The so-called spiral argument has been dealt with many times before. Here is a rather good and concise deconstruction of it:

One common attack used by the apologists of Rome is to assert that a Protestant's ultimate authority is private judgment or, as they sometimes pejoratively label as being "protestant personalism" or a person being his own "mini-pope." Supposedly, this problem of private judgment is solved by referring to an infallible magisterium. In fact, however, the recourse to the infallible magisterium is just further application of private judgment.

This argument against private judgment can take various forms. One form of the argument is a syllogism in the form:

1. If God gave us a way to know the truth, that way would give us knowledge of the truth with reasonable certainty.

2. Private judgment doesn't provide reasonable certainty, because reasonable people differ in the application of private judgment.

3. Therefore, private judgment is not the way God gave us to know the truth.

There are several problems with this argument. The number one problem is that it employs the fallacy of skepticism. The way that it employs the fallacy of skepticism is in establishing the minor premise, i.e. private judgment doesn't provide reasonable certainty.

This is a logical fallacy for a couple of reasons. The most obvious reason is that private judgment is necessarily used to deny that private judgment provides reliable conclusions. If the conclusion is correct (i.e. that private judgment does not provide reliable conclusions) then the conclusion itself is not reliable since it obtained by private judgment.

Second, this is a logical fallacy in the sense of simply being a universal denial of knowability of information. That is to say, this argument lacks uniqueness. It is not particularly a criticism of the "Protestant" position. It is a modus tolens argument that can be applied mutatis mutandis to any epistemology.

Specifically, this same argument can be applied to the epistemology of Catholicism, because Catholicism too requires, at some link in the chain, one to use private judgment. This is illustrated in the attempt of certain of Rome's apologists to escape the apparent circularity of Rome's epistemology.

The apparent circularity is this:

1. The Bible is right because the Church says it is.
2. The Church is right because the Bible says it is.

There, the circularity is obvious. It is bigger than the circle of:

1. The Bible is right because the Bible says it is.

... but it is still a circle.

To try to escape this circle, some apologists for Catholicism use what they call a "spiral argument" that was apparently developed by Karl Keating, one of the more prominent apologists for Catholicism (although I cannot recall him debating anyone from the Reformed side of the Tiber river in a long time).

Here's one presentation of the argument:

A Spiral Argument
Note that this is not a circular argument. We are not basing the inspiration of the Bible on the Church’s infallibility and the Church’s infallibility on the word of an inspired Bible. That indeed would be a circular argument! What we have is really a spiral argument. On the first level we argue to the reliability of the Bible insofar as it is history. From that we conclude that an infallible Church was founded. And then we take the word of that infallible Church that the Bible is inspired. This is not a circular argument because the final conclusion (the Bible is inspired) is not simply a restatement of its initial finding (the Bible is historically reliable), and its initial finding (the Bible is historically reliable) is in no way based on the final conclusion (the Bible is inspired). What we have demonstrated is that without the existence of the Church, we could never know whether the Bible is inspired.
(link to source) (Notice how, unlike Mr. Patrick Madrid, we're not afraid to let the reader see the writings to which we're responding.)

Let's assess this argument. The core of the argument is:

1. [W]e argue to the reliability of the Bible insofar as it is history.
2. From that we conclude that an infallible Church was founded.
3. And then we take the word of that infallible Church that the Bible is inspired.

We could reasonably expand this argument to the following:

1. The Bible is an historically reliable document.
2. The Bible records the founding of a church.
3. The Bible indicates that this church is infallible.
4. This infallible church is the church headed by the pope.
5. This infallible church teaches that the Bible is inspired.

It could probably be expanded even further, but this is enough for the purposes of illustrating the problems with this supposedly spiral argument.

Problem 1: Private Judgment Vastly Multiplied

Rather than simply accepting the Bible as Inspired Word of God based on one private judgment, this "spiral" argument requires one to employ private judgment over and over and over again.

First, one uses private judgment to answer the question of historical reliability.

