What's the difference between $\iota x \in A. \ \phi(x) $ and $ \exists! x \in A . \phi(x) $ ?
( Where $ \phi(x) $ is some property of $ x $, and $ A $ is the universe of discourse ).
I'm talking about iota-notation as it appears in Bertrand Russell's Principia Mathematica. I found these questions:
Is this notation standard? , Element of a Singleton (set with one element) notation . But I still could not form-out a difference.
I know the statement $ \exists! x \in A . \phi(x) $ has a truth value.
However, does the " $ \iota x \in A. \ \phi(x) $ " also have a truth value? do you have another analogy to what the iota quantifier might represent? and eventually, What's the difference between $ \iota $ and $ \exists! $ quantifiers ?
Best Answer
$\iota x \in A. \phi(x)$ is an object, not a proposition. So, if $A = \{1,2,3\}$, then $(\iota x \in A. x-1=1)$ is the number $2$.
On the other hand, $(\exists ! x \in A. x-1=1)$ is a true proposition.