Theory of M-system - FSB Online of M-system MARIO ESSERT ∗, Faculty of Mechanical Engineering and Naval Architecture, University of Zagreb, Ivana Luci

[11:28 23/8/2017 jzx025.tex] Paper Size: a4 paper Job: JIGPAL Page: 836 836–858

Theory of M-system

MARIO ESSERT∗, Faculty of Mechanical Engineering and Naval Architecture,University of Zagreb, Ivana Lucica 5, 10000 Zagreb, Croatia.

IVANA KUZMANOVIC∗∗, Department of Mathematics, J.J. StrossmayerUniversity of Osijek, Trg Lj. Gaja 6, 31000 Osijek, Croatia.

IVAN VAZLER†, Department of Physics, J.J. Strossmayer University of Osijek,Trg Lj. Gaja 6, 31000 Osijek, Croatia.

TIHOMIR ZILIC‡, Faculty of Mechanical Engineering and Naval Architecture,University of Zagreb, Ivana Lucica 5, 10000 Zagreb, Croatia.

AbstractThis article introduces a new formal theory, the theory of M -system, which represents a many-valued logic system. Basicterms and concepts are defined which are the foundation for their further application. Though the theory comes from the fieldof electric circuitry, an attempt will be made to extend it to applications in linguistics. To check the examples used in thisarticle, a Haskell program was made, which can be found at https://www.schoolofhaskell.com/user/ivazler/linguistic.

Keywords: M -system, M -words, many-valued logic, Haskell supported

1 Introduction

This article introduces the M-system theory which was discovered by Sare (see [20]) as a validdescription of physical laws in electrical circuits. Our goal is to extend the application of this theorybeyond the interpretation of serial and parallel RLC (resistor, inductor, capacitor) interconnections inelectrical circuits, and to show that using many-valued logic, which can also be suitably representedusing the theory of M-systems, it can be further efficiently used to cope with a certain class ofproblems in linguistics. To support this hypothesis, the formal theory has been suitably implementedusing Haskell programming language, which allows for its direct verification in concrete examples.One such example is the verification of tautology in which premises and conclusion are written indifferent natural languages (English, French and Croatian). The M-theory based program code willgive the correct answer irrespective of the language used. Additionally, in this article we presentan extension to the truth verification procedure of sentences composed in a natural language, whichis based on mapping the variables from the M-system many-valued logic to the words of naturallanguage that are in semantic array or in ontology. Logic operations have their impact on mutualdependences between words.

∗E-mail: [email protected]∗∗E-mail: [email protected]†E-mail: [email protected]‡E-mail: [email protected]

Vol. 25 No. 5, © The Author 2017. Published by Oxford University Press. All rights reserved.For Permissions, please email: [email protected]:10.1093/jigpal/jzx025 Advance Access published 16 August 2017

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Theory of M-system 837

FIG. 1. Serial connection between a capacitor and a resistor.

In the remainder of this introduction we present a reconstruction of (the most likely) path whichled Sare to his discoveries. The initial goal of his research was to devise a systematic symbolicrepresentation which would uniquely describe an electrical circuit. His desire was to be able to fullyreconstruct all the components of a circuit and their interconnections (serial/parallel configurationof the network) simply by reading such symbolic representation. He named this representation the‘M-word’. The discovery that the reconstruction of the elements in a network and of its configurationwas independent of whether the word is read from left to right or from right to left led Sare to thinkof the M-word as a number. However, since numbers are well defined and since he could not mapM-words to any of them (i.e. whole, real, complex,...), he decided to introduce a new technical termfor the M-word, and that is ‘jorb’ (pronounced: ‘yorb’).1

In Section 2, we present the formal M-system theory as quadruple based on two sets (alphabet andM-words inside M-space), a binary relation (less than ‘<’) and on a binary operation (concatenation).

Section 3 describe unary and binary operators which act as permutations of an M-word. Fromthe logic point of view, particularly interesting are the unary operators defining dual (D), as wellas the operators of negation (K). The procedure of making serial and parallel interconnections ofM-words, which is presented in the introduction on concrete physical model, is further formalizedin this section using binary operations. These operations present generalization for conjunction anddisjunction from logic algebra.

In Section 4, the M-system is presented as a multi-valued logical system, together with the basiclogic relations and their interconnection with classical logic.

In Section 5, we present one of the more possible partitions of logic space, from which, as aspecial case, we reconstruct Dunn/Belnap’s B4 and indicate possible extensions (in universal logic).Then, several examples from linguistics are presented and one tautology which is used to illustratethe efficiency of the formalized algorithm running on a computer.