Second, one uses private judgment to decide the meaning of Scripture as to whether a single, institutionally unitary church or many institutionally separate churches were founded.

Third, one uses private judgment to decide that this single church is taught as being infallible rather than as being fallible.

Fourth, one uses private judgment to identify this single church as the Roman Catholic church instead of, say, the Eastern Orthodox church or the Ethiopian Orthodox Church.

Fifth, one uses private judgment to decide that the Roman Catholic church teaches that the Bible is inspired, as opposed to teaching that the Bible simply contains God's word.

This is a vast multiplication of private judgment over, for example, simply accepting the Bible on historical grounds and then accepting that the Bible says that the Bible is inspired, or simply accepting the Bible's claim of inspiration as true in the first and only step of the process.

If private judgment is inherently bad, the "spiral argument" uses more of it.

Problem 2: Bootstrapping Error

The Spiral argument attempts to avoid the obvious "jump" from nothing to accepting the Bible's claim of inspiration by steps. But each (or at least several) of the steps are "jumps" in themselves.

There is a jump from nothing to accepting the historical method as providing reliable conclusions.

There is a jump from using the historical method to confirm the general reliability of the Bible, to accepting a particular historical account in the Bible.

There is a jump from accepting an account as historical to accepting the doctrine taught in the event as truth.

There is a jump from accepting the general idea that a church was founded to accepting that a particular church is that church.

If making jumps is bad, breaking up a big jump into several smaller jumps doesn't solve the problem, it just distributes it.

Problem 3: Inspiration Smuggled Back In

In fact, the "spiral argument" is circular, because inspiration is smuggled into step 3 of my expanded formulation of the argument, or step 2 of the original formulation of the argument. That is to say, the teaching of an "infallible church" is accepted allegedly because the Bible as a historically accurate document is accepted. But the claim of infallibility is a theological claim, not an historical claim, and it is accepted either because of the authority of the Bible or the person speaking in the Bible.

Problem 4: Church Infallibility Smuggled Back In

Furthermore, the "spiral argument" has a second circularity, in that the infallibility of the church is smuggled back into the argument twice. It is smuggled back once in telling people which church to accept as "the church," and again (more importantly) in interpreting Scripture as teaching an infallible church in the first place.

Specifically, the claim that the Scriptures disclose the founding of "an infallible church" requires loads of eisegesis - of reading into the text, rather than of obtaining teachings from the text.

Problem 5: Scripture Promotes Private Judgment

Worse (for Catholicism) than the issue simply being a matter of silence, Scripture actually encourages the use of private judgment. For example, the Scripture many times and in various ways encourages people to apply personal judgment to arrive at the truth. For example, the Bereans are commended for using private judgment and Scripture to judge Paul, and Paul tells Timothy that the Scriptures are able to make one "wise unto salvation." Furthermore, John tells us that his gospel was written so that we would believe it and have life through faith in Jesus the Messiah, the Son of God.

Conclusion

So, next time a person tries to tell you that the difference between Protestant interpretation and Romanist interpretation is that the Protestant makes himself his own ultimate authority, be prepared to challenge that deceptive claim. Everyone uses private judgment. If private judgment is inherently untrustworthy, the Roman position is actually worse off than the "Protestant" position.

Furthermore, while it might be nice to hand over one's brain to the church, so that one doesn't have to think about the meaning of Scripture, that's just not how God ordained things. The fact that it would be convenient or handy doesn't make it so.

Instead, God provided fallible churches with fallible elders over them. These fallible teachers are to teach the Scriptures to their people, but the unchanging Scriptures serve as the rule and measure of the Christian faith, with the fallible churches serving as guides.

-TurretinFan

The short form, for those busy executives unable to ponder the full article, is that there is no spiral here. RC epistemology, in this form, is actually a set of many circles strung together with a series of unfounded leaps in logic. Remember, in a true spiral, like the famous slinky toy, one expects a smooth progression along the curve, not abrupt jumps when you return to the period of the circle. But as TurretinFan above points out, each step in Keating's so-called spiral requires multiple separate acts of private judgment, and as private judgment is the big scary boogie man the RC epistemology is supposed to discredit, it fails utterly.

