Several professors told me this when I studied philosophy in grad school. It surprised me at the time—why should it be true? From a Bayesian perspective isn’t it much more evidence when there are a bunch of weak arguments pointing in the same direction, than when there is only one argument that is stronger?
Now I am older and wiser and have lots more experience, and this saying feels true to me. Not just in philosophy but in most domains, such as AGI timelines and cause prioritization.
Here are some speculations about why (and how) this saying might be true:
1. Strong evidence is common, and maybe in particular the distribution of evidence strength is heavy-tailed. In other words, maybe one strong argument typically beats many weak arguments for the same reason that one rich man typically has more money than many poor men—in any particular collection of random people/arguments, most of the money/evidence is concentrated in the richest/strongest.
2. If you have maximally high standards and accept only the most rigorous arguments, you’ll only find arguments supporting the true answer and none supporting the false answer. If you lower your standards maximally and accept any old word salad, by symmetry you’ll have as many arguments supporting the true as the false answer. Maybe standards in between continuously interpolate between these two data points: The lower your standards, the higher the ratio of arguments-for-the-true-answer to arguments-for-the-false-answer.
Maybe this isn’t by itself enough to make the saying true. But maybe if we combine it with selection effects and biases… If you are considering a class of weak arguments that contains 60% arguments for the true answer and 40% arguments for the false answer, it’ll be extremely easy for you to assemble a long list of arguments for the false answer, if for some reason you try. And you might try if you happen to start out believing the false answer, or having a hunch that it is true, and then do some biased search process...
There’s also the weighting problem if you don’t know which arguments are already embedded in your prior. A strong argument that “feels” novel is likely something you should update on. A weak argument shouldn’t update you much even if it were novel, and there are so many of them that you may already have included it.
Relatedly, it’s hard to determine correlation among many weak arguments—in many cases, they’re just different aspects of the same weak argument, not independent things to update on.
Finally, it’s part of the identification of strong vs weak arguments. If it wasn’t MUCH harder to refute or discount, we wouldn’t call it a strong argument.
The strongest argument is a mathematical or logical proof. It’s easy to see why a mathematical proof of the Riemann hypothesis beats a lot of (indeed, all) weak intuitive arguments about what the answer might be.
But this is only applicable for well-defined problems. Insofar as philosophy is tackling such problems, I would also expect a single strong argument to beat many weak arguments.
Part of the goal for the ill-defined problems we face on a daily basis is not to settle the question, but to refine it into a well-defined question that has a strong, single argument. Perhaps, then, the reason why one strong argument beats many arguments is that questions for which there’s a strategy for making a strong argument are most influential and compelling to us as a society. “What’s the meaning of life?” Only weak arguments are available, so it’s rare for people to seriously try and answer this question in a conclusive manner. “Can my e-commerce startup grow into a trillion-dollar company?” If the answer is “yes,” then there’s a single, strong argument that proves it: success. People seem attracted to these sorts of questions.
In the real world the weight of many pieces of weak evidence is not always comparable to a single piece of strong evidence. The important variable here is not strong versus weak per se but the source of the evidence. Some sources of evidence are easier to manipulate in various ways. Evidence manipulation, either consciously or emergently, is common and a large obstactle to truth-finding.
Consider aggregating many (potentially biased) sources of evidence versus direct observation. These are not directly comparable and in many cases we feel direct observation should prevail.
This is especially poignant in the court of law: the very strict laws arounding presenting evidence are a culturally evolved mechanism to defend against evidence manipulation. Evidence manipulation may be easier for weaker pieces of evidence—see the prohibition against hearsay in legal contexts for instance.
It is occasionally suggested that the court of law should do more probabilistic and Bayesian type of reasoning. One reason courts refuse to do so (apart from more Hansonian reasons around elites cultivating conflict suppression) is that naive Bayesian reasoning is extremely susceptible to evidence manipulation.
“One strong argument beats many weak arguments.”
