All answer submissions must be a declarative statement about this market itself, for example, "This market will have more than 10 options," or "The percentage of this option is between 5% and 10%." Any answer that is not phrased as a declarative sentence or isn't about this market will not pay out (when I say "pay out", I mean that I will resolve partially to that option). You can, however, include something in addition to the declarative sentence as a tag, e.g., "ABC: This option is higher than 5%", for the purpose of referencing it with other options (e.g, "ABC has a higher probability than DEF."). But don't duplicate someone else's tag - otherwise, your option won't pay out, as a punishment for confusing traders.

I will resolve equally to all options that meet the above criteria, as well as the following:

They are true at close.

I either know that they are true, or can easily verify it. This means that options like "This option's percentage has the same parity as the eleventh Dedekind number," will not pay out, unless I can find some proof that they are true.

They weren't guaranteed to be true when they were submitted. So no, "The probability of this option is between 0% and 100%, inclusive."

The option does not reference the number of traders or the number of trades on the market.

The option only references events that occur before close, or simultaneously to it. This means that, "At least five answers pay out," is not a valid option, since payouts don't occur until after close. However, "At least five answers are true," is valid (more on that later).

The truth of the option is not a matter of significant controversy among well-informed people, a subjective opinion, or a private detail of someone's life.

The option is not horribly offensive in some way.

No user may submit more than five options. If you do, any options you submit aside from the first five will not pay out. If I can't figure out which ones are the first five, I'll use my best judgement.

In the case of ambiguous options, I will either resolve according to what I think is the most straightforward interpretation of the sentence, or, if I think it is so ambiguous that there is no straightforward interpretation, that option will not pay out unless all interpretations that I think are reasonable candidates for the straightforward meaning are true.

Misspellings and grammatical mistakes are okay, as long as it's obvious what they mean.

The option doesn't violate Manifold's rules, such as encouraging spam, nor does it have a significant chance of encouraging behavior that is harmful or illegal in the U.S.

Options that refer to truth will be dealt with using Tarski's truth hierarchy. This is a solution to linguistic paradoxes like the liar's paradox that notes that semantic terms like "true" are inherently ambiguous when used to refer to sentences that are allowed to include semantic terms themselves. This ambiguity is what leads to the seeming contradiction in "This sentence is false," and the indeterminacy of "This sentence is true," and it results from the fact that no completely precise language can include a truth predicate that applies to all sentences in the language, including sentences that contain the truth predicate itself. To get around this, Tarski proposed defining a hierarchy of truth predicates. The first predicate, true1, applies only to object-level sentences, that is, sentences that do not themselves include semantic terms like truth values. So, for example, "Snow is white," is true1, but "'Snow is white' is true1" is not true1. To describe the truth of meta-sentences that include semantic predicates like true1 and false1, we use sentences on the next level of the hierarchy, true2 and false2. So "'Snow is white' is true1" is true2. "Snow is white," itself is also true2. This can continue indefinitely, where the predicate true-α can describe the truth of any sentence that contains only truth predicates of order less than α.

To utilize Tarski's truth hierarchy, I will assume that, unless otherwise specified, any option that uses words like "true" and "false" is referring to true1 or false1. So an option stating, "This option is not true" would count as being true2, since it is indeed not true1 (since true1 doesn't apply to it). However, that particular example wouldn't pay out anyway, since it is guaranteed to be true2. Similarly, "This option is false" is false2, since the predicate false1 doesn't apply to it, so it would not pay out either. But an option like, "At least five options are true" can pay out, as long as five options that don't contain a truth predicate are indeed true.

(This next bit is technical and almost certainly won't matter to you unless you are trying to find a way to break this market.) Note that the Tarskian hierarchies also apply to other semantic properties, if necessary. For example, if someone tries to break the truth hierarchy by submitting "At least one option is true-n, where n is one higher than the largest order of a truth predicate in any option", then I will not include any truth predicates whose order is predicated on the orders of the whole collection of truth predicates when determining the value of n.

Oh, and for the sake of following my own rules, any time the word "true" is used in this description without qualification, it refers to true-ω₁. If you have no idea what that means, don't worry about it. It will only matter to people who know enough about set theory to try to break the criteria I've put in place.

There is one additional caveat that I need to mention when dealing with words like "true". I will restrict their scope to only include options that are valid responses and fit the bullet points above, unless otherwise stated. So for example, if someone submitted an option like, "This option's percentage has the same parity as the eleventh Dedekind number", the truth of that option would not affect whether "At most ten options are true" pays out. This is to avoid options like the latter being penalized as "can't be verified" just because of other bad options.

If none of the options are valid and true at close (somehow), I will resolve the market to Other. That is the only situation in which Other pays out.

Please note that I have the final say on how this market resolves. I will do my best to follow the rules above, but if some game-breaking exploit is found (something that would make it impossible to resolve correctly in a reasonable amount of time), I may add further rules or clarify the existing rules to counter it. You can, however, abuse loopholes in the rules that don't break the game. I have intentionally left some open.

