I found this statement on the internet and I would like to know why this is a fallacy.
Cheese has holes.
More cheese = more holes
More holes = less cheese
More cheese = less cheese
Why is this false? The second and the third statements contradict each other since more cheese can not equal less cheese. But that's just intuitive to me and not mathematically rigorous. The second statement talks about the total quantity of cheese while the third statement considers cheese per fixed quantity. I observed these but really can not translate it to a mathematical language. How do I do that?
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$\begingroup$Technically, the answer to every question of the form 'why is this ostensibly paradoxical set of statements in natural language not a paradox' is always 'it's not formal, and if you made it formal, it would go wrong at some step'. I'm saying this to emphasize that there isn't any one 'correct' answer as to why this doesn't work; rather, it will depend on how you attempt to formalize it, and given any intuitive explanation, one can always debate whether or not it points to the 'real' flaw in the argument.
That said, I think we first have to disentangle 'cheese' as it has two meanings:
- cheese-matter: the amount of edible-cheese-stuff
- cheese-volume: the size of the block-with-both-cheese-matter-and-holes-in-it
(I.e., if you buy cheese from the store, then cheese-volume is the size of the product, whereas cheese-matter is the volume minus the amount of holes in it.)
This already shows that 'more holes $\implies$ less cheese' is problematic. It's only true if you hold the amount of cheese-volume constant. The most natural formal model for this stuff would probably not have this property, so 'more holes $\implies$ less cheese' would probably come out false.
Furthermore, the first implication 'more cheese $\implies$ more holes' ignores half of the effect. If you increase the amount of cheese-volume, you do get more holes, but you also get more cheese-matter.
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