Are the Five Solas considered "Axioms" or "Theorems" in the logic of salvation in Protestant theology?

Axiom, held to be true (emphasis mine):

We hold these truths to be self-evident, that all men are created equal, that they are endowed by their Creator with certain unalienable Rights, that among these are Life, Liberty and the pursuit of Happiness. Preamble to the Declaration of Independence

Theorem, to be proved as true.

The 5 solas are theorems; that is, they are to be proved, rather than held to be true without proof. How are they proved?

They are derived from scripture. Scripture itself is both self-evident, it is God's word spoken through prophets and apostles, and provable, like archaeology or fulfilled prophecy.

Self-evident examples: God spoke through prophets and apostles Christ is the cornerstone Scripture is complete, leaving out nothing necessary for your salvation God loves us

The 5 solas are these: Sola Fide, by faith alone. Sola Scriptura, by Scripture alone. Solus Christus, through Christ alone. Sola Gratia, by grace alone. Soli Deo Gloria, glory to God alone.

This reestablishment of the authority of the written word grew out from the milieu of various de fide assertions and questionable practices (indulgences for example) that were neither self-evident or provable from scripture and historic tradition. The whole point of the five solas was to base Christianity on scripture, what is self-evident and provable, rather than T(t)radition that makes many assumptions and the "just so" statements.

So, to answer the OP, the 5 solas are theorems based on scripture that is both self-evident and provable.

I know what 'The Five Solas' are (being of the Reformed Protestant 'school'). I know what an axiom is (established principle; self-evident truth). I know what a theorem is (proposition not self-evident but to be proved by a chain of reasoning; a truth to be established by means of accepted truths).

There is such a thing as "biblical evidence", derived from the writings of scripture, which shows that there are three basic lines of evidence God provides for humanity to know things about himself that are necessary for our salvation. Nor is it wrong to have evidence that warrants faith, because faith in God is warranted. There's nothing reprehensible about showing in what way faith is warranted. Those ways are detailed, for example, in the opening chapters of the book of Romans. Paul speaks of knowing the eternal power and invisible qualities of God from God's creation, so that those who deny them are without excuse. He also shows that the witnessed resurrection of Jesus Christ is proof that he is, indeed, the Son of God. His claims to be such while on earth are vindicated with his activities after being resurrected. He also gives further revelations to humanity via his role as the Word of God, the written record of God's words and actions being the third line of evidence. All three lines are in agreement. Protestants consider those three evidences as the basis for having come up with the Five Solas long after creation, long after Christ's time on earth, and long after the 'last word' from the Word of God was committed to writing (i.e. the Book of the Revelation).

What God has revealed of himself in creation, in the incarnation, and in his written record all came before Protestants made logical deductions about these Five Solas. They thought them through as formal statements of belief after those various events, deducing them to be self-evident in light of what had gone before. But that in no way reduces the propositions to something less than Divine Revelation, for without the three lines of God's revealing of himself to humanity, nothing about God could ever have been known with any certainty.

Needless to say, those who don't have faith in God, in the person of Christ, or in the authority of God's written word, will dismiss the Five Solas as back-to-front. As a Protestant, I would say that God-given hindsight is no bad thing. There is a logical progression to understanding the things of God, circumstances often necessitating deeper thought on already established matters of faith. The circumstances that gave rise to the Reformation in the 16th century caused deeper analysis about the fundamental (essential) points of Christian faith. It's not that the Church at the time didn't believe in God's grace as the grounds of salvation: the problem (as the Reformers saw it) was how you accessed that grace. What was the God-given foundation for knowing what he required for sinners to be saved? This is where the question's point about "the logic of salvation" comes in.

At that time, Christian people were being told to do various things to access salvation. The Reformers asked questions: could it only be secured by a sacerdotal Church system? Was it only the voice of the Church and its traditions that had to be listened to? Was it through confessing one's sins to a priest? Was it through praying to deceased saints? Was it by participating in the Mass? They found answers to all those questions in the Bible, but those who attached equal (or more) importance to Church traditions and systems than to the Bible disagreed. And therein lies the rub - where is the source of authority?

As for asking if the Five Solas are "merely true under the light of Divine Truth" - I'm astonished! There's nothing 'mere' about Divine Revelation! Or, are you using the word 'mere' in the 19th century meaning of that word, as C.S. Lewis used it in the title of his book "Mere Christianity"? If so, then that's just fine!

In conclusion, axioms and theorems apply with no trouble to algebra and geometry, but the minute beliefs of faith are examined to see whether they are axioms or theorems, there can be no agreement until first an ultimate source of authority is established. That's what the Five Solas are all about, making clear five united perspectives on salvation:

(1) The only authority for what Protestants believe is the Bible.

(2) We can only be saved through Jesus Christ and what he did.

(3) The only means of salvation is faith (in Christ) alone.

(4) The only source of salvation is God's unmerited grace.

(5) The ultimate goal of saving faith is the glory of God.

Actual answer to the actual (main) question: the Five Solas started off as a theorem and are now axiomatic.

I wouldn't call them axioms; indeed the whole point of Sola Scriptura is that nothing outside the scriptures is infallible or definitive.

If there were axioms of Christianity they would be things like:

• God is love
• We are sinners who need a saviour
• The death and resurrection of Jesus

The Solas are more like fundamental theories, the theological counterpart of Newton's Laws of Motion. Protestant Christianity is built on top of and around them, but they're not axiomatic, they are themselves based on the facts and axioms found in the scriptures. And just like Newton's Laws need to be modified for very small things (quantum mechanics) or very fast things (special relativity), so too do the Solas sometimes need nuance and clarification. For example, Prima Scriptura is a clearer and less-easy-to-misunderstand way of talking about Sola Scriptura (although it also has a bit of a muddy history of use.) Sola Fide is frequently mis-defined as "salvation by faith alone" when it's strictly meant to be justification by faith alone. And that justification is the Protestant theological category of justification, which is not equivalent to every sense of the Greek word δικαιόω, and so much care is needed to neither import every sense of the Greek into the theological category, not to force the Biblical texts to fit the concept. We work from the exegesis of the scriptures towards a systematic understanding of a concept like justification, and from there a slogan like "Faith Alone" was derived.

So you can see that they are very much not treated like axioms. In a century's time Protestants will still be teaching the same theology as the Solas, but there's no guarantee that they'll still use the same slogans to do so. But they are still very important. The Solas are the clear unifying theological standards of Protestantism. If you want to debate whether Presbyterians should divide "ruling" and "teaching" elders it will need a long and nuanced debate. There are lots of debates over various approaches to apologetics, but having one preferred approach won't put you outside the boundaries of Protestantism. But if a cult says that all prayers to God must be made only by the leader, then that violates Christ Alone. We can then delve into the details of why that idea is ruled out by the scriptures, but the use of a Sola allows us to state with force that the idea is no small matter, but a serious divergence from Protestant orthodoxy. Likewise if someone was to teach that baptism itself justifies the sinner, we would say No, that violates Justification by Faith Alone. The Solas don't avoid the necessary detailed scriptural explanation for why that idea is wrong, but they do allow you to state that someone has made a fundamental error.