I've tried to clarify the idea of "generative effects" a bit by adding a summary at the end:

This is the first hint of what Fong and Spivak call a "generative effect". To tell whether \\(x\\) and \\(y\\) are in the same part of \\(P \wedge Q\\), it's enough to know if they're in the same part of \\(P\\) and the same part of \\(Q\\):

$$ x \sim_{P \wedge Q} y \textrm{ if and only if } x \sim_P y \textrm{ and } x \sim_Q y. $$

But tell whether \\(x\\) and \\(y\\) are in the same part of \\(P \vee Q\\), it's _not_ enough to know if they're in the same part of \\(P\\) or the same part of \\(Q\\):

$$ \textbf{THIS IS FALSE: } x \sim_{P \vee Q} y \textrm{ if and only if } x \sim_P y \textrm{ or } x \sim_Q y $$

To decide whether \\(x \sim_{P \vee Q} y\\) you need to look at _other_ elements of \\(X\\), too. It's not a "local" calculation - it's a "global" one!

This is the first hint of what Fong and Spivak call a "generative effect". To tell whether \\(x\\) and \\(y\\) are in the same part of \\(P \wedge Q\\), it's enough to know if they're in the same part of \\(P\\) and the same part of \\(Q\\):

$$ x \sim_{P \wedge Q} y \textrm{ if and only if } x \sim_P y \textrm{ and } x \sim_Q y. $$

But tell whether \\(x\\) and \\(y\\) are in the same part of \\(P \vee Q\\), it's _not_ enough to know if they're in the same part of \\(P\\) or the same part of \\(Q\\):

$$ \textbf{THIS IS FALSE: } x \sim_{P \vee Q} y \textrm{ if and only if } x \sim_P y \textrm{ or } x \sim_Q y $$

To decide whether \\(x \sim_{P \vee Q} y\\) you need to look at _other_ elements of \\(X\\), too. It's not a "local" calculation - it's a "global" one!