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"Holding" Contradictions: Good Or Bad?

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A contradiction is a bad thing, but an arbitrary resolution could be worse.

No rational person is going to claim that A is non-A. Real-life contradictions are more indirect. Say...

1) A is true

2) B is true

3) A implies C

4) B implies non-C

Resolving this means figuring where one made the error. Does A imply C? Does B imply non-C? Is C something that is always true or can it be that C is true in some contexts and non-C in others? What are those contexts?

Thinking takes time and effort. What if one does not expect to be encountered with a situation where the one has to decide between A or B? In this case what is the best way to store this in one's mind?

The way I would store it is: A appears true, B appears true, etc. (i.e. as uncertain knowledge and as an unresolved contradiction).

A bad approach would be this: "I know contradictions cannot exist, and I am slightly more certain that A is true, so I'll simply conclude that B is false and ignore the evidence to the contrary."

This invites disintegration.

If one has to act and choose, one must do so to the best of one's knowledge. However, one does not need to corrupt one's mind by accepting uncertain knowledge as being certain. So, be proud to hold your contradictions, as long as you realize that is what they are!

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The way I would store it is: A appears true, B appears true, etc. (i.e. as uncertain knowledge and as an unresolved contradiction).
As abstract propositions it's hard to decide whether you should say that A merely appears true -- what is A? When checking premises, of course you should check whether A really is true, so how did you conclude that A is true? Sometimes A can stand for the perceptually obvious, in which case you cannot conclude that A is uncertain. So the first step has to be asking "on what basis do I conclude A?".
A bad approach would be this: "I know contradictions cannot exist, and I am slightly more certain that A is true, so I'll simply conclude that B is false and ignore the evidence to the contrary."
I agree, but would make an even stronger statement. There can be no degrees of certainty. Either a proposition is certain, or it is uncertain. If it is uncertain then it might be strongly indicated or weakly indicated, but either way, an uncertain conclusion is not a proper basis for deriving other inferences. Incidentally, that corresponds to a condition on the Objectivist formal logic that I've peddled here and there, to the effect that no premise may be introduced into an inference if it is not true. That means that arbitrary, false and uncertain conclusions cannot be terms in a proper logical deduction. The practical consequence of that requirement is that claims must be proven, not merely shown to be plausible, before they can be used to derive true knowledge.

In those cases where you cannot see how to combine two uncertain propositions -- ones which are well supported but not proven -- and draw a third conclusion, I would say that you do not have to hold any contradictions at all, rather you simply have not yet learned what the correct contextual refinements on your base propositions are.

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I can see why false and arbitrary conclusions have no place in proper deductions. I can also see that an uncertain conclusion cannot lead to a certain deduction.

Would it be true to say that a couple of uncertain conclusions could be usefully combined to lead to an uncertain deduction?

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Would it be true to say that a couple of uncertain conclusions could be usefully combined to lead to an uncertain deduction?
I'm not certain. (Just kidding). One thing is that a supported but uncertain conclusion has to contain some element of certainty -- you can't found a deduction on nothing but completely arbitrary claims. Let me present this abstractly, rather than getting bogged down in arcana (since for the moment the only examples that come to mind are obscurities of linguistic theory). Observations lead us to conclude A and B, but we can't rule out the alternatives A' and B' because no conclusive observations have been made in contexts x, y: thus A, B are certain in the contexts ^x, ^y. Combining A and B may lead to a further conclusion C, which is certain in the contexts ^x, ^y, but it is not certain generally. This is the crux of what I understand "usefully certain" to be: actually certain, but in a particular context. The problem with merely "useful" theories is that context-dropping is too easy -- we don't know how to superglue the necessary awareness of context to the conclusions. OTOH it is counterproductive to be a nihilist, to say "we just don't know" when in fact you do know, except in a particular known case. Thus a pair of probable conclusions can yield a probably true conclusion, which could be useful if you remember when it is certain vs. arbitrary.
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