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.*) ;)
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).
Another way to avoid the mistake is to notice that the implication is false, regardless of the premises. In practice, people’s beliefs are not deductively closed, and (in the context of a natural language argument) we treat propositional formulas as tools for computing truths rather than timeless statements.
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.*) ;)
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).
Another way to avoid the mistake is to notice that the implication is false, regardless of the premises.
In practice, people’s beliefs are not deductively closed, and (in the context of a natural language argument) we treat propositional formulas as tools for computing truths rather than timeless statements.