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Hi, S.R.! Good to see you, again!

Your definition of the "undistributed middle" (also known as the "fallacy of four terms", FWIW) is fine. I'll insert comments at the points where we start to disagree. You wrote, in describing Daniel's argument

Syllogism 1) Assert the following to be true:

P1: An infallible magisterium would not be in conflict with Jesus
P2: Israel's magisterium was in conflict with Jesus
Therefore: Israel did not have an infallible magisterium

No problems, so far.

Syllogism 2) Assert the following to be false:

P1: An infallible magisterium is necessary to know what is of God

Let me stop you, here. On the one hand, your analysis is correct: Syllogism 2 is valid (i.e. logically, the conclusion follows from the premises), but it's UNSOUND (i.e. a bad argument) because it started with a false premise--i.e. P1. Well and good, so far.

But: the suggestion that "an infallible Magisterium is necessary to know what is of God" has some problems which Daniel didn't address. First, the phrase "what is of God" is annoyingly vague, and it covers a great many species of items... some of which WOULD require an infallible Magisterium for certainty (e.g. which books are truly Sacred Scripture, and which are spurious), and some of which would NOT require such an authority (e.g. the existence of God, Himself--which can be known through pure reason, unaided by Divine Revelation).

To make matters worse, this definition doesn't make the needed distinction between absolute certainty (i.e. certainty beyond ALL doubt, which is required for a priori conclusions such as the Pythagorean Theorem) and MORAL certainty (which logicians and lawyers would call "certainty beyond REASONABLE doubt, where rejection of the conclusion would entail a violation of sane reason", which is the minimum requirement for a posteriori, sense-data-dependent conclusions such as determining whether the room contains any penguins, or not). In short: this premise is an absolute mess.

If you'll remember from logic, there are three things needed for an argument to be sound (i.e. valid and true): (a) clear definitions, (b) true premises, and (c) a conclusion which logically follows from the premises (i.e. no fallacies). "P1" fails on two counts: it's not sufficiently clear, and--as a general statement, it is false (though it's true for some subsets of "that which is from God").

If the argument progresses, while "limping along" with this inadequate definition, it's going to collapse. That's one key thing which flagged my attention.

P2: Israel did not have an infallible magisterium (see A above)

True enough.

Therefore: It was impossible for Israel to know if Jesus was of God

See above; there was no effort to parse out the subsets of "that which is of God" (much less to place "knowledge that Jesus was of God"--which is vague in its own right, since ALL humans are "from God" in a looser sense), and so there's no clear way to reach any certain conclusion.

Syllogism 3) Assert the following to be false, because it uses the same faulty premise as Syllogism 2

P1: An infallible magisterium is necessary to know what is of God

See above; this is far too murky and insecure to use, at all.

P2: Only Catholicism possesses an infallible magisterium

Therefore: Only Catholicism provides what is necessary to know what is of God

Again, see above.

BTW, it should be pointed out that P2 in Syllogism 3 above is radically flawed as well. Catholicism's claim to an infallible magisterium is unconvincing:

Just as a side note: "unconvincing" and "flawed" are not equivalent. I've run into many people who were "unconvinced" even of a priori truths which were proven beyond all doubt. "Convincing", unfortunately, is a process which usually has to wade through the recipient's EMOTIONS, as well as their reason, to attain its end.

1. It cannot be demonstrated from Scripture that an infallible magisterium was ever created.

It depends on what you mean by "demonstrated"; how strict are your standards? Are you expecting a mathematical, airtight proof? Then no, it cannot be demonstrated... but neither can it be demonstrated mathematically that Jesus ever walked the Earth at all. Are you expecting a proof beyond all reasonable doubt? If so, then I submit for your consideration the fact that this has been proven, over and over again, in hundreds of venues. If you're really ambitious, try to locate the book, "Christianity and Infallibility: Both or Neither", by Fr. Daniel Lyons. (It's out of print, but inter-library loan should be able to find it.)

By the way: is a demonstration from SCRIPTURE the ONLY type of demonstration you'd find convincing? I don't see why, since "sola Scriptura" is a self-contradictory, invalid bit of nonsense which deserves no allegiance from anyone.

