English Version Đóńńęŕ˙ âĺđńč˙
ÎĎČŃŔÍČĹ

Brusentsov Nikolai
MSU, ramil@cs.msu.su

Ternary Dialectical Informatics

The deficiency of modern binary informatics that demonstrative shows up in the impossibility of natural number presentation with binary code sign that is specified with inadequacy of its foundation – binary logic. This so called “classical” formal logic that is based on antecedent “law of the third deletion” doesn’t show the perfect reflection of the reality, is out of keeping with common sense. So opposite to the arithmetic’s, for example, practically it isn’t used for solving real problems. This logic is logic of artificial, discrete world of binary computers. There is no modality; possibility can’t be differentiated from necessity. Even the most fundamental logical relation of content abidance is expressed by “material amplification” the paradoxes of witch are being persistently tried but in vain to be overcome by distinguished logics for many years.

That’s clear why logic can’t become a school subject however the developing of logic thinking is the first as school task as university one where logic is studied but doesn’t lead to the mentality improvement. The situation became worse with computerization of the education. With practically issuelessness of talks about the metallization of computer processing of information and information safety there is suppression of people’s creativity everywhere with stilted binary logic.

The founder of logic is considered to be Aristotle who created a system of demonstrative conclusion – syllogism that is a still unsurpassed mentality instrument. Syllogism is dialectic, there are no paradoxes in it but it can’t be placed in the modern logical calculus. This was a motive to suspicions that there was something wrong in Aristotle’s logic, e.g. that he didn’t acknowledged empty sets. But really there are not only empty sets in syllogism but also fuzzy sets that were invented by L. Zade in 1965 and are not still “mastering” with modern logic. But principal difference of Aristotle logic from modern one – “classical” is that it is not binary but ternary. Contrary to “law of the third deletion” with “necessary is” and “necessary isn’t” there is the third one “possible yes and possible no”. Ternary is appropriate to the relation of abidance that is depleting the determined by Aristotle in the “First analytics” [57bl]:

“… when two objects are related to each other in the way as there is one and the second then if the there is no the second there won’t be the first one, but if there is the second it is not necessary that there will be the first one. But it is impossible that the same is necessary and when there is the other or not”

In syllogism abidance relation is presented as generally affirmative condition “any x is y”, nature of witch is that any x-thing necessary is xy-thing, and any y’-thing necessary is x’y’-thing. But xy’-thing are excluded as x must have y not y’. At the same time x’y-things are excluded, they are possible but not necessary as there are xy-things and x’y’-things. If “any x is y” is performed then “any y is x” then x’y’- and x’y-things will be excluded, so there will be a relation of equivalence – “x is interchangeable to y”, in binary logic – “interchangeable/not interchangeable”. To the relation of abidance three values are needed: “necessary is” / “possible but not necessary” / “impossible”, so there is not enough binary implication for adequate expression of abidance. With xy’-things abidance is impossible and if there are no xy’-things then abidance is not excluded otherwise possible but not necessary. In case of inexistence of x-things or y’-things implication doesn’t express any interconnection between variants and doesn’t form two dimensional relation. In these cases the other term can take any value independently.

To smooth away “paradoxes” of implication it is enough to prevent these cases in variety of its terms. So strict implication of Louise that is an inexistence of xy’-things, i.e. V’y’ is paradoxically performed when x-things exist and y’-things don’t exist, i.e. V’x and V’y’. There won’t be paradoxes if with V’xy’ we demand Vx and Vy’ – the existence of x-things and y’-things. As the result of Louise implication becomes Aristotle necessary abidance VxV’xy’Vy’ [1]. In perfect disjunctive usual form this relation VxyV’xy’Vx’y’ is visible its ternary: with accessory of concerned subset Dekart’s members product {x, x’}x{y, y’} and member x’y is withhold with anti accessory xy’. Withholding expresses the third variety of accessory that is possible but not necessary. Subset that admits possible accessory is careless and presented with its relation is ternary.

