Is there a well-known formalization of the fallacy of composition? More generally, where in mathematics is it true that if a property holds for all of some elements of a set it holds for the whole set, and is there a particular requirement that will make this true?
To try to make this more clear, when is it that for, say, a set $X$ where $|X| = n$, $Px_1$ $\implies$ $\forall x$ $Px$.
On
In a set theoretic setting, it sounds like what you're asking is: For what properties (i.e. formulas with one free variable) do we have, that, for all $X$, if $\forall x\in X(\phi(x)) \to \phi(X)$?
Answer: Any $\phi$ with this property applies to all sets.
Proof: By induction on rank:
Base case: We have vacuously that $\forall x\in\emptyset, \phi(x)$, thus $\phi(\emptyset)$.
Successor case: Suppose that $X\in V_{\alpha+1}$. Then $X\subseteq V_\alpha$. I.e. $\forall x\in X(x\in V_\alpha)$. But, by the induction hypothesis, $\forall x\in X, \phi(x)$. Thus, $\phi(X)$.
Limit case is trivial. QED
Edit: I've realised that this may be a more interesting question when we allow urelements, that is, objects which are not sets. Properties applying to everything will still have the property (vacuously), but there are others. For example, being a pure set is such a property. Of, for any set $U$ of urelements, having no urelements other than those in $U$ in the transitive closure is such a property.
Each of the five chapters of my book is less than 100 pages long, but the whole book is more than 100 pages long.
The rigorous treatment of the part-whole relation is Mereology.