Idea for resolving Liar's Paradox:
In logic, sentences should be assigned circuits* instead of truth values.
Firstly, processing circuits, which execute the action described by the sentence.
"Proof" that there exists a suitable circuit for sentences such as "2+2=4" "2+2=5" and "9^9=387420489", exists in the form of calculators.
Secondly, circuits for checking if a truth value assignment is consistent. They receive "True" or "False" as an input, hereafter referred to as "I", and output "True" if the assignment is consistent. This can be done by having both the processing circuit and "I" as inputs to a gate which returns "True" if they are both equal. (An equals gate.)
This is like turning the problem of evaluating the truth value of "2+2=4" into constructing a circuit that "represents" "2+2==4".
These are necessitated by the existence of sentences which may be assigned multiple truth values. "This sentence is true." Assigning 'true' would be consistent, as would 'false'. (This sentence is unusual in that the consistent assignment circuit has the input, "I", go into both the processing circuit, and the equals gate.)
*Several things work here. In addition to being a good place to hold a lot of technical/logical questions, circuits can be implemented fairly easily.
I came up with this when thinking through something I saw someone write on the resolving the LP (on their blog, they also wrote a book). Now I'm trying to find it again.*
What they said, paraphrased (from memory): You attempt to assign a truth value as follows: you suppose it is true. If it were true, then it would be false. So you suppose it is false. But then it would be true. At this point, they rejected this line of thinking on the grounds that, what it means for something to be true, is for there to exist something in reality that corresponds to it, and there is no operator that satisfies this criteria. So the LP is false. (This is similar to the answer that it's "not true or false, but nonsense" (from xkcd forums results on google search, when I tried to find their blog). The author took the additional step of combining the notions of "falsehood" and "nonsense" under the label of "false".)
When I was thinking through that, while I got their point, it sounded like a NOT-gate. That is, I figured you could assign to a sentence a circuit which takes the truth value you would assign to it, and returns what it would be if that were so. This made sense for both of the self-referential sentences I considered (LP and "This sentence is true."), and valid assignments were fixed points. What makes LP "paradoxical" is that trying to assign it a truth value is a process that corresponds to trying to find the fixed point of a function which doesn't have any fixed points. (It's opposite behaves the opposite way: it has all the fixed points.) When I thought about other sentences that weren't self-referential, this didn't make as much sense, and that was when I came up with the other two types of circuits/(ways of thinking about this).
*EDIT: It's fakenous.net.
There is a general pattern that occurs wherein something is expressed as a dichotomy/binary. Switching to a continuum afterwards is an extension, but this does not necessarily include all the possibilities.
Dichotomies: True/False. Beautiful/Ugly.
Logic handles this by looking for 'all true'.
If 'p' is true, and 'q' is false, 'p and q' is false.
More generally, a sentence could be broken up into parts that can be individually rated. After this, the ratio of true (atomic) statements to false (atomic) statements could be expressed - unless all the sub-statements are true, or all false. This can be fixed be expressing the 'score' as a (rational) number, with two choices of score:
true(sentence) = number of true statements / number of statements
false(sentence) = number of false statements / number of statements
And since every statement is true or false:
true(s) + false(s) = 1
And if we want to express how much truth is expressed, true(s)*num(s) = # of true statements. (These functions don't have the best relationship to each other, they're just meant to be intuitive enough.)
Consider the assumption: every statement is true or false. (Exclusively.)
Instead of diving into paradox, consider functions. Equality(s) returns the sentence "s is true". Negation(s) returns "s is false". These functions don't have a truth value, it's dependent on the variable that's passed in. Concat(s_1, s_2) returns "s_1 s_2", which can be just gibberish. But why is "Equality" named Equality - it preserves truth value but there are other functions with that property? It might be better thought of as a family of functions.
Now consider the function f that, given s, returns "s is true and s is false". And here is a function that is always false. Right?
(This next paragraph* is a framing without examples, and may be rejected or accepted. I'm treating 'paradoxes' in this way because, as the paragraph after it notes, truth seems to come from a system.)