Finally, in the last section we indicate future research paths which will be based on the resultsfrom the book ‘Jorbologija’ [20] of Sare and from the results of this article.

1.1 The physical basis of the M-system

Electrical impedance is the measure of the opposition that a circuit presents to a current when avoltage is applied. Figure 1 shows a serial connections between a capacitor (C) and a resistor (R)in two possible forms: CR and RC connection. For the same values of R and C, both connectionshave the same impedance. For instance, if ZC is the impedance (reactance) of a capacitor and ZR isa impedance (resistance) of a resistor, then the impedance of the CR is ZCR =ZC +ZR and of the RCis ZRC =ZR +ZC . Obviously, it follows that ZCR =ZRC .

1We note that the word ‘jorb’ is the word ‘broj’ spelled backwards. ‘Broj’ in Croatian means ‘number’. Analogously,‘jorb’ would in English be ‘rebmun’.



838 Theory of M-system

FIG. 2. A serial connection (denoted by ∪) between the marked two-pole elements.

Sare tried to find a way to represent the impedance of a serial CR and RC connection through aunique simple representation. Both a capacitor and a resistor are two-pole elements, and both polesof a capacitor and resistor can be marked by characters as shown in Figure 2.

If C two-pole impedance is presented as M -word aa and R two-pole impedance bb, then it couldbe possible to represent a serial CR impedance as aa ∪ bb and a serial RC impedance as bb ∪ aa.These impedances should be equal, so (aa ∪ bb)=(bb ∪ aa).

To make notation even simpler, we introduce A≡ aa and B≡ bb, then (A ∪ B)=(B ∪ A). Thenext idea is to present both (A ∪ B) and (B ∪ A) with a unique representative, for instance, usinga simple joining of characters: AB instead of A ∪ B and also AB instead of B ∪ A. Instead ofAB, BA could also taken, but after that decision, only that representation can be used in the work,because one choice excludes the other.2

Task: to find a rule for a serial connection ∪ between two two-pole impedances that satisfy bothAB ≡ A ∪ B, and AB ≡ B ∪ A.

(1) A ∪ B≡ AB. This is equal to: (aa ∪ bb)≡ aabb. The following question arises: What can beplaced instead of ∪ in order to achieve aabb. The physical example in Figure 1 shows that thereis a wire that connects R and C two-pole elements. Since we already have C and R impedancemarked with aa and bb, respectively, there remain two options for the impedance of ordinarywire: ab or ba.In the first case, the total impedance is aaabbb, and in the second case it is aababb.

However, Sare introduced the following string-compressing rule:

x=xyx (1)

for any x, y∈{a,b} and applied it to the two strings, i.e. impedances:(a) aaabbb → aaabbb (or aaabbb) → abbb (or aaab) → ab(b) aababb → aababb (or aababb) → aabb.

The second (b) solution is the good one. So, ordinary wire has the impedance ba.

(2) B ∪ A≡ AB. This is equal to: (bb ∪ aa)≡ aabb. The physical example in Figure 1 shows thatthere is a wire that connects R and C two-pole elements.If we use ba for the impedance of a wire, as we did in the last experiment, we get bbbaaa. Ifwe apply rule (1) to bbbaaa, we get ba, because a≡aaa and b≡bbb. Unfortunately, we didnot get aabb. What should be done with ba to get aabb? Is it even possible?Sare tried with a new, wrapping rule:

aa x bb→y. (2)

For this, particular, aforementioned jorb x=ba, he joined aa on the left of jorb and bb on theright of the same jorb, getting aababb. Now applying rule (1) to it, he indeed got y as aabbi.e. AB.

2Note: The impedance AB is not equal to BA, which could represent an impedance of other (parallel) type of connectionelements. The chosen impedance should be equal for both serial connections (A ∪ B) and (B ∪ A).




It remains to check whether the application of the same wrapper to the previous experimentwill change its result.We apply this wrapper to aa ∪ bb i.e. to aababb (which is equal to aabb, as we have alreadyshown):

aa aabb bb→aaaabbbb→aabb.

Now, simultaneously using zipping (1) and wrapping (2) rules, we again get aabb, so it didnot spoil the initial result. It can be concluded that the second rule is valid for both B ∪ A andA ∪ B.These two rules show a unique way of transformation how a serial connection of two two-poleimpedances can be represented as a simple M-word.

