He does not seem to understand what we believe

It appears that Turretinfan is declining Dave Armstrong's debate challenge/offer (well, maybe not quite in so many words, but personally I'd say that's the substance of it). I confess to being disappointed, but only in the sense of an unfulfilled prospect for a spectacle. Frankly I don't think debates are worth the time or energy, although some may judge my opinion by the fact that I don't feel particularly qualified to engage in them. In the first place, debates don't "prove" anything, particularly if one participant is especially stronger than the other. But even granting equal capacities on both sides, it seems to me that their usefulness is limited: truth is not contingent upon the outcome of a debate.

Mr. Armstrong is a big boy and can certainly handle the parts of Turretinfan's response that are directed towards him. There are some things there, though, that catch my eye and seem worthy of a paragraph or three here.

Trent denied Sola Gratia by Denying Sola Fide

His objections notwithstanding, this is precisely the sort of thing we are talking about when we say some Protestant or other doesn't understand what we believe. At the same time, too, this statement is dependent upon a question-begging definition of "sola gratia" that not only hasn't been established, but is also entirely unnecessary.

As to the first - whether the Council of Trent "denied Sola Gratia:" I can hardly do better than to quote from Trent itself again, from the Decree on Justification:

The Synod furthermore declares, that in adults, the beginning of the said Justification is to be derived from the prevenient grace of God, through Jesus Christ... (Chapter V, emphasis added)

Okay, so at the beginning of our Justification we find Trent saying that it comes from God's grace alone (now some silly people might object to the ellipsis there; I will come back to that, I promise).

Now they (adults) are disposed unto the said justice, when, excited and assisted by divine grace, conceiving faith by hearing, they are freely moved towards God, believing those things to be true which God has revealed and promised,-and this especially, that God justifies the impious by His grace, through the redemption that is in Christ Jesus... (Chapter VI, emphasis added)

Here we see again Trent declaring that God justifies us by His grace through Christ.

Of this Justification the causes are these: the final cause indeed is the glory of God and of Jesus Christ, and life everlasting...(Chapter VII, emphasis added)

Now we come to the meat of sola gratia. And the very first thing that must be said is that if you don't understand scholastic (and in particular Thomistic) philosophy, or even Aristotelian philosophy at least, and if you don't understand their notion of causation, then you are in no position to judge what Trent says here, and once again we are back to what Turretinfan seems to dislike so much: you don't understand what we believe. And I don't pretend to be an expert on St. Thomas nor on the scholastics, but I can at least see that you don't understand Trent apart from them.

The first and most important cause is the final cause: that is, the end or purpose of the thing that is done. And the purpose of our justification is God's own glory. But whose purpose or end is it? It can be none other than God's, because he is the First Cause, and because he is the one who at the beginning of justification gives grace (see above), and who justifies us by his grace (see above). Now this cause above all is the most important, and it comes from God alone, and it finds expression in all the other causes, all of which are dependent upon this one, and apart from which they would never have come to pass. So here we see at the very outset that according to Trent, it is impossible for our salvation to be caused by anything other than grace, properly speaking: because God purposes it from the outset.

while the efficient cause is a merciful God who washes and sanctifies gratuitously, signing, and anointing with the holy Spirit of promise, who is the pledge of our inheritance;... (ibid., emphasis added)

The efficient cause is that which actually performs the action. Wikipedia puts it this way:

This is the agent which brings something about, for example, in the case of a statue, it is the person chiseling away, and the act of chiseling, that causes the statue. This answers the question, how does it happen? It is the sort of answer we usually expect when we ask about cause; the thing which happened to bring about certain results (emphasis added).

And, from a presumably more academic source:

[T]he efficient cause: the means or agency by which a thing comes into existence (a potter is the efficient cause of a bowl);... (emphasis added)

So here we see again: We are saved by a "merciful God who washes and sanctifies gratuitously" - that is, graciously. But if the cause of our salvation, "the sort of answer we usually expect when we ask about cause," is God's gracious action, then it has nothing to do with us. It is by grace alone: sola gratia.

but the meritorious cause is His most beloved only-begotten, our Lord Jesus Christ, who, when we were enemies, for the exceeding charity wherewith he loved us, merited Justification for us by His most holy Passion on the wood of the cross, and made satisfaction for us unto God the Father;... (Trent, loc. cit.; emphasis added)

Now this really ought to lay to rest all the nonsense about whether Trent taught sola gratia. Because for the Protestant, all their insistence upon sola fide has to do with whether one merits his salvation in any way. But here we see it clearly and unequivocally stated: the meritorious cause of our salvation is Jesus Christ himself. It is not anything I do. It is not anything you do. It is Christ. Alone. He merits our salvation. All clear now? Good.

