On the learning difficulty of visual and auditory modal concepts: Evidence for a single processing system

Abstract

The logic operators (e.g., “and,” “or,” “if, then”) play a fundamental role in concept formation, syntactic construction, semantic expression, and deductive reasoning. In spite of this very general and basic role, there are relatively few studies in the literature that focus on their conceptual nature. In the current investigation, we examine, for the first time, the learning difficulty experienced by observers in classifying members belonging to these primitive “modal concepts” instantiated with sets of acoustic and visual stimuli. We report results from two categorization experiments that suggest the acquisition of acoustic and visual modal concepts is achieved by the same general cognitive mechanism. Additionally, we attempt to account for these results with two models of concept learning difficulty: the generalized invariance structure theory model (Vigo in Cognition 129(1):138–162, 2013, Mathematical principles of human conceptual behavior, Routledge, New York, 2014) and the generalized context model (Nosofsky in J Exp Psychol Learn Mem Cogn 10(1):104–114, 1984, J Exp Psychol 115(1):39–57, 1986).

Appendices

Appendix A

The generalized invariance structure theory model (GISTM)

Generalized invariance structure theory, or GIST (Vigo 2013, 2014), proposes that observers are invariance pattern detectors. In other words, observers detect abstract symmetries inherent in the dimensional structure of a category of objects with the ultimate aim of efficiently determining the degree of diagnosticity of each of the category’s relevant dimensions. The observer is then able to ascertain or assess which dimensions should be used in the formation of concept learning rules. As such, the ability to detect invariance patterns in categorical stimuli is a necessary precursor to concept formation in GIST. The core model of the theory is referred to as the “generalized invariance structure theory model,” or GISTM. The parameterized variant of the model we are employing (see Vigo 2014 and supplementary materials to Vigo 2013) is expressed as follows:

$$ \psi ({\text{X}}) = pe^{{ - k\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\Phi }^{\lower0.5em\hbox{$\scriptscriptstyle{2}$}}_{\alpha } ({\text{X}})}} $$

(1)

where $\psi$ is the degree of perceived learning difficulty of a continuous or dichotomous category X, p is the cardinality or size of the categorical stimulus, D is the number of dimensions used to define X, and $\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\Phi }_{\alpha }$ is the degree of perceived categorical invariance determined by the proportion of categorical invariants H_[d] (X) in X with respect to dimension d (1 ≤ d ≤ D) as follows.

$$ \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\Phi }_{\alpha } ({\text{X}}) = \left[ {\sum\limits_{d = 1}^{D} {\left[ {\alpha_{d} H_{[d]} ({\text{X}})} \right]^{2} } } \right]^{1/2} $$

(2)

Note that this parameterized version of the GISTM includes a discrimination parameter k $(k \ge 0)$ and an invariance detection sensitivity parameter $\alpha_{d}$ per dimension d (where for any d, $0 \le \alpha_{d} \le 1).$ On the other hand, the nonparametric variant of the model (i.e., the GISTM-NP) does not feature any free parameters and takes the following forms (where D₀ = 2 and $\frac{{D_{0} }}{D}$ is a category structure discrimination index determined relative to the smallest number of dimensions of a category structure: namely, two):

$$ \psi ({\text{X}}) = pe^{{ - \frac{{D_{0} }}{D}\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\Phi }^{\lower0.5em\hbox{$\scriptscriptstyle{2}$}} ({\text{X}})}} $$

(3)

$$ \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\Phi } ({\text{X}}) = \left[ {\sum\limits_{d = 1}^{D} {\left[ {H_{[d]} ({\text{X}})} \right]^{2} } } \right]^{1/2} $$

(4)

Essentially, the invariance detection sensitivity parameter reflects the effectiveness of a lower level cognitive mechanism of invariance detection referred to as “dimensional binding.” Dimensional binding requires that similarity assessment be relativized by the process of systematically and completely suppressing each relevant dimension during similarity comparisons. For a formal specification of this mechanism, please refer to Vigo (2013, 2014).

