Representations of negated propositions The following question is quite hypothetical, and just to get an idea.

Assume any model of how positive propositions $p$ are actively (opposed to synaptically) represented in a neural network, e.g. "the cat sat on the mat". Presumably, this will be by some kind of firing of some neurons or neuron assemblies.

I wonder how the corresponding negative proposition $\neg p$ might be represented, e.g. "the cat sat not on the mat". Will there be an extra NOT-neuron firing? Will the same neurons fire in NOT-mode? Will the cat sit on a NON-mat? Or completely different. Or are such propositions not represented at all, i.e. we can only "think positively", and only report verbally, that we do or did not think that $p$ (which would be some kind of a meta-report)?

Which proposals are around for the representation of negated propositions?
A plausible candidate for that is interneurons subnetwork. 

Lets say some entity may be green and cold. It also may be yellow and hot, but not cold and hot. So, we can say that for this type of entities there are 2 independant variables, describing its state. If right now the variable 'color' is set to the value "red" - that means the negation of all other values of that variable (not green, not blue, ... infinity of them). The question is only how to store the construct "variable" in the neural tissue? 

One can gather all possible values of the variable and wire them all with dense lateral inhibition circuit. Activation of one neuron inside that ensemble will kill the activations of the others. 

And look - we've got a good instrument for detection of mistakes in our model of the world now! If the network has wired some entities' representations by lateral inhibition - it means that the network has stated a hypothesis that these entities (in some context) are just different values of the same driving factor. If in the next moment strong feedforward signal has activated 2 principal neurons inside the circle - the hypothesis was not correct. The network has detected "paradox".

This explanation is plausible because of 3 facts: 


There is a famous pinwheel pattern in the visual cortex:
 


The figure above depicts reconstructed map of different values of the variable "spatial orienation" in the visual cortex. They are arranged in non-random order.


Superimposition of inhibitory neurons onto that map shows that one interneuron can work with different "values" of spatial orienation.

There is a scientific proposition, that underlaying algorithm is the same in all parts of the cortex (i.e. differences are more likely in hyperparameters, than in essence). We dont know the algorythm (and I dont pretend that my aswer is correct), but scientists do not see dramatical differences in the microbiology of tissue in different cortex regions. So.. if some "variable - value" relation appears in the most good studied part of the cortex (primary one) than we, probably, have some ground to expect such tendensies in other parts.


source of pictures - Neurobiology: Turning a corner in vision research,1999,Nature