Proving Too Much

[Scott Alexander]

The fallacy of Proving Too Much is when you challenge an argument because, in addition to proving its intended conclusion, it also proves obviously false conclusions. For example, if someone says “You can’t be an atheist, because it’s impossible to disprove the existence of God”, you can answer “That argument proves too much. If we accept it, we must also accept that you can’t disbelieve in Bigfoot, since it’s impossible to disprove his existence as well.”

I love this tactic so much. I only learned it had a name quite recently, but it’s been my default style of argument for years. It neatly cuts through complicated issues that might otherwise be totally irresolvable.

Because here is a fundamental principle of the Dark Arts – you don’t need an argument that can’t be disproven, only an argument that can’t be disproven in the amount of time your opponent has available.

In a presidential debate, where your opponent has three minutes, that means all you need to do is come up with an argument whose disproof is inferentially distant enough from your audience that it will take your opponent more than three minutes to explain it, or your audience more than three minutes’ worth of mental effort to understand the explanation.

The noncentral fallacy is the easiest way to do this. “Martin Luther King was a criminal!” “Although what you say is technically correct, categories don’t work in the way your statement is impl – ” “Oh, sorry, time’s up.”

But pretty much anything that assumes a classical Aristotelian view of concepts/​objects is gold here. The same is true of any deontological rules your audience might be attached to.

I tend to get stuck in the position of having argue against those Dark Artsy tactics pretty often. And the great thing about Proving Too Much is that it can demolish an entire complicated argument based on all sorts of hard-to-tease-apart axioms in a split second. For example, After Virtue gave (though it does not endorse) this example of deontological reasoning:

I cannot will that my mother should have had an abortion when she was pregnant with me, except perhaps if it had been certain that the embryo was dead or gravely damaged. But if I cannot will this in my own case, how can I consistently deny to others the right to life that I claim for myself? I would break the so-called Golden Rule unless I denied that a mother in general has a right to an abortion.

It seemed unfair for me to move on in the book without at least checking whether this argument was correct and I should re-evaluate my pro-choice position. But that would require sorting through all the weird baggage here, like what it means to will something, and whether your obligations to potential people are the same as your obligations to real people, and how to apply the Golden Rule across different levels of potentiality.

Instead I just thought to myself: “Imagine my mother had raped my father, leading to my conception. I cannot will that a policeman had prevented this rape, but I also do not want to enshrine the general principle that policemen in general have no right to prevent rape. Therefore, this argument proves too much.” It took all of five seconds.

Sometimes a quick Proving Too Much can tear apart extremely subtle philosophical arguments that have been debated for centuries. For example, Pascal’s Wager also proves Pascal’s Mugging (they may both be correct, but bringing the Mugging in at least proves ignoring their correctness to be a reasonable and impossible-to-critique life choice). And Anselm’s Ontological Argument seems much less foreboding when you realize it can double as a method for creating jelly donuts on demand.

Interestingly, I think that one of the examples of proving too much on Wikipedia can itself be demolished by a proving too much argument, but I’m not going to say which one it is because I want to see if other people independently come to the same conclusion.

[eigen]

Instead I just thought to myself: “Imagine my mother had raped my father, leading to my conception. I cannot will that a policeman had prevented this rape, but I also do not want to enshrine the general principle that policemen in general have no right to prevent rape. Therefore, this argument proves too much.” It took all of five seconds.

At first I thought of this argument as really poor one, but on subsequent thought of it, I guess it really is a perfect proving too much argument according to the first premise.

[lubinas]

The fallacy of Proving Too Much is when you challenge an argument because, in addition to proving its intended conclusion, it also proves obviously false conclusions. For example, if someone says “You can’t be an atheist, because it’s impossible to disprove the existence of God”, you can answer “That argument proves too much. If we accept it, we must also accept that you can’t disbelieve in Bigfoot, since it’s impossible to disprove his existence as well.”

Wow, I’ve been looking for a name for this thing for sooo long. Thanks so much. The phrasing here is a bit ambiguous, and can lead to confusion I think.

From the whole of the text, it seems that Scott’s view on this is that of the Wiki page, that the fallacy is committed when someone claims a conclusion that is a special case in some category of which there are obviously false instances that would be true if the reasoning was valid. Something like

A) You can (validly) argue that someone else is committing the Proving Too Much fallacy when their argument, were it valid, in addition to proving its intended conclusion, would also prove obviously false conclusions

But

The fallacy of Proving Too Much is when you challenge an argument because, in addition to proving its intended conclusion, it also proves obviously false conclusions

Can also read (my first understanding of it) as:

B) You commit the Proving Too Much fallacy when you (invalidly) challenge an argument because, in addition to proving its intended conclusion, it also proves obviously false conclusions.

I’m leaning towards A, but would appreciate more info on this. Again, I found this extremely useful.

[deepthoughtlife]

Funny to see such a highly up-voted post with only one comment. (Of course, he does have his own rather famous blog, and I assume people commented there.) The following are just some musings I had while reading the original post.

I’ve always had a bit of a problem with claiming things are wrong because they ‘prove too much’. The strange thing with ‘proving too much’ argument, is that it, in itself, also proves too much. Much like it is easy to reduce the use of ‘reductio-ad-absurdum’ to absurd places, any argument form can be used to prove too much, especially in the colloquial sense.

Let’s use logic as an example. In binary logic, they define A->B such that it is only a ‘false’ statement if ‘A’ AND ‘not B’, and that it is ‘true’ otherwise. This means that every if then statement with a false ‘A’ is ‘true’. Thus the statement ‘If I am the king of England, the moon is made of blue cheese’ is ‘true’ (assuming I am not the king of England), and so is the statement ‘If I am the king of England, the moon is NOT made of blue cheese’. Thus, ‘B’ AND ‘not B’. Through the ‘principle of explosion’ we can thus prove anything. Thus the simple construct of ‘If A then B’ proves too much. In practice, we ignore/​ridicule anyone who seriously makes such an argument because we are not first order logic reasoners ourselves, but this same weakness underlies all versions of ‘proves too much’.

The obvious reply is ‘Who cares? If it’s only proven in circumstances inconsistent with the actual state of the world, then what does it matter?’ It doesn’t usually, but we don’t always know in advance that we have contradicting premises. Due to the ‘Gödel’s second incompleteness theorem’ we know that even the natural numbers cannot be fully expressed in arithmetic in a system that can prove it isn’t inconsistent. Natural numbers are even a subset of natural language so… we always embed logically inconsistent premises within any argument form, and all argument forms can prove anything, so all forms of argument prove too much.

Then, why do people claim something proves too much as if that means anything? Oh, because it’s useful...k, thx. Bye. (*Well I look stupid.*) ;)

[Proph3t]

Thus the statement ‘If I am the king of England, the moon is made of blue cheese’ is ‘true’ (assuming I am not the king of England), and so is the statement ‘If I am the king of England, the moon is NOT made of blue cheese’. Thus, ‘B’ AND ‘not B’.

This doesn’t prove B and not B, it proves (~A OR B) AND (~A OR ~B), which is true since A is false (you are not the King of England).