The power of the square of opposition has been proved during millennia, It supplies logic by the ontological language of infinity for describing anything... 6th WORLD CONGRESS ON THE SQUARE OF OPPOSITION http://www.square-of-opposition.org/square2018.html

1. The Square of Opposition
&
The Concept of Infinity:
The shared information structure

2. 6th WORLD CONGRESS ON
THE SQUARE OF OPPOSITION
Orthodox Academy of Crete, 1-5 November 2018
http://www.square-of-opposition.org/square2018.
html

3. Vasil Penchev
Dept of Logical Systems and Models, Institute for
the Study of Societies and knowledge, Bulgarian
Academy of Sciences
vasildinev@gmail.com

4. Project:
ДН 15/14 - 18.12.2017 to the "Scientific
Research" Fund, Bulgaria: "Non-Classical
Science and Non-Classical Logics.
Philosophical and Methodological
Analyses and Assessments"

5. Prehistory and background

6. “Two bits, two bits!” for Mr Finnegan!
One may demonstrate that the
square of opposition shares the least
possible structure of the ordering
which cannot be well-ordered directly
This structure is equivalent to two
bits of information, absolutely
independent of each other

7. The alphabet of “‫”יהוה‬ [YHWH]
That minimal possible information
structure is a kind of the minimal
“alphabet” of the universe
That “alphabet” is necessary for anything
to be “written” unambiguously

8. The YHWH alphabet of four letters
That alphabet consists of four letters
corresponding one-to-one to two
independent bits
It can be found as the four “letters”
of human genome as the four “colors”
enough to be colored any map

9. “Four letters” vs “two letters”
The minimal possible alphabet consists
of only two letters
This is a single bit of information
That alphabet of two letters is used is
our computers
It is enough for any well-ordering to be
written exactly

10. Why our computers do not think (yet)
One may define ‘thinking’ as the ability of
well-ordering of all (which is not ordered in
general)
The alphabet of two letters is insufficient to
be written relevantly anything which is not
well-ordered
So, the alphabet of our computers and that
of thinking are essentially different yet

11. A fundamental loss of information
The record of anything which is not
well-ordered by only two letters implies
the loss of information necessarily
For example, let that not well-ordered be
written by four letters, i.e. as if “two bits”
However, one needs furthermore the
superposition of the order, in which the
two bits are recorded:

12. One bit
Another bit
Their superposition
One quantum bit
(qubit)

13. An ontological example
An exceptionally important exemplification
of that loss of information is quantum
measurement
It reduces a coherent state before
measurement unorderable in principle to a
statistical and thus well-ordered ensemble
after measurement

14. What the alphabet of two letters implies ...
Any record of something not well-ordered
by the alphabet of two letters implies the
well-ordering principle (equivalent to) the
axiom of choice
One may mention Voevodsky’s univalence
axiom (2009) able to link logic to
topology on a similar basis

15. Adding the lost information...
The sense to that application of the axiom
of choice is to add outwardly, by external
choices the information lost before
Namely, that information lost after
recording by that insufficient alphabet of
two letters

16. Restoring the equation of information...
The information equation before and
after recording, respectively before and
after measurement, need be restored
What the axiom of choice does do
actually...

17. ‘Infinity’: the mathematical “God”
The power of the square of opposition has
been proved in logic during a few
millennia
It can be also utilized to be hinted the
most general concept in the framework of
logic, particularly the being of infinity

18. The alphabet of infinity...
A reasonable conjecture in the same
course of thought is:
The concept of infinity admits to be
written by the same alphabet of four
letters or two independent bits
consistently

19. ‘Infinity’ by two bits?
Or even a more strong hypothesis is
possible:
Infinity might be equated to te the
same elementary information
structure of two independent bits

20. The present talk’s intention
The talk intends the discussion of that
hypothesis:
Two bits of information
Infinity
Square of
opposition
Four-letter
alphabet

21. Thesis:

22. Four letters (or two bits) for infinity
1. The concept of infinity can be
written relevantly by the alphabet of
four letters
Thus, infinity shares the universal
alphabet enough for anything in the
universe

23. All you need … are four letters
2. Even more, the concept of
infinity might be equivalent to that
fundamental alphabet
One means the alphabet consisting
of four letters or two independent
bits

24. The four-letter language of nature
3. Metaphorically, anything in the
universe, or in other words, anything
existing is written by the language of
infinity
However, it consists necessarily of
only four letters...

25. An idea for the proof of the thesis

26. The two dimensions of two
independent bits:
The structure of two independent bits
is two-dimensional necessarily
Indeed, it implies a gap between
the two bits due right to their
independence

27. Infinity as .. a “gap”
The gap between any two dimensions
both requires and impies the concept
of infinity to be described explicitly or
“cataphatically”:
Dimension 1 Dimension 2
The gap “of” (or “as” infinity)

28. The ‘gap’ of infinity and the “gap” of bits
In other words, the gap between two
dimensions (such as two independent
bits) describes the same, however in an
implicit, or “apophatic” way

29. The gap of infinity as the gap between
qualities
‘Infinity’ means not any finite entity,
but another quality different from any
finiteness
Quantity suggests homogeneity and
thus, does not a gap
Indeed, ‘two quantities’ means “two
qualities”

30. Any single quality as a quantity
Thus, infinity can be represented
equivalently by two “finitenesses”
Properly: by the gap between them
symbolizing the different qualities
of each of those “finitenesses”

31. The mathematical structure of two
qualities
Thus, the minimal structure of two
independent bits or the alphabet of
four letters, both can describe infinity
That description of infinity coincides
with infinity in a sense, even

32. A symmetry of four elements...
Furthermore, the structure of four
letters is symmetric to the transforms
between elements and triples of
elements
And the triples of elements are all
well-ordering available among four
elements...

33. The structure of four letters as
a tetrahedron
That symmetry of four letters can
be visualized as a “tetrahedron”:
Correspondingly, by its symmetry
to the transformation between
vertices and faces

34. The structure of four elements as a qubit
The structure of four elements is the
minimal one able to represent
equivalently a qubit
It can be explained as the
“superposition” of two bits, as
minimal as possible

35. From a qubit to a string of qubits...
Any entity in the physical world is
necessarily a quantum one (according
to the contemporary stage of
cognition)
Thus, it is representable as a linear
string of qubits (words)

36. Conclusion

37. The power of the square of opposition
The power of the square of opposition
has been proved during millennia
It supplies logic by the ontological
language of infinity for describing
anything...

38. A happy end!
Thank you for your kind attention!
Any questions?
I would be very glad of many questions!