I can't immediately think of any additional issue. It's more that I don't see the lack of well-known disjoint sets of uncorrelated cognitive modules as a fatal problem for Thomson's theory, merely weak disconfirming evidence. This is because I assign a relatively low probability to psychologists detecting tests that sample disjoint sets of modules even if they exist.

For example, I can think of a situation where psychologists & psychometricians have missed a similar phenomenon: negatively correlated cognitive tests. I know of a couple of examples which I found only because the mathematician Warren D. Smith describes them in his paper "Mathematical definition of 'intelligence' (and consequences)". The paper's about the general goal of coming up with universal definitions of and ways to measure intelligence, but in the middle of it is a polemical/sceptical summary of research into g & IQ.

Smith went through a correlation matrix for 57 tests given to 240 people, published by Thurstone in 1938, and saw that the 3 most negative of the 1596 intercorrelations were between these pairs of tests:

• "100-word vocabulary test // Recognize pictures of hand as Right/Left" (correlation = -0.22)
• "Find lots of synonyms of a given word // Decide whether 2 pictures of a national flag are relatively mirrored or not" (correlation = -0.16)
• "Describe somebody in writing: score=# words used // figure recognition test: decide which numbers in a list of drawings of abstract figures are ones you saw in a previously shown list" (correlation = -0.12)

In Smith's words: "This seems too much to be a coincidence!" Smith then went to the 60-item correlation matrix for 710 schoolchildren published by Thurstone & Thurstone in 1941 and did the same, discovering that

the three most negative [correlations], with values -0.161, -0.152, and -0.138 respectively, are the pairwise correlations of the performance on the "scattered Xs" test (circle the Xs in a random scattering of letters) with these three tests: (a) Sentence completion ... (b) Reading comprehension II ... (c) Reading comprehension I ... Again, it is difficult to believe this also is a coincidence!

The existence of two pairs of negatively correlated cognitive skills leads me to increase my prior for the existence of uncorrelated cognitive skills.

Also, the way psychologists often analyze test batteries makes it harder to spot disjoint sets of uncorrelated modules. Suppose we have a 3-test battery, where test 1 samples uncorrelated modules A, B, C, D & E, test 2 samples F, G, H, I & J, and test 3 samples C, D, E, F & G. If we administer the battery to a few thousand people and extract a g from the results, as is standard practice, then by construction the resulting g is going to correlate with scores on tests 1 & 2, although we know they sample non-overlapping sets of modules. (IQ, being a weighted average of test/module scores, will also correlate with all of the tests.) A lot of psychologists would interpret that as evidence against tests 1 & 2 measuring distinct mental abilities, even though we see there's an alternative explanation.

Even if we did find an index of intelligence that didn't correlate with IQ/g, would we count it as such? Duckworth & Seligman discovered that in a sample of 164 schoolchildren, a composite measure of self-discipline predicted GPA significantly better than IQ, and self-discipline didn't correlate significantly with IQ. Does self-discipline now count as an independent intellectual ability? I'd lean towards saying it doesn't, but I doubt I could justify being dogmatic about that; it's surely a cognitive ability in the term's broadest sense.