And what are we left with? That the Ecclesia, like Israel before them, had arrived at an informal consensus, by faith, that certain books were Scripture, well before any councils had so decided; that this consensus developed as a result of the Holy Spirit working amongst all believers everywhere, the truly universal church, or better, Ecclesia; that those today who wish to know and interact with the revelation of the mind and heart of God may do so in the pages of Scripture; that God has given us natural and supernatural gifts which we are obligated to use to apprehend His word to us, which gifts culminate in a private act of faith, that we believe God and His word, and choose to live by it, no matter where it leads; that in the exercise of this necessary private judgment, we are but clay, and prone to error, and so must lean on God's Holy Spirit and on each other to remain honest and faithful stewards of His word.

That is how it was during the early years of the church. The Roman schism did not develop for about 160 years, though there were early stirrings of the faction that would later emerge as political Rome's favorite flavor of Christianity. But generic Christianity, the Ecclesia that Jesus is building, is not irrevocably bound to that or any other human institution. It is God's work, not man's, a work of God's Spirit, and not the arm of flesh. And it is that Ecclesia who was at the beginning, now is and ever shall be, the true steward of God's Holy word. The Lord knows those who are His, and what it takes to feed the sheep of His pasture:

Matthew 4:4 But he answered and said, It is written, Man shall not live by bread alone, but by every word that proceedeth out of the mouth of God.

Peace,

SR

Matthew 4:4

But he answered and said, It is written, Man shall not live by bread alone, but by every word that proceedeth out of the mouth of God.

NIV Matthew 2:5
"In Bethlehem in Judea," they replied, "for this is what the prophet has written:

NIV Matthew 4:1-11
1. Then Jesus was led by the Spirit into the desert to be tempted by the devil.
2. After fasting forty days and forty nights, he was hungry.
3. The tempter came to him and said, "If you are the Son of God, tell these stones to become bread."
4. Jesus answered, "It is written: `Man does not live on bread alone, but on every word that comes from the mouth of God.' "
5. Then the devil took him to the holy city and had him stand on the highest point of the temple.
6. "If you are the Son of God," he said, "throw yourself down. For it is written: "`He will command his angels concerning you, and they will lift you up in their hands, so that you will not strike your foot against a stone.' "
7. Jesus answered him, "It is also written: `Do not put the Lord your God to the test.' "
8. Again, the devil took him to a very high mountain and showed him all the kingdoms of the world and their splendor.
9. "All this I will give you," he said, "if you will bow down and worship me."
10. Jesus said to him, "Away from me, Satan! For it is written: `Worship the Lord your God, and serve him only.' "
11. Then the devil left him, and angels came and attended him.

NIV Matthew 11:10
This is the one about whom it is written: "`I will send my messenger ahead of you, who will prepare your way before you.'

NIV Matthew 21:13
"It is written," he said to them, "`My house will be called a house of prayer,' but you are making it a `den of robbers.' "

NIV Matthew 26:24
The Son of Man will go just as it is written about him.

NIV Matthew 26:31
Then Jesus told them, "This very night you will all fall away on account of me, for it is written: "`I will strike the shepherd, and the sheep of the flock will be scattered.'

NIV Mark 7:6-7
6. He replied, "Isaiah was right when he prophesied about you hypocrites; as it is written: "`These people honor me with their lips, but their hearts are far from me.
7. They worship me in vain; their teachings are but rules taught by men.'

NIV Mark 9:11-13
11. And they asked him, "Why do the teachers of the law say that Elijah must come first?"
12. Jesus replied, "To be sure, Elijah does come first, and restores all things. Why then is it written that the Son of Man must suffer much and be rejected?
13. But I tell you, Elijah has come, and they have done to him everything they wished, just as it is written about him."

NIV Mark 11:17
And as he taught them, he said, "Is it not written: "`My house will be called a house of prayer for all nations' ? But you have made it `a den of robbers.' "

NIV Mark 14:27
"You will all fall away," Jesus told them, "for it is written: "`I will strike the shepherd, and the sheep will be scattered.'

