This is a challenging matter to argue deductively, especially given that a) the critical passage from 2 Timothy doesn't define key terms like "God-breathed" & "scripture", and b) the passage that supports premise 1 is part of the body of text that is being defended by this argument.
I do not agree with all of the objections I will suggest below, but I will attempt to offer a survey of the types of objections I am aware of.
- There are people who believe in inspiration & inerrancy, there are people who believe in inspiration but not inerrancy, and there are people who believe in neither. The latter group would object to anything that follows from premise 1
- The human factor remains--if this were an argument that God personally penned the Biblical manuscripts we could probably construct it in a manner that was as logically air-tight as a geometric proof...but the medium of human authors leaves some grey area. Did God give them the exact words, did He give them the ideas to convey in their own words, did He consider their good-but-imperfect consecrated efforts adequate for His purposes despite their human limitations, etc.?
- The argument will be more persuasive to those who already believe the text is inspired, but it will not be persuasive to those who are investigating the matter and remain unsure, because we are using the Bible to argue for the credibility of the Bible. Those without separate grounds for trusting the Bible would perceive this as circularity
- There is no delimiter here...what is covered under the umbrella of "scripture"? Paul almost certainly had in mind the Tanakh/Septuagint when he wrote this to Timothy. What set of books counts? People who believe in a 39/46/66/73/74+ book canon could all use this argument and still come to strikingly divergent conclusions regarding what writings are inspired
If the target audience is people who already believe in an Omniscient, Omnipotent, Omnibenevolent God, this premise is probably pretty secure.
Those who don't already hold this belief would put no stock in premise 2, and would argue that the Bible is the source for believing premise 2 in the first place, leading again to circularity.
Is communicating some but not all of what one knows truthful? One could argue that God only speaks truth, but deliberately withholds information that would be unhelpful, overwhelming, or unproductive, even if the result is that readers' understanding is (for the moment) incomplete.
Point 4/Conclusion 1
Everything but the last 7 words follows deductively from the first 3 premises. The last 7 words offer a definition that would be split out into a separate premise in a formal proof.
Point 5/Conclusion 2
This point is offering a definition of terms, the logical deduction was complete with point 4. It leaves open (perhaps intentionally if it's outside the intended scope of this argument) whether "errors" contemplates matters of doctrine, matters of science, matters of history, matters of grammar, etc. In what sense are they not in error?
One might also object to the present-tense verbs--is the argument focused on inerrancy of the autographs or inerrancy of a present-day text?
I would classify this argument as abductive rather than deductive--it makes a point that many will find persuasive, even if it is not logically air-tight. As noted by jaredad, using a Biblical passage to argue for Biblical inerrancy probably won't give us a deductive argument.
My own observation is that most of us who believe the Bible do so on the basis of the personal witness of the Holy Spirit, often with supporting arguments derived from history, linguistics, archeology, and so forth. I shared my thoughts on a variety of these argument in this post. That may not get us all the way to a theology of inerrancy, but it certainly appears to have given millions (if not billions) compelling reason to trust the Bible as a reliable source.
It is my personal belief that God is able to get His work done through imperfect people and their imperfect efforts.