Eusebius HE 3.39.15-16 on Papias:
But we are now obliged to append to the words already quoted from him a tradition about the Mark who wrote the gospel, which he [Papias] expounds as follows:
“And the Presbyter used to say this, ‘Mark became Peter’s interpreter [or translator] and wrote accurately all that he remembered, not, indeed, of the things said or done by the Lord. For he had not heard the Lord, nor had he followed him, but later on, as I said, followed Peter, who used to give teaching as necessity demanded, but not making, as it were, an arrangement of the Lord’s logia, so that Mark did nothing wrong in thus writing down single points as he remembered them. For to one thing he gave attention, to leave out nothing of what he had heard and to make no false statements in them.”
This is related about Mark, and about Matthew this was said:
Matthew arranged the logia in the Hebrew dialect, and each translated [or interpreted] them as best he could.
At present, I would interpret Papias’ remarks on the composition of Mark and Matthew as follows:
Papias is using “the logia” to refer to the material about Jesus, both words and deeds, which comprise the gospel. You could substitute the word gospel for the word logia in Eusebius’ quotations of Papias above, though the gospel/the logia/the sacred tradition would be broader than only that subset included in Mark and Matthew.
Papias is using Mark and Matthew to refer to the authors of the Greek gospels we know as “The Gospel According to Mark” and “The Gospel According to Matthew.”
The issue Papias is addressing is why, if both Mark and Matthew are The Gospel, they differ in wording and order.
Papias’ explanation of why Mark and Matthew differ order is that Mark wrote down Peter’s teaching, which was not put in order but given as occasion demanded, and that Mark wrote it down as he remembered it, omitting nothing. Matthew, on the other hand, made an arrangement of the logia (as Mark had not) in the Hebrew dialect.
Papias’ explanation of why Mark and Matthew differ in wording is that: “each [of the evangelists Papias has been discussing, Mark and Matthew] translated them [the logia about the Lord] as best he could.”
I’m also supposing that Eusebius’ quotation of Papias on Matthew followed immediately, or very soon after, his quotation of Papias about Mark, so that “each” may refer to the two evangelists. I can’t of course, prove this, but there’s no evidence against it either. I’m interpreting evidence we actually have.
I’m rejecting at least two common interpretations of Papias:
Papias’ logia were sayings (which would be logoi) or oracles, in the sense of prophecies about the future (I would suggest logia meant something more general, like “divine communications” ).
Papias actually had good historical information about how Mark and Matthew wrote their gospels. On the contrary, I’m suggesting that Papias (or the Elder on whom he depends) did not have good information about how the gospels came to be written, but is trying to deduce the reasons for their differences in wording and order in their extant Greek versions. I’m not particularly going out on a limb in saying this, as he has missed the fact (as I take it to be) that Matthew is dependent on mark in Greek. More radically, I would suggest that Papias had no direct knowledge of a Hebrew Matthew, and, moreover, is not saying that such a document ever circulated, only that Matthew put the Hebrew logia in order (prior to the time he translated it into Greek) unlike Peter who gave teaching as occasion required.