"All men are mortal. Socrates is a man. Therefore Socrates is mortal."
Socrates raised the glass of hemlock to his lips...
"Do you suppose," asked one of the onlookers, "that even hemlock will not be enough to kill so wise and good a man?"
"No," replied another bystander, a student of philosophy; "all men are mortal, and Socrates is a man; and if a mortal drink hemlock, surely he dies."
"Well," said the onlooker, "what if it happens that Socrates isn't mortal?"
"Nonsense," replied the student, a little sharply; "all men are mortal by definition; it is part of what we mean by the word 'man'. All men are mortal, Socrates is a man, therefore Socrates is mortal. It is not merely a guess, but a logical certainty."
"I suppose that's right..." said the onlooker. "Oh, look, Socrates already drank the hemlock while we were talking."
"Yes, he should be keeling over any minute now," said the student.
And they waited, and they waited, and they waited...
"Socrates appears not to be mortal," said the onlooker.
"Then Socrates must not be a man," replied the student. "All men are mortal, Socrates is not mortal, therefore Socrates is not a man. And that is not merely a guess, but a logical certainty."
The fundamental problem with arguing that things are true "by definition" is that you can't make reality go a different way by choosing a different definition.
You could reason, perhaps, as follows: "All things I have observed which wear clothing, speak language, and use tools, have also shared certain other properties as well, such as breathing air and pumping red blood. The last thirty 'humans' belonging to this cluster, whom I observed to drink hemlock, soon fell over and stopped moving. Socrates wears a toga, speaks fluent ancient Greek, and drank hemlock from a cup. So I predict that Socrates will keel over in the next five minutes."
But that would be mere guessing. It wouldn't be, y'know, absolutely and eternally certain. The Greek philosophers—like most prescientific philosophers—were rather fond of certainty.
Luckily the Greek philosophers have a crushing rejoinder to your questioning. You have misunderstood the meaning of "All humans are mortal," they say. It is not a mere observation. It is part of the definition of the word "human". Mortality is one of several properties that are individually necessary, and together sufficient, to determine membership in the class "human". The statement "All humans are mortal" is a logically valid truth, absolutely unquestionable. And if Socrates is human, he must be mortal: it is a logical deduction, as certain as certain can be.
But then we can never know for certain that Socrates is a "human" until after Socrates has been observed to be mortal. It does no good to observe that Socrates speaks fluent Greek, or that Socrates has red blood, or even that Socrates has human DNA. None of these characteristics are logically equivalent to mortality. You have to see him die before you can conclude that he was human.
(And even then it's not infinitely certain. What if Socrates rises from the grave a night after you see him die? Or more realistically, what if Socrates is signed up for cryonics? If mortality is defined to mean finite lifespan, then you can never really know if someone was human, until you've observed to the end of eternity—just to make sure they don't come back. Or you could think you saw Socrates keel over, but it could be an illusion projected onto your eyes with a retinal scanner. Or maybe you just hallucinated the whole thing...)
The problem with syllogisms is that they're always valid. "All humans are mortal; Socrates is human; therefore Socrates is mortal" is—if you treat it as a logical syllogism—logically valid within our own universe. It's also logically valid within neighboring Everett branches in which, due to a slightly different evolved biochemistry, hemlock is a delicious treat rather than a poison. And it's logically valid even in universes where Socrates never existed, or for that matter, where humans never existed.
The Bayesian definition of evidence favoring a hypothesis is evidence which we are more likely to see if the hypothesis is true than if it is false. Observing that a syllogism is logically valid can never be evidence favoring any empirical proposition, because the syllogism will be logically valid whether that proposition is true or false.
Syllogisms are valid in all possible worlds, and therefore, observing their validity never tells us anything about which possible world we actually live in.
This doesn't mean that logic is useless—just that logic can only tell us that which, in some sense, we already know. But we do not always believe what we know. Is the number 29384209 prime? By virtue of how I define my decimal system and my axioms of arithmetic, I have already determined my answer to this question—but I do not know what my answer is yet, and I must do some logic to find out.
