I used context as I insist it must be preserved to maintain true meaning. The two variables are as connected as form, matter, and intent are when dealing with a sacrament.
But this particular line of Trent admits of two meanings. It's ambiguous, in and of itself. If someone says this teaches BOD, they're putting an interpretation on the text. Likewise, if someone claims this sentence excludes BOD, they're putting an interpretation on the text. You saying the "two variables" must be connected is an effect of such an imposition. You're requiring that any "analogy" have a form in which both of the "two variables" are required. You reject the "driving or walking" analogy because it doesn't fit what you appear to want this sentence to say. And yes, of course "driving or walking" do not relate to each other in precisely and exactly the same way that "laver or desire" relate to each other, but that wasn't the point of the analogy. (There's a saying that analogies limp except on the point of the comparison.)
So, OK, we could look at analogies that compare grammatically to
"Justification cannot be had without the laver or the desire".
Note that the subject is justification - a grace, not a sacrament, and not sacramental character.
What we would have is a sentence like
"The graces of a sacrament cannot be had without the sacrament or the desire"
and this sentence would supposedly mean that both the sacrament and the desire are required.
Penance: Can there be no forgiveness without the sacrament of penance in re? No perfect contrition?
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Confirmation: Do the gifts of the holy ghost not exist in the soul without the sacrament of confirmation?
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Communion: Are the fruits of communion possible through "spiritual communion" at least in some limited way?
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Marriage: Since the couple are the ministers, one could view a desire for the sacrament, with appropriate external conditions, as the sacrament itself. So the sacrament and the desire are arguably the same.
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Last rites: not sure but should be similar to penance.
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Holy orders. Obviously, desire does not confer the character or the power to offer mass, but I don't see why other graces could not flow based on a "desire" without reception of the sacrament.
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Your interpretation for baptism wouldn't fit any other sacrament. I can see potential reasons baptism could be entirely different, but it still means analogies with other sacraments do not appear to support your view.
Whose intent depends of course on which sacrament is in question.
Yes, but this discussion is about baptism. If both the sacrament in re, AND the votum, are required for justification, then something should be said about infant baptisms. Do infants express a desire for the sacrament?
And if you say the votum can be in the sponsors, does that mean a third party impacts validity?