In the mathematical logic deviation from Aristotle opinion of common condition “All is B” that made it from content abidance into binary implication. Gilbert and Akkerman proved the deviation with mathematical application logic needs “where taking Aristotle’s opinion as a foundation was pointless” [2, p.79]. They didn’t take into consideration that logic lost the richness of content, thinking that their logical calculus “makes possible successful problem comprehension where simple content logical thinking is principally weak” [2, p.17]

Indeed logic was made insipid 2, 3 thousand years ago by antique stoics who also sought after incredible abstractiveness that was carried out with the help of “propositions” “subordinated” to “law of the third deletion” that allowed only two values “true” – “true / false”. Adequate to reality Aristotle’s syllogism was made dead scholasticism with binary. Mathematical logic presented this “classic” in strict algebraetic forms evidently showed its inadequateness. [3]

Stoics “compensated” the lack of abidance relation in their logic with necessity to implement conclusions according to rules modus ponens and modus tollens. In mathematical logic it is also pointed that implication is not abidance: “The relation “if X then Y” shouldn’t be understood as a form for foundation and effect relation. To the contrary proposition X – Y is always true when X is false or Y is true.” [2, p. 201] But at the same time in mathematical logic even its founders identify binary implication with ternary Aristotle’s abidance as the first and the second ones are associated with the opinion “All A is essence of B”. As a result from mathematical logic point of view modi of syllogisms logic darapti, bamalip, felapton, fesapo are considered to be perfect false in reality [2, p.79] and syllogism of submission of quotient to common is rejected as according to the deviation from Aristotle’s interpretation in “All A is essence B” there is no “Some A is essence B”.

Yan Lukasevitch, created ternary modal logic in 1920, in his detailed book “Aristotle’s syllogism from modern formal logic point of view” [4] proved algebraic with the help of identification of ternary abidance with material implication that Aristotle quoted statement “But it is impossible that the same is necessary and when there is the other or not” is false. It is false from logic point of view that didn’t obey the basic logical law – law of identity. The opinion “All A is essence B” that expresses the abidance relation B from A can’t be identified with “No one A isn’t essence not-B” with the help of witch binary implication relation is presented in natural language.

The problem is that logic can’t be without abidance relation that is in essence is ternary and it doesn’t exist in binary logic. Relation that is called implication like abidance is expressed with the same “If… then…” and marker the same →. It is not striking that implication can be taken as abidance. But if there is no logic without abidance then logic with implication instead of abidance is not logic at all. Then from inexistent things follows “whatever”, from 2x2=4 is that “snow is white”.

This is all defective binary logic that ignores common sense and the result of its application doesn’t meet with expectations. In the book by T. Oppengamer [5] bad influence of computer education in the USA schools is indisputable proved. The author insists on the taking away computers from schools, but it is hardly possible in present situation. The root of all evil is not in computers but in the primitive unnatural logic of discrete binary world that students study. It blocks their ability to master the logic of real world. If there was natural logic in computers the result of computer studying would be opposite.

However, there is no adequate logic in all “science of thinking” even in its part where it doesn’t follow “law of the third deletion” and focuses on invention of not binary logics. Invention is unsuccessful because has an informal character. If the problem had been researched essentially then they would have found out that Aristotle’s logic is ternary and the ternary is necessary but not the main condition for adequate logic. Aristotle’s logic is adequate so there is no sense in invention of not Aristotle’s logic.

A good exception is “Symbolic logic” by L. Kerrol [6] that didn’t get appropriate (like Aristotle’s one) understanding and development. There is neither discontent “true” / “false” propositions nor “law of the third deletion”. His logic researches opinions that expressed interconnections of things that are characterized with combinations of features (peculiarities). “There is a set of things in the universe … Things have features … Any feature or any combination of features we will call a peculiarity of a thing.”

Opinion is considered as a natural language expression of relation that connects peculiarities of things with terms x, y, z… At the same time the essence of Kerrol’s relation visually reflects at his diagram and algebraic “index method” that formally allows getting content conclusion from data opinions, if it exists.

