Confounding use of genitive

[Scarlatti]

De Mendacio by St. Augustine:

"Unde si appareat fieri posse ut aliquis propterea falsum dicat, ne fallatur ille cui dicitur; existit aliud e contrario genus, propterea verum dicentis ut fallat."

Why is the present participle "dicentis" in the genitive? I don't understand why the last clause doesn't just read "Propterea verum dicens ut fallat." I feel like I have gone through every possible solution in my mind to no avail.

1) If it is connected to "verum," whether verum is a noun or adverb, it doesn't make sense. We get "a fact of speaking" or "truly of speaking."

2) Does propterea take the genitive? "For the reason of saying the truth to deceive"? This fits the translation and makes sense, but I can find no proof that propterea ever takes the genitive, and it seems to fly in the face of "propterea quod."

3) Is "dicentis" actually modified by "genus" from the preceding clause? That fits the translation best of all, perhaps. I don't know, my brain has frozen. But can a word from one clause modify a word from a subsequent clause without a relative pronoun?

I wonder if I am just messed up on the preceding clause and that is why I can't figure it out. "Existit aliud ex contrario genus..." Genus agrees with aliud, right? It means "There exists on the contrary another category [ of person ]."

[Scarlatti]

Here's another use of a genitive present participle in the same book that I don't get. Clearly this has something to do with the present participle!

Porro si mendacium est enuntiatio falsum enuntiare volentis ut fallat.

I can't translate this like I can other genitives. Mater Dei. Mother of God. To God belongs the mother. Liber puerorum. Book of the children. To the children belongs the book. Falsum enuntiare volentis. The willing of speaking an untruth. To the willing belongs the speaking of an untruth? Why? Or is there an implied PERSON WHO IS WILLING? This is how it is translated.

I'm sorry if I'm not making sense, something is not clicking for me with this genitive present participle and I don't know why. If there is anything special about it, or if it needs specific explanation, I'd be obliged.

[Alatius]

3) Is "dicentis" actually modified by "genus" from the preceding clause? That fits the translation best of all, perhaps. I don't know, my brain has frozen. But can a word from one clause modify a word from a subsequent clause without a relative pronoun?

I think this is the correct explanation. There exists a different kind, [that] of the one speaking thruth in order to deceive. Note that we are not dealing with two different clauses, only one: dicentis is obviously not a finite verb.

[Alatius]

Porro si mendacium est enuntiatio falsum enuntiare volentis ut fallat.
... Or is there an implied PERSON WHO IS WILLING? This is how it is translated.

Yeah, participles are often treated as if they were nouns:
volens = someone who wants
falsum enuntiare volens ut fallat = someone/the one who wants to speak false in order to deceive.
enuntiatio falsum enuntiare volentis ut fallat = the declaration of someone who... etc.