The square of opposition: Four colours sufficient for the “map” of logic

The square of opposition:
Four colours sufficient for
the “map” of logic
From the “four-colours theorem” to
the “four-letters theorem”

Vasil Penchev
Bulgarian Academy of Sciences: Institute for the Study of
Societies and Knowledge: Logical Systems and Models
vasildinev@gmail.com
“The Square of Opposition”,
5th World Congress
Rapanui (Easter Island), Chile,
10-15, November 2016
http://www.square-of-opposition.org/Rapanui2016.html

How many “letters” does the “alphabet of
nature” need?
• Nature is maximally economical, so that that number would
be the minimally possible one. What is the common in the
following facts?
o (1) The square of opposition
• (2) The “letters” of DNA
o (3) The number of colors enough for any geographic al map
• (4) The minimal number of points, which allows of them not
be always well-ordered

A note: the well-ordering of cyclic orderings
• Here and bellow, the term of well-ordering as to cyclic
orderings means the option for any point in those to be able
to be chosen as the “beginning”, i.e. as the least element in
well-ordering
o This means that a cyclic ordering is well-ordered iff it
contains a single cycle. Indeed, it can be opened anywhere
transforming into a normal well-ordering
• This corresponds to the prohibition of „vicious circle” in
logic, which can be also always opened

Four!
• The number of entities in each of the above cases is four though
the nature of each entity seems to be quite different in each one
o The first three facts share that to be great problems and thus
generating scientific traditions correspondingly in logic, genetics,
and mathematical topology
• However, the fourth one (4) is almost obvious: triangle do not
possess any diagonals, quadrangle is just what allows of its
vertices not to be well-ordered in general just for its diagonals
o Four elements seem to be necessary where one would describe a
structure, which is not well-ordered, i.e. the general case of
structure

From Three to Four?
• Thus, the limit of THREE as well as its transcendence by
FOUR seems to be privileged philosophically, ontologically,
and even theologically
o It is sufficient to mention Hegel’s triad, Peirce’s or Saussure’s
sign, Trinity in Christianity, or Carl Gustav Jung’s discussion
about the transition from Three to Four in the archetypes in
“the collective unconscious” in our age
• One can describe the dilemma “three or four” as the
alternative between a single well-ordering (i.e. a single linear
hierarchy) and a set of arbitrarily many well-orderings (one
might say “a democracy of hierarchies”), which is to be
described relevantly

Our suggestion
• The base of all cited absolutely different problems and
scientific traditions is just (4)
o Thus the square of opposition can be related to those
problems and interpreted both ontologically and differently
in terms of each one of the cited scientific areas as well as in
a few others
• This means that the number of four is privileged as the least
number of the elements of a set, which admit not to be well-
ordered therefore being able to designate any set, which is
not well-ordered

The square
of opposition
Four elements and their
unordered topological
structure
A
C G
T
A
C G
T
Four letters enough
to encode anything,
e.g. DNA
Four colours enough
for any map

A few arguments “pro” the hypothesis

A few arguments: Argument 1
• Logic can be discussed as a formal doctrine about correct
conclusion, which is necessarily a well-ordering from
premise(s) to conclusion(s)
o To be meaningful, that, to which logic is applied, should not
be initially well-ordered just for being able to be well-
ordered as a result of the application of logical tools
• Any theorem being a correct conclusion from the premises
can be sees as a well-ordering from the premises to the
statement of the theorem
o Then any logic being a set of true theorems will be therefore
a set of well-orderings, irreducible to each other, but all
reducible to the axioms

Comments to Argument 1
• The usual viewpoint to a given logic pays attention first of all
to the rules of conclusion, which are different for each logic
o Therefore a set of true well-ordering turns out to be
supplied by a certain algebraic structure, usually a lattice
• Then one can described that logic exhaustedly by
corresponding algebraic operations interpretable as valid
operations to the elements of the set of true well-orderings
such as a propositional calculation
o Thus the usual focus of logical investigation addresses the
corresponding rules of conclusion and an algebraic structure
as well as eventually in relation to other logics, but almost
never the set of all logic(s)

The standard approach
to any given logic
The rules of conclusion defining implicitly
a set of well-orderings (the true conclusions)
A set of well-
orderings meant
Implicitly (as a featuring property)
Explicitly (as elements, i.e. well-orderings)
The problem of how that explicitly
given set can be “coloured”
An algebraic structure
(usually lattice) on that set

