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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?

wizzwizz4
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dictatemetokcus
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    If you are down-voting, at least comment why you're doing so. – dictatemetokcus Jul 10 '21 at 11:50
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    (+1) This question gets at logic, quantitative modeling, and ambiguity (or even misuse) of language in ways that seem both important to clarify and within the scope of MSE. – Andrew D. Hwang Jul 10 '21 at 11:56
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    Define "cheese". Is it just the solid matter, or even the part which are holes? Once you do that, the error in reasoning should become apparent. A "hole" is just the amount of solid matter that is missing. – Prasun Biswas Jul 10 '21 at 12:00
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    Unsurprisingly, the issue is vagueness. Think about "More holes = Less cheese." This is a silly way of saying (roughly) that given two pieces of cheese with the same density whose convex hulls have the same volume, if one piece has more holes than the other then that piece has less mass. Even that's not exactly right, but it's close enough that it shows the issue: the "more cheese = more holes" is absolutely not about two pieces of cheese with the same size in some loose sense. Always remember that with dairy products it's crucial to clarify. – Noah Schweber Jul 10 '21 at 12:05
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    That said, I don't really think this is appropriate for MSE. – Noah Schweber Jul 10 '21 at 12:07
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    Evidently, the law of identity is important :) – AlvinL Jul 10 '21 at 12:10
  • The second "equality" (they are rather implications, to me) is true provided we are considering a fixed total volume. There are hidden hypothesis so the whole reasoning seems logical. – nicomezi Jul 10 '21 at 12:13
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    I think Abbott and Costello presented it better. https://youtu.be/Lh2E0z42RGA – Gerry Myerson Jul 10 '21 at 13:09
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    @GerryMyerson Good ol' memories! I feel refreshed. – dictatemetokcus Jul 10 '21 at 14:41
  • Did you just prove that less is more?! ☺ – Kjetil S. Jul 10 '21 at 22:43
  • What is wrong is the use of equality. – Mauro ALLEGRANZA Jul 12 '21 at 12:32

1 Answers1

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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.

silver
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    Re: the first sentence, that is indeed always the whey it goes - it curd not be otherwise, and this is as goudan explanation as any. ... Something something wensleydale. – Noah Schweber Jul 10 '21 at 12:19