Finally, a serial connection ∪ between two two-pole impedances can be generalized as follows:

(d.1) define an M-word as a result of a serial connection, i.e choose AB.(d.2) suppose that characters ‘a’ and ‘b’ are sorted, i.e. say that a is a less-valued character than

b.(d.3) between aa and bb or bb and aa put a greater-value character on the left and a less-value

character on the right (like ba) to finally achieve aa ba bb or bb ba aa.(d.4) apply the second (wrapping) rule as: put two less-value characters (like aa) to the left of

aa ba bb or bb ba aa and put two greater-value characters (like bb) to the right of aa ba bbor bb ba aa. This will produce aa aababb bb or aa bbbaaa bb.

(d.5) after applying the first rule3 (1) to both previous results aa aababb bb or aa bbbaaa bb, thesame jorbs will appear: aabb or aabb, i.e. AB or AB, respectively.

which can also be expressed as a simple structure:A serial impedance of CR

(d.1) aabb ≡ aa ∪ bb(d.2) a < b(d.3) aa ba bb(d.4) aa aababb bb(d.5) aabb ≡ AB

A serial impedance of RC

(d.1) aabb ≡ bb ∪ aa(d.2) a < b(d.3) bb ba aa(d.4) aa bbbaaa bb(d.5) aabb ≡ AB

2 M-system theory

The notion of the M -system is based on two sets:

(1) set �, called an alphabet, which is a finite totally ordered set (chain) of symbols ordered bybinary relation <� ,

(2) set or space M� over � which consists of all words made of an even number of symbols from�, called M-words.

A binary operation of joining (sticking) two symbols from � end-to-end is called concatenationand it is denoted by ’·’. For practical reasons, we often write ab instead of a·b.

3Note: the first rule (1) will mean: a≡aaa and a≡aba, b≡bbb and b≡bab.




If n is a cardinal number of the set �, then the base �0 of the space M� is the set consisting of n2

elements, containing all words obtained by a concatenation of two symbols from �, which are alsocalled atoms.

The concatenation on the set M� is defined in a similar way to the concatenation of symbols, thatis for x=x1 ···x2n,y=y1 ···y2m ∈M� , x ·y=x1 ···x2ny1 ···y2m . It is easy to see that a word created bya concatenation of words from the set M� has an even number of symbols so it also belongs to M� .Every M -word is an atom or it is obtained as a concatenation of one or more atoms. For concatenatedM -words x and y, notation xy is also used instead of x ·y.

Finally, the M -system is a quadruple (�, <� , M� , ·).

The set � may have an arbitrary number of symbols, but, as a finite totally ordered set, it isbounded by some starting (α) and ending (ω) symbol. The order between the symbols of the chain isdefined by the binary relation <� , so that any two symbols are comparable in size (order). Usually(but not necessarily), elements of the alphabet � are lowercase letters of the English alphabet ordigits. Regardless of whether the symbols in the alphabet are digits of the decimal number system(0,...,9) or letters of some usual alphabet, the distance between adjacent symbols in the chain � isalways defined to be 1. The distance between two arbitrary symbols a and b from � is defined as

δ(a,b)=|val(a)−val(b)|,that is, as an absolute value of the difference of valuations of symbols, where the valuation of thesymbol is its order in the chain � defined by the relation <� . In the realization of M -systems incomputer programs, <� can be defined by ASCII sorting and the distance can then be calculated bytaking the difference of the integer values associated with ASCII equivalents of the symbols from �.

Alphabet centre � is equally distant from the alphabet’s starting and ending symbols (α and ω). Ithas an associated symbol if the number of symbols in � is odd, while in the case of an even numberof symbols in the alphabet, the center has no associated symbol.

Two symbols a1 and a2 are mutually dual if they are equally distant from the alphabet edges, thatis, if δ(a1,α)=δ(a2,ω), or, equivalently, if δ(a1,ω)=δ(a2,α). For word a=a1a2 ...a2n from M� itsdual word is a′ =a′

1a′2 ...a

′2n where a′

1,a′2,...,a

′2n are dual symbols corresponding to a1,a2,...,an.

The starting symbol of the M -word x, is denoted by l(x), and the ending symbol by r(x). For thesake of simplicity, we will sometimes denote them by lx and rx. The shell q(x) of an M -word x isdefined as the concatenation of the starting symbol l(x) and the ending symbol r(x):

q(x)= l(x)·r(x).

The dual symbol of the starting symbol of M-word x will be denoted by l′(x) or l′x, while a dualsymbol of the ending symbol will be denoted by r′(x) or r′

x.Unlike the usual notion of word length, which refers to the number of symbols/letters in a word,

the length of an M -word, denoted by λ, characterizes its ‘waviness’ and it is defined as the alternatingsum of distances between adjacent letters in the M -word by the following formula:

λ(a1a2a3 ...a2n)=−δ(a1,a2)+δ(a2,a3)− ...−δ(a2n−1,a2n).

