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2 Fixed mistake by clarifying that constructive logic is to be used.

Mulling this over, I thought: "Compared to what?". So another question is "Why are more properties closed under intersection than under union"? While that may be arguable, there's something to be said for looking at it (e.g. a union of two non-trivial subgroups of the same subgroup is not a subgroup).

My intuition is that it has to do with the trivial facts fact that, constructively, $(P \rightarrow X \wedge Y) \Leftrightarrow (P \rightarrow X) \wedge (P \rightarrow Y)$ but $(P \rightarrow X \vee Y) \nLeftrightarrow (P \rightarrow X) \vee (P \rightarrow Y)$. Here $P$ restricts the properties $X$ and $Y$ to an interesting 'sub-universe'.

I'm sure someone with more knowledge of lattice or even category theory can restate this more succinctly.

1

Mulling this over, I thought: "Compared to what?". So another question is "Why are more properties closed under intersection than under union"? While that may be arguable, there's something to be said for looking at it (e.g. a union of two non-trivial subgroups of the same subgroup is not a subgroup).

My intuition is that it has to do with the trivial facts that $(P \rightarrow X \wedge Y) \Leftrightarrow (P \rightarrow X) \wedge (P \rightarrow Y)$ but $(P \rightarrow X \vee Y) \nLeftrightarrow (P \rightarrow X) \vee (P \rightarrow Y)$. Here $P$ restricts the properties $X$ and $Y$ to an interesting 'sub-universe'.

I'm sure someone with more knowledge of lattice or even category theory can restate this more succinctly.