NIV Luke 1:1-4
1. Many have undertaken to draw up an account of the things that have been fulfilled among us,
2. just as they were handed down to us by those who from the first were eyewitnesses and servants of the word.
3. Therefore, since I myself have carefully investigated everything from the beginning, it seemed good also to me to write an orderly account for you, most excellent Theophilus,
4. so that you may know the certainty of the things you have been taught.

NIV Luke 4:17-19
17. The scroll of the prophet Isaiah was handed to him. Unrolling it, he found the place where it is written:
18. "The Spirit of the Lord is on me, because he has anointed me to preach good news to the poor. He has sent me to proclaim freedom for the prisoners and recovery of sight for the blind, to release the oppressed,
19. to proclaim the year of the Lord's favor."

NIV Luke 7:27
This is the one about whom it is written: "`I will send my messenger ahead of you, who will prepare your way before you.'

NIV Luke 10:26
"What is written in the Law?" he replied. "How do you read it?"

NIV Luke 18:31-33
31. Jesus took the Twelve aside and told them, "We are going up to Jerusalem, and everything that is written by the prophets about the Son of Man will be fulfilled.
32. He will be handed over to the Gentiles. They will mock him, insult him, spit on him, flog him and kill him.
33. On the third day he will rise again."

NIV Luke 20:17-18
17. Jesus looked directly at them and asked, "Then what is the meaning of that which is written: "`The stone the builders rejected has become the capstone ' ?
18. Everyone who falls on that stone will be broken to pieces, but he on whom it falls will be crushed."

NIV Luke 21:22
For this is the time of punishment in fulfillment of all that has been written.

NIV Luke 22:37
It is written: `And he was numbered with the transgressors' ; and I tell you that this must be fulfilled in me. Yes, what is written about me is reaching its fulfillment."

NIV Luke 24:44-47
44. He said to them, "This is what I told you while I was still with you: Everything must be fulfilled that is written about me in the Law of Moses, the Prophets and the Psalms."
45. Then he opened their minds so they could understand the Scriptures.
46. He told them, "This is what is written: The Christ will suffer and rise from the dead on the third day,
47. and repentance and forgiveness of sins will be preached in his name to all nations, beginning at Jerusalem.

NIV John 2:17
His disciples remembered that it is written: "Zeal for your house will consume me."

NIV John 6:31
Our forefathers ate the manna in the desert; as it is written: `He gave them bread from heaven to eat.' "

NIV John 6:45
It is written in the Prophets: `They will all be taught by God.' Everyone who listens to the Father and learns from him comes to me.

NIV John 12:14-16
14. Jesus found a young donkey and sat upon it, as it is written,
15. "Do not be afraid, O Daughter of Zion; see, your king is coming, seated on a donkey's colt."
16. At first his disciples did not understand all this. Only after Jesus was glorified did they realize that these things had been written about him and that they had done these things to him.

NIV John 15:25
But this is to fulfill what is written in their Law: `They hated me without reason.'

NIV John 20:30-31
30. Jesus did many other miraculous signs in the presence of his disciples, which are not recorded in this book.
31. But these are written that you may believe that Jesus is the Christ, the Son of God, and that by believing you may have life in his name.

NIV Acts 1:20
"For," said Peter, "it is written in the book of Psalms, "`May his place be deserted; let there be no one to dwell in it,' and, "`May another take his place of leadership.'

NIV Acts 7:42
But God turned away and gave them over to the worship of the heavenly bodies. This agrees with what is written in the book of the prophets: "`Did you bring me sacrifices and offerings forty years in the desert, O house of Israel?

NIV Acts 13:29
When they had carried out all that was written about him, they took him down from the tree and laid him in a tomb.

