The square of opposition: Four colours sufficient for

the “map” of logicFrom 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 Modelsvasildinev@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

A hypothesis

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 DNAo (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 oneo 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 diagonalso 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 theologicallyo 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 theirunordered topological

structure

A

C G

T

A

C G

T

Four letters enoughto encode anything,e.g. DNA

Four colours enoughfor 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 theoremo 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 logico 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 calculationo 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 approachto 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

A few arguments: Argument 2• 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 Aristotleo 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”

Comments to Argument 2• Indeed one can involve a certain general structure of a set of

well-orderings of the elements of an initial seto 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”

A few arguments: Argument 3

• 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

Comments to Argument 3• 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 besufficient 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 ito 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 mathematicsAn interpretation as the “four-

letters theorem”

The “four letters theorem”on the bridge between

The infinity of set theory

The finitenessof arithmetic A human

proof

Enumerating a huge thoughfinite number of cases

Software programs for proving in any case

A “computer proof”

A few arguments: Argument 4• 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 alphabeto 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 fora geographical mapo 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 …

Comments to Argument 4• Any logic is defined as a set of well-orderings and thus it can

be in turn well-ordered in a second dimensiono 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

. . .. . .

. . .. . .

Lo

gic

The axiom of choice allows ofall theorems to be always well-

ordered

If the number of theorems isfinite, the axiom of choice is not

necessary

Then still two additional coloursare sufficient for any neighbouringtheorem to be coloured differently

If any well-ordered string can beunambiguously encoded as binary,

any partial ordering needs four “letters” or “colours”1

2

G

T . . .

. . .. . .. . .

. . .. . .

. . .Another theorem

. . .. . .

. . .. . .. . .. . .

. . .. . .. . .. . .

. . .. . .. . .. . .

. . .. . .. . .. . .. . .. . .. . .

Any logic is a partial orderingneeding only four “colours”

A few arguments: Argument 5

• 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

Comments to Argument 5• 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 principleo 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 themto a finite and well-ordered set, but

always randomly chosen

A statistical ensemble (mix) of the randomly chosen well-orderings

That statistical mix is equivalentand even identical

to the “things themselves”according quantum mechanics

It can be considered as apartially 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 conclusiono 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 seto 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 wholeo 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 caseo 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 conditionso 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 codableby four colours as those necessary four letters for anything

The approach for a proof…

The approach for the four-letters theoremto be proved

• The idea consists in representing any partial ordering as a well-ordered set of well-orderings therefore involving two dimensions of well-orderingo 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 transfinitewell-ordering

The axiom of inductiongenerating a finite well-ordering of finitewell-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 orderingo 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” theoremА: 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

The structure of the paper instead of 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 choiceo 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

The structure of the paper instead of 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

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Thank you very muchfor your kind attention!