In February I posted on David Buss being convinced that "true love" exists, even though he’s never seen it and his job is to study love. Similarly, Michael Murrin tells (p.11) how Europeans were immune to being told by Marco Polo how real unicorns (rhinoceroses) differed from what they had imagined:
I will use only the strictest first-order logic to form my deductive chain, which is modeled on the proof that "P and ~P implies Q". First, the source material from which I take my axioms:
"God is not a man, that he should lie; neither the son of man, that he should repent: hath he said, and shall he not do it? or hath he spoken, and shall he not make it good?" -- Numbers 23:19 (KJV)
Interpreted literally, we can state that "God can not change his mind." Let this be Axiom A.
"And the LORD repented of the evil which he thought to do unto his people." -- Exodus 32:14 (KJV)
Interpreted literally, we can state that "God can change his mind." Let this be Axiom B.
Update: Before we go any further, I'd like to address some potential points of contention. The KJV is a horrible, horrible translation -- as Jon notes below, when properly translated, the first quote I mention doesn't lead to Axiom A. But there are plenty of places in the Bible that lead to contradiction, either directly or indirectly. I'm just using this particular pair of quotes as an example -- I'm sure somehow more well-versed in the Old and New Testaments could find a much better pair. [more?]
1. Introduce Axiom A: God can not change his mind.2. Introduce Axiom B: God can change his mind.3. Enter hypothetical on the base of "Unicorns do not exist."1. First line of hypothetical: Unicorns do not exist.2. By statement 1: God can not change his mind.Conclusion of hypothetical: If unicorns do not exist, then God can not change his mind.4. Contrapositive of statement 3: If it is not true that God can not change his mind, then it is not true that unicorns do not exist.5. Cancel double negative: If God can change his mind, then it is not true that unicorns do not exist.6. Cancel double negative: If God can change his mind, then unicorns exist.7. By statements 2 and 6: Unicorns exist.
The result is that in any system where the statements "God can change his mind" and "God can not change his mind" are both true, unicorns must exist. (Incidentally, the same logic can be used to prove that unicorns do not exist. Slippery beasts, eh?) In fact, any statement of fact can be used instead of "Unicorns (do not) exist", including "1 equals 2" or "Tim is God" or "Thou shalt kill" -- that is the power of direct contradiction. Thus it is shown that if you believe the Bible (or Quran or Torah) word-for-word, you'll believe anything.
You guys may be interested in looking up Hempel's "all crows are black" paradox: if you see a non-black non-crow, e.g. a red apple, is that evidence for the the statement "all crows are black"?
In the philosophical literature, there are different solutions proposed for this. Personally, I'm ok with saying "yes, it is".
Mobile,Though the two statements are logically equivalent, you have to be very careful with how you update your prior. If it is very rare to see a unicorn given that they exist, then seeing something that is not a unicorn will only slightly increase your belief in the proporition and it will still be approximately equal to your prior belief.
Let U be the hypothesis that unicorns exist. Let X be the event of seeing something that is not a unicorn. Then, through bayes formula,
but P(X|U^c)=1, and P(X|U) approximately equals 1... thusP(U^c|X) approximately equals P(U^c), your prior, though only SLIGHTLY bigger. And if in your environment, it is impossible to see a unicorn, then P(X|U)=1 and your posterior equals your prior.
If you spend the rest of your life searching for unicors in the ocean and observe many non-unicorns, should that cause an increase in your posterior probability that unicorns do not exist? Think about it... under your reasoning above, it should.
The proposition "unicorns do not exist" is equivalent to the proposition "everything that exists is not a unicorn". To acquire new knowledge of something that exists that is not a unicorn should therefore cause you to increase your prior belief in the proposition.
Mencius Moldbug just pointed out to me Charles Kingley's "Water-Babies", a book in which he attacks those who disbelieve in souls (or water babies) because they have not seen them. I suspect it is evidence of my confirmation bias that I felt more smugly materialist after reading that passage.
There is a difference between the need to believe the ideal exists somewhere and the belief in the possibility that it could exist under other conditions.
Also, just because you don't have any evidence that it exists is not the same as having evidence that it doesn't exist or can't exist. This seems to be an example of the generalizability of your conclusions. Just because we haven't found examples of ideals UNDER OUR STUDY CONDITIONS does not mean that we couldn't find an example under other conditions. If your prior is that an ideal is impossible, then this doesn't matter, but if your prior belief is that certain conditions could increase the possibility of an ideal, then an infinite number of examples under one condition shouldn't change your view about the probability of an ideal under other (more ideal :) ) conditions.