NIV Acts 13:32-33
32. "We tell you the good news: What God promised our fathers
33. he has fulfilled for us, their children, by raising up Jesus. As it is written in the second Psalm: "`You are my Son; today I have become your Father. '

NIV Acts 15:15-18
15. The words of the prophets are in agreement with this, as it is written:
16. "`After this I will return and rebuild David's fallen tent. Its ruins I will rebuild, and I will restore it,
17. that the remnant of men may seek the Lord, and all the Gentiles who bear my name, says the Lord, who does these things'
18. that have been known for ages.

NIV Acts 23:5
Paul replied, "Brothers, I did not realize that he was the high priest; for it is written: `Do not speak evil about the ruler of your people.' "

NIV Acts 24:14
However, I admit that I worship the God of our fathers as a follower of the Way, which they call a sect. I believe everything that agrees with the Law and that is written in the Prophets,
and I have the same hope in God as these men, that there will be a resurrection of both the righteous and the wicked.

NIV Romans 1:17
For in the gospel a righteousness from God is revealed, a righteousness that is by faith from first to last, just as it is written: "The righteous will live by faith."

NIV Romans 2:24
As it is written: "God's name is blasphemed among the Gentiles because of you."

NIV Romans 3:4
Not at all! Let God be true, and every man a liar. As it is written: "So that you may be proved right when you speak and prevail when you judge."

NIV Romans 3:10-12
10. As it is written: "There is no one righteous, not even one;
11. there is no one who understands, no one who seeks God.
12. All have turned away, they have together become worthless; there is no one who does good, not even one."

NIV Romans 4:17
As it is written: "I have made you a father of many nations." He is our father in the sight of God, in whom he believed--the God who gives life to the dead and calls things that are not as though they were.

NIV Romans 4:23-24
23. The words "it was credited to him" were written not for him alone,
24. but also for us, to whom God will credit righteousness--for us who believe in him who raised Jesus our Lord from the dead.

NIV Romans 8:36
As it is written: "For your sake we face death all day long; we are considered as sheep to be slaughtered."

NIV Romans 9:13
Just as it is written: "Jacob I loved, but Esau I hated."

NIV Romans 9:33
As it is written: "See, I lay in Zion a stone that causes men to stumble and a rock that makes them fall, and the one who trusts in him will never be put to shame."

NIV Romans 10:15
And how can they preach unless they are sent? As it is written, "How beautiful are the feet of those who bring good news!"

NIV Romans 11:7-10
7. What then? What Israel sought so earnestly it did not obtain, but the elect did. The others were hardened,
8. as it is written: "God gave them a spirit of stupor, eyes so that they could not see and ears so that they could not hear, to this very day."
9. And David says: "May their table become a snare and a trap, a stumbling block and a retribution for them.
10. May their eyes be darkened so they cannot see, and their backs be bent forever."

NIV Romans 11:26-27
26. And so all Israel will be saved, as it is written: "The deliverer will come from Zion; he will turn godlessness away from Jacob.
27. And this is my covenant with them when I take away their sins."

NIV Romans 12:19
Do not take revenge, my friends, but leave room for God's wrath, for it is written: "It is mine to avenge; I will repay," says the Lord.

NIV Romans 14:11
It is written: "`As surely as I live,' says the Lord, `every knee will bow before me; every tongue will confess to God.'"

NIV Romans 15:3-4
3. For even Christ did not please himself but, as it is written: "The insults of those who insult you have fallen on me."
4. For everything that was written in the past was written to teach us, so that through endurance and the encouragement of the Scriptures we might have hope.

NIV Romans 15:7-12
7. Accept one another, then, just as Christ accepted you, in order to bring praise to God.
8. For I tell you that Christ has become a servant of the Jews on behalf of God's truth, to confirm the promises made to the patriarchs
9. so that the Gentiles may glorify God for his mercy, as it is written: "Therefore I will praise you among the Gentiles; I will sing hymns to your name."
10. Again, it says, "Rejoice, O Gentiles, with his people."
11. And again, "Praise the Lord, all you Gentiles, and sing praises to him, all you peoples."
12. And again, Isaiah says, "The Root of Jesse will spring up, one who will arise to rule over the nations; the Gentiles will hope in him."

