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Notes -
I think "the general factor g of intelligence" is just "the principle component if you do principle component analysis on things that purport to measure intelligence". Like it definitely exists in the sense that PCA does indeed spit out a large first component if you do that. As long as all of the intelligence tests you're feeding into the statistical process meet some specific criteria (e.g. linearity, similar scale), it very strongly demonstrates that all of the tests are mostly measuring the same thing as each other.
To develop an intuition for what "the PCA spit out a large primary component" means in practice, let's consider an example housing dataset which includes attributes like price, number of beds / baths / parking spaces, presence of AC/heat, etc. If you do a PCA on that dataset, you get a primary component which explains 25% of the variance (and the next components explain 12%, 9%, 9%). Let's call this primary component the "general factor h of house-goodness". Sale price, number of bedrooms, and presence of air conditioning, in particular, are very strongly h-loaded, though every thing that would be nice to have in a house is positively h-loaded.
It's pretty clear that h reflects an actual thing, and that actual thing is probably approximated by "how good the house is". It's sensible talk about things like "high h houses that score low on the price test", or, as we say in plain language "nice houses that are cheap".
What the presence of h does not mean is anything like "this house scores higher on the number of bedrooms test because it has a high h factor".
You should view "the general factor g of intelligence" through that lens.
(cc @RococoBasilica, this seems relevant to your questions)
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