Can one think like a crab?
February 27, 2009
We express ourselves, among other things, through language, which is supposed to reflect our logical structure. In our language, we have created a bivalent structure, with YES and NO, TRUE and FALSE, leading to "Aristotelian thought," according to which any statement (a logician would call it a "proposition") can only be TRUE or FALSE. This is called
the principle of the excluded middle.
Unfortunately, experience does not follow theory, and our phrasing is full of undecidable propositions, which are neither true nor false, such as
"I am lying."
For a good century, logicians have shown great imagination trying to build non-bivalent logics. Let us give an example of a trivalent logic, the
so-called fuzzy logic,
whose truth values are
True Indeterminate False
a logic that has proven its operational character in automatism, process controls (in engineering).
Attempts to construct a tetralogic have also been made, the most classical one having the following truth values:
True
False
True and False
Neither true nor false
An extension attempt that has not proven fruitful.
In his book:
To contact the author directly:
Erratum
The author informs us that there is an erratum in one of the tables presented in his book. It is the one on page 29, whose color version is on page 135.
First of all, thank you for your interest in this work, and for having chosen to purchase the book.
These things happen... There is a nice typo! Third row and column, instead of a 1 there is mistakenly a 0. This correction will be sent to everyone in a few days.
On the other hand, the signs = and || are found in the diagonals: these double bars, viewed along a diagonal, give the sign =, and along the other diagonal give || which should be understood as "different," where they are found.
We hope this will allow you to continue reading effectively, and once again, we offer our warmest thanks (and our apologies as well!), and remain at your disposal if you are ever again confronted with a doubt... Or a new typo!
Figure 2.2, to be replaced by the table above
Denis Seco de Lucena invites us to a strange exploration, from which the reader may not emerge unscathed. Let us start with an examination of language, which is the approach of any logician. The author proposes to introduce what he calls the concept of
transversality.
From this perspective, any proposition, regardless of its nature, would be susceptible of a declination in four forms, symmetric two by two, consisting of "two symmetric pairs." Many examples exist in language, but the "fourth proposition" is sometimes difficult to formulate, or does not correspond to any existing qualifier in the language.
Let us first give examples where this "transversality" is clearly expressed. Take, for example, the concept of movement. There are thus four ways of "moving":
Forward
Backward
Stagnate
Move
It is immediately clear that the pairs with their symmetries emerge. Backward is the opposite of forward, and vice versa. Move is the opposite of Stagnate, and vice versa.
If we refer to topology, we would introduce four adverbs or adverbial expressions:
Outside
Inside
On the border
Elsewhere
February 29, 2010
: My friend Jacques Legalland suggests that the fourth proposition would be better phrased by writing:
Outside
Inside
On the border
Nowhere
If we refer to colors:
White
Black
Gray
Tinted
February 27, 2010
: Jie suggests:
White
Black
Gray
Transparent
By playing with time:
Before
After
Now
Never
The adverb "never" is the temporal equivalent of the adverbial expression "nowhere" (see above).
This way of seeing things reminds me of the Ummite text on logic, which, if I recall correctly, mentioned four truth values:
True
False
True and False
Indecipherable
If we take the truth values of the classical tetralogic:
True
False
True and False
Neither true nor false
February 27, 2010
: The fourth value should be reinterpreted as "does not correspond to this type of classification":
True
False
True and False
Does not correspond to this type of classification
Let us take the real numbers. We have:
Positive
Negative
Zero (in the sense of positive and negative)
The fourth proposition could be:
Positive
Negative
Zero (in the sense of positive and negative)
Imaginary
By moving to implication:
Implies
Is implied by
Contingent with respect to
Unrelated to
We can see that four ways of "saying" emerge, which are different from the classical tetralogic mentioned above. The symmetry of the last two propositions is different. The author proposes to say that these pairs of propositions, these qualifiers, are "transverse."
The way we present things does not correspond to the one used by the author in his book, which I recommend you read. But at first, you will ask yourself "what is behind this?" This question will take you far. The author, a scientist, found his starting point in a letter I received in 1992 from mysterious correspondents named "Ummites," a letter addressed to me from Riyadh, Saudi Arabia. For those who do not know this story, it is good to recall the context. Among the documents brought back from Spain since the mid-1970s, the authors of these texts emphasized the need to abandon Aristotelian logic and move to a
tetralogic
.
For years, I struggled with various attempts. In 1992 I had a first-generation Mac Intosh running at 2 Mhz, and obviously completely devoid of a modem or any means of communication with the outside. In this computer, I recorded reflections that were known only to me. Prompted by Gödel's theorem, I remembered that it was based on arithmetic (the manipulation of natural integers), axiomatized at the end of the last century by the mathematician Peano. The mathematician Gauss invented what is now called "Gauss integers," that is, complex numbers with integer values.
I noticed that these Gauss integers were classically considered as pairs of natural integers (a, b), and no axiomatization had been attempted to construct them, other than deciding to give them "two integers."
A few days after putting these few reflections on my hard drive, I was surprised to receive a letter addressed to me from Saudi Arabia mentioning the same reflections.
The content of this letter
It turns out that Denis, who is a scientist, found in this strange letter the starting point for a ten-year endeavor, which he has reported in the book he has just published. Given the exotic, to say nothing of the controversial, nature of this source, one can understand why he decided to publish it under a pseudonym.
I don't know if you remember a book by Jules Verne: Journey to the Center of the Earth, where the heroes play with a mysterious message left by Aarne Saknudsen. By combining its elements, they eventually discover the path to follow to reach the center of the Earth. Atten...