TITLE: Are tautologies and contradictions analogous to universal sets and empty sets, respectively?
QUESTION [5 upvotes]: Already read: $\wedge,\cap$ and $\vee,\cup$ between Logic and Set Theory always interchangeable? 
I am learning logic for the first time, about six months after finishing my undergraduate degree. I notice that there seem to be some similarities between set theory and logic. 
For example, if $A$ is a set, $a$ is a statement form, and $\mathbf{t}$ and $\mathbf{c}$ are a tautology and contradiction respectively, letting $U$ be a universal set [yes, I know this leads to a paradox] and $\varnothing$ be the empty set, I find that $U$ and $\mathbf{t}$ have similar properties, as do $\mathbf{c}$ and $\varnothing$. 
Is my hunch wrong? For example, $A \cap \varnothing = \varnothing$, and $a \wedge \mathbf{c} = \mathbf{c}$.

REPLY [6 votes]: Yes. In fact, for any set $U$, you can consider its boolean algebra of subsets. Then, when you're speaking about $U$ specifically, it is quite reasonable (and rather traditional!) to define "proposition" to mean "subset of $U$", and then define $\bot,\top,\vee,\wedge$ to mean $\varnothing,U,\cup,\cap$ and so forth.
When we're working within $U$, $U$ is the 'universal set'.
This can even be extended to typed first-order logic; from the right point of view, you can even view some kinds of set theory as being basically the same thing as some kinds of higher order logic.