Bertrand Wittgenstein's Ghost

Unfortunately I can't really grasp what you are trying to do from your question, that's one of the main issues. Other than that, $+,×$ are all well defined binary operations on congruence classes $[a]_n+[b]_n=[a+b]_n$, but I don't know if that's what you are looking for. Good luck.

Regardless of how you are phrase it, the end result is that you are trying to define a binary operation on set of set, it doesn't work unless you give it some nice properties. Call it $-3A=-A-A-A$, or call it $-3f(Dom(f))=-f(Dom(f))-f(Dom(f))-f(Dom(f))$ these are all sets, they are not integers.

Your question is very - very - confusing. What do you mean by $f(5m+1)$? What does it map to? $f(5m+1)=[1]=\{6,11,16,....\}$? This doesn't make sense because, again, this is not a function but a set. The same idea applies here. You can't add sets unless you define what $+$ means.

Equivalence classes are sets, a collection, not a singular element like some arbitrary integer. You are comparing apples to oranges. How do you solve this equation: $3+\mathbb{Q}$? What does $+$ even mean in this context?

Couple of errors I came across: 1. $p| r^2$ does not follow from anything you have said prior to that, Notice $5 | 15$ but $5 \not | 12+3$. Then you said $0<r<p$ so $p \not | r^2$, again this is wrong. Notice, $0<3<9$ and $9\not | 3$ but $9|3^2$. Everything you said after that is clearly wrong too.

@KasiReddySreemanReddy Epiphenomenalists are not eliminative materialists, so obviously they won't say brain states are isomorphic to mental states... that would make them - wait for it - eliminative materialists. They are Epiphenomenalists because they think mental states are not isomorphic to brain states, I mean that's their distinguishing feature.

@KasiReddySreemanReddy "phenomen(on) that are not subject to laws of nature" I believe you are a smart individual, you can connect the dots now, right? mental states are not isomorphic to brain states. Latter is reducible to fundamental physical interaction, i.e. subject to laws of nature, so, former being non-isomorphic to latter implies what? Which implies we have at the very least one group of people that can consistently believe in the supernatural without denying induction and science.

@KasiReddySreemanReddy I asked a specific question, what do you mean by supernatural?

@KasiReddySreemanReddy what do you mean by supernatural?

Epiphenomenalists are, for all intents and purposes, supernaturalists who don't deny induction and science.

@NoahSchweber Thank you for that, I will try wrapping my head around it and see if I can form an intuition of sorts.

@AndreaMarino Thanks for the information. I am not familiar with much of what you said, but it's interesting. Can you refer me to some text that can further elaborate on that?

@NoahSchweber Thanks for that! You are correct, I shouldn't have called it an identity function. That said, I have fixed the question. If not, I will try retyping all the relevant definitions in here but that might take some time.

@AlexKruckman I am sorry, I realized that right now. It should be fixed. If not I will retype all the relevant definitions from that chapter in here, but that will take some time.

@NoahSchweber Yes. That's the book!

@NoahSchweber That citation you gave seems very very similar, I believe they are the same thing except that the book I am referring to is for propositional logic and the other for predicate.

Sorry, I try to put it in my own words but it seems like it doesn't work.

@NoahSchweber Hi, thank you for that, I added the relevant excerpts from the book. Feel free to have a look.

@AlexKruckman Hi, I added the relevant excerpts feel free to have a look.

@Speakpigeon I know that you know what I said and meant. Let's drop the semantic hoop−jumps. The question is subjective because every answer to it is subjective (which, btw, is one of the listed option one can choose to flag the question for closure, but that's besides the point). I don't know a single logician that has claimed Material conditional is THE logic for conditional. I will tell you this though it's the most pragmatic approximation.

That's of course my subjective opinion. xD.

@Speakpigeon I think this question is very subjective. Do I believe MC is necessary? No. Is it pragmatic? Sure.

You are also forgetting, you don't only use MC for proofs of the form: (if a, then b) & (a); therefore, (b). Every subsequent step given an assumption is conditional: Suppose P, deduce Q; therefore, Q.

@MauroALLEGRANZA They do exist, but working with them, in any practical sense, is a pain. I think they might work for the foundational end of the spectrum, but I don't see anyone abandoning material conditionals for proofs like OP mentioned any time soon.

@user4894 I am messing with ya. You hit the nail on the head. I would call all these number games just that: arm−chair postulates. Useful? Sure. Meaningful... Well that's for the realists to decide.

@user4894 I see that you don't like the real numbers. Wait till you hear about ℂ, hyperReals, and the arm−chair postulates. xD

That should be sufficient as a survey.

You can also get an intro to philosophy of mind book.

Alright, i gtg. Any ways, if you want to read this stuff I can name you the pertinent theories: Behaviorism, type Identity theory, functionalism, panpsychism. Have fun!

Only eliminative materialism does. Also, wikipedia is not entirely accurate at times

Plus reductionism doesn't reject it.

They reject is based on mere hope.

"Magical" since it hasn't been explained.

As far as our experience is concerned it is magical.

The subjective feeling of being alive, feeling, tasting.

Qualia is how an entity experiences existence.

Yes.

Identity theorists wouldn't call the computer conscious. Same for the behaviorist (to an extent, but it can work).

Depends on the theory of mind.

It's missing Qualia.

Qualia.

What you are presenting now is called the behaviorism of logical positivists. X is conscious iff X behaves as if its conscious.

Then that is behaviorism.

Why would they have it?

Carbon-based Biological neurons.

Neurons

I will write it one more time:

What?

As for particle themselves being conscious than that is panpsychism.

Identity theory.