Credit to @SophiaLaird for creating the market that inspired this and to @Joshua for originally coming up with a list of rules to make a version with more precise resolution criteria that wouldn't lead to paradoxes. I have kept or modified some of Joshua's rules and added a bunch of rules of my own to prevent game-breaking exploits that were possible even under his rule set.

Ok, it seems like all the questions about the spreadsheet have been resolved, maybe with the exception of the .42069 option. But I do think the most straightforward interpretation of that one is to assume it means an actual probability of .42069, not .42069%. The alternate interpretation requires me to assume that the person who submitted the option misspoke and meant to say percentage, or that they thought the word "probability" refers to percentage.

The main reason this interpretation was questioned is that I also interpreted a different option as meaning .2% when it refers to a probability of .2, when a probability of .2 is actually 20%. But the reason I took this interpretation is that, for that option, it was impossible for the criteria to be fulfilled if it really meant a probability of .2 and not a percentage. So in that case, it seemed more reasonable to interpret it as a mistake in wording, and assume that the creator really meant a probability of .2%, not .2. With the .42069 option, there is nothing to indicate that it was a mistake in wording.

Does everyone agree with this interpretation, or do you think I am interpreting one of the answers wrong?

Okay, here is the Google sheet with all the truth values. I will leave this open for a while, though, in case anyone wants to check it over to make sure I haven't made a mistake.

https://docs.google.com/spreadsheets/d/1DiQUwwxG9N62YbdGlmv_92g-41Y2wmIjetc7Hm-ytsM/edit?usp=sharing

@JosephNoonan I also made a substack post about this market. It explains the reasoning behind the resolutions for some of the options. https://plasmabloggin.substack.com/p/the-free-response-free-for-alls

@JosephNoonan LOL I am happy that I created (AA) that could be manipulated last minute (by adding another entry). I thought you resolved it wrong, but I read your reasoning and it makes sense. I don't think anyone manipulated it though

@PC I don't think anyone manipulated it. There was a good chance of there being no median if the number of options was even.

@FlorisvanDoorn row 42 is inconsistent with row 46. But that's probably okay, because that was likely the way they were both intended.

For row 20 (in the spreadsheet), did you forget to count row 40?

Good question. My interpretation is that, "the number of true-N options is greater than sqrt(N), where N is the number of responses," is true-ω, but it's not true-α for any α<ω. The reason is that the sentence uses the term true-N, where N can in principle refer to any natural number, and therefore it can't have a truth-n value for any natural number n.

Since ω isn't a successor ordinal (there is no α such that α+1 = ω), having an options that's true-ω but not true-α for α<ω doesn't actually meet the requirement for the option in Row 20.

row 42 is inconsistent with row 46

Hmm, yeah I didn't notice that. I think the interpretation of Row 42 is clear, since it would be impossible for A, B, C, D, E, F, G, and H to all have displayed probabilities greater than 20%, but it's less clear for Row 46. I guess I should ask @JohnBrown what he meant by, "The displayed probability of this answer is less than 0.42069." Was it supposed to mean 0.42069%, or 42.069%?

@JosephNoonan That is not how it should work.

If I write "X is true-42", then that statement is true-43 (or false-43, but let's assume X is true-1)

If I write "X is true-N for N=42" then that statement is clearly true-(N+1). But since N=42, this means that it is also true-43 (and true-ω).

If I write "X is true-N for N = <some complicated formula>" then that is true-(N+1). Once I determine that N=42, then it means that this statement is true-43.

(silly exception: if the formula for N mentions true-k for k > 42 then the previous bullet doesn't hold anymore, of course).

@FlorisvanDoorn I can see the argument for why it should be considered true-69, but I don't think it works. The problem is that the sentences uses N as a free variable whose scope is all natural numbers. It can be rephrased as, "There exists some natural number N such that there are N options on this question, and the number of true-N options is greater than sqrt(N)." This sentence can't be written even in the meta^68 language because, even though N happens to be 68 in reality, the sentence can't be correctly interpreted unless you also have the concepts of true-N for N>68 in the language. The closest analogue we can form in the meta^68 language is, "There exists some N≤68 such that there are N options on this question, and the number of true-N options is greater than sqrt(N)." That has the same truth value, since the number of options is 68, but it isn't the same proposition, since it would have a different truth value if the number of options were greater than 68. So the option cannot be considered true-69, since the proposition it expresses can't be written in the meta^68 language. The smallest order language it can be written in is the meta^(<ω) language, which contains all truth predicates of order <ω, and hence it is true-ω, but not true-n for any n<ω.

@JosephNoonan I see. If each level of the hierarchy comes with its own language and allowed symbols, then indeed, to make this statement you have to be on level omega. I was thinking more something along the lines of you have a fixed language, and then you carve out subsets of sentences that you mark true-1, then true-2, and so on, and then then this statement could be considered true-69. I don't know what Tarski's formulation is, you're probably right.

It reminds me of the difference between dynamic typing and static typing.

@FlorisvanDoorn Tarski's formulation is to use a hierarchy of languages. The fixed language approach probably also works without logical contradiction, but I think I will go with the language hierarchy formulation since that is more like what Tarski originally did.

