Final-Fan said:
"Confusion"
The reason I've used varying terminology is to try to avoid misunderstanding; using different words to convey the same idea lowers the chance IMO that a key phrase will cause misinterpretation of meaning. It would seem that this attempt has failed spectacularly.
What I meant by the math comment was that the person who thought that 4 was the only answer to X times 0 equals 0 according to the rules of mathematics as we know them would have to reevaluate his position based on the fact that his position can be demonstrated to be wrong (that is, at least ONE of the TWO beliefs in his belief set is clearly incorrect as they contradict one another). This is not a rebuttal to your point but (once again) me using your example to illustrate an example of my own.
"Sophistry"
No problem. The first dictionary I checked said it involved deliberate deception, but the second, more trusted one didn't mention that as necessarily the case, so you're off the hook. 
"My Fundamental Issue"
A.
"(No I don’t agree… my fundamental point is that sense data doesn’t indicate anything by itself so you’d have to arbitrarily choose an interpretation to make it contradict some other belief you’ve taken [for a reason still unknown to me].)"
And MY point is that (1) a BELIEF is an arbitrary choice. One can arbitrarily believe that sense data means such and such, and some beliefs would be that a given type of sense data can ONLY mean ONE thing. (2) Thus, if sense data is inputted to a person with a belief that it means a certain thing, and that certain thing in this instance contradicts another belief they have been holding, then their beliefs contradict each other. (3) Therefore, their BELIEF SET (beliefs taken together in a system that ought to be internally self-consistent) is self-contradicting and has disproved itself, with the inclusion of the sense data (that can only be interpreted one way by the belief set).
The only way to deny this IMO is to deny that a person can possibly have a belief that a particular type of sense data can only be interpreted one way.
Even aside from the above argument however, I am puzzled that you would say you DISAGREE with the statement "Belief sets exist that can be contradicted by sense data" when you have just said, "So yes, of course your scenario is possible in the sense of illegitimate beliefs. I never meant to imply such illegitimate beliefs didn’t exist."
If illegitimate belief sets exist (as you agree -- or at least specifically deny implying otherwise) and they can be self-contradictory in the face of sense data (as in my example, which you agree is possible) then how can you deny that belief sets exist that can be contradicted by sense data? HOW?!
B.
With the piano example, the empirical belief set includes the possibility of illusion, because the senses are not always 100% accurate. But they DID detect things that have the appearance of piano keys, etc. The person did not approach and find that the piano had turned into a bear. The world did not explode. You misunderstood what I meant by "one general picture of the world", and I struggle to continue to believe it was a genuine mistake. How often does a piano really turn out to not be a piano? The senses are generally reliable, by which I mean they are reliable enough that they form a coherent world. The fake piano revealed itself in proper parallax, wouldn't you agree? There was no inaccuracy there?
MOREOVER, your reply completely ignores the very next sentence: "It may not be a correct picture, but the picture exists." When I said "picture", I didn't mean like a Polaroid. I meant it more generally, to include the passage of time. So in fact, the fake piano is consistent with this picture, because as I said it tricks one, and reveals its trick, in a way consistent with the empirical understanding of your senses and powers of observation and time. I thought you would understand this because I talked about output 'affecting the picture' and the picture 'reacting' which implies successive images, not just one -- and therefore, by implication, the entire observable world from supposed cradle to assumed grave. (I suppose this assumes the existence of memory but as this part of the discussion is at some point about practicality, that must be a given IMO.)
Also, this part of the discussion is not about absolute truth. If you mean possibilities like "this could be a fake piano like the one in that exhibit" then that is a possibility accounted for in empiricism.
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Confusion
I hope you don’t think I’m doing this on purpose but again, I really can’t understand what you’re trying to communicate much of the time. E.g.
(that is, at least ONE of the TWO beliefs in his belief set is clearly incorrect as they contradict one another).
What does ‘one of the two beliefs’ refer to here? And what does contradictoriness have to do in relation to the point I was making using the example… if it relates to my point.
This is not a rebuttal to your point but (once again) me using your example to illustrate an example of my own.
Ok so apparently you’re not addressing the use of my example in relation to the point I was making. Well, I still don’t understand what you’re using the example for.
Other things that confuse me are below.
My Fundamental Issue
You still seem to be making a point about a belief(s) and a belief on sense data contradicting each other. Still. And I don’t know why. I really don’t.
I thought I was clear that my point was about legitimately establishing a belief (method of truth). And that since you seemed not concerned with that, you were establishing something else (input/output, regular belief… I still don’t know what that means). I needed to understand what you mean still.
So not only do I not understand what you’re trying to establish in terms of “illegitimate belief” but I can’t find the relationship it would have with your point of showing how you can contradict an (illegitimate) belief with an (illegitimate) belief on sense data. (You made it clear that these were illegitimate beliefs, in my terms, when you emphasized “arbitrarily”.)
What I just said here should clear up what my problem is but just to address your specific statement with reiteration:
“how can you deny that belief sets exist that can be contradicted by sense data?”
Again I never denied you can have illegitimate beliefs (from arbitrary assumption or of sense data) that contradict each other. I was speaking of legitimate beliefs (method of truth) or what you call absolute knowledge. Everytime I said "it wasn't possible" I was referring to legitmacy. Thus your point was always irrelevent to me as I was under the impression we were talking about estblishing legitmate beliefs.
Now A Side Note – My Fundamental Issue
Again, as I’m under the impression that you’re not addressing legitimate beliefs I still am hoping you did see the point I was making for empiricism. And it seems you did in regards to the math example: that a person could be shown how he was wrong to believe that only 4 could represent Y. Yes, when someone writes this equation for you (whether related to something in life or not) and has a specific number in mind, you have no way of figuring it out by itself. Truth is in his mind (like reality) and the evidence of it presented to you can represent, certainly clearly in this case, an infinite number of possibilities.
B.
Here it almost sounds like your dealing with legitimate beliefs again. I don’t know how to respond if you’re not because again I still haven’t understood what exactly you’re trying to establish if not legitimate beliefs. I can’t help but think you are inevitably speaking of legitimate beliefs because… well you’re explaining the reality of the matter in regards to whatever point you’re trying to make.
I did hear one familiar response that, if you were discussing legitimate beliefs, would make sense in an argument on empiricism. You said in effect that sense data reveals it’s own trick with the piano by power of observation and time. I’d respond to that but I’m too nervous I’ll hear “I’m not talking about the ‘really real truth’ ” again, which sends me into that dark corner of monopolized confusion. lol
Conclusion
Please explain clearly what you are trying to establish if not legitimate beliefs (method of truth). And also explain how whatever you’re trying to establish relates to arbitrary assumptions (illegitimate belief) – and why I would be “interested” with what you can do with arbitrary assumptions.
