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Excellent explanation.

 ... it's utterly lame on all counts.  Trent's narrative describing the process of justification is clearly describing the process as it leads to justification.  It doesn't use the expression "may" at every step.  Then the soul "may" conceive contrition for past sins and the soul "may" made the determination to receive the Sacrament.  That's nonsense.  Obviously if any step described by Trent doesn't happy, then there's no justification.

As for the "very dense, verbose explanation", that's simply due to Trent wanting to rule out 1) all the Protestant errors AND 2) all the Protestant strawmen in mischaracterizing Catholic teaching.

I think I made a mistake here about the 'negative'. The statement is negative not because of 'without' but because of 'anathema'. I think I'm starting to confuse myself.

Really, the problem with the Canons is that they're not explicit about wihich (benefits of the) Sacraments are available  or accessible through votum.  We know that it taught this about the Sacrament of Confession.  But the Canons lump all the Sacraments together.

What's at issue is with "without A or B" construct.  It's ambiguous on its own and could be read the BoD way or could be read the non-BoD way.

But my chief arguments for the non-BoD way are:

1) the citation from Sacred Scripture from Our Lord used as proof text, where He teaches that water AND the Holy Ghost are necessary
2) the logical corollary of the BoD reading would be that justification can be received WITHOUT the Sacrament of Baptism, and that expression would promote the exact same heresy that Trent is condemning

These two considerations render the BoD understanding of the passage impossible.

"No X without A or B" ... this can mean "No X without (A or B)." or it can mean "No X without A or without B."  We've seen examples of both.

I can't write a letter without a pen or a pencil.  Means that either one suffices.  (equivalent of the BoDer reading of Trent)
No wedding without a bride or a groom.  Means that both are necessary and that there can be no wedding if either one is missing.  (equivalent of the non-BoDer reading of Trent)

Context would be required to determine the meaning.  Dimond Brothers sent this passage to a Latin scholar at Oxford, who concurred that either meaning is possible, and that the correct meaning can be known only from context or an individual's prior knowledge.

I know that "I can't write a letter without a pen or a pencil." means that I need only one or the other (not both) because of my prior knowledge of letter writing and how pens and pencils relate to that.  Same with the bride and groom reading.  I know that both are required for the wedding because I have prior knowledge of what a wedding is.

There's also context.  So take this example.  Assume that I know absolutely nothing about baseball.  Never heard of it before.

"We can't play baseball without a bat or a ball."  Hmmm.  Does this mean that I can play if I have EITHER a bat OR a ball or does it mean that I can't play unless I have both of them?  I don't know, because I don't have knowledge or context.

Now let's add some context.  "We can't play baseball without a bat or a ball, since Jim told me that we need a bat and a ball to play baseball."

But this is precisely what Trent is doing.  Trent immediately disambiguates the passage by citing the text from the Gospel as "proof text" (that's what the phrase, "as it is written" means).

[paraphrase] "Justification can't happen without the laver (water) or the votum (Holy Ghost), because Jesus taught that both water and the Holy Ghost are required."

We could no more read this as meaning "Justification can't happen without EITHER laver/water OR the votum." than we could read "We can't play baseball without a bat or a ball, since Jim told me that we need a bat and a ball to play baseball." as meaning that we can play baseball with either a bat or a ball.

So this is reason 1 why the BoDer reading doesn't work, the disambiguating proof text provided by Trent.

For reason number 2, if you say that ...
Justification cannot happen without EITHER the laver (the Sacrament) OR ELSE the desire for it, the logical corollay is:  "Justification can happen WITHOUT the laver, without the Sacrament of Baptism."  If I say, "I can't write a letter without a pen or a pencil." ... can I write a letter without a pen?  Yes, yes I can ... if I have a pencil.  This expression means that I CAN write a letter without a pen.  And I CAN write a letter without a pencil.

That's just plain heretical, to assert that I CAN be justified WITHOUT the Sacrament, and this is in fact the same heresy that Trent is condemning by teaching dogmatically that the Sacrament is necessary for salvation.  Even with BoD, if you believe in it, justification does not and cannot happen WITHOUT the Sacrament of Baptism.  Even in BoD, the Sacrament remains the instrumentcal cause of justification.  Otherwise, you're a Pelagian who holds that the votum itself, without the Sacrament, can justify, effectively ex opere operantis, i.e. that you can will your own justification and salvation.

So for both of these very compelling reasons, the BoDer reading of Trent is absolutely untenable.