A simple kind of duality in logic is between implication and set-theoretic inclusion, which explains why the horseshoe $\supset$ is found in both contexts. The most natural way to think about it, is the reverse of the actual usage:

A $\supset$ (implies) B if A $\subseteq$ (is a subset of) B

So for instance,

x is a person implies x is a mammal, since the set of people is a subset of the set of mammals.