TITLE: Is an empty conjunction in propositional logic true?
QUESTION [11 upvotes]: Consider an illustratory formula
$\psi \equiv \bigwedge_{i \in \emptyset} false$, does $\psi$ valuate to $true$?
Is such a formula ill-formed?
If not, is there a symbol for an empty formula?
The reason why I ask is that I have a definition where I can have a conjunction over a set of constraints, and it would be really helpful if I could interpret an empty set of constraints as true.

REPLY [7 votes]: With Stefan, I think empty conjunction is equivalent to $\top$ for the following reason within classical propositional logic:
$\Gamma\vDash\Delta\qquad$ iff $\qquad\vDash\bigwedge\Gamma\supset\bigvee\Delta$
Now for the special case $\Delta = \emptyset$,
$\Gamma\vDash\emptyset\qquad$ iff $\qquad\Gamma\vDash\bot\qquad$ iff $\qquad\vDash\bigwedge\Gamma\supset\bot$
and similarly, for the special case $\Gamma = \emptyset$,
$\emptyset\vDash\Delta\qquad$ iff $\qquad\top\vDash\Delta\qquad$ iff $\qquad\vDash\top\supset\bigvee\Delta$