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gjm10y00

Actually, a countable union of countable sets is countable

Technical note: strictly this requires the Axiom of Choice, or at least some weaker version of it. For each of your countable sets, there is at least one way of counting it; but to count the whole lot you need to pick one way of counting each set. This is exactly the kind of thing that can fail to happen without Choice. You don't need "very much" Choice; e.g., the axiom of choice for countable collections is enough; but it turns out that the axiom of choice for countable collections of countable sets is not enough.

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19 Respond to what they probably meant

by Adam Zerner

17th Jan 2015

4 min read

19

Edit: I didn't realize this before writing the post, but what I'm referring to is The Principle of Charity.

Story

I was confused about Node Modules, so I did a bunch of research to figure out how they work. Explaining things helps me to understand them, and I figured that others might benefit from my explanation, so I wrote a blog post about them. However, I'm inexperienced and still unsure of exactly what's going on, so I started the blog post off with a disclaimer:

Disclaimer

I'm a bit of a noob. I just graduated from a coding bootcamp and am still trying to wrap my head around this stuff myself (that's actually why I'm writing this article). I tried to do my research, but I can't guarantee that everything is correct. Input from more knowledgeable people is very welcome.

My friend said that it's a bad idea to do that. He said:

You're literally discrediting yourself in the first sentence of the article. Stand by what you've written!

I interpreted what he said literally and basically responded by saying:

Why should I "stand by what I've written"? What I mean to communicate to the readers is that, "I'm x% sure of what I'm about to say." To "stand by what I've written" is to assign a higher confidence to the things I've written than what my true confidence is. It wouldn't even be a stretch to interpret "stand by what you've written" as meaning "claim that you're 100% sure of what you've written". Why would I do that?

This was stupid of me. He didn't mean "claim that you're 100% sure of what you've written". He didn't mean "pretend that you're way more confident in what you've written than what you really are". He meant, "I think that it comes across as you being less confident than you actually are. And so I think you should reword it to better communicate your confidence."

I shouldn't have interpreted what he said so literally. I should have thought about and responded to what I thought he meant to say. (Although, he also should have been more precise...)

Thesis

People often interpret and respond to statements literally. Instead of doing this, it's often useful to think about and respond to what the other person probably meant.

For example, "If I interpret what you said literally, then A. But you probably meant X, so B. If you meant Y, then C."

Depending on how confident you are in your interpretation, you should probably respond to a variety of possibilities. Like if you're < 80% sure that you know what they meant, you should probably respond to possibilities that have at least a 5% chance of being what they meant. I'm not sure whether 80 and 5 are the right numbers, but hopefully it communicates the point.

Why don't people do this?

I see two likely reasons:

The whole "argument is a war that I must win" attitude.
Habit.

1 - "You said X! Gotcha! That's stupid! You're wrong!". This clearly isn't a productive approach.

2 - I think that a lot of people - myself included - have a bad habit of interpreting things too literally. Well actually, that by itself isn't what's bad. What's bad is stopping after your literal analysis, and not considering alternatives that are likely to be what they actually meant. This bad habit isn't ill-intentioned - that's why I distinguish it from reason 1). It's just an analytical impulse.

Practical considerations

In "low friction" situations (like when you're talking to someone face-to-face), it's probably a better idea to just say, "I think that what you're trying to say is X. Is that true?". Ie. instead of responding to what you think they mean... you could just ask them to clarify.

In higher friction situations, there's a cost (in time and/or effort) to having one person stop talking and another person start talking. Like in online discussions, you might have to wait a while before they respond. So if you're 95% sure that you know what they meant, you could just say, "I think that you meant X, so A. But if you meant Y, then B". The alternative is to respond by saying, "I think you meant X but I'm not sure. Did you mean X", and then having to wait for a reply.

I'm having trouble thinking of other "higher friction situations". Perhaps a (semi)formal debate where you have to speak for a certain length of time would be a good example. In this situation you're expected to just keep speaking, so you can't pause to ask people what they meant - you just have to think about and respond to the possibilities on the spot.

Another practical point to make is that the flow of the conversation has to be taken into account. Stopping to address every possible interpretation of what the other person said is obviously impractical - it'd take too long, and it's hard for everyone to follow the logic of the conversation.

