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Hi is the translation for "if and only if" (iif) the same as "All and only" i.e. both requires the bicondtional?

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    Yebs, "if and only if" is the bi-conditional (also: "exactly when"). – Mauro ALLEGRANZA Nov 23 '20 at 6:59
  • "All and only" sounds a little bit weird: iff is used with sentences. – Mauro ALLEGRANZA Nov 23 '20 at 6:59
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    A if and only if B will be A ↔ B. All and only Ps are Qs will be ∀x (P(x) ↔ Q(x)). – Conifold Nov 23 '20 at 8:26
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Standard translations for each do require the biconditional.

We translate "P, if Q" as "Q -> P"
We translate "P only if Q" as "P -> Q"
So when we combine "P, if Q" and "P only if Q" we get "P <-> Q"

We translate "All As are Bs" as "ALL(x)(Ax -> Bx)"
We translate "Only As are Bs" as "ALL(x)(Bx -> Ax)"
So when we combine "All As are Bs" and "Only As are Bs" we get "ALL(x)(Ax <-> Bx)"

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