O...kay. It looks like you just decided to post the first thing on your head without concern for saying anything useful.
You come up with fractional values for truth, but don’t think it’s necessary to say what a fractional truth value means, let alone formalize it.
You propose the neato idea to use fractional truth values to deal with statements like “this is tall”, and boost it with a way to adjust such truth values as height varies. Somehow you missed that we already have a way to handle such gradations; it’s called “units of measurement”. We don’t need to say, “It’s 0.1 true that a football field is long”; we just say, “it’s true that a football field is 100 yards long.
Anyway, I thought I’d use this opportunity to say something useful. I was just reading Gary Drescher’s Good and Real (discussed here before), where he gives the most far-reaching, bold response to the claim that Goedel’s theorem proves limitations to machines, and I’m surprised the argument doesn’t show up more often, and that he didn’t seem to have anyone to cite as having made it before.
It goes like this: people claim that formal systems are somehow limited in that they can’t “see” that Goedel statements of the form “This statement can’t be proven within the system” are true. Drescher attacks this at the root and says, that’s not a limitation, because the statement’s not true.
He explains that you can’t actually rule out falsehood of the Goedel statement, as many people immediately do. Because it’s falsity still leaves room for the possibility that “This statement has a proof, but it’s infinitely long.” But then the subtle assumption that “This statement has a proof” implies “This statement is true” becomes much more tenuous. It’s far from obvious why you must accept as true a statement whose proof you can never complete.
Silas, a suggestion which you can take or leave, as your prefer.
This comment makes some sound points, but IMHO, in an unnecessarily personal way. Note the consistent use of the critical “you”-based formulations (“you just decided”, “you come up with”, “you propose”, “you missed that”). Contrast this with Christian’s comment, which is also critical, but consistently focuses on the ideas, rather than the person presenting them.
I have no idea why you feel the need to throw about thinly-veiled accusations that Warrigal is basically an idiot. (How else could he or she possibly have missed all these really obvious problems you so insightfully spotted?). Maybe you don’t even intend them as such (though I’m baffled as to how could you possibly miss the overtones of your statements when they’re so freakin’ OBVIOUS). But the tendency to belittle others’ intellectual capacities (rather than just their views) is one that you’ve exhibited on a number of prior occasions as well, and one that I think you would do well to try to overcome—if only so that others will be more receptive to your ideas.
PS. For the avoidance of doubt, that final para was intended in part as an ironic illustration of the problem. I’m not that un-self-aware.
I agree that I’ve been many times unnecessarily harsh. But seriously, take a look at a random sampling of my posts and see how many of them are that way. It’s not actually as often as you’re trying to imply.
I do it because some people cross the threshold from “honest mistake” into “not even trying”. In which case they need to know that too, not just the specifics of their error. Holding someone’s hand through basic explanations is unfair to the people who have to do the work that the initial poster should have done for themselves.
And FWIW, if anyone ever catches me in that position—where I screw up so bad that I didn’t even appear to be thinking when I posted—I hope that you treat me the same way, so that I learn not just my specific error, but why it was so easily avoidable. Arguably, that’s the approach you just took.
Now a suggestion for you: your comment was best communicated by private message. Why stage a degrading, self-congratulatory “intervention”? Unless...
Holding someone’s hand through basic explanations is unfair to the people who have to do the work that the initial poster should have done for themselves.
What’s obvious to one person is seldom obvious to everybody else. There are things that seem utterly trivial to me that lots of people don’t get immediately, and many more things that seem utterly trivial to others that I don’t get immediately. That doesn’t mean that any of us aren’t trying, or deserve to be belittled for “not getting it”. (I can’t quite tell if your second paragraph is intended as justification or merely explanation; apologies if I’ve guessed wrongly).
Why stage a degrading, self-congratulatory “intervention”?
It wasn’t intended to be self-congratulatory; it was intended to make a point. Oh well. As for being degrading, I was attempting, via irony, to help you to understand the impact of a particular style of comment. It’s a style that I would normally try to avoid, and I agree that in general such comments might be better communicated privately, and certainly in a less inflammatory way. (In this case, it honestly didn’t occur to me to send a private message. Not sure what I would have done if it had. I think the extent to which others’ here agree or disagree with my point is useful information for us both, but information that would be lost if the correspondence were private.)
It’s not actually as often as you’re trying to imply.
I’m not sure what you think I was trying to imply, but I had two specific instances in mind (other than this one), and honestly wasn’t trying to imply anything beyond that.
