what happens in a universal implication when the premise is false

Question

I have just started learning Mathematical logic and couldn't figure out the answer to the above question .

my question is what happens to the truth value if the premise in a universal implication is false eg: ∀x ( (Purple(x) ∧ Mushroom(x)) ⇒ Poisonous(x) )

if in the universe x is not purple or not a mushroom .what happens to the implication ?

Answer 1

Since the variable $x$ is bound by the quantifier, the truth value of the sentense does not depend on any choice of value for $x$. That's what the quantifier says: $\forall x(\cdots)$ is true if "$\cdots$" is true no matter what we bind to $x$.

For those particular choices of $x$ that are not purple mushrooms, the formula to the left of $\Rightarrow$ is false, and therefore the implication is automatically true -- see its truth table.

Effectively this means that the choices of $x$ that are not purple mushrooms do not contribute to the truth value of $\forall x(\cdots)$ -- formally they do contribute, of course, but in a way that makes no difference, because one more "true" in he pile doesn't change the outcome of "are they all true?".

Answer 2

My question is what happens to the truth value if the premise in a universal implication is false eg: $\forall x : ((Purple(x) \land Mushroom(x)) \implies Poisonous(x))$ (assuming outer brackets)

If in the universe, x is not purple or not a mushroom, what happens to the implication?

Then the implication would tell you nothing. $x$ may or may not be poisonous. If $x$ was the only element of your universe, the implication would be true, however. Otherwise, it may or may not be true.