Polo saw rhinoceroses in Indonesia and described them minutely. He ended his discussion with the remark that such unicorns do not resemble at all European notions, nor do they allow themselves to be captured by a virgin. Yet despite the fact that the Divisament dou monde had wide circulation and multiple translations, even while Polo was alive, his attempt to dispel or correct such European fantasies failed.
If you know that there exists a massively-built quadruped with a horn, would this cause you to lower your estimate of the possibility of there existing a slightly-built quadruped with a horn?
Ok, but politicians, celebrities, and scientists are not unicorns. (Perhaps we wish particular ones of them didn't exist.) I'm not sure if "believing that an ideal exemplar exists" is a bias, or perhaps just a conceptual misunderstanding. It probably depends upon how the term ideal is being used - an ideal specimen might be one that has all the typical features of its type, but it might also be one that has all the best features that things of its type can have. But I take it that there's a third sense of ideal, where an idea is an "organizing principle" that informs action, decisions, and understanding, such that we can always travel in the direction of the ideal, but by definition, never attain it. (I suppose some overachievers will become annoyed by being told that there's somehting of which they can have some, and even a lot, but not all...)
I'm about to travel so far afield, and quickly, so write me off as a nutjob if it serves. The phrase that popped into my mind as I read this was "In the beginning was the Word..." the opening of the Gospel of John. The level of abstraction that language makes possible,or perhaps 'accessible' in humans, combined with the "bias" that seems, at least in some of us to require white unicorns and true love is the "logic pit" that makes us religious animals. Since Godel tells me (I hope I understand the basics anyway) that logic leaves us those unprovable truths, and we crave the abstraction of unicorns, perfect love, absolute courage, ultimate truth, hence there is another way of "knowing," totally outside of logic. Anyway, there's a beautiful paradox in there somewhere...as I said, off topic, but you were pointing in a direction that my head has been traveling of late.
I will use only the strictest first-order logic to form my deductive chain, which is modeled on the proof that "P and ~P implies Q". First, the source material from which I take my axioms:
"God is not a man, that he should lie; neither the son of man, that he should repent: hath he said, and shall he not do it? or hath he spoken, and shall he not make it good?" -- Numbers 23:19 (KJV)
Interpreted literally, we can state that "God can not change his mind." Let this be Axiom A.
"And the LORD repented of the evil which he thought to do unto his people." -- Exodus 32:14 (KJV)
Interpreted literally, we can state that "God can change his mind." Let this be Axiom B.
Update: Before we go any further, I'd like to address some potential points of contention. The KJV is a horrible, horrible translation -- as Jon notes below, when properly translated, the first quote I mention doesn't lead to Axiom A. But there are plenty of places in the Bible that lead to contradiction, either directly or indirectly. I'm just using this particular pair of quotes as an example -- I'm sure somehow more well-versed in the Old and New Testaments could find a much better pair. [more?]
1. Introduce Axiom A: God can not change his mind.2. Introduce Axiom B: God can change his mind.3. Enter hypothetical on the base of "Unicorns do not exist."1. First line of hypothetical: Unicorns do not exist.2. By statement 1: God can not change his mind.Conclusion of hypothetical: If unicorns do not exist, then God can not change his mind.4. Contrapositive of statement 3: If it is not true that God can not change his mind, then it is not true that unicorns do not exist.5. Cancel double negative: If God can change his mind, then it is not true that unicorns do not exist.6. Cancel double negative: If God can change his mind, then unicorns exist.7. By statements 2 and 6: Unicorns exist.
The result is that in any system where the statements "God can change his mind" and "God can not change his mind" are both true, unicorns must exist. (Incidentally, the same logic can be used to prove that unicorns do not exist. Slippery beasts, eh?) In fact, any statement of fact can be used instead of "Unicorns (do not) exist", including "1 equals 2" or "Tim is God" or "Thou shalt kill" -- that is the power of direct contradiction. Thus it is shown that if you believe the Bible (or Quran or Torah) word-for-word, you'll believe anything.
Mobile, Joe,
You guys may be interested in looking up Hempel's "all crows are black" paradox: if you see a non-black non-crow, e.g. a red apple, is that evidence for the the statement "all crows are black"?
