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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 ?

hazelnut_116
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1 Answers1

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$\iota x \in A. \phi(x)$ 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.

Trebor
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  • If I'm writing $ \iota x \in A. \phi(x) $ , but such object $ x $ does not exist, then what can I say about $ x $? am I saying that such an object $x $ does not exist? can I also say that the object $ \iota x \in A. \phi(x) $ is ill-defined ( or does this work only in the context of talking about function domains )? – hazelnut_116 Jul 05 '21 at 08:56
  • @hazelnut_116 You can either define it to be the empty set (or any fallback value you want) or call it undefined. – Trebor Jul 05 '21 at 08:58
  • @hazelnut_116 Just think about our attitude towards 0/0. – Trebor Jul 05 '21 at 08:58
  • And If I were asked to prove that the object $ \iota x \in A. \phi(x) $ exists. Then I'd have to prove the proposition $ \exists! x \in A . \phi(x) $ ? – hazelnut_116 Jul 05 '21 at 09:02
  • @hazelnut_116 Yes, but no one will phrase the problem that way. – Trebor Jul 05 '21 at 09:25
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    You prove that $\exists x Px$ and this is fine, but if you want to rintroduce a "name" for an object, like e.g. the empty set, starting with the axiom: $\exists x \forall y \lnot (y \in x)$ you have first to prove that there is only one such set and only after that you can add to the theory a name for the (unique) set with no members. – Mauro ALLEGRANZA Jul 05 '21 at 10:23