Therefore, there are only two ways in which you can arrive at a numerical probability estimate for a common-sense belief:
Translate your vague feeling of certainly into a number in some arbitrary manner. This however makes the number a mere figure of speech, which adds absolutely nothing over the usual human vague expressions for different levels of certainty.
Perform some probability calculation, which however has nothing to do with how your brain actually arrived at your common-sense conclusion, and then assign the probability number produced by the former to the latter. This is clearly fallacious.
On point #2, I agree with you. On point #1, I had the same reaction as xv15. Your example conversation is exactly how I would defend the use of numerical probabilities in conversation. I think you may have confused people with the phrase “vague figure of speech,” which was itself vague.
Vague relative to what? “No idea / kinda sure / pretty sure / very sure?”, the ways that people generally communicate about probability, are much worse. You can throw in other terms like “I suspect” and “absolutely certain” and “very very sure”, but it’s not even clear how these expressions of belief match up with others. In common speech, we really only have about 3-5 degrees of probability. That’s just not enough gradations.
In contrast, when expressing a percentage probability, people only tend to use multiples of 10, certain multiples of 5, 0.01%, 1%, 2%, 98%, 99% and 99.99%. If people use figures like 87%, or any decimal places other than the ones previously mentioned, it’s usually because they are deliberately being ridiculous. (And it’s no coincidence that your example uses multiples of 10.)
I agree with you that feelings of uncertainty are fuzzy, but they aren’t so fuzzy that we can get by with merely 3-5 gradations in all sorts of conversations. On some subjects, our communication becomes more precise when we have 10-20 gradations. Yet there are diminishing returns on more degrees of communicable certainty (due to reasons you correctly describe), so going any higher resolution than 10-20 degrees isn’t useful for anything except jokes.
The numbers are still used as mere figures of speech, but “vague” is not a correct way to describe them, since the information has been transmitted in a more precise way than if A had just used verbal qualifiers.
Yes. Gaining the 10-20 gradations that numbers allow when they are typically used does make conversations relatively more precise than just by tacking on “very very” to your statement of certainty.
It’s similar to the infamous 1-10 rating system for people’s attractiveness. Despite various reasons that rating people with numbers is distasteful, this ranking system persists because, in my view, people find it useful for communicating subjective assessments of attractiveness. Ugly-cute-hot is a 3-point scale. You could add in “gorgeous,” “beautiful,” or modifiers like “smoking hot,” but it’s unclear how these terms rank against each other (and they may express different types of attraction, rather than different degrees). Again, it’s hard to get more than 3-5 degrees using plain English. The 1-10 scale (with half-points, and 9.9) gives you about 20 gradations (though 1-3, and any half-point values below 5 are rarely used).
I think we have a generalized phenomenon where people resort to using numbers to describe their subjective feelings when common language doesn’t grant high enough resolution. 3-5 is good enough for some feelings (3 gives you negative, neutral, and positive for instance), but for some feelings we need more. Somewhere around 20 is the upper-bound of useful gradations.
I mostly agree with this assessment. However, the key point is that such uses of numbers should be seen as metaphorical. The literal meaning of a metaphor is typically nonsensical, but it works by somehow hacking the human understanding of language to successfully convey a point with greater precision than the most precise literal statement would allow, at least in as many words. (There are other functions of metaphors too, of course, but this one is relevant here.) And just like it is fallacious to understand a metaphor literally, it is similarly fallacious to interpret these numerical metaphors as useful for mathematical purposes. When it comes to subjective probabilities, however, I often see what looks like confusion on this point.
It is wrong to use a subjective probability that you got from someone else for mathematical purposes directly, for reasons I expand on in my comment here. But I don’t think that makes them metaphorical, unless you’re using a definition of metaphor that’s very different than the one I am. And you can use a subjective probability which you generated yourself, or combined with your own subjective probability, in calculations. Doing so just comes with the same caveats as using a probability from a study whose sample was too small, or which had some other bad but not entirely fatal flaw.
I will write a reply to that earlier comment of yours a bit later today when I’ll have more time. (I didn’t forget about it, it’s just that I usually answer lengthy comments that deserve a greater time investment later than those where I can fire off replies rapidly during short breaks.)
But in addition to the theme of that comment, I think you’re missing my point about the possible metaphorical quality of numbers. Human verbal expressions have their literal information content, but one can often exploit the idiosyncrasies of the human language interpretation circuits to effectively convey information altogether different from the literal meaning of one’s words. This gives rise to various metaphors and other figures of speech, which humans use in their communication frequently and effectively. (The process is more complex than this simple picture, since frequently used metaphors can eventually come to be understood as literal expressions of their common metaphorical meaning, and this process is gradual. There are also other important considerations about metaphors, but this simple observation is enough to support my point.)
Now, I propose that certain practical uses of numbers in communication should be seen that way too. A literal meaning of a number is that something can ultimately be counted, measured, or calculated to arrive at that number. A metaphorical use of a number, however, doesn’t convey any such meaning, but merely expects to elicit similar intuitive impressions, which would be difficult or even impossible to communicate precisely using ordinary words. And just like a verbal metaphor is nonsensical except for the non-literal intuitive point it conveys, and its literal meaning should be discarded, at least some practical uses of numbers in human conversations serve only to communicate intuitive points, and the actual values are otherwise nonsensical and should not be used for any other purposes—and even if they perhaps are, their metaphorical value should be clearly seen apart from their literal mathematical value.