 Kerrol’s diagram is on the surface identified to Pirs’ truth table that is used to identify Boolean functions. But expressed with diagram it interpreted not as extensional but intentional as set of things or Kart’s subset product of in pairs opposite peculiarities. Besides, diagram cells get not one from two but one from three values – along with cells that have “0” or “1” empty cells are permitted, they denote nonthingness of things that belong to them that is presented at the subset diagram. But Kerrol understood value “1” as existence, value “0” as inexistence of a thing and empty cells don’t denote both.

 For example, Louise’ strict implication relation V’xy’ at Kerrol’s two terms diagram only one value is presented “0” in xy’-cell. Karol expressed this relation in three ways: common negative opinion “No one xy’ exists” or “no one x is y’” or “No one y’ is x”.

Common affirmative opinion “All x is essence y” includes particular affirmative in Kerrol’s “Some x is essence y” that is equal to existence opinion “Some xy exist”. Kerrol called “All x is essence y” double opinion that was equal to two opinions: “No one x is y’” and “Some x is essence y”, i.e. V’xy’Vxy – at the diagram “0” in the cell xy’ and “1” in the cell xy.

It is clear that this is not implication (one of its paradoxes is eliminated) but it is full Aristotle abidance. Nonpositiveness of abidance is not taken into consideration – false step as it is peculiar to all efforts to make syllogism algebraic. As compensation [7] “Symbol logic” of Kerrol becomes a well-composed and perfect statement of Aristotle’s syllogism – a foundation of dialectic logic.

The most important component of logic content appeared an identified dialectic principle in the Aristotle’s syllogism – principle of contrasts existence [8]. Nonpositiveness of common affirmative opinion as symmetry of relation that is expressed with common negative opinion is the visible demonstrations of this principle. The fact is that initial peculiarities x, x’, y, y’, z, z’ … expressed with terms x, y, z … get sense as the result of things comparison that have opposite peculiarities, for example x-thing and x’-thing.

In other words principle of co-existence of contrasts means that subset of Dekart’s product {x, x’} x {y, y’} that reflects content relation has all in pairs opposite peculiarities – VxVx’VyVy’. At Karol’s diagram VxVx’VyVy’ reflected with token data of “1” existence on every of four interior walls that denotes unemptyness of classes x, x’, y, y’.

These equivalent realities Aristotle’s Universe (AU) – the foundation of content logic [8].init Louise’ implication V’xy’ and Kerrol’s VxV’xy’ become full ambiances

 

(V¢xy¢)(VxVx¢VyVy¢º VxyV¢xy¢Vx¢y¢ 

(VxV¢xy¢)(VxVx¢VyVy¢º VxyV¢xy¢Vx¢y¢

 

Inexistence of any possible at diagram things, e.g. xy-thing in AU it means the existence of two adjoining with it things:

 

(V¢xy)(VxVx¢VyVy¢º V¢xy º V¢xy¢Vx¢y VxyVxVy

The existence of xy’-things according to the principle of contrasts co-existence entails the existence of its antipode – x’y-thing. So particular affirmative and particular negative conditions of syllogism become double and they are two not four:

Ixy º AxyAyx º VxyVx¢y¢

Oxy º ExyEx¢y¢ º Vxy¢Vxy¢

At the same time common conditions are not two but four that becomes one by term invertening: Exy º Axy¢, Ex¢y¢ º Ax¢y, Ayx º Ax¢y¢.

Syllogism algebra that corresponds to Kerrol’s interpretation of its diagram with token data symbolized existence and inexistence of things compared with cells and “walls” in similar to “index method” but instead of indexes it uses prefix functor of existence V – “disjunctive” ( integral disjunction similar to integral sum S) and its inversion V’ – symbol of inexistence. The presented in the diagram relation is reflected with conjunction of disjunctives, not inverted and inverted, and members of conjunction that corresponds to empty cells are not given. Every trit takes one of three values: “+” – existence, “-“– inexistence, “0” – silence. For example, abidance (xÞyº VxyV¢xy¢Vx¢y¢ is reflected with value of four trit scale: +-0+, a particular negative condition Oxy are coded with value: 0++0.