What’s obvious to one person is seldom obvious to everybody else.
You’re preaching to the choir here. But when Warrigal announces some grand new idea, but just shrugs of even the importance of spelling out its implications, that’s well beyond “not noticing something that’s obvious to others” and into the territory of “not giving a s---, but expecting people to do your work for you.”
As for being degrading, I was attempting, via irony, to help you to understand the impact of a particular style of comment.
Right. I “got” that the first time around (even before PS), thanks. That wasn’t what I was referring to as “degrading”; it was actually pretty clever. Good work!
The degrading bit was where you do the internet equivalent of calling someone out in public, and then going through your accumulated list of their flaws, so anyone else who doesn’t like the resident “bad guy” (guy who actually says what everyone else isn’t willing to take the karma hit for) can join the pile-on.
In this case, it honestly didn’t occur to me to send a private message.
Sure, because what you were trying to accomplish (self-promotion, “us vs. them”)wouldn’t have been satisfied by a private message, so of course it’s not going to occur to you.
Other people seem to manage to PM me when I’m out of line (won’t name names here). But that’s generally because they’re actually interested in improving my posting, not in grandstanding.
I see no “accumulated list of [your] flaws” in what conchis has posted here. I see some comments on what you said on this particular occasion; and I see, embedded in something that (as you say you understood, and I’m sure you did) was deliberately nasty in style in order to make a point, the claim that you’ve exhibited the same pathology elsewhere as is on display here. No accumulated list; a single flaw, and even that mentioned only to point up the distinction between criticizing what someone has written and criticizing them personally.
Also: You’re being needlessly obnoxious; please desist. I am saying this in public rather than by PM because what I am trying to accomplish is (some small amount of) disincentive for other people who might wish to be obnoxious themselves. I am interested in improving not only your posting but LW as a whole.
And, FWIW, so far as I can tell I have no recollection of your past behaviour on LW, and in particular I am not saying this because I “don’t like” you.
I’m willing to apologise for publicly calling you out. While I’m still not totally convinced that PMing would have been optimal in this instance, it was a failing on my part not to have considered it at all, and I’m certainly sorry for any hurt I may have caused.
I’m also sorry that you seem to have such a poor impression of me that you can’t think of any way to explain my behaviour other than self-promotion and grandstanding. Not really big on argumentative charity are you?
I apologize for loading up on the negative motives I attributed to you. I appreciate your feedback, I would just prefer it not be done in a way that makes a spectacle of it all.
He cites “Goedel, Escher, Bach”, in which Hofstadter makes the same argument. Hofstadter doesn’t apply it to the silly why-we-aren’t-machines argument, though. (And Drescher doesn’t actually say that a Goedel sentence isn’t true, just that we can’t really know it’s true.)
An infinitely long proof is not a proof, since proofs are finite by definition.
The truth value of a statement does not depend on the existence of a proof anyways, the definition of truth is that it holds in any model. It is just a corollary of Goedel’s completeness theorem that syntactic truth (existence of a (finite) proof) coincides with semantic truth if the axiom system satisfies certain assumptions.
With that definition of truth, a Goedel sentence is not “true”, because there are models in which it fails to hold; neither is its negation “true”, because there are models in which it does. But that’s not the only way in which the word “true” is used about mathematical statements (though perhaps it should be); many people are quite sure that (e.g.) a Goedel sentence for their favourite formalization of arithmetic is either true or false (and by the latter they mean not-true). There’s plenty of reason to be skeptical about the sort of Platonism that would guarantee that every statement in the language of (say) Principia Mathematica or ZF is “really” true or false, but it hardly seems reasonable to declare it wrong by definition as you’re doing here.
many people are quite sure that (e.g.) a Goedel sentence for their favourite formalization of arithmetic is either true or false (and by the latter they mean not-true).
Those people seem a bit silly, then. If you say “The Godel sentence (G) is true of the smallest model (i.e. the standard model) of first-order Peano Arithmetic (PA)” then this truth follows from G being unprovable: if there were a proof of G in the smallest model, there would be a proof of G in all models, and if there were a proof of G in all models, then by Godel’s completeness theorem G would be provable in PA. To insist that the Godel sentence is true in PA—that it is true wherever the axioms of PA are true—rather than being only “true in the smallest model of PA”—is just factually wrong, flat wrong as math.
