Why does "by an elementary valid argument" exist where the sentence is not passive voice?
How can I understand "by an elementary valid argument"?

We can define a formal proof, that a given argument is valid as a sequence of statements each of which follows from preceding statements of the sequence by an elementary valid argument.

Question

Why does "by an elementary valid argument" exist where the sentence is not passive voice?
How can I understand  "by an elementary valid argument"?

We can define a formal proof, that a given argument is valid as a sequence of statements each of which follows from preceding statements of the sequence by an elementary valid argument.

Answer

It exists because it is the *always true thing* that is linking one statement to the next, i.e. each statement connects from the preceding statement through this basic (=elementary) valid (=true) argument.