NIV Romans 15:21
Rather, as it is written: "Those who were not told about him will see, and those who have not heard will understand."

NIV 1 Corinthians 1:19
For it is written: "I will destroy the wisdom of the wise; the intelligence of the intelligent I will frustrate."

NIV 1 Corinthians 1:31
Therefore, as it is written: "Let him who boasts boast in the Lord."

NIV 1 Corinthians 2:9
However, as it is written: "No eye has seen, no ear has heard, no mind has conceived what God has prepared for those who love him" --

NIV 1 Corinthians 3:19-20
19. For the wisdom of this world is foolishness in God's sight. As it is written: "He catches the wise in their craftiness" ;
20. and again, "The Lord knows that the thoughts of the wise are futile."

NIV 1 Corinthians 4:6
Now, brothers, I have applied these things to myself and Apollos for your benefit, so that you may learn from us the meaning of the saying, "Do not go beyond what is written." Then you will not take pride in one man over against another.

NIV 1 Corinthians 9:9
For it is written in the Law of Moses: "Do not muzzle an ox while it is treading out the grain." Is it about oxen that God is concerned?

NIV 1 Corinthians 10:7
Do not be idolaters, as some of them were; as it is written: "The people sat down to eat and drink and got up to indulge in pagan revelry."

NIV 1 Corinthians 10:11
These things happened to them as examples and were written down as warnings for us, on whom the fulfillment of the ages has come.

NIV 1 Corinthians 14:21
In the Law it is written: "Through men of strange tongues and through the lips of foreigners I will speak to this people, but even then they will not listen to me," says the Lord.

NIV 1 Corinthians 15:45
So it is written: "The first man Adam became a living being" ; the last Adam, a life-giving spirit.

NIV 1 Corinthians 15:54
When the perishable has been clothed with the imperishable, and the mortal with immortality, then the saying that is written will come true: "Death has been swallowed up in victory."

NIV 2 Corinthians 1:13-14
13. For we do not write you anything you cannot read or understand. And I hope that,
14. as you have understood us in part, you will come to understand fully that you can boast of us just as we will boast of you in the day of the Lord Jesus.

NIV 2 Corinthians 4:13-14
13. it is written: "I believed; therefore I have spoken." With that same spirit of faith we also believe and therefore speak,
14. because we know that the one who raised the Lord Jesus from the dead will also raise us with Jesus and present us with you in his presence.

NIV 2 Corinthians 8:15
as it is written: "He who gathered much did not have too much, and he who gathered little did not have too little."

NIV Galatians 3:10
All who rely on observing the law are under a curse, for it is written: "Cursed is everyone who does not continue to do everything written in the Book of the Law."

NIV Galatians 3:13
Christ redeemed us from the curse of the law by becoming a curse for us, for it is written: "Cursed is everyone who is hung on a tree."

NIV Galatians 4:22
For it is written that Abraham had two sons, one by the slave woman and the other by the free woman.

NIV Galatians 4:27
For it is written: "Be glad, O barren woman, who bears no children; break forth and cry aloud, you who have no labor pains; because more are the children of the desolate woman than of her who has a husband."

NIV Hebrews 10:7
Then I said, `Here I am-- it is written about me in the scroll-- I have come to do your will, O God.'"

NIV 1 Peter 1:15-16
But just as he who called you is holy, so be holy in all you do; for it is written: "Be holy, because I am holy."

NIV 2 Peter 3:16
He writes the same way in all his letters, speaking in them of these matters. His letters contain some things that are hard to understand, which ignorant and unstable people distort, as they do the other Scriptures, to their own destruction.

NIV 1 John 2:12-14
12. I write to you, dear children, because your sins have been forgiven on account of his name.
13. I write to you, fathers, because you have known him who is from the beginning. I write to you young men, because you have overcome the evil one. I write to you, dear children, because you have known the Father.
14. I write to you, fathers, because you have known him who is from the beginning. I write to you, young men, because you are strong, and the word of God lives in you, and you have overcome the evil one.