The people I’m thinking of—I was one of them, once—would not say either “G is true in PA” or “G is true in such-and-such a model of PA”. They would say, simply, “G is true”, and by that they would mean that what G says about the natural numbers is true about the natural numbers—you know, the actual, real, natural numbers. And they would react with some impatience to the idea that “the actual, real, natural numbers” might not be a clearly defined notion, or that statements about them might not have a well-defined truth value in the real world.
I think most people who know Goedel’s theorem say “G is true” and are “unreflective platonists,” by which I mean that they act like the natural numbers really exist, etc, but if you pushed them on it, they’d admit the doubt of your last couple of sentences.
Similarly, most people (eg, everyone on this thread), state Goedel’s completeness theorem platonically: a statement is provable if it is true in every model. That doesn’t make sense without models having some platonic existence. (yes, you can talk about internal models, but people don’t.) I suppose you could take the platonic position that all models exist without believing that it is possible to single out the special model. (Eliezer referred to “the minimal model”; does that work?)
You are right: you may come up with another consistent way of defining truth.
However, my comment was a reaction to silas’s comment, in which he seemed to confuse the notion syntactic and semantic truth, taking provability as the primary criterion. I just pointed out that even undergraduate logic courses treat semantic truth as basis and syntactic truth enters the picture as a consequence.
You propose the neato idea to use fractional truth values to deal with statements like “this is tall”, and boost it with a way to adjust such truth values as height varies. Somehow you missed that we already have a way to handle such gradations; it’s called “units of measurement”.
Units of measurement don’t work nearly as well when dealing with things such as beauty instead of length.
I think an important distinction between units of measurement and fuzzy logic is that units of measurement must pertain to things that are measurable, and they must be objectively defined, so that if two people express the same thing using units of measurement, their measurements will be the same. I see no reason that fuzzy logic shouldn’t be applicable to things that are simply a person’s impression of something.
Or perhaps it would be perfectly reasonable to relax the requirement that units of measurement be as objective as they are in practice. If Helen of Troy was N standards of deviation above the norm in beauty (trivia: N is about 6), we can declare the helen equal to N standards of deviation in beauty, and then agents capable of having an impression of beauty could look at random samples of people and say how beautiful they are in millihelens.
If there’s a better way of representing subjective trueness than real numbers between 0 and 1, I imagine lots of people would be interested in hearing it.
Or perhaps it would be perfectly reasonable to relax the requirement that units of measurement be as objective as they are in practice. If Helen of Troy was N standards of deviation above the norm in beauty (trivia: N is about 6), we can declare the helen equal to N standards of deviation in beauty, and then agents capable of having an impression of beauty could look at random samples of people and say how beautiful they are in millihelens.
That’s still creating a unit of measurement, it just uses protocols that prime it with respect to one person rather than a physical object. It doesn’t require a concept of fractional truth, just regular old measurement, probability andinterpolation.
Why don’t you spend some time more precisely developing the formalism… oh, wait
how can this be treated formally? I say, to heck with it.
I don’t think it’s fair to demand a full explanation of a topic that’s been around for over two decades (though a link to an online treatment would have been nice). Warrigal didn’t ‘come up with’ fractional values for truth. It’s a concept that’s been around (central?) in Eastern philosophy for centuries if not millenia, but was more-or-less exiled from Western philosophy by Aristotle’s Law of the Excluded Middle.
Fuzzy logic has proven itself very useful in control systems and in AI, because it matches the way people think about the world. Take Hemingway’s Challenge to “write one true [factual] sentence” (for which you would then need to show 100% exact correspondence of words to molecules in all relevant situations) and one’s perspective can change to see all facts as only partially true. ie, with a truth value in [0,1].
The statement “snow is white” is true if and only if snow is white, but you still have to define “snow” and “white”. How far from 100% even reflection of the entire visible spectrum can you go before “white” becomes “off-white”? How much can snow melt before it becomes “slush”? How much dissolved salt can it contain before it’s no longer “snow”? Is it still “snow” if it contains purple food colouring?
The same analysis of most concepts reveals we inherently think in fuzzy terms. (This is why court cases take so damn long to pick between the binary values of “guilty” and “not guilty”, when the answer is almost always “partially guilty”.) In fuzzy systems, concepts like “adult” (age of consent), “alive” (cryonics), “person” (abortion), all become scalar variables defined over n dimensions (usually n=1) when they are fed into the equations, and the results are translated back into a single value post-computation. The more usual control system variables are things like “hot”, “closed”, “wet”, “bright”, “fast”, etc., which make the system easier to understand and program than continuous measurements.
