Yet another excerpt from The Book of Real and Imaginary Drugs. I achieved a new high in self promotion.

The Double Negator

If you are a geometer, you draw pictures. But what do you do if you are a logician? I guess you can still draw pictures, like proof trees and so on, but these wouldn’t be pictures of the objects you work on. Intuitively, this is obvious. We say “I can draw a triangle.” not “I can draw a triangle picture. ” On the other hand one cannot draw a tautology.

So we have a natural question here: In logic, what is the verb that corresponds to draw? I think it is easy to answer this question once we look at some standard terminology in mathematical logic: sentence, term, syntax, parsing, . . . As the reader has hopefully guessed, the verb I am talking about is write.

Now let us make things more interesting. It is a well known historical fact that Maurice Princet, a mathematician and an associate of Picasso, had some role in the birth of cubism.  For the ones who ask for visual evidence, here is a page from a book by Maurice Princet,

and here is a cubist painting by Braque.

The resemblance is obvious.

Now suppose that we put a logical writing instead of Princet’s geometrical drawing. What sort of literary work can we put instead of Braque’s painting? Honestly, I do not know an example. I know mathematical writings, for instance produced bu Oulipo. There are self referential books too. But non of the texts I have read so far is linked to a branch of mathematical logic like Braque’s painting is linked to higher dimensional geometry.

What I know, though, is a drug called double negator which enhances ones creativity in this hypothetical literature. Just as LSD affects the way you see colors, double negator affects the way you reason. In order to explain the details we need some material from mathematical logic, namely the Goedel-Gentzen translation.

Goedel-Gentzen translation is a transformation on the set of first order sentences. It is defined inductively as follows. If  $\varphi$  is an atomic sentence, then its translation, denoted by  $\varphi^N$, is defined to be $\neg\neg\varphi$. So if  $\varphi$  is “I love you.” then  $\varphi^N$  is “It’s not true that I don’t love you.”  Conjunction, implication, negation and universal quatification commute with the translation, that is we have

$(\varphi \wedge \psi)^N$   is    $\varphi^N \wedge \psi^N$,

$(\varphi \rightarrow \psi)^N$   is    $\varphi^N \rightarrow \psi^N$,

$(\neg \varphi )^N$   is   $\neg (\varphi^N)$,

$(\forall x \varphi(x))^N$   is   $\forall x (\varphi (x)^N)$.

For disjunctions we have

$(\varphi \vee \psi)^N$   is   $\neg(\neg\varphi^N \vee \neg\psi^N)$.

Finally,  the existential quantifier is translated as follows:

$(\exists x \varphi(x))^N$   is   $\neg \forall x \neg (\varphi(x)^N)$.

A person who uses double negator perceives the Goedel-Gentzen translation of each sentence. As we have seen in the “I love you.” example, this induces a kind of paranoia. However, this does not cause contradictory conclusions or delusions because of the following theorem: If a set  $\Sigma$  of sentences proves a sentence  $\varphi$  in classical logic then  $\Sigma^N$  proves  $\varphi^N$  in intuitionistic logic.

If you use LSD, you look at objects with bright colors to have a vibrant experience. And if you us double negator, you read a text with a rigid logical structure.  This is exactly what we are going to do.

Here is a proposition from Spinoza’s Ethica Ordine Geometrico Demonstrata, an ethics book written in the style of Euclid.

PROPOSITION V. There cannot exist in the universe two or more substances having the same nature or attribute.

Ignoring the modality  coming from can we can write this sentence like this.

$\neg\left( \exists x\exists y [\neg x=y \wedge [ {\rm A}(x)={\rm A}(y) \vee {\rm N}(x)={\rm N}(y) ]] \right)$

where ${\rm A}$ stands for attribute and ${\rm N}$ stands for nature. Here is the Goedel-Gentzen translation of this sentence.

$\neg\neg\left( \forall x\forall y \neg[\neg\neg\neg x=y \wedge \neg [\neg\neg\neg{\rm A}(x)={\rm A}(y) \vee \neg\neg\neg{\rm N}(x)={\rm N}(y) ]]\right)$

And here is the same text perceived by someone who used the double negator.

PROPOSITION V. It is not true that the following sentence is false: For any two substances it is not true that both the the sentence “it is not true that the substances are not unequal” and the negation of the sentence “either it is not true that the attributes of the substances are not unequal, or it is not true that the natures of the  substances are not unequal” hold.