Quo negato & contrario illius admisso

Question

How would I translate the major premise of this syllogism:

1. Dogma, quo negato, & contrario illius admisso, omnium adhortationum ad perseverandum in fide, comminationum si non perseveremus, promissionum si perseveremus, quas Deus in verbo suo proponit & serio urget , vis atque efficacia perit, planeque evertitur, illud verissimum & certissimum est, ac proinde religiose credendum.
2. Vere fideles a fide & justitia, sive obedientia posse totaliter atque finaliter deficere, est tale dogma, quo negato etc.
3. Ergo.

My best attempt at translation:

Literal:

1. The dogma, wherein having been denied and its contrary admitted, the force and efficacy of all the exhortations to perseverance in faith which God sets forth in His word--threats if we do not persevere and promises if we do persevere--vanishes and is plainly overthrown, is most true and certain, and should hence be religiously believed.

Reworked and rearranged:

That dogma, the denial of which and the admission of its contrary plainly overthrows and brings to nothing the force and efficacy of all the exhortations to perseverance in faith--threats if we do not persevere and promises if we do persevere--which God sets forth and seriously presses in His word, is most true and certain, and should hence be religiously believed.

2. That true believers can totally and finally defect from faith and righteousness, or obedience, is such a dogma, the denial of which etc. . .

How should that phrase, "Dogma, quo negato, & contrario illius admisso," be translated? Literally, it seems to be something like: "The dogma, in which it is denied and [in which] its contrary is admitted. Am I more or less accurate here?

Answer 1

Quo is an ordinary relative pronoun (neut. sg. abl.) referring to dogma. It introduces a relative clause whose verbs are perit and evertitur. Within that clause, quo negato and contrario admisso are ablatives absolutes. If we interpret them temporally, we could say: "when it [the dogma] has been denied and the opposite has been granted."

(Small quibble: illius is a really strange choice here in my opinion. I would have written cuius.)

We cannot translate this relative construction literally because relative clauses can do things in Latin that they simply cannot do in English. But you could say something like:

A dogma that is such that, when it has been denied and its opposite granted, the force and efficacy of all the exhortations to perseverance in faith which God sets forth in His word, and earnestly urges, of the threats if we do not persevere and the promises if we do persevere, vanishes and is plainly overthrown, that dogma is most true and certain, and should hence be religiously believed.