7), (8), (9), (10) //=

=// Joe Blow: (0), (1), (2), (3), (4), (5), (6), (7), (8), (9) //=

With the first, “one to ten” ballot, the voter has exactly eleven options; (1) to (10), plus abstention.

With the second, “zero to nine” ballot, the voter has exactly eleven options; (0) to (9), plus abstention.

With the first kind of ballot the significance of abstention is obvious. With the second kind, the significance of abstention remains to be determined by officials. Plus, as mentioned above, “one to ten” ballots are significantly easier for voters to reason about.

Have I answered the question?