But just as (self referential) sentences can be constructed that are 'paradoxical' - neither 'false' nor 'true', sentences may also be constructed which 'are both'. This may be resolved by pointing out that the first are "nonsense", and resolving that "nonsense is false", and saying that it doesn't matter what value is assigned to the second as there is no consequence. (For such a sentence may be false, or it may be true, but not both at once.) But these resolutions are at odds. Are not both kinds "nonsense"? Or if they are different, they seem different from both 'statements which can only be true' and 'statements which can only be false'.
To get back to our 'functions' (which take sentences as input, and return a sentence as output), consider the sentence "1+1=2". Is this true? In many systems yes, "base 3, base 4, base 5, ...", but not in "base 2", where "2" is not defined, "1+1=10". These systems may be converted between, and we may even say that while something is expressed one way in one system, and another way in another system, they're the same "fact" (or falsehood).
But having different systems enables much confusion, two (or more) people might disagree on what color the sky is currently, even if they both have eyes that work fine, and without any unusual atmospheric phenomena that change what the sky looks like if you take a few steps to the right, or the left, if only they disagree on what colors the words for colors mean. If you call X "red", and I call X "blue" we may still both see X.
To get back to truth(s) which can return "2/3" (meaning s contains 3 statements, 2 of which are true, one of which is false), why return one number? Why not two: 2,1: 2 true statements, 1 false. But there could be more statements than those two kinds. And here the path splits in two.
1. A particular methods of assigning one value to a sentence may 'fail on the paradox', or choose to call it false.* One method, one answer - every statement is true or false, exclusively.
2. A set for each possibility: It is true, it is false, it can be true or false, it cannot be either, etc. There's still a binary aspect to this: "is it true" receives the answer "yes" or the answer "no" exclusively. But, independently, "is it false" may also receive either answer.
Following the 2nd path, what does it mean for something to be true and false? Neither?
One way is this: "The sky is blue, and the clouds are red." Part of it is true, and part of it false. That which holds neither truth not falsehood, is nonsense.
How does this generalize? For that another dichotomy will be required.
Beautiful/Ugly***. While this may be subjective, the quaternary** view can be seen as claiming the binary view is false, some things are both beautiful and ugly, and some things are neither. Perhaps here this view will be less controversial, after all, if a thing is judged to be beautiful by one person, and ugly by another, "subjectively", then "objectively" might not the object be both? Perhaps something ugly and beautiful could be created by cutting something beautiful in half, and something ugly in half, and combining them? This may be trickier than combining a true statement and a false statement, but perhaps if something is both beautiful and ugly, both aspects can be seen, where something that is true and false might be swiftly proclaimed 'all wrong' (or all right).
Perhaps this has all just been confusing, or perhaps it will be useful. The notion of 'logical counterfactuals/counter-logicals' has seemed strange to me - it is not that "it could be that 2+3 = 4" but that must be a different system. What such a thing could mean in conjunction with a world, say, if you put 2 things in a container, and then three, and what results is 4, seems unclear. (Even making them creatures doesn't make sense, for if one eats another, why won't that happen later?) If it holds for a class of objects, then that changes the relationship between numbers and objects - an apple and an orange are together are two things, but even if all things have the property that under certain circumstances they react to produce or eliminate another of the same type, then unless this holds between classes, no more might one speak of an apple and an orange being 2, because they don't react with each other.
*Paradoxes working this way may be avoided by system design.
**One may eliminate one of these categories, and say, that nothing is neither beautiful nor ugly. Then the category still 'exists' though it has no members - a broader view may include things that are not, but absent a process for creating new categories, the more expansive view may be better before examining reality. And if someday that person finds something which is neither, then the bucket will be ready for this new object unlike anything seen before.
***This is one area where things may not be fixed, in a way that we don't see in math or logic. A view in which things don't have properties may be more useful - but it is harder to see this for things/properties like "numbers" which 'seem to exist'. "The tree falls in the forest" argument may also be had about beauty.