Bart Kosko’s book on the topic is Fuzzy Thinking. He makes some big claims about probability, but he says it boils down to fuzzy logic being just a different way of thinking about the same underlying math. (I don’t know if this gels with the discussion of ‘truth functionalism’ above) However, this prompts patterns of thought that would not otherwise make sense, which can lead to novel and useful results.
O...kay. It looks like you just decided to post the first thing on your head without concern for saying anything useful.
You come up with fractional values for truth, but don’t think it’s necessary to say what a fractional truth value means, let alone formalize it.
You propose the neato idea to use fractional truth values to deal with statements like “this is tall”, and boost it with a way to adjust such truth values as height varies. Somehow you missed that we already have a way to handle such gradations; it’s called “units of measurement”. We don’t need to say, “It’s 0.1 true that a football field is long”; we just say, “it’s true that a football field is 100 yards long.
Anyway, I thought I’d use this opportunity to say something useful. I was just reading Gary Drescher’s Good and Real (discussed here before), where he gives the most far-reaching, bold response to the claim that Goedel’s theorem proves limitations to machines, and I’m surprised the argument doesn’t show up more often, and that he didn’t seem to have anyone to cite as having made it before.
It goes like this: people claim that formal systems are somehow limited in that they can’t “see” that Goedel statements of the form “This statement can’t be proven within the system” are true. Drescher attacks this at the root and says, that’s not a limitation, because the statement’s not true.
He explains that you can’t actually rule out falsehood of the Goedel statement, as many people immediately do. Because it’s falsity still leaves room for the possibility that “This statement has a proof, but it’s infinitely long.” But then the subtle assumption that “This statement has a proof” implies “This statement is true” becomes much more tenuous. It’s far from obvious why you must accept as true a statement whose proof you can never complete.
Take that, Penrose!
Silas, a suggestion which you can take or leave, as your prefer.
This comment makes some sound points, but IMHO, in an unnecessarily personal way. Note the consistent use of the critical “you”-based formulations (“you just decided”, “you come up with”, “you propose”, “you missed that”). Contrast this with Christian’s comment, which is also critical, but consistently focuses on the ideas, rather than the person presenting them.
I have no idea why you feel the need to throw about thinly-veiled accusations that Warrigal is basically an idiot. (How else could he or she possibly have missed all these really obvious problems you so insightfully spotted?). Maybe you don’t even intend them as such (though I’m baffled as to how could you possibly miss the overtones of your statements when they’re so freakin’ OBVIOUS). But the tendency to belittle others’ intellectual capacities (rather than just their views) is one that you’ve exhibited on a number of prior occasions as well, and one that I think you would do well to try to overcome—if only so that others will be more receptive to your ideas.
PS. For the avoidance of doubt, that final para was intended in part as an ironic illustration of the problem. I’m not that un-self-aware.
PPS. Also, I didn’t vote you down.
I agree that I’ve been many times unnecessarily harsh. But seriously, take a look at a random sampling of my posts and see how many of them are that way. It’s not actually as often as you’re trying to imply.
I do it because some people cross the threshold from “honest mistake” into “not even trying”. In which case they need to know that too, not just the specifics of their error. Holding someone’s hand through basic explanations is unfair to the people who have to do the work that the initial poster should have done for themselves.
And FWIW, if anyone ever catches me in that position—where I screw up so bad that I didn’t even appear to be thinking when I posted—I hope that you treat me the same way, so that I learn not just my specific error, but why it was so easily avoidable. Arguably, that’s the approach you just took.
Now a suggestion for you: your comment was best communicated by private message. Why stage a degrading, self-congratulatory “intervention”? Unless...
What’s obvious to one person is seldom obvious to everybody else. There are things that seem utterly trivial to me that lots of people don’t get immediately, and many more things that seem utterly trivial to others that I don’t get immediately. That doesn’t mean that any of us aren’t trying, or deserve to be belittled for “not getting it”. (I can’t quite tell if your second paragraph is intended as justification or merely explanation; apologies if I’ve guessed wrongly).
It wasn’t intended to be self-congratulatory; it was intended to make a point. Oh well. As for being degrading, I was attempting, via irony, to help you to understand the impact of a particular style of comment. It’s a style that I would normally try to avoid, and I agree that in general such comments might be better communicated privately, and certainly in a less inflammatory way. (In this case, it honestly didn’t occur to me to send a private message. Not sure what I would have done if it had. I think the extent to which others’ here agree or disagree with my point is useful information for us both, but information that would be lost if the correspondence were private.)
