Yes, they actually do. IQ tests evaluate not just mental processing speed but also a number of orthogonal cognitive faculties. You cited an opinion article from the New York Times regarding the divergence between rationality and intelligence. I'm not talking about rationality, and in fact I would not immediately believe rationality and intelligence can be adequately tested using the same examination. One relates to decision making processes and the other relates to learning and aptitude processes. There is some overlap, but they're categorically different things.

I could direct you to a survey of studies that would more accurately describe the breadth of IQ assessments [1], but I think using my own IQ examination as an example might be more instructive. I've taken a formally administered IQ assessment under a qualified clinician. This assessment included the usual tests, such as the Wechsler Adult Intelligence Scale and Wechsler Memory Scale (among others).

The Wechsler Adult Intelligence Scale includes composite domains for "Full Scale IQ", "Verbal Comprehension", "Perceptual Reasoning", "Working Memory" and "Processing Speed." Note that mental processing speed is merely one of the measured dimensions of intelligence here. Each of these is given its own percentile score.

Within the Verbal Comprehension Scale domain there are subtests for "Similarities", "Vocabulary" and "Information." Under Percentual Reasoning there are subtests for "Block Design", "Matrix Reasoning" and "Visual Puzzles." Working Memory includes tests for "Digit Span" and "Arithmetic", and finally Processing Speed includes "Symbol Search" and "Coding."

During the verbal comprehension battery, you'd be asked questions such as, "How are an apartment and a house alike", or "What is the definition of this word?" You'll also be assessed on your knowledge of various culturally relevant facts.

Perceptual reasoning will test spatial orientation, integration performance and fine motor skills. This might include arranging blocks of various colors and shapes into different patterns and being asked to find the missing portion in a complex visual pattern.

The working memory examination will test for how much you can focus on simultaneously, and explicitly does not test for your processing speed. You'll be asked about sequences of numbers and tested for both long and short term recollection, as well as how accurately you can mentally perform operations on sequences while keeping them in your mind. That's the digit span test; the arithmetic tests are self-explanatory.

The only portion of the examination which explicitly tests for processing speed are the ones focusing on symbol search and coding. Therein you're asked to quickly scan a group of symbols and find a match of one or two targets, or to copy numbers lexicographically coded to symbols as quickly as possible.

I am interested in what measure of analogous "Turing Completeness" in human intelligence should thus be measured that is not already reflected in the examination I've just outlined. If you'd like to define that as rationality, then that's fine. But as I said before, I don't believe rationality should be coupled with intelligence. And moreover, I don't think "Turing Completeness" is a good analogy for rationality, since computers are (almost definitionally) neither rational nor irrational, regardless of their computational capabilities.

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1. http://www.gwern.net/Iodine