Thanks to this question, I recently started thinking about how progress on open problems in math  could be made faster, at least with regard to low hanging fruit. I made a comment there about modeling the problem (how can progress be made faster) and a possible solution.
This brings me to a few questions:
 They have a certain formal/empirical quality which makes things simpler. It also might be easier to use this as a metric for 'how good is our X  at advancing research (progress)'?
 Anything that could make a difference - a Platform, Organization, Program, a set of Math courses...
Two models of how feedback is useful, for making corrections.
1) Post quality*:
*Intended generally. May also apply to books, etc.
** This is a subset of (possible) responses, because it is still about communicating the same idea/s, rather than doing something new.
Some ideas on structure:
"Posts" are usually "about themselves". For example, SSC has posts with no comments section. For counter-examples, see posts like Reasonable Explanations - the author is interested in comments that fit a certain format, the body of the post has the rules for (top-level) comments, and the author posts a comment (that fits the format) as a starter.
This is a format for a "Discussion". If an author includes both the rules and the starter in the "post" body, it's still a "Discussion". If the OP expounds an idea, and includes examples (usually of a certain form) and suggests people comment other things they think might be examples, that is both a "Post" and a "Discussion".
The Monthly threads are of course "Discussion"-like, though they're more free form - a "Discussion" with no rules*.
*Since this is LessWrong, both LessWrong's rules apply, and ways people here prefer things be discussed - this is why "The Sequences" are emphasized. The second type, not being laid out in a short explicit set of rules are at times "broken", leading to conflict.
The value of structure, including linearity. (A lens.)
Some (if not all) sites are about ideas. This site does it by "one-off" methods for presenting ideas. These may be contrasted with systems that present an idea, but (may) continually change the presentation (non-automatically), such as wikis. What might a site look like if it tried for more integration?
(Bending the format:)
What if comments sections, rather than staying fixed in their attachments to (a) post, roamed around?
ETA: What is an actually good way of getting combinations of ideas/comparing how similar problems are solved in different fields?
From the other direction - Ideas are in posts. This is part of why re-runs exist - to send the idea out again, to reflect, and to bring comments on the idea to life again.
When a post is run the first time there are comments. When a post is re-run (unchanged), the idea may already be out there (it's possible all the readers have read it), but there are new comments. In this way, the comments section on the re-run is still about the same thing, absent changes resulting from time, it's just comments 2.0. It's also fresh - when I read The Sequences, I did not read all the comments.
(Bending the format.)
Setting up a comments section so that is possible would require a redesign, and probably work against the reasons they were set up the way they are. (Which is why The Sequences were made into a book instead.) I haven't seen a lot of sites do this intentionally. There are blogs with no comments sections anywhere, but making a set readable by making it empty is trivial.
Things that becomes "finished" ('Posts') versus Things that don't ('Lists'):
Finished: Posts/Comments are (individually) created, then submitted.
Common Exceptions: "Update" may be appended to the end, followed by content. Alternatively, changes may be made, and described in a section added to the end marked "Edit".
Un-Finished: All Posts, All Questions, All Sequences (The Library). These lists keeps changing.
While a List may come to an end, if all Lists die (and stay dead), that's a sufficient condition for the site to be considered dead.
That's not to say the site would be dead if there stopped being new posts for a time - if people started revising their posts, and submitting those changes, discussion of ideas (and the life of the site) could continue - but then the currently existing posts would "living lists" while "all posts" would be dead.
The pattern seem to be "Lists", which can go on forever, contain "items" which have a short life.
Two ways on looking at things:
1) See what this website calls, say, "Posts". Look for patterns. (Practice -> Theory.)
2) Consider different Ideas, and look at what 'implements' them. (Theory -> Practice.)
This is why what the site designates "comments" are often referred to by users as "posts" - they implement the same idea.
Perfection is achieved, not when there is nothing more to add, but when there is nothing left to take away.
