I. Introduction to Bryan's Argument
Now, Mr. Bryan Cross has suggested that we look specifically to his "argument." Section IV(A) of Bryan's post contains a section titled "the argument." I reproduce that section here:
1. According to solo scriptura, Scripture is the only ecclesial authority. [def](source)
2. If solo scriptura is true, then each individual is his own final interpretive authority concerning what he considers to be essential. 
3. According to sola scriptura, Scripture is the only infallible ecclesial authority. [def]
4. If sola scriptura entails that each individual is his own final interpretive authority concerning what he considers to be essential, then in this respect there is no principled difference between solo scriptura and sola scriptura.
5. If apostolic succession is false, then no one’s determination of the marks of the Church is any more authoritative than anyone else’s.
6. If no one’s determination of the marks of the Church is any more authoritative than anyone else’s, then each individual is his own final interpretive authority concerning what he considers to be essential.
7. If apostolic succession is false, then each individual is his own final interpretive authority concerning what he considers to be essential. [(5),(6)]
8. The doctrine of apostolic succession is false. [A]
9. If sola scriptura is true, then each individual is his own final interpretive authority concerning what he considers to be essential. [(7),(8)]
10. There is no principled difference between sola scriptura and solo scriptura. [(4),(9)]
II. Most Obvious Problem - Improper Generalization to obtain 10
Right off the bat, one will notice that conclusion 10, does not follow from the premises. Conclusion 10 is an invalid generalization.
Premise 4 stated: "If sola scriptura entails that each individual is his own final interpretive authority concerning what he considers to be essential, then in this respect there is no principled difference between solo scriptura and sola scriptura." (bold emphasis supplied)
Conclusion 10, however, drops the necessary qualification "[with respect to] each individual [being] his own final interpretive authority concerning what he considers to be essential." Notice that this qualification significantly narrows "no principled difference" and even further narrows "no principled difference with respect to the ultimate holder of interpretive authority."
III. Some Less Obvious Problems
Even if we correct Bryan's argument to make 10 read: "There is no principled difference between sola scriptura and solo scriptura with respect to each individual being his own final interpretive authority concerning what he considers to be essential," there remain some less obvious problems.
A. "Final Interpretive Authority"
No matter who one's final authority is, one is necessarily the final interpretive authority of that authority. That is true whether one uses oneself, Scripture, tea leaves, or the Roman Catholic magisterium as one's final authority. If one looks to oneself for guidance, one has to read one's own mood. If one looks to Scripture for guidance, one has to interpret Scripture. If one looks at tea leaves - one has to interpet their significance. Even if one has a "living magisterium" one must interpret what that magisterium tells someone. That's so, because we must interpret information in order to understand information. That's just the way that humans work.
In some cases, of course, the interpretation involved seems trivial. If one's authority is an oracle (like the Urim and Thummim or - to pick a more familiar example - a magic eight ball) it may be that you ask a specific question and the oracle provides you with a yes/no answer. In that case, the interpretation involved seems trivial, particularly if one has chosen one's question well (As in the cases of Achan and Jonah).
Even if such an oracle provided an example of an authority that needed no interpretation, that's not one of the available options today. Even the Roman Catholic church does not claim that one get an infallible answer to a yes/no question posed to the magisterium. So, all of the available options require the individual to be the "final interpreter" or, as Bryan prefers to express it, "the final interpretive authority."
This issue ends up plaguing premises 2, 4, 6, 7, and 9. In each of 2, 6, and 9, there is a statement that is in the form "If [x] is true, then each individual is his own final interpretive authority [as to y]." In 7, the same form is used, except that it is introduced by "if [z] is false ... ." Premise 4 begins with "If each individual is his own final interpretive authority [as to y] ... ."
The impact of the fact that each individual is always his own final interpretive authority as to every source of information that he receives is significant with respect to these premises. Each of the premises is in the form If A, then B. However, since B is always the case, premises 2, 6, 7, and 9 are trivial. Furthermore, premise 4 is in the form If B, then C. Premise 4, therefore, is simply reduced to an assertion of C, which ends up being modified 10 (as modified above).
B. "Concerning what he considers to be essential"
This expression ends up creating ambiguity. I believe that Bryan is trying to avoid conceding that the Reformed person can know what is essential and what is not essential. As such, the whole phrase: "each individual is his own final interpretive authority concerning what he considers to be essential" probably is intended to mean that each person is the final interpreter on those points where the matter is an essential matter (in the judgment of the individual), in contradistinction to those points where the matter is a non-essential matter (again, in the judgment of the individual). The alternative sense, however, is that the individual is interpreting his own opinion of what is essential, i.e. the individual is simply the one who decides what is essential. The lack of clarity as to the intended sense could have been avoided if the expression had simply been "concerning essential doctrines."
That would mean that a modified 10 would be "There is no principled difference between sola scriptura and solo scriptura with respect to each individual being his own final interpretive authority concerning essential doctrines."
Since, as we noted above, the individual finally interprets all information he receives, essential doctrines are no exception. That's true regardless of whether one employs the category of essential doctrines. That suggests that the "essential doctrines" issue is really a red herring.