There are 8 double place relations in syllogism [10]:

Axyº Ay¢x¢ º Exy¢º Ey¢x º +-0+

Ayx º Ax¢y ¢º Eyx¢º Ex¢yº +0-+

Exy º Eyx º Axy¢º Ayx¢º-++0

Ex¢y¢ º Ey¢x¢ºAx¢y ºAy¢xº 0++-

Ixy º Ix¢y¢º Iyx ºIy¢x¢ º Oxy¢º Oyx¢ º Ox¢y º Oy¢x º +00+

Oxy ºOy¢x¢ º Oyx ºOx¢y¢º Ixy¢º Iy¢x º Ix¢y º Iyx¢ º 0++0

x Û y ºAxyAyx º Exy¢Eyx¢ º +--+

x Û y¢º ExyEyx ºAxy¢Ayx¢ º -++-

Computerizing proof of conclusions (true modi of syllogism) is carried out by two terms condition presentation with three term scales from witch crossing searching conclusion is taken by middle term elimination, if it exists. For example, modus Barbara:  AyzAxyÞ Axz in three term x, y, z-scales is realized:

Ayz º+-0++-0+

Axy º++--00++

Ayz ÇAxy º+---000+

Elimination y gives x, z-scale +-0+, i.e. Axz.

Subordination of particular conditions is proved with common crossing of coding these conditions scales. So the subordination Þ Ixy, is equal to AxyIxy = Axy, proved with crossing +-0+ìü +00+ = +-0+.

In syllogism that is based on co-existence of contrasts all doubtful modi from classical logic point of view and also a row of modi that are missed with traditional syllogism. For example, from conditions of doubtful modus bamalipreally there is not only particular but also common conclusion:

Azyº +0-++0-+

Ayxº ++00--++

AzyìüAyxº +0-0---+

Eliminating y we have +0-+ ºAzxºAx¢z¢, ň. ĺ. AzyAyxÞAzx

The correction of traditional theory is proof of denying with it modus of the first figure IyzAxyÞIxz:

Iyzº +00++00+

Axyº ++--00++

IyzìüAxyº +0--000+

That with y exception is +00+, ň.ĺ. Ixz.

By the same way the true of the next missed modus is proved of the first figure IyzExyÞOxz and similar and similar modi of other figures

 

 

 

 

Material implication

Extensional interpretation

(Pirs’ table)

Content abidance

Intentional interpretation

(Kerrol’s diagram in Aristotle’s Universe (AU)

 

 

 

Referrences

 


  1. N.P Brusenzov Ternary interpretation of Aristotle syllogism // Historical mathematical researches, issue ą8 (43) – M, 2003, pp.317 – 327

  2. D. Gilbert., V. Akkerman Foundations of theoretical logic, - M., 1947

  3. A.F. Losev Critical notes about bourgeois mathematical logic // Historical mathematical researches, issue ą8 (43) – M., 2003, pp.339 – 401

  4. Y. Lukasevitch. Aristotle’s syllogism from modern formal logic point of view – M., 1959

  5. T. Oppenheimer The Flickering Mind: The False Promise of Technology in the Classroom and How Learning Can Be Saved. – New York, Random House, 2003. 512 p.

  6. L. Kerrol Symbolic logic // L.Kerrol Story with bundles, - M., 1973, pp 189 – 361

  7. N.P. Brusenzov. Diagrams of L. Kerrol and Aristotle syllogism // “Calculating technology and cybernetics questions”, issue 3, - M., 1977, pp. 164 – 182


8. N.P. Brusenzov Wandering around three pines (dialectics adventures in informatics), M., 2000, (http://www.computer-museum.ru/books/archiv/3pines.zip)

9. N.P. Brusenzov, U.S. Vladimirova Turnery computerization of logic //Mathematical methods of form identification: the 12th Russian conference: a report digest, - M., 2005, pp. 40 – 42

10. N.P. Brusenzov reanimation of Aristotle syllogism // Logic restoration, - M., 2005, pp. 140 – 145

 

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