I’m not sure what you think I was trying to imply, but I had two specific instances in mind (other than this one), and honestly wasn’t trying to imply anything beyond that.
You’re preaching to the choir here. But when Warrigal announces some grand new idea, but just shrugs of even the importance of spelling out its implications, that’s well beyond “not noticing something that’s obvious to others” and into the territory of “not giving a s---, but expecting people to do your work for you.”
Right. I “got” that the first time around (even before PS), thanks. That wasn’t what I was referring to as “degrading”; it was actually pretty clever. Good work!
The degrading bit was where you do the internet equivalent of calling someone out in public, and then going through your accumulated list of their flaws, so anyone else who doesn’t like the resident “bad guy” (guy who actually says what everyone else isn’t willing to take the karma hit for) can join the pile-on.
Sure, because what you were trying to accomplish (self-promotion, “us vs. them”)wouldn’t have been satisfied by a private message, so of course it’s not going to occur to you.
Other people seem to manage to PM me when I’m out of line (won’t name names here). But that’s generally because they’re actually interested in improving my posting, not in grandstanding.
I see no “accumulated list of [your] flaws” in what conchis has posted here. I see some comments on what you said on this particular occasion; and I see, embedded in something that (as you say you understood, and I’m sure you did) was deliberately nasty in style in order to make a point, the claim that you’ve exhibited the same pathology elsewhere as is on display here. No accumulated list; a single flaw, and even that mentioned only to point up the distinction between criticizing what someone has written and criticizing them personally.
Also: You’re being needlessly obnoxious; please desist. I am saying this in public rather than by PM because what I am trying to accomplish is (some small amount of) disincentive for other people who might wish to be obnoxious themselves. I am interested in improving not only your posting but LW as a whole.
And, FWIW, so far as I can tell I have no recollection of your past behaviour on LW, and in particular I am not saying this because I “don’t like” you.
I’m willing to apologise for publicly calling you out. While I’m still not totally convinced that PMing would have been optimal in this instance, it was a failing on my part not to have considered it at all, and I’m certainly sorry for any hurt I may have caused.
I’m also sorry that you seem to have such a poor impression of me that you can’t think of any way to explain my behaviour other than self-promotion and grandstanding. Not really big on argumentative charity are you?
Apology accepted! :-)
I apologize for loading up on the negative motives I attributed to you. I appreciate your feedback, I would just prefer it not be done in a way that makes a spectacle of it all.
Apology likewise accepted! ;)
He cites “Goedel, Escher, Bach”, in which Hofstadter makes the same argument. Hofstadter doesn’t apply it to the silly why-we-aren’t-machines argument, though. (And Drescher doesn’t actually say that a Goedel sentence isn’t true, just that we can’t really know it’s true.)
An infinitely long proof is not a proof, since proofs are finite by definition.
The truth value of a statement does not depend on the existence of a proof anyways, the definition of truth is that it holds in any model. It is just a corollary of Goedel’s completeness theorem that syntactic truth (existence of a (finite) proof) coincides with semantic truth if the axiom system satisfies certain assumptions.
With that definition of truth, a Goedel sentence is not “true”, because there are models in which it fails to hold; neither is its negation “true”, because there are models in which it does. But that’s not the only way in which the word “true” is used about mathematical statements (though perhaps it should be); many people are quite sure that (e.g.) a Goedel sentence for their favourite formalization of arithmetic is either true or false (and by the latter they mean not-true). There’s plenty of reason to be skeptical about the sort of Platonism that would guarantee that every statement in the language of (say) Principia Mathematica or ZF is “really” true or false, but it hardly seems reasonable to declare it wrong by definition as you’re doing here.
Those people seem a bit silly, then. If you say “The Godel sentence (G) is true of the smallest model (i.e. the standard model) of first-order Peano Arithmetic (PA)” then this truth follows from G being unprovable: if there were a proof of G in the smallest model, there would be a proof of G in all models, and if there were a proof of G in all models, then by Godel’s completeness theorem G would be provable in PA. To insist that the Godel sentence is true in PA—that it is true wherever the axioms of PA are true—rather than being only “true in the smallest model of PA”—is just factually wrong, flat wrong as math.
This thread needs a link to Tarski’s undefinability theorem.
Also, you’re assuming the consistency of PA.
