score:2
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.
Premise 1:
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Premise 2
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.
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Premise 3
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.
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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.
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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?
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Conclusion
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.
Upvote:3
This is not a valid deductive argument. 4 does not follow from 1, 2, and 3. Even the proposed edit to 4 that removes the phrase "and are inerrant" would not make it valid.
4 does not follow from 1, 2, and 3 because it is only stated in 3 that lying entails falsehood, whereas other acts may also entail falsehood, and nothing is said of these in relation to God. For instance, if God could Himself be deceived, He could then put falsehoods into Scripture without lying. Hence, 4 does not follow.
To amend this, 2 ought to be adjusted to say "God can neither intentionally deceive (that is, lie) nor can He Himself be deceived (that is, He cannot be mistaken about a matter of fact)." Then, 3 ought to be adjusted to say "Falsehoods can only be propagated by those who either are deceived, or lie, or both." From this, it follows that no Word of God can be false.
This edit still does not cause 4 to follow from the premises unless we precisely define what "breathed by God" means. Does it mean that God spoke and the authors of Scripture wrote what they heard? Does it mean that the Holy Spirit guided the authorship of Scripture, but that the authors were ultimately in control of their own wills? If the latter, it is evident that 4 does not follow from the premises, since the authors of scripture both can lie and can themselves be deceived.
Assuming that we take the former position, and say that God literally spoke every word of Scripture Himself, then this is a valid argument. Opponents of biblical inerrancy might then level attacks against the very weak premise that God literally spoke every word of Scripture, undermining its soundness.
This is also not to say that proponents of biblical inerrancy need to believe that God literally spoke every word of Scripture. I personally believe, and my church teaches, that the Holy Spirit inspired the authorship of Scripture while permitting the authors to retain controls of their own wills, and yet that the words of Scripture are factually inerrant. However, this takes faith in those authors, and in the wisdom of God, to not permit the Scriptures to be corrupted by wicked men. A reasonably sound deductive argument probably does not exist to support biblical inerrancy.