However, I think that my core point is applicable for all types of conversations. The goal of communication is for each person to interpret and respond to the others' statements. Interpreting things literally instead of thinking about what the other person probably meant to say is a failure to interpret, and it impedes communication.

Steelmanning

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[-]TheAncientGeek10y80

Isn't this the principle of charity?

[-]Adam Zerner10y60

Yes it is. I had never heard of it before, thanks for commenting.

[-]btrettel10y50

Yes, and "steelmanning" is the stronger form of the idea.

[-]Dahlen10y30

The principle of charity more closely resembles a fairly complex skill with a lot of subskills. (That may be why it's so difficult and scarcely applied.) This is but one of them. (Others might be: assuming a common basis of reason and morality; do not ruin your relationship with the other person along the way; remember that there is such a thing as a state of mind (theirs) in which those beliefs are natural and obvious; etc.)

[-]Ben Pace10y80

I see a few problems with this: if we try to assign beliefs and statements to people that aren't what they say they believe, but what we think they really believe or meant to say, then we will be biased to interpreting their beliefs and statements to be like the closest cliches we know, or closer towards ideas we take seriously. We might think they didn't mean X, because we think X is ridiculous, and when they protest that we've not responded directly, there'll be an impasse. They may even accuse you of deliberately misinterpreting what they've said. Also, if we respond to what people actually say, then there'll be an incentive for people to be clearer in their words, which seems a net positive. Fin

Interpreting words further than simply understanding what people actually say seems like it could be rife with errors. This doesn't mean it's not a useful skill, but I wouldn't make it my general strategy.

[-]jimmy10y20

That's why its important to check for understanding if you're at all unsure. It's the ideological turing test thing.

[-]Unknowns10y00

Very frequently people do not find at all if you interpret them to be saying something different from what they actually meant to say, as long as you interpret them to be saying something true. They mainly mind if you interpret them to be saying something false, whether or not they actually meant to say that thing.

[-]Adam Zerner10y00

I'll assume that you agree that it depends on the confidence you have in your interpretation. That if you're 99% sure that they meant X, it's worth it for you to address X. You seem to be arguing that our biases make it difficult to reach such high levels of confidence.

I think that biases can sometimes make it difficult, but not always. There are still many situations where it's pretty easy to be confident in your interpretations. And in these situations, it makes sense to address what they probably meant. For example, in the story I referenced, I think it would have been quite easy to infer what my friend really meant if I thought to do so.

[-]dspeyer10y60

This advice can backfire. If you're applying it and I'm trying to say something you didn't expect, I'm going to get very frustrated -- trying phrasing after phrasing until you hear what I'm actually saying (or I'll walk away).

The advice can also work well.

I think it's necessary to keep per person probabilities for meaning what they say and meaning something novel.

[-]Adam Zerner10y00

This advice can backfire, but it can also work well.

True, but to what extent? My impression is that it's very unlikely that it backfires. I imagine it being used primarily in situations where it's pretty obvious (if you thought about it) that the person means one thing, but that they're just being imprecise with their words.

I'm going to get very frustrated -- trying phrasing after phrasing until you hear what I'm actually saying (or I'll walk away).

I think that only one attempt would ever be made. In a face-to-face conversation, you'd probably just ask them to clarify. In an online one, if someone says, "I think you meant A, so X." and they're wrong, you'd just respond by saying, "no, I meant B".

[-]BoilingLeadBath10y30

To paraphrase adamzerner...

My impression is that the expected cost of using this technique online - the probability of it backfiring multiplied by the average cost in the case that it does - is low.

While most of my communication experience is from my past role as a moderator of a youth-dominated engineering forum, and so is somewhat unusual, I believe that the expected value is in fact highly positive.

I think this is mostly because:

It's a pretty cheap technique to implement - you can simply paraphrase the person you are responding to, rather than directly quoting. (As I did in this post)
In the case that you, in good faith, misunderstand the other member, they are going to have to re-explain their position anyways; it is far better to catch this early on, before anyone gets frustrated and before any more time is wasted.

Same function and justification as checksums, I suppose...

On the other hand, if you are only 50% sure what the other person meant, I found it was better to simply let them know that they were obscure.

[-]Adam Zerner10y00

To paraphrase adamzerner...

My impression is that the expected cost of using this technique online - the probability of it backfiring multiplied by the average cost in the case that it does - is low.

That's a great way of wording it. I had been trying to think about how to word it in terms of expected value, but my thoughts were too jumbled to post. Thank you for clarifying!