The people I’m thinking of—I was one of them, once—would not say either “G is true in PA” or “G is true in such-and-such a model of PA”. They would say, simply, “G is true”, and by that they would mean that what G says about the natural numbers is true about the natural numbers—you know, the actual, real, natural numbers. And they would react with some impatience to the idea that “the actual, real, natural numbers” might not be a clearly defined notion, or that statements about them might not have a well-defined truth value in the real world.
In other words, Platonists.
I think most people who know Goedel’s theorem say “G is true” and are “unreflective platonists,” by which I mean that they act like the natural numbers really exist, etc, but if you pushed them on it, they’d admit the doubt of your last couple of sentences.
Similarly, most people (eg, everyone on this thread), state Goedel’s completeness theorem platonically: a statement is provable if it is true in every model. That doesn’t make sense without models having some platonic existence. (yes, you can talk about internal models, but people don’t.) I suppose you could take the platonic position that all models exist without believing that it is possible to single out the special model. (Eliezer referred to “the minimal model”; does that work?)
You are right: you may come up with another consistent way of defining truth.
However, my comment was a reaction to silas’s comment, in which he seemed to confuse the notion syntactic and semantic truth, taking provability as the primary criterion. I just pointed out that even undergraduate logic courses treat semantic truth as basis and syntactic truth enters the picture as a consequence.
Units of measurement don’t work nearly as well when dealing with things such as beauty instead of length.
Then neither does fuzzy logic.
I think an important distinction between units of measurement and fuzzy logic is that units of measurement must pertain to things that are measurable, and they must be objectively defined, so that if two people express the same thing using units of measurement, their measurements will be the same. I see no reason that fuzzy logic shouldn’t be applicable to things that are simply a person’s impression of something.
Or perhaps it would be perfectly reasonable to relax the requirement that units of measurement be as objective as they are in practice. If Helen of Troy was N standards of deviation above the norm in beauty (trivia: N is about 6), we can declare the helen equal to N standards of deviation in beauty, and then agents capable of having an impression of beauty could look at random samples of people and say how beautiful they are in millihelens.
If there’s a better way of representing subjective trueness than real numbers between 0 and 1, I imagine lots of people would be interested in hearing it.
That’s still creating a unit of measurement, it just uses protocols that prime it with respect to one person rather than a physical object. It doesn’t require a concept of fractional truth, just regular old measurement, probability andinterpolation.
Why don’t you spend some time more precisely developing the formalism… oh, wait
That’s why.
I don’t think it’s fair to demand a full explanation of a topic that’s been around for over two decades (though a link to an online treatment would have been nice). Warrigal didn’t ‘come up with’ fractional values for truth. It’s a concept that’s been around (central?) in Eastern philosophy for centuries if not millenia, but was more-or-less exiled from Western philosophy by Aristotle’s Law of the Excluded Middle.
Fuzzy logic has proven itself very useful in control systems and in AI, because it matches the way people think about the world. Take Hemingway’s Challenge to “write one true [factual] sentence” (for which you would then need to show 100% exact correspondence of words to molecules in all relevant situations) and one’s perspective can change to see all facts as only partially true. ie, with a truth value in [0,1].
The statement “snow is white” is true if and only if snow is white, but you still have to define “snow” and “white”. How far from 100% even reflection of the entire visible spectrum can you go before “white” becomes “off-white”? How much can snow melt before it becomes “slush”? How much dissolved salt can it contain before it’s no longer “snow”? Is it still “snow” if it contains purple food colouring?
The same analysis of most concepts reveals we inherently think in fuzzy terms. (This is why court cases take so damn long to pick between the binary values of “guilty” and “not guilty”, when the answer is almost always “partially guilty”.) In fuzzy systems, concepts like “adult” (age of consent), “alive” (cryonics), “person” (abortion), all become scalar variables defined over n dimensions (usually n=1) when they are fed into the equations, and the results are translated back into a single value post-computation. The more usual control system variables are things like “hot”, “closed”, “wet”, “bright”, “fast”, etc., which make the system easier to understand and program than continuous measurements.
Bart Kosko’s book on the topic is Fuzzy Thinking. He makes some big claims about probability, but he says it boils down to fuzzy logic being just a different way of thinking about the same underlying math. (I don’t know if this gels with the discussion of ‘truth functionalism’ above) However, this prompts patterns of thought that would not otherwise make sense, which can lead to novel and useful results.
I voted up your post for its conclusions, but would request that you make them a bit friendlier in the future...