in the sense of cognition having the ability to "move up" in layers of abstraction by having larger networks or recursive networks.

The 'hierarchical notion of perceptual inference' would relate the sense of cognition having the ability to "move up" in layers of abstraction.

As you requested a 'formal theory' I will refer to another reference

...this hierarchical notion of perceptual inference seems able to capture something central about perceptual experience, which sets it apart from mere categorization or labelling, namely that perception is always from a first-person perspective. It is not just that we see a car but that we see it, as a car, from our own perspective. Different levels of our perspectival experience change in concert as the movement of eyes, head, or body changes our perspective on the world. Perceptual content is embedded in the cortical, perceptual hierarchy and there can be dramatic changes in this content as our first-person perspective changes. This teaches us that what we say about how things are, how we end up categorizing them, depends on how things more transiently seem to us in during perspectival changes. I now set out the notion of a perceptual hierarchy and attempt to capture these aspects of our perceptual phenomenology.

Hohwy, Jakob. The Predictive Mind (pp. 26-27). Oxford University Press. Kindle Edition.

So yes I would view Quinian bootstrapping as an emergent property of neural networks embedded within a 'cortical, perceptual hierarchy'.