But wait. There's more.

the instrumental cause is the sacrament of baptism, which is the sacrament of faith, without which (faith) no man was ever justified...(ibid.; emphasis added)

The instrumental cause has to do with the tool used to perform an action. A hammer is the instrumental cause when driving a nail; for the Protestant, the instrumental cause of salvation would be faith. It is the instrument by which an effect is brought about by the efficient cause for the sake of the fulfillment of the final cause. And even here, we see that only in an instrumental sense can human action be said to be involved: because it is God who makes use of Holy Baptism as the instrument by which we are saved. So once again we see that our salvation is wrought by God alone.

lastly, the alone formal cause is the justice of God, not that whereby He Himself is just, but that whereby He maketh us just, that, to wit, with which we being endowed by Him, are renewed in the spirit of our mind, and we are not only reputed, but are truly called, and are, just, receiving justice within us, each one according to his own measure, which the Holy Ghost distributes to every one as He wills...(ibid.; emphasis added)

The formal cause is like the blueprint or model for the thing. Again, see here and here. And once again, we see that this cause has nothing to do with us. It is all from God, and consequently all of grace.

Now it ought to be obvious that anything said in Trent about man's activity within the process of our justification simply has to be understood with the foregoing in the background. Anything less would be to completely misunderstand what the Fathers at Trent meant. You can't have human merit being a factor apart from grace, because Christ is the meritorious cause of our salvation. You can't have human merit be the start of our salvation, because the final cause of our salvation is God's glory. You can't have merely human righteousness as sufficient cause for salvation, because the formal cause of our salvation is the justice of God whereby "He maketh us just" (ibid). It's grace, from start to finish.

Once upon a time I came across Turretinfan making this sort of argument:

Legalism => Salvation by works
Denial of Sola Fide => Salvation by works

(Or - in the form he gave it later in the same combox:)

The comment ought to have read:

Salvation by works => Legalism
Denial of Sola Fide => Salvation by works

Ergo
Denial of Sola Fide => Legalism

It ought to be obvious that this begs the question, and (as his interlocutor at the time pointed out) it really amounts to nothing more than a bald assertion. Now we may perhaps wish to be more charitable than Fred was, and say that what it amounts to is a rather limited notion of the alternatives to sola fide. For the only way that such an argument could possibly work is if it were actually the case that only works salvation is an alternative to sola fide. As we have seen, however, Trent has absolutely no difficulty in acknowledging God's sovereign grace as the entire basis of our salvation, while at the same time affirming that our salvation calls for us to obey God.

Towards the end of his post declining Mr. Armstrong's debate challenge, Turretinfan says:

What is more significant, though, is that we reject the gospel of Rome, because it is not the Gospel of Christ: the answer to "What must I do to be saved," is not simply "Believe on the Lord Jesus Christ, and thou shalt be saved, and thy house" in official Roman Catholic soteriology.

This is handy-dandy proof-texting, but he's going to have a hard time dealing with St. Peter's Pentecost sermon. When he was asked, "Brethren, what shall we do?" (Acts 2:37, emphasis added), we find perhaps the most spectacular occasion in practically the whole Bible for making the case for "sola fide." It's the first sermon of the Christian era. His listeners are Jews - Jews who legalistically suppose that they must earn their salvation by their works. And they've asked the question, "What shall we do?" Now if there was ever a perfect time and opportunity for a sola fide response, it was right then. If there was ever a better setup question, a better moment for declaring the Protestant gospel than this, I don't know what it would have been.

But St. Peter didn't give them that. He didn't say anything like this: "You don't have to do anything. Just have faith in Christ, and you will be saved, because the only way you can be saved is by faith in Christ alone." No. He didn't say that. Instead, he told them what they must do.

Repent and be baptized every one of you in the name of Jesus Christ for the forgiveness of your sins; and you will receive the gift of the Holy Spirit (Acts 2:38; emphasis added).

And notice what he did not include in this list of things to do: he doesn't even mention faith. Now of course it would be absurd to suggest that St. Peter was preaching a gospel of works, and I don't mean to say that he is. And of course faith is necessary. But the real question is this: is it not entirely incongruous for St. Peter, on an occasion like that, to say nothing explicit about salvation by faith alone if the gospel really is "sola fide"? In my judgment it is. It's preposterous even to suggest.

We are saved by grace alone, as the Catholic Church has always taught. Anyone who says otherwise just doesn't understand what we believe, and they demonstrate their incomprehension just by denying that Trent taught something other than sola gratia.