C. Fallacy of Affirming the Consequent (Almost ...)
Bryan's syllogisms form three chains:
5 (If A, then B)
6 (If B, then C)
∴ 7 (If A, then C)
2 (If D, then C)
9 (If E, then C)
Where A = "apostolic succession is false"
B = "no one’s determination of the marks of the Church is any more authoritative than anyone else’s"
C = "each individual is his own final interpretive authority concerning what he considers to be essential"
D = "solo scriptura is true"
E = "sola scriptura is true"
In view of aleph, bet, and gimel, premise 4 is constructed, which in essence asserts that in principle A, D, and E are the same, because they all entail the same thing. But, of course, the following is an invalid syllogism:
7 (If A, then C)
2 (If D, then C)
9 (If E, then C)
∴ A = D = E
After, as we've noted above, we could construct a further item:
11 (If notA, then C)
which would then yield the odd result:
∴ A = notA
The reason we could construct 11 is that even if apostolic succession is true, each individual is still going to be the final interpreter of his source(s) of authority.
I will note, however, that Bryan does not explicitly state that 4 is supposed to derive from 2, 7, and 9. Instead, he simply states the premise. However, 4 is based on the reasoning that if D and E both entail C then "in this respect there is no principled difference between" D and E, to which we might as well add A, though Bryan neglects to do so. Since we've proven above that C is always the case, when apply the rationale behind 4, we obtain not just our modified 10, but the further addition that there is no principled difference as to C with respect to notA (apostolic succession is true), D(solo is true), and E (sola is true).
This demonstration is what Bryan mislabels a tu quoque objection. Section V(A) of his article attempts to address this objection, but fails. I explain why in the following part.
IV. Bryan's Response to III(C)
Of course, Bryan hasn't read this article yet (at the time of my writing it) but I can reasonably anticipate that he'll respond to III(C) of my comments above, by referring to V(A) of his own article. That's where he attempts to argue that an individual following apostolic succession avoids C ("each individual is his own final interpretive authority concerning what he considers to be essential"). In symbolic terms, Bryan is disputing my:
11 (If notA, then C)
12 (If noA, then notC)
In that section of the article he claims:
But when a person finds the Magisterium, and recognizes it for what it is, he immediately ceases to be his own final interpretive authority. He recognizes that his interpretation of Scripture ought to be conformed to the teaching and interpretation of the Magisterium, and that to reject the teaching of the Magisterium would be to reject Christ ... .(source)
What Bryan has overlooked is that nothing has fundamentally changed about the way in which the person mentally functions. The choice to submit to "the Magisterium" does not change the fact that the person will still have to interpret what the magisterium says.
Bryan essentially anticipates this objection describing it thus:
This objection can also take the following form. Even if the Church possesses final interpretive authority, yet because the individual must nevertheless interpret the Church’s dogmatic pronouncements, therefore, the individual must be the final interpretive authority of the Church’s dogmatic pronouncements.Bryan replies to the objection he has stated by responding:
This objection conflates two senses of the term ‘final.’ ‘Final’ can mean the terminus of a movement or of a series of movements, as an airplane has a final destination, the terminus of a series of flights for the day. ‘Final’ can also mean the terminus in an order or hierarchy, as the Commander in Chief is for the military. In a communication, the individual receiving that communication is, by definition, the terminus of the movement whereby knowledge is transmitted. He is, in that sense, the final interpreter. But he is not thereby the final interpretive authority in the sense of a terminus in an order or hierarchy. He may be the terminus of the motion of the communication, while remaining subordinate in the order of interpretive authority. The exercise of interpretive authority by the Magisterium, say, at an ecumenical council, does not prevent believers from interpreting Scripture or any other communication. Nor does it withhold from them the skill by which to interpret Sacred Scripture. On the contrary, the exercise of this teaching and interpretive authority provides a supernatural light by which the believer ought to interpret Scripture. We ignore or disregard that interpretive authority at our peril, because it is God-given authority, for our good.(footnotes omitted)
There are four rebuttals to Bryan's response:
1) In sola scriptura and Mathison's position, the individual is only the "final" interpreter in the sense of communication. Scripture itself is the final authority for the sola scriptura position and scripture plus the ecumenical creeds (in essence) is for Mathison. Even in solo scriptura, it is not necessarily the case that the individual grants himself any authority over the text (though it appears that Mathison thinks this does happen in solo scriptura). So to the extent that Bryan's escape from lack of principled distinction works for the RC position, it proves too much, in that it undermines premises 2, 7, and 9.
2) Mathison explicitly selects the ecumenical creeds as his extrinsic grid to which he submits. That's the same as Bryan's example. If submitting to the outcome of an ecumenical council liberates you from matching the solo position, Mathison is liberated in the exact same way.
3) The key weakness (and perhaps I ought to have placed it first, so bored readers would find it) to Bryan's response is that he is comparing the whole governing authority of the solo person to a part of the governing authority of the RC person. In other words, where he ought to compare (Scripture, Oral Tradition, and the Magisterium) to (Scripture) he compares (Scripture,
4) Does Bryan really mean to say that the Magisterium is above Scripture in a military-like hierarchy? Catechism of the Catholic Church (CCC) 86 states:
"Yet this Magisterium is not superior to the Word of God, but is its servant. It teaches only what has been handed on to it. At the divine command and with the help of the Holy Spirit, it listens to this devotedly, guards it with dedication and expounds it faithfully. All that it proposes for belief as being divinely revealed is drawn from this single deposit of faith." (Quoting from Dei Verbum, Section 10, paragraph 2)Interestingly, Bryan's analogy is the usual Reformed criticism of the RC position, whereas CCC 86 is the usual response by Roman Catholic proselytizers and apologists.
Bryan's argument is plainly invalid on its face since conclusion 10 is an invalid generalization. Furthermore, even when corrected, Bryan's conclusion can be just as legitimately expanded to include the RC position. And if the RC position is permitted to escape by distinguishing between finality of communication and finality of authority, the sola scriptura position (whether in the classical reformed sense or in the Mathisonian form - and even solo scriptura) also escapes.