-Antoine de Saint-Exupéry
"The No True Scotsman fallacy" is often cited when people do things like defining X not as Y, but Y when Y works.* This is the (explicit) ideal (that people may admit to). While those asking "What is X" are probably interested in "When does Y work?", if X/a group that defines itself based on X (and refers to itself with the label 'X'), then since their goal is to achieve that ideal, they themselves would very much like to know/and are working on "what is necessary to make Y work/happen?". Thus 'Y is not working' may be (seen as) a criticism of (the group) X - and spark some debate. (The ideal may be a motte and bailey, or fake.)
To make this more concrete here is an example: "Rationality is about winning." (I'm still waiting for the "X is not about Y" article "Rationality is not about winning".) What other things are (or can be) defined in terms of 'when they work'?
*Or more specifically (see the wikipedia article) 'people who like X' think of 'the examples of X they like' when they hear 'X'.
This thread get Meta (about LW) here, TLDR here. (I discuss a feature I think would be useful, that might affect user interaction, in hopes of starting a discussion. This may become a post for that purpose later.)
Comment types (on posts):
--Comments (I love this post/I hate this post/etc.)
--bolding [X] might increase readability
--[X] would be good in a summary of the posts
General Styling question: Does this site support bulleted lists which contain bulleted lists?
Relatedly, it's useful to know what users (esp. authors of lots of posts) like what kinds of feedback.
Obviously there can be nuance:
- Especially grammar and spelling.
- Or prefer commentary include points of agreement (or overall impression) as well as disagreement.
While explicit site rules/norms can guide interaction methods overall, tools for this purpose might enable both more improvement and lower friction - users getting feedback/interaction they find valuable, even when different users want different kinds of feedback/interaction.
But tools are expensive. A giant list (in one place) might capture some of the value, by enabling such information being searchable common knowledge.
Retrieving such information might constitute a "trivial inconvenience", but such a document would be easy to create, and have a shorter feedback cycle than a tool that would require more investment.
Past discussions of possible features suggests that a sequence of such lists might be useful.
Benefits of tools over lists: it would be good to have the information accessible (searchable) in multiple ways, but easy for a user to change their info across all the different places quickly.
If someone just wants to know what interaction style a user prefers, then a list (in a google doc/the new LW editor when it comes out) with users and styles can be searched (using a keyboard shortcut, and typing the user's name). But if more information gets stored this way, it might be helpful if all such information concerning a user could also be accessed.
This could be done in a google spreadsheet. (I'll add a link to an example (with fictional users) when it's complete.) I don't think that'd be a good long term solution, but it illustrates what features are necessary.
Some information about users (interaction style, preferences on feedback type or format) might be useful to have available when interacting with them. I propose a document to be created, that users can edit with their information in this regard. Such a list may require the google-docs-like-LW-editor to be completed so it can be edited by users to stay current.* I also outline (above) possible implementation and what features it would be useful to have if it expands to incorporate more information.
*Since the key is "everyone is able to edit it" maybe it could go on the wiki.
There are other aspects of feedback as well, see Comment, Don't Message for one.
Bookmarking comments (This is a list.)
Also pages (https://www.lesswrong.com/allComments) and sequences (https://www.lesswrong.com/s/yai5mppkuCHPQmzpN).
Better stored tag-wise (Issue appeared here: https://www.lesswrong.com/posts/po8guXNhXzXYo5yFM/shortform#YzXqCuWTZqF8ZTToX)
Great things about Greaterwrong:
[On LW] if a comment is automatically minimized and buried in a long thread, then even with a link to it, it's hard to find the comment - at best the black line on the side briefly indicates which one it is. This doesn't seem to be a problem in greaterwrong.
Example: Buried comment, not buried.
Discussions of tagging (and within subreddit tagging versus shared taggging)
Implementing AIXI (question).
(What makes a proverb different from a meme?)
My own (long?):
https://www.lesswrong.com/posts/9FNHsvcqQjxcCoJMJ/shortform-vs-scratchpad-or-other-names#GKuKJFnvDuinjWGYd Some names.
Google docs are good for saving content on the fly.