• Consequently, the initial “map” (to which logic is applied)
should be “coloured” at least by four different types of
propositions, e.g. those kinds in the “square of opposition”
o They are generated by two absolutely independent binary
oppositions: “are – are not” and “all – some”, thus resulting
exactly in the four types of the “square”
• In fact, those “colour” oppositions are chosen in tradition:
the tradition, which can be traced back to Aristotle
o Any two logically meaningful oppositions (therefore
internally disjunctive) independent of each other (therefore
externally disjunctive) would be relevant as “four colours”
for the “map of logic”

• Indeed one can involve a certain general structure of a set of
well-orderings of the elements of an initial set
o It can be also considered as a partly ordered set, in which all
(maximal) well-orderings are separated as a special class of
subsets
• Any logic and any geographical map share the same
mathematical structure
o Then and particularly, one can defined any logic as that
description of a corresponding “map” of e.g. propositions,
which is inventoried by the characteristic property of the set
of all linear neighbourhoods in the map (a rather
extraordinary way for a map to be depicted)

A partially
ordered set
A set of well-orderings
(i.e. well-ordered
subsets of another set)
Any
geographical map
Any logic A “map” of propositions
needing not more than four
colours to be coloured such as
those of the “square of
oppositions”The “four-colours theorem”

• Five or more types of propositions would be redundant from
the discussed viewpoint since they would necessary iff the
set of four entities would be always well-orderable, which is
not true in general
o Consequently, the “four-colours theorem” might be
alternatively interpreted by means of the following
formulation: three colours is the maximal number of colours,
which are not enough to colour any map
• The three elements of a set are always well-ordered being
incapable to constitute different cycles more than one

• Consequently, one can unify and therefore generalize the
problems how a map should be uniquely coloured or a logic
described, by the following question:
o How many “letters” are necessary for any partially ordered
set to be described unambiguously?
• The usual confusion preventing that fundamental and
generalizing problem question to be asked consists in the
following:
o The “map” misleads to be interpreted right topologically
complicating redundantly the problem by enumerating all
possible topological cases

How many “letters” would be
sufficient to be described all
in the universe?

Still a few comments to Argument 3
• That number of topological cases though finite is so huge
that only computers can manage it
o In fact, that non-human approach is not necessary if one
generalizes all topological cases to a partially ordered set and
proves the theorem about it
• This means that the four-colours theorem should be
interpreted in a non-topologically to be proved in a “human
way”, ant its “obvious” topological definition is seeming and
misleading
o Then, any logic can be described in the same way

Both approaches for proving the “four-colours
theorem” illustrated
Topological As a problem in
the foundation of mathematics
An interpretation as the “four-
letters theorem”
The “four letters theorem”
on the bridge between
The infinity
of set theory
The finiteness
of arithmetic A human
proof
Enumerating a huge though
finite number of cases
Software programs
for proving in any case
A “computer proof”

• Logic can be discussed as a special kind of encoding namely that
by a single “word” thus representing a well-ordered sequence
of its elementary symbols, i.e. the letters in its alphabet
o The absence of well-ordering needs at least four letters to be
relevantly encoded
• The four letters are just as many (namely four) as the “letters”
in DNA or the minimal number of colours necessary for
a geographical map
o Two “letters” such as “0” and “1” are sufficient to encode any
linear string: then, the string, which is not well-ordered, needs
at least two dimensions …

• Any logic is defined as a set of well-orderings and thus it can
be in turn well-ordered in a second dimension
o Consequently any logic can be represented as a well-
ordered set of binary strings
• Two different letters are necessary for any binary string
o Still two different letters are necessary for any two
neighbouring strings to be designated differently
• The present argument addresses the core of the proof of the
four-letters theorem: the axiom of choice should be applied
in a way to conserve the partial ordering so not to call a total
linear well-ordering

1
A theorem
2
A
C
. . .. . .
. . .. . .
Logic
The axiom of choice allows of
all theorems to be always well-
ordered
If the number of theorems is
finite, the axiom of choice is not
necessary
Then still two additional colours
are sufficient for any neighbouring
theorem to be coloured differently
If any well-ordered string can be
unambiguously encoded as binary,
any partial ordering needs
four “letters” or “colours”1
2
G
T . . .
. . .. . .. . .
. . .. . .
. . .
Another
theorem
. . .. . .
. . .. . .. . .. . .
. . .. . .. . .. . .
. . .. . .. . .. . .
. . .. . .. . .
. . .. . .. . .. . .
Any logic is a partial ordering
needing only four “colours”

• The alphabet of four letters is able to encode any set, which is
neither well-ordered nor even well-orderable in general, just to
be well-ordered as a result eventually involving the axiom of
choice in the form of the well-ordering principle (theorem)
o It can encode the absence of well-ordering as the gap between
two bits, i.e. the independence of two fundamental binary
oppositions (such as both “are – are not” and “all – some” in
the square of opposition)
• If one represents infinity as a gap such as that between two
dimensions, four letters are sufficient to encode any infinite set
including the finite subsets

