In the way where a man who has parallel families leads a “double life”, we could say you have as many lives as there are groups of people you know where you intentionally avoid telling anyone in one group about anyone in any of the others. For instance, if you don’t tell your drug dealer about your therapist and you don’t tell your therapist about your drug dealer, then that’s a minimum setup for you to live a double life. Or if you don’t tell your far-right activist friends about your gay lover, and you don’t tell your gay lover about your far-right activist friends, that’s another double life.
I think it doesn’t count if you tell group A about group B but don’t tell group B about group A, since then it’s clear that your “real life” is more with group A. Call your life with group B a “fake life” in this case; call it “integration” when you have only one real life (with indefinitely many fakes) and “strong integration” when you have one real life and zero fakes.
I suppose you can have overlapping double lives. For example:
- you don’t tell your drug dealer about your therapist
- you don’t tell your therapist about your drug dealer
- you don’t tell your Jewish lover about your Nazi friends
- you don’t tell your Nazi friends about your Jewish lover
In the example scenario, it’s possible that you tell your Nazi friends and Jewish lover about both your drug dealer and your therapist, and you tell both your drug dealer and your therapist about both your Nazi friends and your Jewish lover. So it’s not like you have four lives – it seems cleaner to say you have four lives if you have four groups where none of the four knows about the other three. But you have overlapping double lives in different respects.
Please think of the cleanest and most natural way to set up a mathematical model of these ideas.
The cleanest model is:
the primitive thing is a directed “disclosure graph,” not a single number of lives.
A single scalar like “how many lives?” throws away too much structure. Your examples really involve two different phenomena:
- mutual compartmentalization (“these two worlds are kept apart”), and
- one-way incorporation (“this world is part of a more real world, but not vice versa”).
A directed graph captures both at once.
1. Primitive data: a directed graph of disclosure
Let \(G\) be the set of your social compartments.
A compartment can be a literal group of people, or just a “social world”/role/context. It is better to think of vertices this way, because real groups can overlap.
For \(x,y\in G\), write
\[
x \to y
\]
to mean:
people in compartment \(x\) are intentionally told about compartment \(y\).
Assume \(x\to x\) for all \(x\): each compartment “knows itself.”
So the whole social setup is a directed graph \(D=(G,\to)\).
2. The three basic pairwise relations
For distinct \(x,y\), exactly one of these can happen:
Mutual disclosure
\[
x\leftrightarrow y
\]
meaning \(x\to y\) and \(y\to x\).
This is the pairwise form of integration.
One-way disclosure
\[
x \Rightarrow y
\]
meaning \(x\to y\) but \(y\nrightarrow x\).
Then \(y\) is a fake life relative to \(x\).
Mutual concealment
\[
x \parallel y
\]
meaning \(x\nrightarrow y\) and \(y\nrightarrow x\).
This is the pairwise form of a double life seam.
So the intuitive English is already sitting in the edge-types.
3. “Real life” and “fake life” from knowledge-sets
For each compartment \(x\), define its visible world
\[
K(x)={y\in G: x\to y}.
\]
This is the set of compartments that \(x\) is allowed to know about.
Now compare compartments by how much of your overall life they see:
\[
x \succeq y \quad \text{iff} \quad K(x)\supseteq K(y).
\]
Interpretation: \(x\) is at least as “real” or “integrated” a life as \(y\), because \(x\) sees everything \(y\) sees, and maybe more.
Then:
- \(x\) is a real life if it is maximal under (\succeq)
(no other compartment sees strictly more than it does). - \(y\) is a fake life if it is not maximal
(some other compartment strictly contains it).
This matches your intuition that if \(A\) knows about \(B\) but not conversely, then \(A\) is the more real life and \(B\) is fake.
4. Integration and strong integration
With that setup:
Integration
You have integration if there is exactly one maximal visible world.
In other words, there is one “top-level” life that contains all the others informationally, and everything else is subordinate/fake relative to it.
Strong integration
You have strong integration if every compartment has the same visible world:
\[
K(x)=G \quad \text{for all } x\in G.
\]
Equivalently: for every pair \(x,y\), both \(x\to y\) and \(y\to x\).
So strong integration is just the complete bidirected graph.
5. Double lives and “how many lives?”
Now define the incompatibility graph \(I\) on the same vertex set \(G\), where
\[
{x,y}\in E(I) \quad \text{iff} \quad x\parallel y.
\]
So \(I\) forgets one-way fake-life relations and remembers only mutual concealment.
This gives a very natural meaning of “\(n\) lives” in your strict sense:
you have at least \(n\) lives iff \(I\) contains a clique of size \(n\).
That is, there are \(n\) compartments such that no one in any of them is told about any of the others.
So the strict “number of lives” is
\[
\omega(I),
\]
the clique number of the incompatibility graph.
That is exactly the quantity you seemed to want when you said it feels clean to say “four lives” only when there are four groups none of which knows about the other three.
6. Your overlapping example
Let the compartments be
- \(D\): drug dealer
- \(T\): therapist
- \(J\): Jewish lover
- \(N\): Nazi friends
and suppose:
- \(D\parallel T\)
- \(J\parallel N\)
but all cross-pair disclosures happen both ways:
- \(D\leftrightarrow J\), \(D\leftrightarrow N\)
- \(T\leftrightarrow J\), \(T\leftrightarrow N\)
Then the incompatibility graph \(I\) has exactly two edges:
- \(D\!-\!T\)
- \(J\!-\!N\)
and no others.
So:
- you definitely have double lives, because \(I\) has edges;
- these are overlapping double lives, because there is more than one edge-pattern of mutual concealment;
- but you do not have four lives in the strict sense, because \(I\) has no 4-clique;
- in fact \(\omega(I)=2\), so the strict life-count is only 2.
That seems to line up with your intuition almost perfectly.
7. Why this is the right model
It separates the things that should be separated:
- Mutual concealment is modeled by missing edges both ways.
- Fake life / real life is modeled by one-way visibility and inclusion of knowledge-sets.
- Integration is not “no secrecy at all,” but “one top-level world.”
- Strong integration is full mutual openness.
- Overlapping double lives are just nontrivial structure in the incompatibility graph, without forcing a misleading scalar count.
So the best answer is:
Model the situation by a directed disclosure graph.
From it derive
- a preorder by visible-world inclusion \(K(x)\supseteq K(y)\) for real/fake/integration, and
- an incompatibility graph for double-life structure and strict life-count.
That gives you one clean object and two natural derived notions, instead of trying to force everything into one number.
If you want, I can turn this into a compact definition/proposition style writeup, as if for a paper or notebook.
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