Exercise 1150Logic, sets and proofs › Quantifiers and negationL1Added 3 Nov 2026

Quantifier absorption

#quantifier absorption#vacuous quantifier

Statement

When is $\forall x,\ P$ equivalent to $P$ , where $P$ does not involve $x$ ?

Hint

If $P$ is independent of $x$ , the quantifier is vacuous.

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Solution

Claim. If $x$ does not occur free in $P$ , then $\forall x,\ P\equiv P$ and $\exists x,\ P\equiv P$ .

Justification. $\forall x,\ P$ asserts " $P$ holds for every $x$ "; if $P$ doesn't depend on $x$ , this is just " $P$ holds" (assuming the domain is nonempty). Similarly for $\exists$ .

Conclusion. Vacuous quantifiers can be dropped.

Quantifier absorption

Statement

Solution

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