Why does the property “exactly three different vowels” remain unchanged when negating “at least two siblings” in a quantified statement?

Question

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Context and Motivation
I am an undergraduate student reviewing basic logic and quantifier manipulation in the context of a Calculus 1 course. Our Unit 1 Practice Problems include exercises on how to negate statements involving “for all,” “there exists,” and various numerical thresholds. One practice problem states (in an informal way):

“I have a friend, all of whose ex-boyfriends had at least two siblings with exactly three different vowels in their name.”

My first (incorrect) attempt at negating this was:

“Each of my friends has an ex-boyfriend who had at most one sibling with a number of different vowels in their name different from three.”

However, I learned that the correct negation should preserve the property “exactly three vowels” and merely change the threshold “at least two” into “fewer than two.” In other words:

“All of my friends have at least one ex-boyfriend who has fewer than two siblings with exactly three different vowels in their name.”

Despite understanding how to write the correct negation, I am struggling to grasp why the property (“exactly three different vowels”) remains the same in the negation, while the numerical condition (“at least two”) is what flips to (“fewer than two”).

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What I Understand So Far

1. The original statement can be parsed with quantifiers (schematically) as:
   $$ \forall (\text{friend } F) \; \bigl[\forall (\text{ex-boyfriend } B \text{ of } F) : \#\{\text{siblings of }B \text{ with exactly 3 vowels}\} \ge 2 \bigr]. $$

2. The negation of $\forall x\, P(x)$ is $\exists x \, \lnot P(x)$.
   So it becomes:
   $$ \exists (\text{friend } F) \; \bigl[\exists (\text{ex-boyfriend } B \text{ of } F) : \lnot(\#\{\text{siblings of }B \text{ with exactly 3 vowels}\} \ge 2) \bigr]. $$

3. Logically, $\lnot(A \ge 2)$ is $A < 2$. This is how “at least two” becomes “fewer than two,” leaving the property “exactly three vowels” intact.

4. My confusion: Why do we not negate “exactly three vowels”? Why do we treat “exactly three vowels” as a fixed part of the statement, instead of flipping it to “not exactly three vowels”?

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What I’ve Tried to help myself understand

• I rewrote the statement as “Let $x$ be the number of siblings who have exactly three different vowels in their name. The original statement says $x \ge 2$.”
• When negating, I replaced “$x \ge 2$” with “$x < 2$.”
• Intuitvely, the negation needs to have the opposite truth value of the original statement. For this specific example, if I negate the property “exactly three vowels”, it would fail this requirement.

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What I want to clarify
In general, when we have a statement of the form “$\#(\text{objects that satisfy property } P) \ge k$,” why is the correct negation “$\#(\text{objects that satisfy property } P) < k$” rather than “$\#(\text{objects that do *not* satisfy property } P) \ge k$” (or similarly, altering $P$ itself)?

Is there a systematic or formalized rule of thumb to decide which parts of a complex statement (the property vs. the threshold) remain unchanged during negation, and which parts flip?

Any step-by-step explanation or standard logical approach might help me understand where my initial thinking went wrong. I want to gain a solid understanding so I can apply it in future problems involving nested quantifiers and inequalities. Thank you for your time and expertise!

Answer 1

Revised edition

The sentence is quite convoluted, mainly due to the need to manage differet sorts of objects: persons, names, vowels.

In order to do this, we can use suitable predicates: $P(x), N(x), V(x)$ as well as a relation $In(v,n)$ between vowels and names, $Fr(x,y)$, $Exb(x,y)$ and $Sib(x,y)$ for "to be friend of", "to be an ex-boyfriend of" and "to be sibling of".

Thus, for

“I have a friend, all of whose ex-boyfriends had at least two siblings with exactly three different vowels in their name.”

we have to start with a constant "I" and the fact that there is a friend of mine: $\exists x [P(x) \land Fr(x,I) \land \forall y (P(y) \land Exb(y,x) \to \ldots)]$

Now we can consider the part with the numerical quantifiers.

"At least two..." is $\exists w_1 \exists w_2 [P(w_1) \land P(w_2) \land Sib(w_1,y) \land Sib (w_2,y) \ldots \land (w_1 \ne w_1) ]$

Finally, for "Exactly three..." we need three variables for names and we have to express that they are the name of the $w_1,w_2,w_3$ siblings.

For each name $n$ we have: $\exists v_1 \exists v_2 \exists v_3 [V(v_1) \land V(v_2) \land V(v_3) \land In (v_1,n) \land In(v_2,n) \land In(v_3,n) \land (v_1 \ne v_2) \land (v_1 \ne v_3) \land (v_2 \ne v_3) \land \forall v (V(v) \land In(v,n) \to (v=v_1 \lor v=v_2 \lor v=v_3))]$

Why does the property “exactly three different vowels” remain unchanged?

This is not true, mainly because - as you have seen in previous formulas - it is not a property but a complex sentence,

In general, the rule of thumb to manage negation is simple: move the negation inside step-by-step.

We start negating the first existential quantifier to transform $\lnot \exists x$ into $\forall x \lnot$, and so on until we arrive at atomic formulas: either predicates, relations, or equality.

Thus, when we arrive at the negation of "at least two..." we will get:

$\forall w_1 \forall w_2 \lnot [P(w_1) \land P(w_2) \land Sib(w_1,y) \land Sib (w_2,y) \ldots \land (w_1 \ne w_1)]$

and moving it inside we arrive at the negation of the atoms. Negating "at least two...", we expect to produce "at most one..." and thus we will express it with:

$\forall w_1 \forall w_2 [ \ldots (w_1 = w_2)]$.

Similar wrt the negation of "Exactly three..."

$\forall v_1 \forall v_2 \forall v_3 [(V(v_1) \land \ldots ) \to \exists v (V(v) \land In(v,n) \land \lnot(v=v_1 \lor v=v_2 \lor v=v_3))]$

Also in this case, we will arrive at the negation of the atoms: $\lnot (v = v_1)$ etc.