I am unsure about the transcription of a predicate $\psi$ applied to a definite description $℩x(\phi x)$, in some places it is transcribed as,
- $\exists !x(\psi (x) \land \phi (x))$
, whereas in others it is transcribed as
- $\exists x(\phi (x) \land \forall y(\phi(y) \rightarrow y=x) \land \psi(x))$.
Definition 2 seems like the correct transcription as it says 'There is exactly one object with the property $\phi$ and that object also has the property $\psi$' however definition 1 says 'There is exactly one object that is both $\psi$ and $\phi$.'
Could you please tell me which transcription is correct?
Also, the book I am reading says a definite description is a term. Since a term is a name or a variable, a definite description could be a name or a variable, but wouldn't it just be a name?
$\exists !x~(\phi(x)\wedge\psi(x))$ is equivalent to $\exists x~((\phi(x)\wedge\psi(x)\wedge(\forall y~(\phi(y)\wedge\psi(y)\to y=x))$. This says "there is only one entity which satisfies $\psi$ and $\phi$". This assertion does not deny the existence of entities that satisfy $\phi$ yet not $\psi$.
$\exists x~(\phi(x)\wedge\forall y~(\phi(y)\to y=x)\wedge\psi(x))$ says "the only entity which satisfies $\phi$ also satisfies $\psi$". This assertion indicates that any entity which satisfies $\phi$ must also satisfy $\psi$, because that would definitely be the only entity which satisfies $\phi$.
So your two statements are not equivalent, and the second does look to say what you require.