• Quantum mechanics offers a relevant conception for how any
unorderable in principle entity may be anyway studied and
therefore represented by partially ordered sets (i.e. logically)
o Any coherent state before measurement is unorderable in
principle for the theorems about the absence of hidden
variables in quantum mechanics
• Nevertheless, it is ordered after measurement, but by a
randomly chosen ordering as an unconditional principle
o Thus any unorderable entity can be represented equivalently
as a statistical ensemble of well orderings corresponding to
certain partial orderings equivalent to logics

The “things by themselves”
A coherent superposition
of all possible states
and thus unorderable in principle
Any measurement reduces them
to a finite and well-ordered set, but
always randomly chosen
A statistical ensemble (mix)
of the randomly chosen
well-orderings
That statistical mix is equivalent
and even identical
to the “things themselves”
according quantum mechanics
It can be considered as a
partially ordered structure
Then it is encodable by four letters
(“colourable by four colours”)

Logical and mathematical introduction into
the problem
• All logics seem to be unifiable as different kinds of rules for
conclusion
o Thus any logic is a set of correct well-orderings (i.e. sequences
from the premise to the conclusion)
• The axiomatic description of logic consists in explicating the
characteristic property of that set so that one can decide for any
well-ordering whether it belongs or not to that set
o To be a well-ordering ‘correct’ means just that it belongs to the
set defined by its characteristic property as a certain kind of
logic

The set of all logics and its property
• Then, the characteristic property of the set of all logics seems
to be the set of all sets of well-orderings in a class identifiable
as language as a whole
o The advantage of that definition is that one can “bracket” (in
a Husserlian manner) the latter class being too fussy, unclear,
and uncertain
• It is substituted by the set of all natural numbers perfectly
sufficient for representing all well-orderings. Indeed, this is
the sense of the well-ordering principle equivalent to the
axiom of choice

Language: the enumerated
• The initial class of language can be interpreted as what is
enumerated, then “bracketed” and “forgotten”
o This follows the essence (though not literally) of Gödel’s
approach for the arithmetical “encoding” of all meaningful
statements being true, false, or undecidable
• However, the enumeration suggests a single dimension such
as that of the well-ordered natural numbers: their order is a
single one
o However, if that was the case, the words or terms in a
language would be also well-ordered, which is no true even
to the artificial, computer languages created intendedly by
humans to be unambiguous

The map of a logic
• If all logics as that set of all sets of well-orderings of natural
numbers are granted, one can define the concept of the
‘map’ of any given logic as the graph of all correct
conclusions in the logic at issue
o The vertices of the graph are natural numbers
• Just four colours are enough to be coloured that graph so
that any two neighbouring vertices to be coloured differently
according to the direct corollary from the “four-colours”
theorem
o Then the maps of all logics share the same property

Colours, letters and … amino acids
• One can choice any four certain and disjunctive “colours” for
all maps, e.g. those of the square of opposition according to
the tradition, or the “A-C-G-T” alphabet of DNA
o Nature always simplifying maximally has also “proved”
the “four-colours” theorem as to DNA
• One may speak rather of the “four-letters” theorem than of
“four-colours” theorem in that case
o The sense is: the DNA itself can be encoded by four letters
practically realized by the four amino acids designated as A,
C, G, and T: adenine, cytosine, guanine, and thymine

A generalization of
the “four-colours theorem”
• The “four-colours” theorem seems to be generalizable as
follows:
o The four-letters alphabet is sufficient to encode
unambiguously any set of well-orderings including a
geographical map or the “map” of any logic and thus that of
all logics or the DNA (RNA) plan(s) of any (all) alive being(s)
• Then the corresponding maximally generalizing conjecture
would state: anything in the universe or mind can be
encoded unambiguously by four letters

Formulating the “four-letters theorem”
• That admits to be formulated as a “four-letters theorem”,
and thus one can search for a properly mathematical proof
of the statement
o It would imply the “four-colours theorem”, the proof of
which many philosophers and mathematicians believe not to
be entirely satisfactory for it is not a “human proof”, but
intermediated by computers unavoidably since the necessary
calculations exceed the human capabilities fundamentally
• It is furthermore rather unsatisfactory because it consists in
enumerating and proving all cases one by one

The “four-colours” theorem: a corollary from
the “four-letters theorem”
• Sometimes, a more general theorem turns out to be much
easier for proving including a general “human” method, and
the particular and too difficult for proving theorem to be
implied as a corollary after certain simple conditions
o The same approach will be followed as to the four colours
theorem, i.e. to be deduced more or less trivially from the
“four-letters theorem” if the latter is proved
• Indeed, anything in the universe is codable by four letters,
then of course, the mutual position in a map is also codable
by four colours as those necessary four letters for anything