[-]philh10y50

Why don't people do this?

Two more reasons: In an involved debate, if I have three different plausible interpretations of what they said, I don't want to put in the time and effort to respond to all of them. It might also come across as patronising if I did, like "yes, obviously what I said was not literally true, do you think I'm stupid or something?"

Under low friction, getting clarification before responding is good advice. In high friction, I might say "I think you meant X, but I'm not sure. If you did, then A." A more subtle variant would be to attempt to summarise X and respond to it, and if X is not what they meant then they hopefully catch this. But these are still potentially wasteful and patronising if I get it wrong.

[-]DanArmak10y20

Another reason to do this is to explicitly point out that what the other person means and what they are literally saying are incompatible. Whether this helps or impedes the conversation depends on how serious it is and how clear it was to everyone what was actually meant.

Example:

Speaker 1: "Pi has infinitely many digits".

Speaker 2: "So does every real number (e.g. 0 = 0.000....). What you mean is there's no way to write Pi with finitely many digits, in any basis."

[-]Plasmon10y70

What you mean is there's no way to write Pi with finitely many digits, in any basis."

pi=1 in base pi

... but that's not what you meant :)

[-]Kindly10y110

As long as we're nitpicking, it should be 10.

[-]DanArmak10y00

A basis is normally defined as an integer. But you're right; there doesn't seem to be any reason not to extend this to reals. (Although what good is a non-computable basis to anyone I don't know...)

[-]Vaniver10y20

Wiki's discussion of base pi.

[-]CCC10y00

Wiki raises an interesting point, a little further down, talking about base sqrt(2). Any number that can be expressed in a finite number of digits in base 2 can also be expressed in a finite number of digits (albeit twice as many) in base sqrt(2); also, some irrational numbers (like sqrt(2)) can also be expressed in a finite number of digits in base sqrt(2).

This leads me to wonder whether there is any number base in which any number, rational or irrational, can be fully described as either a finite (potentially extremely large) number of digits or a repeating pattern (like 1/7 in decimal) or not...

[-]DanArmak10y80

There isn't. You can see this by considering the cardinality. The set of all (finite) sequences of digits in any finite alphabet (e.g. 0 to 9) is only countably large. So it can't map the irrational (real) numbers.

[-]g_pepper10y70

The set of all (infinite) sequences of digits in any finite alphabet (e.g. 0 to 9) is only countably large

Actually, the set of all infinite sequences of digits in any finite alphabet with two or more symbols is uncountable - this can be shown via a diagonalization argument. I suspect you meant to say that the set of all finite sequences of digits is countably large.

[-]DanArmak10y20

Yes, that's right. Thanks!

[-]CCC10y00

I thought I might be able to get around that limit by permitting infinite repeating sequences of digits - but it turns out that that's equivalent to introducing a single new symbol to the notation (with the further restriction that it can only be used not more than once per number), and therefore the set of infinite repeating sequences is also countable; thus, in order to represent all real numbers, it remains insufficient.

[-]gjm10y30

Yup. Just for fun, some other ways to see that there are only countably many repeating sequences:

A countable union of finite sets is countable. For any m,n there are only finitely many sequences that repeat everything from place m to place n.
- (Actually, a countable union of countable sets is countable but we don't need that here.)
A repeating sequence can be generated by a computer program. There are only countably many computer programs, so there are only countably many repeating sequences.
- (So there are only countably many real numbers that can be completely and explicitly specified by a computer program. They include, e.g., all the rational numbers; all the numbers that satisfy algebraic equations with integer coefficients; things like e, pi, gamma, etc.; every number that is explicitly considered in any mathematics textbook, with a few exceptions for things like Chaitin's Ω. These are the "computable numbers"; they form e.g. an algebraically closed field. But I don't think I know of any actual useful or interesting applications.)
Take your repeating sequence and just treat it as the decimal expansion of a number. That number is always rational. There are only countably many rational numbers.

[-]CCC10y00

A repeating sequence can be generated by a computer program. There are only countably many computer programs, so there are only countably many repeating sequences.

That's an interesting point. A computer program is itself a finite number of symbols chosen from a finite list (with certain restrictions that further reduce the number of programs that make sense).

The same can be said for the English language. In fact, the same can be said for any language that can be translated into English, or into any other language that has a finite alphabet.

