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When training requires that only two categories be identified, then the classifier (and therefore the organism) need only identify some difference that distinguishes them, and nothing more. The result is a tailor-made classifier: Tailored by the specifics of the binary training paradigm, and optimized solely for that dichotomous discrimination.
As a psychologist I use binary classification tasks quite often. My approach (and the approach that I recommend) is to let SVM algorithm do the classification. If it performs above chance then I look at the minima and maxima (i.e. the typical TRUE or FALSE stimulus) of the SVM classifier. If I find that the SVM exploits some stimulus ideosynchracies (what you call "specifics of the binary training paradigm") I improve the stimulus. I iterate this process until I obtain stimuli where SVM performs at chance. Then I conclude that there can't be any "tailor-made classifier" as SVM - a state of the art classifier - doesn't find any.
However, it is difficult to assess what proportion of correct responses are genuine classifications and what proportion are merely lucky guesses. Accuracy of 70% on a binary test could mean that the subject is guessing at random more than half the time.
The usual way to asses this is to compute the (say 95 %) confidence interval of accuracy with binomial distribution. Then you get statement like "accuracy is 70 [40-90] %" or "accuracy is 70 [69-71] %" (in "x [l,u]" x is the mean, l is the lower bound and u is the upper bound) . The latter case excludes the possibility of guessing.
Regarding your simulations, I assume that the airplane or motorbike images also had background. Then the category "background" overlaps with other categories and the poor performance of background category is not surprising. I assume behavioral experiments would not use overlapping categories. If we exclude the "background" category, I do not think that the results in figure 1 left support your conclusion.
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