| Description: An "all some
one" statement is equivalent to its universal part conjoined
with the claim that exactly one 𝑥 satisfies both 𝜑 and 𝜓.
In other words, given ∀𝑥(𝜑 → 𝜓), requiring exactly one 𝑥
to satisfy 𝜑, which is what df-alseu 50656 requires, and requiring
exactly one 𝑥 to satisfy (𝜑 ∧ 𝜓) come to the same thing.
Read 𝜑 as "is a king" and 𝜓 as
"is hungry": if every king is
hungry, then "there is exactly one king" and "there is
exactly one hungry
king" say the same thing, so either of them, together with
"every king is
hungry", gives "the king is hungry".
The universal conjunct is what makes that work, and it cannot be dropped.
∃!𝑥(𝜑 ∧ 𝜓) on its own is strictly weaker than
∀∃!𝑥(𝜑 → 𝜓), since it is satisfied when many
things are 𝜑
and just one of those is 𝜓, as in a region with five kings
exactly
one of whom is hungry; see alseueu 50672 for the one direction that does hold
without it. Uniqueness attaches to the antecedent, not to the
conjunction. Russell's analysis of a definite description is built the
same way: its uniqueness clause constrains the description predicate
alone, while the predication is a separate conjunct. See his worked
example of "the father of Charles II was executed", [Russell1905] p. 482.
(Contributed by David A. Wheeler, 21-Jul-2026.) |
