Quantifiers duality, extending De Morgan's law to uncountable discourse domains

Question

I'd like to prove that $\lnot \forall$ is equivalent to $\exists \lnot$ (and conversely that $\lnot \exists$ is equivalent to $\forall \lnot$) . The finite case is somewhat easy when we write $\forall x\le n$, P(x) as $P(0) \land P(1) \land ... P(n)$ (and $\exists x\le n$, P(x) as $P(0) \lor P(1) \lor ... P(n)$). The countable infinite case, can be proved by inference. But for the infinite case.. I've got no clue how to find it.

Finite case:

This approach only works when the domain of discourse is finite. It consists in writing $\forall x\le n$, P(x) as $P(0) \land P(1) \land ... P(n)$. Then by applying the negation to it, and by De Morgan's law, we have $\overline {P(0)} \lor \overline {P(1)} \lor ... \overline {P(n)} := \exists x\le n, \lnot P(x)$

Infinite countable:

By induction and using the truth tables or the logic AND and the logic OR we get the result.

PS: This is not the first question on this, I came across this one but could not understand Gödel's argument (especially when he says «elementarily»).

EDIT :

To make things more clear I call $(1) $ and $(2)$ the equivalences I want to prove

$$ \lnot \forall ⇿\exists \lnot \ \tag{1}$$

$$\lnot \exists ⇿ \forall \lnot \tag{2} $$

Answer 1

The proof for this is just from pure formal semantics.

Formal semantics states that $\forall x, \phi(x)$ is true for some interpretation $I$ (which we write as $I \vDash \forall x, \phi(x)$) if and only if all objects from the domain of that interpretation 'satisfy' $\phi(x)$, i.e. that the formula $\phi(x)$ is true 'for' all objects $d$ from the domain $D_I$ of interpretation $I$

We can make hard the notion of an object $d$ 'satisfying' a formula $\phi(x)$ (or the formula $\phi(x)$ being 'true for' some object $d$) as follows:

An object $d$ from some domain $D_I$ of interpretation $I$ satisfies a formula $\phi(x)$ if and only if the statement $\phi(c)$ is true under interpretation $I[d/c]$ where $\phi(c)$ is the result of substituting a 'new' constant $c$ (that is: a constant symbol $c$ from our language that does not already occur in the formula $\phi(x)$) for all free occurrences $x$ in formula $\phi(x)$, and where $I[d/c]$ is the same as interpretation $I$ except that it also interprets the constant symbol $c$ as object $d$. Thus, if we write $I \vDash \phi[d]$ for an object $d$ satisfying a formula $\phi(x)$ under some interpretation $I$, then we have:

$I \vDash \phi[d]$ iff and only if $I[d/c] \vDash \phi(c)$

and as such we also have that $I \vDash \forall x, \phi(x)$ if and only if for all $d \in D_I$ we have that $I \vDash \phi[d]$

NOTE: we cannot say 'for all $d \in D_I$ we have that $I \vDash \phi(d)$' because $d$ is an object, and not a symbol of our language, and hence the expression '$\phi(d)$' makes no mathematical sense. But this is basically what you do when you start talking about $P(1)$, $P(2)$, etc. ... you are confusing $1,2,..$ as objects of the intended domain of natural numbers with the symbols as defined in the language of formal logic (to be exact: you are confusing the mathematical numerals with logical symbols). So this is why we have to do the whole thing with introducing new constants and extending interpretations.

Using this notation we can now also define the semantics of an existential: $I \vDash \exists x, \phi(x)$ if and only if there is at least $d \in D_I$ such that $I \vDash \phi[d]$

OK, with all that in place, we can prove what you want to prove:

Consider any interpretation $I$ and any formula $\phi(x)$. We have:

$I \vDash \neg \forall x, \phi(x)$

if and only if (by semantics of $\neg$)

it is not the case that $I \vdash \forall x, \phi(x)$

if and only if (by semantics $\forall$)

it is not the case that for all objects $d \in D_I$: $I \vDash \phi[d]$

if and only if (by pure logic!!!)

there is some object $d \in D_I$ for which it is not the case that $I \vDash \phi[d]$

if and only if (by semantics $\neg$)

there is some object $d \in D_I$ for which $I \vDash \neg \phi[d]$

if and only if (by semantics $\exists$)

$I \vDash \exists x, \neg \phi(x)$

Obviously, the second equivalence between $\neg \exists$ and $\forall \neg$ can be proven similarly ... a good exercise for you to do.

Now, even if the technical details go a little over your head, I think that you will still understand that what I just did should works for any interpretation defined over any domain, whether finite or not, and whether countable or not, because what I did made no assumptions about the nature of the domain whatsoever.

Also note that if you were to use an uncountable domain, you couldn't even use a unique constant symbol for each individual object from the domain, since there are uncountable objects in the domain, but classical logic assumes only countably many symbols. So that's another reason to clearly separate between the objects of the domain on the one hand, and the language of logic that you can use to talk about that domain on the other hand.

And finally note the purely logical step we do in the middle of the proof. This means that at its core, the equivalence between $\neg \forall$ and $\exists \neg$ boils down to the purely logical and conceptual equivalence between 'not all' and 'some not'. So, if you were to question that, we'd be stuck: there is no further justification for that other than our purely logical concepts and intuitions. But, once you do accept those intuitions, then we can also prove the logical equivalence between $\neg \forall$ and $\exists \neg$.

In sum, we define $\forall$ to mean the same thing as 'all' means, we define $\exists$ to mean the same thing as 'some' means, and the equivalence between $\neg \forall$ and $\exists \neg$ simply boils down to our understanding of 'all', 'some, and 'not' and how those concepts interplay. It's really that simple.

If that sounds like a let-down (isn't the point of formal logic and mathematics in general that we justify every little thing?!) please note that obviously we can't forever ask 'but why is that?' ... at some point we just have to make some assumptions we feel that we can live with (and often that means: the assumptions work when we apply them to the world around us!).