Negating "Bob visits at least one of his siblings every year"

Question

The sentence is "Bob visits at least one of his siblings every year." My negation of this sentence is "Bob doesn't visit all of his siblings every year." Is this correct? if not, what would be a correct answer and why? I'm familiar with the sentences that are "mathy" or start with a quantifier. That's why I'm not sure about this one.

Answer 1

The sentence is "Bob visits at least one of his siblings every year."

Although the given sentence ("Bob visits at least one sibling every year") and "Bob visits some sibling every year" ostensibly have the same literal meaning, I feel that the former unquestionably means $$\forall y \exists s \;V(s,y)$$ whereas the latter suffers from hanging quantifiers (is there a particular sibling whom Bob unfailingly visits yearly, or can the visitee vary across years?). As such, the required negation is $$\text{In some year, every sibling is not visited by Bob}\\\exists y \forall s \;\lnot V(s,y).\tag1$$

My negation is "Bob doesn't visit all of his siblings every year." Is this correct?

This is ambiguous—again due to hanging quantifiers—and could mean either $$\text{It's not that every year, every sibling is visited by Bob}\\\lnot \forall y \forall s \;V(y,s)\tag2$$ or the stronger statement $$\text{Every year, it's not that every sibling is visited by Bob}\\\forall y \lnot \forall s \; V(s,y).\tag3$$

Statements 1, 2 and 3 have different meanings from one another.

Answer 2

I disagree with ultralegend; I do not view your response as correct. In math, we like to start with the quantifier, so we would say "Every year, Bob visits at least one sibling."

Hence, the negation of this would be "There exists a year where Bob visited none of his siblings."

The important idea is we actually are negating the "every year" aspect as well, which is where the "there exists" comes from.