In the philosophical literature, there are different solutions proposed for this. Personally, I'm ok with saying "yes, it is".
Mobile,Though the two statements are logically equivalent, you have to be very careful with how you update your prior. If it is very rare to see a unicorn given that they exist, then seeing something that is not a unicorn will only slightly increase your belief in the proporition and it will still be approximately equal to your prior belief.
Let U be the hypothesis that unicorns exist. Let X be the event of seeing something that is not a unicorn. Then, through bayes formula,
P(U^c|X)= P(X|U^c)P(U^c) / [ P(X|U^c)P(U^c) + P(X|U)P(U) ].....
but P(X|U^c)=1, and P(X|U) approximately equals 1... thusP(U^c|X) approximately equals P(U^c), your prior, though only SLIGHTLY bigger. And if in your environment, it is impossible to see a unicorn, then P(X|U)=1 and your posterior equals your prior.
If you spend the rest of your life searching for unicors in the ocean and observe many non-unicorns, should that cause an increase in your posterior probability that unicorns do not exist? Think about it... under your reasoning above, it should.
The proposition "unicorns do not exist" is equivalent to the proposition "everything that exists is not a unicorn". To acquire new knowledge of something that exists that is not a unicorn should therefore cause you to increase your prior belief in the proposition.
Mencius Moldbug just pointed out to me Charles Kingley's "Water-Babies", a book in which he attacks those who disbelieve in souls (or water babies) because they have not seen them. I suspect it is evidence of my confirmation bias that I felt more smugly materialist after reading that passage.
There is a difference between the need to believe the ideal exists somewhere and the belief in the possibility that it could exist under other conditions.
Also, just because you don't have any evidence that it exists is not the same as having evidence that it doesn't exist or can't exist. This seems to be an example of the generalizability of your conclusions. Just because we haven't found examples of ideals UNDER OUR STUDY CONDITIONS does not mean that we couldn't find an example under other conditions. If your prior is that an ideal is impossible, then this doesn't matter, but if your prior belief is that certain conditions could increase the possibility of an ideal, then an infinite number of examples under one condition shouldn't change your view about the probability of an ideal under other (more ideal :) ) conditions.
Polo saw rhinoceroses in Indonesia and described them minutely. He ended his discussion with the remark that such unicorns do not resemble at all European notions, nor do they allow themselves to be captured by a virgin. Yet despite the fact that the Divisament dou monde had wide circulation and multiple translations, even while Polo was alive, his attempt to dispel or correct such European fantasies failed.
If you know that there exists a massively-built quadruped with a horn, would this cause you to lower your estimate of the possibility of there existing a slightly-built quadruped with a horn?
Matthew, by "ideal" I mean having a best feature combination.
Ok, but politicians, celebrities, and scientists are not unicorns. (Perhaps we wish particular ones of them didn't exist.) I'm not sure if "believing that an ideal exemplar exists" is a bias, or perhaps just a conceptual misunderstanding. It probably depends upon how the term ideal is being used - an ideal specimen might be one that has all the typical features of its type, but it might also be one that has all the best features that things of its type can have. But I take it that there's a third sense of ideal, where an idea is an "organizing principle" that informs action, decisions, and understanding, such that we can always travel in the direction of the ideal, but by definition, never attain it. (I suppose some overachievers will become annoyed by being told that there's somehting of which they can have some, and even a lot, but not all...)
I'm about to travel so far afield, and quickly, so write me off as a nutjob if it serves. The phrase that popped into my mind as I read this was "In the beginning was the Word..." the opening of the Gospel of John. The level of abstraction that language makes possible,or perhaps 'accessible' in humans, combined with the "bias" that seems, at least in some of us to require white unicorns and true love is the "logic pit" that makes us religious animals. Since Godel tells me (I hope I understand the basics anyway) that logic leaves us those unprovable truths, and we crave the abstraction of unicorns, perfect love, absolute courage, ultimate truth, hence there is another way of "knowing," totally outside of logic. Anyway, there's a beautiful paradox in there somewhere...as I said, off topic, but you were pointing in a direction that my head has been traveling of late.
Much like the implanted memory of a unicorn in Deckards memory, he dreams of a white unicorn, it's there for a reason.(Blade Runner, Ridley Scott).