PEIRCE'S PREMISE: The object of reasoning is determining from what we know something not known.

PEIRCE'S CONCLUSION:Reasoning is good if it gives true conclusions from true premises, and not otherwise.

The conclusion implies that good reasoning applies to things that are true, in modern terms, to things having the truth-value TRUE. But, there is no such implication in the premise.

Incidentally, "what we know" includes not only propositional knowledge but also operational knowledge--skills, know-how--and also objectual knowledge--knowledge of objects including people, poems, songs, tools, etc.

In order for a premise-conclusion argument to be valid it is necessary and sufficient for every implication of the conclusion to be an implication of the premises.

This is a very good question with more than one possible answer and with quite a number of surprising research paths to follow.

To continue this discussion, a distinction is made between truth-preservation and  correctness-preservation in

O.T Hjortland, The Structure of Logical Consequence: Proof-Theoretic Conceptions, Ph.D. thesis, University of St. Andrews, 2009:

http://research-repository.st-andrews.ac.uk/bitstream/10023/892/6/Ole%20Thomassen%20Hjortland%20PhD%20thesis.PDF

See Def. 6.13, page  196 (correctness).   Non-falsity-preservation is introduced in Def. 6.28, p. 211 (see, also, Def. 6.29, p. 212).

Truth-preservation is contrasted with degree-preservation in

Handbook of the 5th World Congress on Paraconsistency, Indian Statistical Institure, 2014:

http://www.paraconsistency.org/book/Handbook-WCP5.pdf

See Sect. 6.12, starting on page 48.

It is observed that modelling consequence in terms of truth-preservation is substantive and debated question in

S. Yalcin, A counterexample to modus tolens, J. Philos. Logic 41, 2012, 1001-1024:

http://yalcin.cc/resources/yalcin.2012.counterexampleMT.pdf

One shouldn't give an argument about what "good reasoning" is. One should just stipulate a definition, and leave it at that.

For what it's worth, however, perhaps Pierce didn't take himself to be giving an argument about what "good reasoning" is, and his "premise" was just his unorthodox way of stipulating a definition of "good reasoning," so that, by "the object of reasoning," he meant "that which would make reasoning good." On this reading, the purpose of his conclusion is simply to point out a CONSEQUENCE of his definition of "good reasoning" -- namely, that valid reasoning is necessary and sufficient for good reasoning. On the face of it, THIS ARGUMENT -- from the definition to the consequence of the definition -- might seem to be invalid; yet it's hard to tell without knowing what he meant by "determining," etc.

I don't know if this helps (because I think the question entails Peirce's developed ideas, beyond what we find in that—very early—"Fixation" essay), but here's what Terry Moore (U. of Tenn.) writes from a "Richard Smyth perspective" on Peirce:

I don’t know whether it would do any good or not to send him my/Smyth’s answer (see below the listed questions). I certainly see why he thinks Peirce had a problem. Feel free to forward it if you want, but I don’t have time right now to follow up with him. (You can send him a pointer to https://dl.dropboxusercontent.com/u/17460644/Smyth/RPR-24Aug96.pdf).

My brief answers to his questions are in-line below.

Q1 Did Peirce ever retract PTPA?

No, I don’t think so. If you believe that it’s part of his general idea that “all valid reasoning is of one general form,” then it’s there from his earliest published essays and I don’t think he ever changed his mind about it. By way, the rest that first paragraph of section II of Fixation shows that the PTPA is closely connected with Peirce’s anti-psychologism.

It’s true that “P, therefore P” doesn’t seem to be going from known to not known; but it’s a limiting case. The point is that it’s not making you any stupider.

Q2 Has PTPA been discussed in the literature?

If I’m right, Smyth’s discussion of “A Reduction Thesis for Reasonings” counts on this front. So yes.

Q3 Did Peirce ever recognize consequence-preservation as a desideratum of reasoning?

Not sure what “consequence preservation” means.

Q4 Did Peirce ever recognize knowledge-preservation as a desideratum of reasoning?

“P, therefore P” would seem to be knowledge preserving, if P is assumed to be known.

Q5 Did Peirce ever retract the premise or the conclusion of PTPA?

I don’t think so.

Smyth has an answer to the problem he’s raising:

“Proposition 10 (A Reduction Thesis for Reasonings): There are analogues of the deduction-theorem for each form of valid reasoning so that, if one assumes that any reasoning (deductive or non-deductive) is valid, then from that assumption one can derive the truth of a corresponding generalization, and when that (putative) truth is added, as a new principle, to our original stock of assumptions the given reasoning can always be replaced by a valid deduction. (p. 141 of RPR_pdf)”

This is Smyth’s way of explaining Peirce’s claim that “all valid reasoning is of one general form” in Some Consequences:

“Finally, if the conclusion differs from either of its premisses, both in subject and predicate, the form of statement of conclusion and premiss may be so altered that they shall have a common term. This can always be done, for if P is the premiss and C the conclusion, they may be stated thus:

The state of things represented in P is real, and

The state of things represented in C is real.

In this case the other premiss must in some form virtually assert that every state of things such as is represented by C is the state of things represented in P.

All valid reasoning, therefore, is of one general form; and in seeking to reduce all mental action to the formulæ of valid inference, we seek to reduce it to one single type. (my emphasis)”

In other words, if you add a premise to any abduction or induction, assumed to be valid, a hypothetical proposition that expressed the connection between the premise(s) and conclusion of that abduction or induction in, then you get a valid deduction.