# How to use implication with universal quantifier?

So im trying to learn some basic logic and I got stuck on how we use implications with the universal quantifier.

My problem is the following:

$$\forall x {\in}\varnothing\; P(x)$$ is the same as saying

$$\forall x\; \big(x\in\varnothing \to P(x)\big)$$

However if I've been given a statement of the form $$\forall x\in A \; \big(x\in\varnothing \to P(x)\big)$$ should I read this as $$\forall x\; \big(x\in A \to x \in \varnothing \to P(x)\big)$$

or instead $$\forall x\; \big(x\in A \wedge x \in \varnothing \to P(x)\big)$$.

• $(P \land Q) \to R$ is equiv to $P \to (Q \to R)$ Mar 22, 2021 at 19:35
A statement of the form $$\forall x\in A[Q(x)]$$ always means $$\forall x[x\in A\rightarrow Q(x)],$$ by definition (the first statement is just an abbreviation for the second). So, $$\forall x\in A[x\in\emptyset \rightarrow P(x)]$$ means $$\forall x[x\in A\rightarrow (x\in\emptyset\rightarrow P(x))].$$ This is equivalent to $$\forall x[(x\in A\wedge x\in\emptyset)\rightarrow P(x)]$$ though since $$x\in A\rightarrow (x\in\emptyset\rightarrow P(x))$$ and $$(x\in A\wedge x\in\emptyset)\rightarrow P(x)$$ are equivalent. I'm not sure if this is what you meant with your second statement since you left out the parentheses.