@LordWilmgaddark Any idea how I can find out what the New York Times headline from yesterday is? The best thing I can find is this from Internet Archive: https://web.archive.org/web/20230924004417/https://www.nytimes.com/

In which case option Y should not pay out.

@JosephNoonan

https://static01.nyt.com/images/2023/09/24/nytfrontpage/scan.pdf

However, doesn't it violate the rule "all answers must be about the market itself"?

@FlorisvanDoorn I was going to count it because it also is about what the first letter of the top option will be. Doesn't matter, though, because based on that scan the first letters aren't the same.

@JosephNoonan It might matter for options that talk about the truth of other options, since options that are disqualified are not counted, but options that are false are counted.

@JosephNoonan For example, I bet YES on "(V) A majority of the answers are true" since many options that don't pay out are disqualified, and therefore are not counted for the resolution of this option.

@FlorisvanDoorn I was actually planning to resolve options like based on the proportion of all options that are true, not the proportion of valid options that are true. Although rereading that part of the description, I can see why you thought it was the latter. When I said I was restricting the scope of the word true, I just meant that I wouldn't include, e.g., unknowable options as true, because then we could get a problem where a valid option gets the unknowability of that option transferred to it.

Okay, I'm looking at the options, and a few things I've noticed:

The only single-letter label that hasn't been used yet is (W), so (Z) will pay out as long as someone uses that one.

(U) and (Ñ) aren't self-resolving, so they can't pay out.

A fake version of (A) has been created, "(A) This market will have > 80 unique traders." That means that, "This answer resolves true if someone creates a lettered option that's a duplicate of a different lettered option (I.G: there are two option (B))," will pay out. The

*real*version of (A) is, "This option does not have highest probability.""At the time of close, if (N) is true then the number of responses is no more than 50," is now equivalent to "not (N)", since the consequent is false.

It could be argued that (X) is subjective, but I'm going to allow it, because if the visual errors really are obvious, they should be obvious to everyone and so wouldn't be subjective. However, there are currently no obvious visual errors.

I have no idea how to verify, "At close there is a finite-state automaton with less than P states which accepts all true options and rejects all other options, where P is the highest percentage of a single option," so that won't pay out unless I find some way to determine its truth. The option is also ambiguous: What exactly is the Turing machine supposed to accept: the position of the option in the ordering that appears in the market, the number assigned to each option (the order submitted), or a binary encoding of the text of each option?

I think, "At least 5 users will admit to adding an invalid option to this option and not reading the rules carefully in the comments and are telling the truth," will probably end up being true, because more than five people have submitted invalid options. At least two have already admitted to doing so in the comments (me and Bohaska), and I think some others have as well, but I'm not sure how many. Currently some of the comments are missing. To make this easier to resolve, I would appreciate if some people responded to this comment admitting that they accidentally submitted an invalid option (only if you're telling the truth).

I took more than 10 minutes looking over the answers to write this comments, so "@JosephNoonan will use 10 minutes or longer to determine the resolution of this market," will pay out. (I said it was invalid before, but that was because I wasn't thinking of the possibility of using ten minutes before the market closes, and you can see my reply to Floris below that I would consider that valid.)

@JosephNoonan I also said that I mistakenly added an invalid option because I didn't read the rules carefully.

@JosephNoonan Here's a fun one for you; I submitted my response 'test' to see if I could buy it and make a profit on buying NO against it (but this market loads so slow for me that I was sniped), AND I did not read the section on Tarski's truth hierarchy in the market description carefully. So "At least 5 users will admit to adding an invalid option to this option and not reading the rules carefully in the comments and are telling the truth" does apply to me, b/c it uses 'and' not 'and so'. (Also, I am telling the truth). I think I count towards resolving that option YES.

@JosephNoonan Assuming my previous comment ("Eeek, didn't read the rules carefully enough. Nuts.") somehow did not qualify, let it be said that I admit that added the QED option.

Here is a visual graph error, that hopefully becomes obvious once I point it out (and a similar error probably still exists at close). The screenshot below suggest that option (C) has been constant at 11% for the last 24-48 hours. This is false: this option hasn't been above 6% for the last 36 hours. Proof: The second screenshot shows my limit order on this option that is (as of writing this comment) not yet completely filled nor canceled.

The last part of a graph often shows a bunch of constant lines (lately), which are clear errors in the graph.

@FlorisvanDoorn I'm not sure it counts as obvious when you have to use a specific user's unfilled limit order to prove that its an error. My interpretation is that you can just look at the graph and immediately tell that there's something wrong with it.

@JosephNoonan The graph is now showing that no trades have significantly changed the value of any of the options in the last two days... If that's not an obvious error, that seems like a really high bar to me.

And also: you can just look at the graph (at the current moment) and at what the top option is, and see that they don't match. I just wanted to give extra proof that the graph was not just wrong now but consistently and significantly wrong for >24 hours.

@FlorisvanDoorn Yeah, I agree that having it flatline like that is an obvious visual error, so I think I will have X pay out if it stays like that.