And I don't know how a language with an infinite alphabet would work, but if it can be explained in English, then that implies that it can be translated to English.

This therefore implies that there must exist numbers which cannot be precisely specified at all.

[-]gjm10y80

there must exist numbers which cannot be precisely specified at all.

Almost all of them! But, y'know, I can't tell you what any of them are :-).

[-]Richard_Kennaway10y00

This therefore implies that there must exist numbers which cannot be precisely specified at all.

And among these, there is the smallest undefinable number.

[-]gjm10y20

I was going to say "Congratulations, you just proved the halting theorem." -- but actually I think the paradox you're gesturing at fails to "work" for shallower reasons (e.g., the reals not being well-ordered -- trivially not by the usual ordering, and less trivially not by anything computable, because it's consistent with ZF for there to be no well-ordering of the reals).

[-]Epictetus10y00

And among these, there is the smallest undefinable number.

There isn't. The collection of positive, undefinable numbers is bounded below by zero, but doesn't actually have a smallest element (due to the reals not being well-ordered).

[-]CCC10y00

the smallest undefinable number.

If you mean the smallest undefinable positive number, then isn't that epsilon?

[-]gjm10y00

Actually, a countable union of countable sets is countable

[-]CCC10y20

That was quick. Thank you.

[-]helltank10y10

For me, the problem with this is that if I'm speaking to an autistic person(and a very large number of LWers identify themselves as on the autistic spectrum), they tend to use literal meanings very often. In fact, some of them(including me) get offended or confused when they say something literal and it is interpreted as sarcastic or subtext.

Suppose I am speaking to an autistic person, and he says, "I am 87% confident that X is true." The issue with this statement is that a lot of people use this sort of statement in a metaphorical sense(ie. they just pull the number out of their butt to make it oddly specific and get a cheap laugh) but an autistic or rationality-trained person may literally mean that they are 87% sure it is true, especially if they are good at measuring their own confidence levels. In this case, the usual situation- the number being picked randomly - is false.

There are also, however, a large number of statements that are almost always meant sarcastically or in a non-literal way. The statement "I, for one, welcome our new alien overlords" is almost always sarcastic as it is 1) invoking a well-known meme which is intended to be used in this manner and 2) it is extremely unlikely that the person I am speaking to is actually someone who wants aliens to take over the world. These statements are, for want of a better word, "cached non-literal statements"(as in, it is an automatic thought that these statements are not literal), or CNLS for short.

It might be useful to append the guideline "All statements have a probability of being literal that is worth considering, except in the case of CNLSes. This probability is adjusted up if the person you are speaking to is known for being extremely literal and adjusted down if they are known for using figurative speech(although that last sentence should be fairly obvious, I throw it in for the sake of completeness)" to your thesis.

This actually got me thinking if there is a methodical, objective and accurate way to find out if someone's statement is literal or not, perhaps by measuring their posture, tone of voice. The only difficulty is to try to weasel some quantifiable data out of context. If it can be done, that would be a great resource to people who have trouble understanding the non-literal meanings of statements everywhere.

[-]Ixiel10y70

Give me all the bacon and eggs you have.

Wait, wait. I'm worried what you just heard was, "Give me a lot of bacon and eggs." What I said was, "Give me all the bacon and eggs you have". Do you understand?

-Ron Swanson

[-]Adam Zerner10y10

Great example :)

[-]Adam Zerner10y00

This probability is adjusted up if the person you are speaking to is known for being extremely literal and adjusted down if they are known for using figurative speech(although that last sentence should be fairly obvious, I throw it in for the sake of completeness)" to your thesis.

Exactly. I think my thesis covers that in saying "Depending on how confident you are in your interpretation". I don't explicitly talk about how do the interpretation because:

It's usually easy enough to assign a good confidence level to your interpretation. With a good confidence level you could cater your response properly. Ie. If you're 99% confident, say, "I think you're trying to say this". If you're 50% confident you could say, "I'm not sure what you're trying to say, but I think that it may be A, B or C.".
It's outside of the scope of this article. Perhaps I could mention a few guidelines, but a) it's really a very intuitive thing, b) I don't know much about that topic (how to interpret), and c) I sense that it'd be rather involved to go over how to interpret, and that the benefit isn't worth it because it's something that most people could do well enough intuitively.