Therefore, regardless of our disagreement about subjective probabilities (of which more in my planned reply), this is a separate important point I wanted to make.
On point #2, I agree with you. On point #1, I had the same reaction as xv15. Your example conversation is exactly how I would defend the use of numerical probabilities in conversation. I think you may have confused people with the phrase “vague figure of speech,” which was itself vague.
Vague relative to what? “No idea / kinda sure / pretty sure / very sure?”, the ways that people generally communicate about probability, are much worse. You can throw in other terms like “I suspect” and “absolutely certain” and “very very sure”, but it’s not even clear how these expressions of belief match up with others. In common speech, we really only have about 3-5 degrees of probability. That’s just not enough gradations.
In contrast, when expressing a percentage probability, people only tend to use multiples of 10, certain multiples of 5, 0.01%, 1%, 2%, 98%, 99% and 99.99%. If people use figures like 87%, or any decimal places other than the ones previously mentioned, it’s usually because they are deliberately being ridiculous. (And it’s no coincidence that your example uses multiples of 10.)
I agree with you that feelings of uncertainty are fuzzy, but they aren’t so fuzzy that we can get by with merely 3-5 gradations in all sorts of conversations. On some subjects, our communication becomes more precise when we have 10-20 gradations. Yet there are diminishing returns on more degrees of communicable certainty (due to reasons you correctly describe), so going any higher resolution than 10-20 degrees isn’t useful for anything except jokes.
Yes. Gaining the 10-20 gradations that numbers allow when they are typically used does make conversations relatively more precise than just by tacking on “very very” to your statement of certainty.
It’s similar to the infamous 1-10 rating system for people’s attractiveness. Despite various reasons that rating people with numbers is distasteful, this ranking system persists because, in my view, people find it useful for communicating subjective assessments of attractiveness. Ugly-cute-hot is a 3-point scale. You could add in “gorgeous,” “beautiful,” or modifiers like “smoking hot,” but it’s unclear how these terms rank against each other (and they may express different types of attraction, rather than different degrees). Again, it’s hard to get more than 3-5 degrees using plain English. The 1-10 scale (with half-points, and 9.9) gives you about 20 gradations (though 1-3, and any half-point values below 5 are rarely used).
I think we have a generalized phenomenon where people resort to using numbers to describe their subjective feelings when common language doesn’t grant high enough resolution. 3-5 is good enough for some feelings (3 gives you negative, neutral, and positive for instance), but for some feelings we need more. Somewhere around 20 is the upper-bound of useful gradations.
I mostly agree with this assessment. However, the key point is that such uses of numbers should be seen as metaphorical. The literal meaning of a metaphor is typically nonsensical, but it works by somehow hacking the human understanding of language to successfully convey a point with greater precision than the most precise literal statement would allow, at least in as many words. (There are other functions of metaphors too, of course, but this one is relevant here.) And just like it is fallacious to understand a metaphor literally, it is similarly fallacious to interpret these numerical metaphors as useful for mathematical purposes. When it comes to subjective probabilities, however, I often see what looks like confusion on this point.
It is wrong to use a subjective probability that you got from someone else for mathematical purposes directly, for reasons I expand on in my comment here. But I don’t think that makes them metaphorical, unless you’re using a definition of metaphor that’s very different than the one I am. And you can use a subjective probability which you generated yourself, or combined with your own subjective probability, in calculations. Doing so just comes with the same caveats as using a probability from a study whose sample was too small, or which had some other bad but not entirely fatal flaw.
I will write a reply to that earlier comment of yours a bit later today when I’ll have more time. (I didn’t forget about it, it’s just that I usually answer lengthy comments that deserve a greater time investment later than those where I can fire off replies rapidly during short breaks.)
But in addition to the theme of that comment, I think you’re missing my point about the possible metaphorical quality of numbers. Human verbal expressions have their literal information content, but one can often exploit the idiosyncrasies of the human language interpretation circuits to effectively convey information altogether different from the literal meaning of one’s words. This gives rise to various metaphors and other figures of speech, which humans use in their communication frequently and effectively. (The process is more complex than this simple picture, since frequently used metaphors can eventually come to be understood as literal expressions of their common metaphorical meaning, and this process is gradual. There are also other important considerations about metaphors, but this simple observation is enough to support my point.)
Now, I propose that certain practical uses of numbers in communication should be seen that way too. A literal meaning of a number is that something can ultimately be counted, measured, or calculated to arrive at that number. A metaphorical use of a number, however, doesn’t convey any such meaning, but merely expects to elicit similar intuitive impressions, which would be difficult or even impossible to communicate precisely using ordinary words. And just like a verbal metaphor is nonsensical except for the non-literal intuitive point it conveys, and its literal meaning should be discarded, at least some practical uses of numbers in human conversations serve only to communicate intuitive points, and the actual values are otherwise nonsensical and should not be used for any other purposes—and even if they perhaps are, their metaphorical value should be clearly seen apart from their literal mathematical value.
Therefore, regardless of our disagreement about subjective probabilities (of which more in my planned reply), this is a separate important point I wanted to make.