Several professors told me this when I studied philosophy in grad school. It surprised me at the time—why should it be true? From a Bayesian perspective isn’t it much more evidence when there are a bunch of weak arguments pointing in the same direction, than when there is only one argument that is stronger?
Now I am older and wiser and have lots more experience, and this saying feels true to me. Not just in philosophy but in most domains, such as AGI timelines and cause prioritization.
Here are some speculations about why (and how) this saying might be true:
1. Strong evidence is common, and maybe in particular the distribution of evidence strength is heavy-tailed. In other words, maybe one strong argument typically beats many weak arguments for the same reason that one rich man typically has more money than many poor men—in any particular collection of random people/arguments, most of the money/evidence is concentrated in the richest/strongest.
2. If you have maximally high standards and accept only the most rigorous arguments, you’ll only find arguments supporting the true answer and none supporting the false answer. If you lower your standards maximally and accept any old word salad, by symmetry you’ll have as many arguments supporting the true as the false answer. Maybe standards in between continuously interpolate between these two data points: The lower your standards, the higher the ratio of arguments-for-the-true-answer to arguments-for-the-false-answer.
Maybe this isn’t by itself enough to make the saying true. But maybe if we combine it with selection effects and biases… If you are considering a class of weak arguments that contains 60% arguments for the true answer and 40% arguments for the false answer, it’ll be extremely easy for you to assemble a long list of arguments for the false answer, if for some reason you try. And you might try if you happen to start out believing the false answer, or having a hunch that it is true, and then do some biased search process...
There’s also the weighting problem if you don’t know which arguments are already embedded in your prior. A strong argument that “feels” novel is likely something you should update on. A weak argument shouldn’t update you much even if it were novel, and there are so many of them that you may already have included it.
Relatedly, it’s hard to determine correlation among many weak arguments—in many cases, they’re just different aspects of the same weak argument, not independent things to update on.
Finally, it’s part of the identification of strong vs weak arguments. If it wasn’t MUCH harder to refute or discount, we wouldn’t call it a strong argument.
Some previous LW discussion on this: https://www.lesswrong.com/posts/9W9P2snxu5Px746LD/many-weak-arguments-vs-one-relatively-strong-argument
(author favors weak arguments; plenty of discussion and some disagreements in comments; not obviously worth reading)
The strongest argument is a mathematical or logical proof. It’s easy to see why a mathematical proof of the Riemann hypothesis beats a lot of (indeed, all) weak intuitive arguments about what the answer might be.
But this is only applicable for well-defined problems. Insofar as philosophy is tackling such problems, I would also expect a single strong argument to beat many weak arguments.
Part of the goal for the ill-defined problems we face on a daily basis is not to settle the question, but to refine it into a well-defined question that has a strong, single argument. Perhaps, then, the reason why one strong argument beats many arguments is that questions for which there’s a strategy for making a strong argument are most influential and compelling to us as a society. “What’s the meaning of life?” Only weak arguments are available, so it’s rare for people to seriously try and answer this question in a conclusive manner. “Can my e-commerce startup grow into a trillion-dollar company?” If the answer is “yes,” then there’s a single, strong argument that proves it: success. People seem attracted to these sorts of questions.
In the real world the weight of many pieces of weak evidence is not always comparable to a single piece of strong evidence. The important variable here is not strong versus weak per se but the source of the evidence. Some sources of evidence are easier to manipulate in various ways. Evidence manipulation, either consciously or emergently, is common and a large obstactle to truth-finding.
Consider aggregating many (potentially biased) sources of evidence versus direct observation. These are not directly comparable and in many cases we feel direct observation should prevail.
This is especially poignant in the court of law: the very strict laws arounding presenting evidence are a culturally evolved mechanism to defend against evidence manipulation. Evidence manipulation may be easier for weaker pieces of evidence—see the prohibition against hearsay in legal contexts for instance.
It is occasionally suggested that the court of law should do more probabilistic and Bayesian type of reasoning. One reason courts refuse to do so (apart from more Hansonian reasons around elites cultivating conflict suppression) is that naive Bayesian reasoning is extremely susceptible to evidence manipulation.