[-][anonymous]10y00

Hey Adam,

What you're describing sounds to me like it might essentially be social intelligence... almost everybody does this in some form (realizing that not everything is literal, but context dependent), and some highly socially intelligent people are much better at figuring out what people mean than others.

An additional step you might want to add in to your process is confirming fhat what you think they mean is what they actually mean by restating it in your own words, and saying something like, "so you're saying that..." This skill is essentially active listening.

One step further than this is also reaffirming their emotion (wow, it sounds like you're really (happy/angry/passionate/etc)... and one step further than that is reaffiriming WHY they said it (It seems like you really care about how this post comes across).

Of course, I'm just inferring what you mean here, without confirming, so I could be totally off base :).

[-]Dahlen10y30

Yes, social intelligence is the core of it, but a lot of the time it's not so much a matter of innate ability than of mindset and values. For a long while I had this strange idea that the overly-literal way of discussing, which assumed the form but not the function of rationality, was the right way of discussing, and the huge majority of people were just doing it wrong, and needed to be informed of the rules of ~rational~ debates. Looking back, I might have been rather frustrating to talk to. :)

But then I changed my mind and reframed debate as an action undertaken by some agents with primary motivations other than debating for the hell of it, like voices in the dark. And started to look at the motivations behind what has been said, with the goal of having the most accurate model possible of how and why the other person actually generates those ideas. The chains of reasoning that output the goals the other person wants out of the debate are not terribly hard to go through, even for a self-identified social "extraterrestrial".

Those particular debate tactics aren't, however, the best idea for improving the dialogue. I've had them used on me, or rather against me, even here on LW, mostly for disingenuous purposes. They're very easy to corrupt. In particular, "so you're saying that..." seem to be the code words that herald the arrival of the Straw Man. Okay, I get it, the other person might be genuinely trying to understand your alien (to them) mind. However, when there's disagreement, in trying to assemble a coherent mental picture of you, they choose, out of the traits you could possibly have, the ones that depict you as farther away from where they come, rather than closer to that. The bottom line seems to be "... so that's why I disagree with you, because I'm too reasonable to believe all the crazy things implied by your expressed beliefs". If you're looking for feedback, it may be best if you ask rather than propose interpretations.

As well as that, telling people what their emotional states seem to be is more often than not a bit offensive. For some reason people like to come across as less emotional and more cool-headed in debates rather than the opposite. The natural reaction would be mildly defensive ("Who, me? Nah, I'm totally chill"). Even if they might be feeling a certain way, emotions aren't often consciously credited as origins of one's behavioral output. If you're using this, then a "buffer" statement might be needed, to reassure the listener that you're not considering their thoughts less legitimate because they have emotional sources, or their emotions less legitimate because they're intense ("of course, I see why you might be happy/angry/passionate about this, and that's totally okay").

[-][anonymous]10y00

The core is empathy... you have to show that you genuinely accept and make space for the other persons idea and emotions, it's the mindset behind the strategy of active-listening. If you simply use the strategy of "so you're saying that" without genuinely trying to understand the other person, it comes off as straw-manning. Similarly, if you are telling someone else what their emotions are, instead of making space for their emotion, it comes off as judgmental.

I think something else you're getting at here is that this is something that's very hard to do with a stranger over the internet - the requisite body language, rapport, voice tone etc all get lost (and therefore more useful in most real-world cases of trying to change someone's mind)

Edit: one final thought on this is that active listening isn't much used in debate. It's much more useful for dialouge and rapport building than trying to convince someone with facts.

[-]Adam Zerner10y20

I agree that what I'm describing is a subset of social intelligence (I'm sure that's what you meant). But I'd like to distinguish the ability to infer what other people mean from the sense to try to do so. The former is definitely a type of social intelligence, but I'm not sure if the latter is. My thesis really is that it's beneficial to do the latter.

As for reaffirming the emotion, I think that that's a good point - it's probably a good and underutilized thing to do. I haven't thought much about it though and don't think I have any useful thoughts on it.

[-][anonymous]10y00

It's almost what I meant... i actually meant that the sense of WHEN you should do it is a subset of social intelligence... on the extreme edge of this, people who are autistic are notoriously bad at understanding sarcastic humor - they have no ability to seperate content from subtext, so take everything literally.

[-]Adam Zerner10y10

i actually meant that the sense of WHEN you should do it is a subset of social intelligence

Ok, I agree with that.

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