Similarly, if I form the uncertain empirical generalization "Humans are vulnerable to hemlock", and the uncertain empirical guess "Socrates is human", logic can tell me that my previous guesses are predicting that Socrates will be vulnerable to hemlock.
It's been suggested that we can view logical reasoning as resolving our uncertainty about impossible possible worlds—eliminating probability mass in logically impossible worlds which we did not know to be logically impossible. In this sense, logical argument can be treated as observation.
But when you talk about an empirical prediction like "Socrates is going to keel over and stop breathing" or "Socrates is going to do fifty jumping jacks and then compete in the Olympics next year", that is a matter of possible worlds, not impossible possible worlds.
Logic can tell us which hypotheses match up to which observations, and it can tell us what these hypotheses predict for the future—it can bring old observations and previous guesses to bear on a new problem. But logic never flatly says, "Socrates will stop breathing now." Logic never dictates any empirical question; it never settles any real-world query which could, by any stretch of the imagination, go either way.
Just remember the Litany Against Logic:
Logic stays true, wherever you may go,
So logic never tells you where you live.
Is this a problem of logic or of definitions? If one remembers to reason about the objects themselves, instead of the words of the definition, then you don't have the problem where you've created a simplified model of something right off the bat (dropping half the information).
This is an area where AIs will have an advantage over us, because they should be able to reason about objects directly without having to compress them in to a definition first.
The linguistic tradition is testament to the fact that we don't usually reason about the objects.
You have to see him die before you can conclude that he was human.
Given the story of Croesus and Solon, I think many ancient Greeks would have been quite comfortable with that.
If we accept that hemlock would kill any mortal, and someone consumes hemlock, and they don't die, either they aren't mortal, it wasn't actually hemlock, they didn't actually consume it, or we were wrong about hemlock being lethal.
But complaining about defined 'truths' is silly. It's not as though the word 'man' has some objective meaning written down in the high heavens by the meaning-granting deity himself. We can use it to mean whatever we please. But when we do so, we must always remember that our application of the word to reality isn't necessarily correct. So we define 'man' to, among other things, be a mortal entity. But asserting that a person is a man is just that - an assertion, and one that can be incorrect.
Definitions cannot be incorrect. They can only be inconsistent, either with other definitions or themselves.
Yes, the probability that the Socrates syllogism would be valid, given that Socrates is mortal, is the same as the probability that it would be valid given that he immortal.
On the other hand, the probability that "I observe a valid syllogism for the conclusion that Socrates is mortal, starting from statements that I believe to be true," given that Socrates is mortal, is not the same as its probability given that he is immortal-- at least if your beliefs have more than zero correlation with reality. So observing a valid syllogism for some conclusion from statements that you believe to be true is indeed Bayesian evidence that the conclusion is true.
This is an important point, because Eliezer seems to have misled himself in the past by noticing the first point but not the second point, for example in his use of these parables in arguing against Robin Hanson regarding disagreement.
What is the easiest way to get a RSS with ONLY the overcomingbias posts that more narrowly discuss BIAS in a way that is engaged with the literature, thereby filtering out the many posts that do NOT and that instead do try to reinvent philosophical wheels without caring about the preexisting relevant literature or terminology? Seriously. I like the gems but I can't justify the time it takes me to see to wade through all the other posts in the RSS to get to the gems. You guy's should do like other high volume blogs and offer some niche RSS channels, like a "best bias posts of the week" one.
Tom, I'm well aware that the point above is not original, but I don't think I can be accused of reinventing the wheel, just re-explaining it. The post above looks pretty literature-informed to me - it says "logically valid" rather than "logically true" and talks about impossible possible worlds. I didn't use the phrase "problem of induction", but then I didn't exactly set out to signal academic in-ness because that would get in the way of explaining. This blog is directed at a wider audience at least half the time, according to its policy. I'm not sure how else you think this post should have been written.