The approach for the four-letters theorem
to be proved
• The idea consists in representing any partial ordering as
a well-ordered set of well-orderings therefore involving two
dimensions of well-ordering
o The problem is not so the well-ordering itself as it to be stopped
before to reduce all to a single well-ordering for
the axiom of choice is valid
• That approach needs a certain “gap” such as that between two
dimensions, over which the axiom of choice not to be able to
transfer its ordering
o However, the boundary between a subset and the set of
corresponding subsets used above is not reliable enough as that
“gap” serving rather for illustrating the idea

The approach for a proof (continuation)
• A gap reliable enough and furthermore utilized already in the
dual foundation of mathematics by both arithmetic and set
theory is that between ‘finiteness’ (after the natural
numbers in arithmetic) and ‘infinity’ (after the infinite sets in
set theory)
o Indeed, the axiom of induction implies that all natural
numbers are finite (1 is finite, adding 1 to a finite natural
number, one obtains a finite number again)
• Set theory (e.g. ZFC for certainty) does not include the axiom
of induction, but the axiom of infinity postulating the
existence of infinite sets as well the axiom of choice able to
order well any infinite set

The approach for a proof …
• Then the two “bases” of mathematics, both arithmetic
and set theory, is not quite simple to be reconciled as to
finiteness and infinity
o The Gödel incompleteness theorems (1931) might be
considered as the demonstration of those difficulties
• A visualization of how arithmetic and set theory can be
reconciled by the axioms of induction and of choice is
suggested on the next slide

The approach for a proof… visualized
The natural numbers:
finite arithmetic
Sets infinite in
general: set theory
The axiom of
choice reducing
to a single, but transfinite
well-ordering
The axiom of induction
generating a finite
well-ordering of finite
well-orderings, i.e. a
two-dimensional,
but finite one
Two letters:
“0” and “1”
Four letters:
“A”, “C”, “G”, and “T”

The approach for a proof …
• From the viewpoint of the finiteness of the natural numbers (i.e.
by the axiom of induction), one will observer a finite well-ordered
set of also finite well-ordered sets divided by gaps as the infinite
mathematical universe, which will be represented as a partial
ordering
o From the viewpoint of the infinite mathematical universe of set
theory (i.e. by the axiom of choice), one will observes a single
well-ordering of all
• Then the former partial ordering will need four letters for its
description in two dimensions forced by the gaps, unlike the only
two letters necessary for the latter, single well-ordering of all
because of the absence of any gaps

The three “whales” of the new gestalt
necessary for a simple proof of the “four-
letters” theorem
А: A generalization from the four-colours theorem to the four
letters theorem
Б: A set-theoretical and arithmetical rather than topological
approach
В: A viewpoint from the well-ordering to the partial orderings
to be revealed the partial orderings as two-dimensional well-
ordering rather than reducing an any-dimensional partial
ordering to a two-dimensional one

The structure of the paper instead of
conclusions
• Section 2 exhibits a general plan for the method, in which the four-
letters theorem might be proved, including all successive steps
considered one by one in detail in the next sections
o Section 3 discusses the separable complex Hilbert space and its
interpretation in quantum mechanics and in theory of (quantum)
information
• Section 4 demonstrates the correspondence between classical
information and quantum information as the correspondence
between the standard and nonstandard interpretation (in the sense
of Robinson’s analysis) of one and the same structure

conclusions
• Section 5 elucidates the link between that last structure and
Skolem’s “relativity of ‘set’” (1922) as the one-to-one mappings
of infinite sets into finite sets under the condition of the axiom
of choice
o Section 6 deduces the “four-letters theorem” and interprets the
theorem as to the physical world after any entity in it might be
considered as a quantum system
• Section 7 interprets the theorem as to mind seen as the set of all
logics by means of representing the well-orderings in the
separable complex Hilbert space

conclusions
• Section 8 discusses the unification of the physical world and
mind under the denominator of the “four-letters theorem”.
o Section 9 deduces the four-colours theorems from the four
letters theorem including the case of an infinite number of
domains by attaching ambiguously a wave function to any
map (the axiom of choice may be excluded for any finite
number of domains)
• The last, 10th section summarizes the paper, suggests
conclusions and direction for future work

You may find or download the complete paper
(2017) or this presentation by taping the title,
The square of opposition: Four colours sufficient
for the “map” of logic, in any search engine such
as Google, Bing, etc.
Thank you very much
for your kind attention!

The square of opposition: Four colours sufficient for the “map” of logic

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The square of opposition: Four colours sufficient for the “map” of logic