Universal Negation

We all know about negation. We can apply it to numbers and propositions. When we negate a number we get a number. Except for the number 0 the negative of a number differs from the number itself. Negating 2 results in -2, negating -3 results in 3, and so on. In the case of propositions there are no exceptions. The negative of a propostion is always a proposition that differs from the original proposition. Take for example the proposition "The cat is on the mat". Negating it leads to the different proposition "It is not the case that the cat is on the mat". Can we also negate properties? Yes, this seems to be the case as well. And the result is also a property if we accept a rather broad account of properties. Take for example the property of being red. The negation of this property is the property of not being red. Sets (collections, classes) can be negated as well. Take the set of all women. When we negate this set we get the set of all things that are not a woman. Again, we get a set when we negate a set. Or take a state of affairs. The negation of it is another state of affairs, namely the state of affairs of the original state of affairs not obtaining.

So, numbers, propositions, properties, sets, and states of affairs can be negated. These observations lead to an interesting question. Are there more kinds that can be negated? Or for that matter, can everything be negated? Is negation maximally universal? Is the domain of negation wholly unrestricted? Is there a natural universal negation function N that maps every x onto its negative N(x)?

In the case of numbers, propositions, properties and sets, N maps items in the actual world to items in the actual world if we assume that numbers, propositions, properties and sets exist as abstract objects and that abstract objects are actual. Note that some properties might not have instances in the actual world or other possible worlds though. In the case of states of affairs N maps actual states to merely possible states and merely possible states to actual states. Merely possible states do not exist as abstract objects in the actual world. So the domain and range of N are not confined to what is actual (i.e., to what is part of the actual world).

It seems to me that N needs to adhere to at least the following nine conditions:

1. N(x) = -x for all numbers x,
2. N(x) = ~x for all propositions x,
3. N(x) = "the property of not having x" for all properties x,
4. N(x) = "the state of affairs of x not obtaining" for all states of affairs x,
5. N(x) = { y | y is not in x } for all sets x,
6. N(x) = "the function that maps y onto N(x(y))" for all functions x,
7. N(N(x)) = x for all x,
8. N(x) is of the same kind as x,
9. N(x)=N(y) if and only if x=y.

Let I(x)=x be the identity function. Then (7) can be written as N^2=I. Also, from (6) it follows that N(I) is a function that maps every x onto N(x). That is to say, N(I)(x)= N(x) for all x. Hence N(I)=N. And other entailments can be derived from (1)-(9) as well. It would be interesting to further explore this.

Now, suppose that there is indeed such a N. What's the result of applying N to itself? That is to say, what results when we negate negation itself? What is N(N)? Is it a being? If so, does this show how being could come from non-being? In any case it seems to me that a further reflection on universal negation might have interesting consequences for metaphysics.

So, what is N(N)? Well, from (7) it follows that N(N(N))=N. We also saw that N(I)=N. But then it follows that N(N(N))=N(I). From (9) it then follows that N(N)=I. Therefore we may conclude that when we apply universal negation onto itself we obtain the identity function I. Since N^2=I it also follows that N(N)=N^2.

The question remains whether there is an universal negation operator N at all. Let's explore this. Suppose there is an universal negator N. We have seen that N(I)=N. Here I is the identity function that maps every x to x. Functions are relations. I is the relation { (x,x) | for all x }. Relations are sets. So I is a set. But then according to (5) it follows that N(I) is { y | there is no x such that y = (x,x) }.

Given that N(I)=N it follows that N = { y | there is no x such that y = (x,x) }. No number can be written as a dupel (x,x) for some x. Hence N contains all numbers. But this is false since N = { (x,N(x)) | x } and thus N only contains dupels. We arrive at a contradiction. It follows that for there to be universal negation, functions are ontologically not to be considered relations or relations are ontologically not to be considered sets (or both). Those who are not willing to accept such ontological anti-reductionism have to accept that universal negation is metaphysically impossible.

Functions and relations cannot be reduced to sets if universal negation is to be possible. This has an impact on formal frameworks, ontologies and programming languages that want to reduce everything to sets.