Figure 5 shows the process of detecting invariants using a simple category structure (small black triangle; small black circle; large white circle) consisting of three objects and three binary dimensions. In the original structural account of the model (Vigo 2009), a differential operator generates the degree of partial invariance by perturbing dimensions of categorical stimuli. These perturbations are dimensional transformations that determine the number of invariants per dimension. The number of invariants per dimension equals the number of common objects between the original and perturbed categories. Thus, upon the shape transformation in Fig. 5 we see that the small black circle and the small black triangle remain after the perturbation. Upon the color and size transformations, however, no objects are common to the original and perturbed sets.

Fig. 5

Summary of the process of detecting categorical invariants using a simple category structure (small black triangle; small black circle; large white circle) consisting of three objects and three dimensions

This differential operator is interpreted as a cognitive mechanism or cognitive operator H_[d](X) via the process of dimensional binding mentioned above in Vigo’s (2013) generalization of the model. The invariance detection operator generates one structural kernel (SK) per dimension where SKs are the proportion of invariant objects to the total number of objects in the category. The structural manifold of the category is found by computing the proportion of categorical invariants (with respect to each dimension) to the total number of objects and arranging these proportions as a vector.

In general, this process determines how relatively essential a given dimension is in terms of characterizing category membership. Simply, objects either remain or are eliminated after a perturbation. Dimensions with a relatively greater number of eliminated objects after perturbation are more essential for determining category membership. Alternatively, dimensions with a relatively greater number of objects that remain after perturbation are relatively non-essential for determining category membership. Therefore, the structural manifold obtained in our Fig. 5 example indicates that color and size are essential for classification, whereas shape is relatively non-essential. At this stage, the observer has the information that is necessary for forming an efficient classification rule.

To determine the degree of learning difficulty of the category in Fig. 5, we use the GISTM-NP (Eqs. 3 and 4) as follows. First, using Eq. 4, we compute the global degree of categorical invariance $\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\Phi}$ using the structural manifold (.67, 0, 0) of the category (we shall refer to the category as X). Recall that H_[d=1] (X) = 2/3 ≈ .67, H_[d=2] (X) = 0, and H_[d=3] (X) = 0. We then get the following:

$$ \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\Phi } ({\text{X}}) = \left[ {\sum\limits_{d = 1}^{D} {\left[ {H_{[d]} ({\text{X}})} \right]^{2} } } \right]^{1/2} = [[2/3]^{2} + [0]^{2} + [0]^{2} ]^{1/2} \approx .67 $$

(5)

We can now compute the degree of learning difficulty ψ of X using Eq. 3 above and get:

$$ \psi ({\text{X}}) = pe^{{ - \frac{{D_{0} }}{D}\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\Phi }^{\lower0.5em\hbox{$\scriptscriptstyle{2}$}}({\text{X}})}} = 3e^{{ - \frac{2}{3}(.67)^{2} }} \approx 2.22 $$

(6)

Appendix B

See Figs. 6 and 7.

Fig. 6

Plotted are supplemental analyses for the Bayesian paired t test results reported in the “Analyses of the 16 structure type instances” in “Results” section. The top row displays plots for the t test result that was closest to providing evidence against the null hypothesis of no statistical difference between three structure type instances of conjunction, inclusive disjunction, and conditional. The bottom row displays plots for the t test result most favorable to the null hypothesis of no statistical difference between each of these three structure instances and biconditional. The plots were created using JASP (JASP Team 2016)

Fig. 7

Plotted are supplemental analyses for the Bayesian independent samples t test results reported in the “Analyses of the six structure types” in “Results” section. The plots in the first row represent the evidence in favor of type 2₂[4] being easier to learn than type 2₂[0]. The plots in the second row represent the t test result that was closest to providing evidence for the null hypothesis of no statistical difference between type 2₂[2]-II and the other three non-trivial structure types (2₂[1], 2₂[2]-I, and 2₂[3]). The plots in the third row represent the lone difference between structure types 2₂[1], 2₂[2]-I, and 2₂[3]. The plots were created using JASP (JASP Team 2016)