The instantiation of the statement ∀x(P(x))⇒∀x(Q(x)) and its contrapositive.

Question

1. ∀x(P(x)⇒Q(x))
2. ∀x(P(x))⇒∀x(Q(x))

Let P stands for the category of breathing people and let Q stands for the category of alive people.

In a world where statements 1 and 2 are both true, there's a difference in the information we can deduce using each statement. In other words, we can't deduce the same information using the two statements. To see such a difference, we should obtain the contrapositive of 1 & 2. When we do so, statements 1 & 2 become 3 & 4 respectively as follows,

3. ∀x(~Q(x)⇒~P(x))
4. ∃x~(Q(x))⇒∃x~(P(x))

Now, we have the two statements in a form that will allow us to see the difference. I claim that the difference is that statement 3 is precise, whereas statement 4 is ambiguous. Specifically, we can use statement 3 to precisely deduce an information about an object, whereas if we were to use statement 4 to deduce information about the same object, then we will face ambiguity in the information deduced.

Statement 3 is saying, "any object that is not alive is not breathing". If we were to use that statement to infer something about an object called John, we will have,

" If John is not alive, then John is not breathing "

So, we can see that we can get precise information about John using statement 3. There is no ambiguity when we tried to infer something about John using statement 3.

Now, let us tackle statement 4. That statement is saying, "if there exists at least one thing that isn't alive, then there exists at least one thing that isn't breathing". I think that by just translating statement 4, as we just did, we can already see the ambiguity. Specifically, the object of discourse in the antecedent isn't necessarily the same object of discourse in the consequent. In other words, you have no guarantee that the object that isn't alive is the same object that isn't breathing. For example , if you were to say that John isn't alive, you have no 100% guarantee that John isn't breathing. The consequent of statement 4 is asserting that there's at least one object not breathing, and the antecedent of statement 4 is asserting that there's one object not alive. But we have no guarantee that both object are the same. So, if we were to instantiate statement 4 with respect to John, then we can have two possibilities:

(a) If John isn't breathing, then John isn't alive.
(b) If John isn't breathing, then Sara isn't alive.

In possibility (a), statement 4 is true and it happens that John is the same object of the antecedent and consequent. In possibility (b) statement 4 is true, but we couldn't infer anything about John. Since we have two possibilities, then statement 4 is ambiguous. Another way to see this is that if we were to instantiate statement 4, then the object of instantiation used for the antecedent mustn't be the same object of instantiation for the consequent. Because we have two existential quantifiers and according to the rules of instantiation, we mustn't instantiate two existential quantifiers using the same instance.

My question : Am I allowed to instantiate the antecedent and consequent of statement 2 with the same instance? Statement 2 has two universal quantifiers, so there shouldn't be any problem using the same instance. But if I were allowed to do so, then this means that statement 2 will become a precise statement and this is in contradiction with what I have argued above.

Answer 1

I would not think about "instantiation": I would think about the rules of logic relating concrete statements to quantified statements. There are two of them:

• $\forall x(R(x)) \implies R(y)$.
• $R(y) \implies \exists x(R(x))$.

Using the first rule, given a statement of the form $\forall x (R(x))$, you can pick any particular object $y$ and assert $R(y)$ about it. That's what we're doing with statement 1: we're setting $x = \text{John}$ and concluding $P(\text{John}) \implies Q(\text{John})$. That's also what we're doing with statement 3: we're setting $x = \text{John}$ and concluding ${\sim}Q(\text{John}) \implies {\sim}P(\text{John})$.

Existential statements work the opposite way: given $R(y)$ for a specific $y$, you can conclude that $\exists x(R(x))$. This is what we want to do with statement 4:

1. We know (as a general rule of logic) that ${\sim}Q(\text{John}) \implies \exists x,{\sim}Q(x)$. "If John is not alive, then there exists at least one entity that isn't alive."
2. We are given that $\exists x,{\sim}Q(x) \implies \exists x,{\sim}P(x)$.
3. We can combine these two implications to conclude: ${\sim}Q(\text{John}) \implies \exists x, {\sim} P(x)$. "If John is not alive, then there exists at least one entity that is not breathing."

We can't go any further with this. We have more instances of the general rule of logic telling us things like $P(\text{John}) \implies \exists x, {\sim} P(x)$, but they don't help us make additional deductions.

With statement 2, we can use the rule for instantiating universal statements, and here is what it will look like:

1. The general rule is: $\forall x(Q(x)) \implies Q(\text{John})$. "If everything is alive, then John is alive."
2. We are given $\forall x(P(x)) \implies \forall x(Q(x))$.
3. Combining these, we get $\forall x(P(x)) \implies Q(\text{John})$. "If everything is breathing, then John is alive."

Again, we can't go any further (and what we've ended up with is the contrapositive of what we've gotten from statement 4). We can also say things like $\forall x(P(x)) \implies P(\text{John})$, but they don't let us make additional deductions from what we have so far.