When you negate a quantifier, you 'bring the negation inside', e.g. $\neg \forall x P(x)$ is equivalent to $\exists x \: \neg P(x)$, where P(x) is some claim about $x$.
If you have two quantifiers, that still works the same way, e.g. $\neg \forall x \exists y P(x,y)$ is equivalent to $\exists x \neg \exists y P(x,y)$, which in turn is equivalent to $\exists x \forall y \neg P(x,y)$. And once you see that, you can understand that you can move a negation through a series of any number of quantifiers, as long as you change the quantifier: each $\forall$ becomes a $\exists$ and vice versa.
Also, since these are all equivalences, you can also bring negations outside, if that's what you ever wanted to, again as long as you change each quantifier that you move the negation through. For this reason, this is sometimes called the 'dagger rule': you can 'stab' a dagger (the negation) all the way through a quantifier, thereby changing the quantifier.
First, I know that you know that the inference rules aren't fractions, but still ... please replace 'Numerator' and 'Denominator' with something more appropriate ... such as 'premise' and 'conclusion' respectively.
OK, the rules, and some more intuitive explanation:
Universal Instantiation
'Typical' Form:
$\forall x P(x)$
$\therefore P(a)$ for any constant $a$
Explanation:
I all things have property $P$, then of course each individual thing has property $P$, whether this is $a$, $b$, ... This is why there are no restrictions here.
Universal Generalization
'Typical' Form:
$P(a)$ ... where $a$ has been introduced as some arbitrary object!
$\therefore \forall x P(x)$
Explanation:
Suppose we have a constant that we are using to denote a specific object, e.g. suppose we use the constant $c$ for 'Charlie', and suppose we have as a given that $Dog(c)$, since we know that Charlie is a dog. Now, clearly we should not be able to infer that everything is a dog just because Charlie is a dog. And that is why we mandate the constant $a$ in the rule to be a temporary name that we use to denote "some arbitrary object from our domain ... let's call it $a$" In fact, many systems require you to explicitly introduce this constant ... it would be the formal logical equivalent to the mathematician's "consider any object $a$".
I must say that in your description off the rule this requirement is not clear. ... so if you don't understand the rule as you yourself stated, I can understand that!
Here is a formal proof example:
$\forall x P(x)$ Premise
$\forall x Q(x)$ Premise
$\qquad a$ (here is where we introduce $a$ ... so we have to make sure that $a$ is not used earlier in the proof, i.e. it is a 'new' constant. Again, this is the equivalent of saying "let's consider any arbitrary object $a$. I use the indentation to create a temporary context for the use of this $a$ ... some systems use subproofs to do this)
$\qquad P(a)$ Universal Instantiation 1 (as we saw, this works for any constant, so also for $a$)
$\qquad Q(a)$ Universal Instantiation 2
$\qquad P(a) \land Q(a)$ Conjunction 4,5
$\forall x (P(x) \land Q(x))$ Universal Generalization 6 (or: 3 through 6) (so why can we do this? Because $a$ was used as an arbitrary constant!)
Existential Generalization
'Typical' Form:
$P(a)$
$\therefore \exists x P(x)$
Explanation:
Like Universal Instantiation, Existential Generalization should really be without any restrictions: If $a$ has property $P$, then there is something that has property $P$, whether $a$ is used to denote a specific or arbitrary object.
So here I am not sure why there is this restriction stated in your description of the rule...
Existential Instantiation
'Typical' Form:
$\exists x P(x)$
$\therefore P(a)$ ... for a new constant $a$
Explanation:
OK, so in this rule we do have to treat $a$ very carefully! Think about it: you know that something has property $P$ .. but do you know what it is? No. So, what the $a$ is representing here, is "some object that has property P ... which we know exists ... but we don't know what specific object it is ... so let's call it $a$". And again, like Universal Generalization, it is best to contrast the correct use of this rule with an incorrect one: Again, suppose we use constant $c$ to denote a specific individual: Charlie. Now, suppose we know that $\exists Dog(x)$ ... can we now infer $Dog(c)$? No! Because even though we know something is a dog, we don't know whether Charlie is a dog. So, like Universal Generalization, the $a$ represents an unknown object, but this time, we do know that $a$ has property $P$. And that also means that $a$ is not a completely arbitrary object .. meaning that we can't use it for a Universal Generalization.
Example:
$\exists x P(x)$ Premise
$\forall x (P(x) \rightarrow Q(x))$
$P(a)$ Existential Elimination (OK use of rule, since $a$ is a new constant)
$P(a) \rightarrow Q(a)$ Universal Instantiation 2
$Q(a)$ Modus Ponens 3,4
$\exists x Q(x)$ Existential Generalization 5
Note that we had to do line 3 before line 4, because if we would have first instantiated the universal with $a$, then we could not have instantiated the existential with that same $a$, since the $a$ is on longer a new constant!
Best Answer
We have always problem in formalizing natural language statements.
The first step is how to translate : whenever.
We assume that it has the same meaning of "when".
Thus, the statement is of the form :
and we symbolize it with the connective : $\rightarrow$ ("if ..., then _") :
Now we need quantifiers for analyzing the two clauses :
there is an active alert
all queued messages are transmitted.
The first one will be :
while for the second we have :
Putting all together :