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

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 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 wellordered therefore being able to designate any set, which is not well-ordered

Four elements and their unordered topological structure

Four letters enough to encode anything, e.g. DNA C

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A C

G

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A

The square of opposition

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 wellordered 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) Implicitly (as a featuring property) A set of wellAn algebraic structure orderings meant (usually lattice) on that set Explicitly (as elements, i.e. well-orderings) The problem of how that explicitly given set can be “coloured�

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 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”

Comments to Argument 2 • 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 Any geographical map

A set of well-orderings (i.e. well-ordered subsets of another set) Any logic

The “four-colours theorem”

A “map” of propositions needing not more than four colours to be coloured such as those of the “square of oppositions”

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 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 Enumerating a huge though finite number of cases Software programs for proving in any case A “computer proof”

As a problem in the foundation of mathematics An interpretation as the “fourletters theorem” The “four letters theorem” on the bridge between

The finiteness of arithmetic A human proof

The infinity of set theory

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 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 …

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 dimension o Consequently any logic can be represented as a wellordered 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

C

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Logic

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A theorem

Another theorem

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The axiom of choice allows of all theorems to be always wellordered 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” Any logic is a partial ordering needing 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 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 Then it is encodable by four letters (“colourable by four colours”)

That statistical mix is equivalent and even identical to the “things themselves” according quantum mechanics It can be considered as a partially ordered structure

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 a proof…

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

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 “fourletters” 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 wellordering 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 fourletters 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 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

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

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