I'm not really sure what the point of the post is.
Logic is always conditional. If the premises are true then the conclusion is true. That means we could reach the wrong conclusion with false premises.
Eliezer, are you saying we should stop or diminish our use of logic? Should I eat hemlock because I might be wrong about it's lethality?
You mean by those last two lines that logic offer's no 'grounding' to reality and only empirical probability does? Since truth does not depend upon us, what does it depend upon? Well, the truth depends on circumstance if utilizing probability theory and empiricism. Since their is no absolute way of knowing their is also no absolute way of knowing how unlucky or lucky our circumstances are in favoring truth.
Sure, reality is non-dependent. But the nature of our circumstances are very dependent...upon that which we cannot measure. Our position in the universe.
And caledonian, if you were arguing with a theist you would have lost by now.
Since no one seems to grasp what you intended to convey, this post clearly wasn't adequate to your purpose... assuming that was your purpose.
You have to be careful where you put your nails, lest you rip reality.
Eliezer, My post was more harsh than I on reflection would have wanted it to be. Let me say some positive things to balance that. I find the productivity of the bloggers here quantitatively and qualitatively really impressive. You post a lot of fresh, inspiring and provoking stuff. I like your engaged way of writing.
Still, I look to this blog for posts on bias, not philosophy in general. So I want to stick to my intended main (and only constructive) point: please implement some more limited RSS feeds. Specifically, a feed that includes all and only posts on bias aimed at academic folks. Or perhaps, to make it even more narrow, only such posts that you the team of bloggers judge to be of extra high quality.
If the blog is targetted at several types of readers then it makes sense to offer a special RSS feed for each target type. Very likely, most academics reading your current RSS feed have a large number of other feeds that they also try to keep up with so helping such readers with the filtering would be great.
As a sideshow, I would note the death of Rasputin, whom some were not so certain was really a man either, although rather than a demi-god possibly like Socrates, some of those doubters thought that he was more like a demon, and I am also unaware of anybody getting involved in such a debate when he refused to die according to the usual causes.
In any case, he was killed by a group of tsarist nobles who were upset about his apparent control over Tsar Nikolai II and his family. So, they invited him to dinner. He was poisoned, he was shut, he was beaten and knifed. None of this did the trick. It required taking him outside and forcing him into an icy river where he presumably both drowned and froze to finally do him in.
I have created an exercise that goes with this post. Use it to solidify your knowledge of the material.
This post is pretty confused and under researched.
Firstly, there seems to be a confusion about the topic: are we talking about definitions or syllogisms? No one (certainly not Aristotle) will claim that definitions, or anything of the form “All humans are mortal” is logically valid. Only inferences can be valid, and definitions are not inferences.
Secondly, there seems to be a confusion between logical validity and truth. No one has ever claimed that the validity of a syllogism can lend certainty or even support any given empirical claim. If the claim is the conclusion of a valid syllogism containing true premises, then it must be true, but it is not empirical. And the question of its certainty rests entirely on our certainty of the premises.
Lastly, whatever claims this post makes about Aristotle or people ‘of Aristotle’s time’ seem to be pretty unfounded. Aristotle’s views on logic, definitions, and necessity are pretty complex, and nothing of them is represented here. Aristotle certainly didn’t believe that everything falling under a definition has every attribute described in that definition. Aristotle’s definition for human being was probably “rational animal” and he did not think every human being was rational.
Here's a short story I remember once reading about a similar problem.
Thanks for writing this, it is what I really needed in my life. I've been struggling so much lately with what to believe and not believe concerning models of reality.
It took me a while to fully understand your point in this post. I think that adding a obviously wrong example that’s identical in structure to "All men are mortal. Socrates is a man. Therefore Socrates is mortal.", will help. My example is “All chickens are mortal. Socrates is a chicken. Therefore, Socrates is mortal.” It’ll help show that the original example given in the post is wrong.