# Contrapositive of a quantified statement

I have the following statement: Let $a,b \in \mathbb{R}$. If $a \leq b_1$, for every $b_1>b$, then $a \leq b$. I have put it into logical notation in the following way: $\forall a,b,b_1 \in \mathbb{R} \, ((b_1>b \to a \leq b_1) \to a \leq b)$.

My question is: if I want to write this implication as its contrapositive, how would the quantifiers change and why? I know that the contrapositive is $a>b \rightarrow (b_1>b \land a > b_1)$, but I am not sure how this would be quantified to get a logically equivalent statement.

## 1 Answer

My question is, if I want to write this implication as its contrapositive, how would the quantifiers change and why?

They don't change.   An implication is equivalent to its contrapositive (in first order classic logic); meaning that they always have the same truth value for the same atoms.   Thus when one is universally true, so too will be the other.

$$\forall a\forall b\forall c~\big(P(a,b,c)\to Q(a,b,c)\big) ~\iff~ \forall a\forall b\forall c~\big(\neg Q(a,b,c)\to\neg P(a,b,c)\big)$$