Let's say we have some given schema within a logical system which is such that there are an indefinite number of variables given (whether this is an axiom-schema or a wff-schema or something else is irrelevant).
So, for example, let's say we had a schema, something like: ($x_1$,$x_2$,$x_3$,...,$x_n$) and it was specified that the $x_i$'s are such and such (let's just say they are positive integers).
The question is how does one define what an instance of this schema is/would it even be necessary?
I can't even recall the first time I had seen the use of a letter like 'n' in $x_n$ come into play nor can I recall of any mathematical/logical texts detailing precisely what exactly its role is..even though we all understand it.
The point being that while we understand that (2,82,523,5,32) would be an instance of the schema, we understand this only because we already know that the '...,$x_n$' in the schema tells us that an instance will be such that there are a finite number of positive integers, each of which are separated by a comma...but as to how many there will be, well this is undetermined. (Although, a question could arise of whether the schema says that there must be at least 3 positive integers)
Would this information need to be laid out in the presentation of the particular system and if so, what would be the most rigorous way? Any example texts?
Also, while trying to look up an answer to this question I came across this: https://books.google.com/books?id=aJTTBwAAQBAJ&pg=PA101&lpg=PA101&dq=logic+algorithm+schema&source=bl&ots=Q2W1MA7U_e&sig=yipc1SAAGnGPrPtV9fXA6zV5V0U&hl=en&sa=X&ved=0ahUKEwjW0LaC68HOAhUGbB4KHcsPAzsQ6AEIITAA#v=onepage&q=logic%20algorithm%20schema&f=false
On page 102 the author talks of "schema-variables" and "wff-schemas." Are these common terms/are there any other texts on them anywhere?
Basically, the question can be summed up like this:
Let's assume we have a formal system where a wff will be defined in the following way (note, that these wffs may seem useless is irrelevant, the point has to do with how to define an instance of a particular kind of schema):
"Any instance of the following schema is a wff
where the $x_i$'s are positive integers."
The question is whether that would be an adequate definition of a wff for the system or whether one would need to define "instance" more precisely, and if so how would that be done?
And my reference request has to do with texts dealing with a systemic logical approach to the usage of letters when presenting an undermined number of variables.