Can you explain an intuition behind Coherent Spaces? Linear Logic is interpreted using Coherent spaces, and they feature prominently in Girard's papers. I know all the three main ways to formally define them, and they don't really pose any problem to use and prove stuff about, but I just can't understand what they mean.

It really feels like there is some kind of a way to understand them. First of all, there're some examples about them which use functions on booleans (like at a wiki somewhere). And it hints at something interesting and meaningful behind the formal definition. However, bool is a very simple coherent space, with no clique of size &gt; 1. Can someone elaborate?

Another thing Girard says somewhere that every point of a coherent space represents a specific "sequence of questions/answers", with two points being coherent if they "bifurcate negatively (i.e., on different questions)", and incoherent if they bifurcate on different answers[1]. It seems like an easy to grasp idea but I just can't invent an example so it means I don't really get it...

Could someone please help me with that?

[1] J-Y Girard, The phantom of transparency. URL: http://iml.univ-mrs.fr/~girard/longo1.pdf
The intuition behind coherence spaces is that the elements of a coherence space represent observations of some underlying data, and the coherence relation tells you whether two observations could have come from the same piece of data.

Concretely, suppose we have a set of animals

Animals = {cat, duck, fish}


Now, we can have a set of observations:

Observations = {warm-blooded, swims, water-breathing, furry}


Let us say that two observations are compatible if they could both be made of the same animal. Every observation is compatible with itself, and in addition:


warm-blooded $\stackrel{{\large\frown}}{{\small\smile}}$ furry
warm-blooded $\stackrel{{\large\frown}}{{\small\smile}}$ swims
warm-blooded $\stackrel{{\large\frown}}{{\small\smile}}$ furry
swims $\stackrel{{\large\frown}}{{\small\smile}}$ water-breathing


We know that being warm-blooded is compatible with swimming, because ducks are both warm-blooded and swim. But being warm-blooded and water-breathing are not compatible, since we have no animals which are both warm-blooded and water-breathing.

At this point, we have defined a coherence space (Observations, $\stackrel{{\large\frown}}{{\small\smile}}$), where Observations is the web, and $\stackrel{{\large\frown}}{{\small\smile}}$ is the coherence relation.