5.1 A. g: Betwixt Horn and Carroll
The CHC model presented in Figure 1e reveals a quandary (i.e., g?) for users of CHC theory.  Namely, “to g (Carroll) or not to g (Horn)?” To properly evaluate the relative merits of the g vs. no-g positions would require extensive reading of the voluminous g literature.  No less than three books (Brand, 1996; Jensen, 1998; Nyborg, 2003) have been devoted exclusively to the topic of g during the past decade.  The existence and nature of g has been debated by the giants in the field of intelligence (since the days of Spearman) with no universal resolution. The essence of the Cattell-Horn versus Carrol g conundrum is best summarized by Hunt (1999) who stated:
Carroll notes that abilities in the second-order stratum (e.g., Gc and Gf) are positively correlated. This led Carroll to conclude that there is a third, highest-level stratum with a single ability in it: general intelligence. Here Carroll differs with the interpretations of Cattell and Horn. Cattell and Horn acknowledge the correlation, but regard it as a statistical regularity produced because it is hard to define a human action that depends on just one of the second- order abilities. Carroll sees the same correlation as due to the causal influence of general intelligence. It is not clear to me how this controversy could be resolved (p.2).

Even if no such “thing” as g exists, applied psychologists need to be cognizant of the reality of the positive manifold among the individual tests in intelligence batteries which is practically operationalized in the form of the global composite IQ score (Daniel, 2000).  Also, the positive manifold among cognitive measures often must be included in research designs to test and evaluate certain hypotheses.   Researchers using the CHC model must make a decision whether g should be included in the application of the model in research.  Brief summaries of the respective Horn and Carroll positions are presented below
[Note.  See Daniel (2000) for a discussion of the various issues involved in calculating practical composite IQ scores from intelligence batteries comprised of different measures.]
Horn on g:
Horn (Horn & Masunaga, 2000) typically presents two lines of evidence against the g “as a unitary process” position.  Structurally, Horn and Masunaga (2000) argue that “batteries of tests well selected to provide reliable measures of the various processes thought to be indicative of general intelligence do not fit the one common factor (i.e., Spearman g) model.  This has been demonstrated time and time again…” (p. 139).  The statement also challenges Jensen’s (1984, 1993) g argument in the form of the “indifference of the indicator” (see Horn, 1998).  Horn (Horn & Noll, 1997; Horn & Masunaga, 2000) further argues that Carroll’s (1993)  research reveals no fewer than eight different general factors, with the general factor from one battery or dataset not necessarily being the same as the general factor in other batteries or datasets.  More specifically, Horn and Noll (1997) argue that “the probem for theory of general intelligences is that the factors are not the same from one study to another…the different general factors do not meet the requirements for the weakest form of invariance (Horn & McArdle, 1992) or satisfy the conditions of the Spearman model.  The general factors represent different mixture measures, not one general intelligence” (p. 68).  That is, the general factors fail to meet the “same factor requirement” (Horn, 1998).
Second, and what is probably the most convincing portion of Horn’s argument, is the fact that research reveals that “the relationships that putative indicators of general intelligence have with variables of development, neurological functioning, education, achievement, and genetic structure are varied” (Horn & Masunaga, 2000, p.139).  That is, the broad CHC abilities demonstrate differential relations with:  (1) different outcome criteria (e.g., in the area of academic achievement see McGrew, 1993; McGrew & Hessler, 1995, McGrew & Knopik, 1993; Evans, Floyd. McGrew & Leforgee, 2002; Floyd, Evans & McGrew, 2003), (2) developmental growth curves, (3) neurological functions, and (4) degree of heritability.  “The many relationships defining the construct validities of the different broad factors do not indicate a single unitary principle” (Horn & Masunaga, 2000, p. 139).  See Horn and Noll (1997) for additional information.
Carroll on g:
As presented earlier in this chapter, Carroll (2003), in his final publication, tested the “g vs. no-g vs. g- is-Gf” models in the WJ-R norm data.  Carroll (2003) concluded that “researchers who are concerned with this structure in one way or another….can be assured that a general factor g exists, along with a series of second-order factors that measure broad special abilities” (p. 19). Carroll (2003) further stated that “doubt is cast on the view that emphasizes the importance of a Gf factor….these data tend to discredit the limited structural analysis view and the second- stratum multiplicity view” (p. 17). 
The primary basis for Carroll’s belief in g stems not necessarily from the positive correlations among dissimilar tasks, but rather “from the three-stratum model that, for a well-designed dataset, yields factors at different strata, including a general factor” (Carroll, 1996;  p. 12-13).   Carroll (1996) believed that for each factor in his three-stratum theory, there was a specific “state or substate” (e.g., “structured patterns of potentialities latent in neurons”; Carroll, 1998, p. 10) that exists within an individual that accounts for the performance on tasks that require a specific latent ability—“we can infer that something is there” (Carroll, 1998, p. 10).  By extension, the emergence of a g-factor in his EFA-SL analyses must reflect some form of specific state or subtrate within an individual. 
Carroll (2003) further argued that the different g factors he reported (Carroll, 1993) do represent the same construct given the underlying assumptions and procedures of Schmid-Leiman factor analysis.   In response to Horn's arguments, Carroll (2003) stated that Horn “conveniently forgets a fundamental principle on which factor analysis is based (a principle of which he is undoubtedly aware)--that the nature of a single factor discovered to account for a table of intercorrelations does not necessarily relate to special characteristics of the variables involved in the correlation matrix; it  relates only to characteristics or underlying measurements (latent variables) that are common to those variables. I cannot regard Horn's comment as a sound basis for denying the existence of a factor g, yet he succeeded in persuading himself and many others to do exactly this for an extended period of years” (p. 19). 
     Finally, in a personnel communication received just prior to his passed away, Carroll provided the following comments regarding the "proof” of g (Carroll, personal communication, 6-30-03) :
It is important to recognize that in my paper published in the Nyborg book occurs the first modern, real, scientific proof of g - in contrast to the many unacceptable "proofs" claimed by Spearman, Burt, Pearson, and others.  It used the features of a complete proof advanced by LISREL technologies. Jöreskog has discussed these features in his many writings…of particular interest are the proofs of the status of g, Gc, and Gf, as provided in the Nyborg chapter…in the sense g, Gc and Gf could be independently established, plus several other factors, (e.g. Gv, Ga).  It was truly marvelous that enough data from these factors had accumulated to make their independence specifiable.

The "general factor" appears to pertain only to very general items of general knowledge- - e.g., items of knowledge that are common to most people, present only as specified by parameters of "item difficulty."  g thus appears not to pertain to the many items of knowledge incorporated in Gf or Gc.  These items of knowledge are in some way special- classified under Gf or Gc (or some combination of these).  It appears that a human being becomes a "member of society" only by acquring apsects of special knowledge (either fluid or crystallized, or some combination of them).