Part I: clearing up some preliminary matters in A37’s favor:
While the insistence on an eternal distinction between those saved and those hopelessly lost is typically focused on by Calvinists as part of their version of the doctrine of election (namely that God chooses from eternity whom He will and whom He will not even act to save), this is not what A37 is trying to claim. He is appealing to God’s omniscience, not to God’s omnipotence, in knowing ahead of time (as it were) who He will save and who He will give up on saving (or perhaps be unable to save due to some power, or due to some love He has for those He refuses to continue trying to save.)
Those God knows He will finally succeed in saving, are thus (according to this school of interpretation) those written in the Lamb’s Book of Life. Those God knows He will finally fail at saving for whatever reason, or those God knows He will eventually and finally give up trying to save, are (according to this school of interpretation) not found there.
Consequently, A37 as an Arminian is not making a contradictory appeal to a specifically Calvinist-and-not-Arminian line of thinking, and shouldn’t be critiqued as doing so.
Furthermore, whether there is literally a physical book (or scroll rather) is beside the point; if the scene testifies to any relevant truth on the matter, that truth remains whether the imagery should be taken literally or figuratively. A37’s argument does not depend on taking the imagery literally, and should not be critiqued as doing so.
A’s argument, however, is not overly coherent in its presentation at all points. To some extent this is only a fault of inept composition, and can be easily corrected without affecting the integrity of his argument.
Specifically: he stresses in his initial presentation and often afterward, the idea that no one’s name can be added to the Lamb’s book of life. Yet in the first paragraph of his initial presentation he writes, “When you accept Jesus Christ as Lord and Savior your name is written in the book of life.” Grammatically this is a statement implying that when X happens then Y happens, i.e. when we accept Jesus Christ then our names are written in.
I think it is clear that he doesn’t actually mean this, since he goes on to ask, “So, when does this happen?” He can hardly mean, “So, when do you accept Jesus Christ as Lord and Savior?” (Unless he is promoting a doctrine of pre-existence of souls!–which I have never once seen him do, and which never shows up in this argument elsewhere that it might be expected.) He must mean, “So, when do our names get written into the Lamb’s Book of Life?”
Consequently, he should have written something like: “When you accept Jesus Christ as Lord and Savior, you discover your name has already been written in the book of life.” Although it would be more accurate to his theology to put it around the other way again: “Your name has already been written in the book of life because God knew you would someday accept Jesus Christ as Lord and Savior.”
Correcting his presentation back around doesn’t hurt his overall argument, and removes ground for spurious criticism.
Slightly more problematic is his gung-ho insistence that no one’s name can be added to the book of life. After all, this contradicts directly with his insistence that people’s names have been written to, i.e. added to, the book of life!–otherwise those people’s names would not be found there at all! It must in fact be possible to add names to the book of life.
But A only means that no one’s name is ever added to the book of life other than those names which (speaking in terms of creation’s history) are found there during the judgment of the lake of fire. Tightening up his presentation on this matter would not hurt the validity of his argument, only remove ground for another spurious critique.
In any case, as I proceed along I will not be adducing such rebuttals against him.