I'd also add: St. Peter, who was the rock on whom Christ built His Church (with the promise that the gates of Hades would not prevail against Her), was given the keys to the Kingdom of Heaven--with an unquestioned power to bind and loose. Compare this to Isaiah 22, and it makes an unmistakable connection: St. Peter is the "prime minister" (al ba'it) of the King of Kings, who is empowered to speak and act with the King's authority. (The others of the Twelve were also given the power to "bind and loose"--cf. Matthew 18:18--though they were not given the keys, and their authority was therefore contingent upon union with St. Peter.) Jesus also said to the Twelve that "he who hears you, hears Me" (Luke 10:16); the Holy Spirit (through St. Paul) calls the Church the "pillar and foundation of truth" (1 Timothy 3:15), and those who spurn the voice of the Church are to be treated as the heathen or tax collector (cf. Matthew 18:17). The early Church fathers were unanimous in their acknowledgement of the Bishop of Rome as being identified with Peter, and having his authority. Church history holds this idea to be absolutely unquestioned by all Church authorities, Fathers, and doctors of the Church, at least until the terrible schism between East and West (at roughly the turn of the millenium) approached.

Beyond this (which I cut very short, for the sake of time), common sense requires that God should have permitted us an infallible interpreter of His Revelation, lest the sinful and darkened intellect of man distort that Revelation beyond recognition. As Fr. Lyons says (and I paraphrase): "To presume that our God did not provide a secure, living, immortal, and accessible authority Who could preserve both His Revelation and the teachings dependent upon it from error and deceit, is to presume that God is indifferent to the damnation of the vast majority of His children." If Revelation is necessary for salvation, and if there is no certain way to know the contents of revelation, then there is no certain way to attain salvation.

2. It cannot be demonstrated from Scripture that an infallible magisterium was perpetuated past the apostolic age.

How about the fact that St. Peter and the other Apostles felt compelled to replace Judas Iscariot? Even more compelling are the reasons they cite: "Let another take his office" (Acts 1:20, citing Psalm 109:8). Why? Why not leave only 11 Apostles? Apparently, there were reasons (I'd argue that the Apostles needed to represent the Twelve Tribes of Israel, among other things) for keeping the "office" alive, even when an officeholder had vacated (through death, etc.). This is strengthened by the fact that St. Paul talks repeatedly about ordaining more bishops (episkopoi) and priests (presbyteroi), and the fact that all of Church history has not only acknowledged, but EXPECTED, that the Apostolic succession must continue. If nothing else, Jesus requires it when He commands them to "make disciples of all nations" (which could hardly have been done in the lifetime of the Apostles). For those who suggest that Apostolic authority died with the Apostles, I'm afraid history and Scripture are against them.

3. It cannot be demonstrated from primary historical sources that any consolidated, unified magisterium, fallible or infallible, ever existed in Rome under one bishop until nearly the end of the Second Century.

What do you mean by "primary" historical sources? Do you require that the original manuscripts still be in existence? And what sort of "consolidated" Magisterium would flag your attention? St. Peter was undeniably the head of the Apostles; even a cursory examination of the Scriptures (to say nothing of early Church history) would show that plainly; but would that satisfy your definition of "consolidated Magisterium"? I do wonder if some definitions might not weed out legitimate evidence, simply because the conclusions are not pleasing to the viewers...

4. It can be demonstrated from Scripture that even a divinely ordained magisterium can fall into catastrophic error.

You'd have to be much more specific than that--both with examples, and with explanations as to what TYPE of error. Was it error in doctrine? Error in personal behavior? Error in matters of faith and morals? Errors in matters not related to those? (And on what basis do you use the word "catastrophic"? It's a very dramatic-sounding word... but it can also be abused.)

5. It can be demonstrated from Scripture that even a divinely ordained magisterium can be sent reformers from God for the purpose of correcting its catastrophic error.

I'll take a guess, here, and say that you may be thinking of Galatians 2:11-14? If so, you'll have to explain what genus the "error" was (e.g. doctrinal, behavioral, etc.), and why you attach the word "catastrophic" to it (aside, perhaps, from emotional emphasis).

More later; must dash!