The length of an M -word defined in this way allows us to define the relation ≡zip on M� ofcompression of the M -word in the following way:

a1a2 ···a2n ≡zip b1b2 ···b2k for k <n




if and only if

(a1 =b1)∧(a2n =b2k )∧(λ(a1a2 ...a2n)=λ(b1b2 ...b2k )).

It is easy to see that ≡zip is a relation of equivalence on the set M� .For two M -words to be equivalent (with respect to the relation ≡zip ) it is necessary (but not

sufficient) that they both have the same shell.Two simpler operations on � will be used - infimum and supremum, which will be denoted by

’↓’ and ’↑’, respectively. The first operation is defined as the smaller of the two, and the other oneis defined as the greater of the two symbols with respect to the relation <� .

Example 1.4

For an alphabet �={a,b,c,d,e,f ,g}, and an order a<� b<� c<� d <� e<� f <� g we can concludethe following:

• the starting symbol is α=a, the ending symbol is ω=g, while the center of the alphabet is�=d;

• the base �0 consists of 49 atoms:

�0 ={aa,ab,ac,...,ag,ba,bb,bc,...,bg,...,ga,gb,...,gg};• the M -word x=abbcdddc=aBcDdc is the concatenation of 4 atoms from �0: x=ab·bc ·dd ·dc;• the dual word of x is x′ =gffeddde;• the starting and ending symbols of x are l(x)= lx =a and r(x)=rx =c, and of its dual are

l′(x)= l′x =g and r′(x)=r′x =e;

• shell: q(x)= l(x)·r(x)=a·c=ac;• the length of x is given by λ(abbcdddc)=−δ(a,b)+δ(b,b)−δ(b,c)+δ(c,d)−δ(d,d)+δ(d,d)−

δ(d,c)=−1+0−1+1−0+0−1=−2;• compression: as λ(ac)=−δ(a,c)=−2, it follows that λ(abbcdddc)=λ(ac), so x=abbcdddc≡zip

ac;• infimum and supremum: d ↓ f =d, f ↑g =g, l′(x)↓r′(x)=e.

3 Unary and binary operators on M�-space

Several unary operators on M�-space will be defined, which are indeed permutations of M -words.Some of them affect all symbols in the M -word, and some affect only the beginning and endingsymbols of the M -word and its dual. Operators which affect all symbols in the M -word are denotedby D, E and F . Operator D is called dual, because it replaces each symbol in the M -word with itsdual symbol, i.e., it replaces the M -word with its dual word:

(∀a1,a2,...,a2n ∈�) (D(a1a2 ...a2n)=a′1a

′2 ...a

′2n).

Operator E flips the M -word, changing the order of symbols from left to right and vice versa:

(∀a1,a2,...,a2n−1,a2n ∈�) (E(a1a2 ...a2n−1a2n)=a2na2n−1 ...a2a1).

4This and other examples mentioned in the article can (only) be downloaded from: https://www.schoolofhaskell.com/user/ivazler/linguistic. Unfortunately, since 2016 ‘School of Haskell’ has disabled the execution of online programs(interactive code snippets).


https://www.schoolofhaskell.com/user/ivazler/linguistic



TABLE 1. Cayley table for ({1,D,E,F},◦)

◦ 1 D E F1 1 D E FD D 1 F EE E F 1 DF F E D 1

Operator F is the composition of operators D and E

(∀x∈M�) (F(x)=D(E(x))=E(D(x))).

It is easy to see that all these operators keep the λ-length of the M -word unchanged:

(∀∈={D,E,F}) (∀x∈M�) (λ((x))=λ(x)).

For example,

λ(E(a1a2 ...a2n−1a2n))=λ(a2na2n−1 ...a2a1)

=−δ(a2n,a2n−1)+δ(a2n−1,a2n)−···+δ(a2,a3)−δ(a1,a2)

=−δ(a1,a2)+δ(a2,a3)−···+δ(a2n−2,a2n−1)−δ(a2n−1,a2n)

=λ(a1a2 ...a2n−1a2n).

Notice that operators D, E and F with identity (denoted by 1), make Klein four-group (see [12])(see Table 1), under the composition operation (◦).

EXAMPLE 2To visualize the symmetries, we define an alphabet �={ ,|,⊥} with the following order: <� |<� ⊥. The operators D, E and F act on M -words over this alphabet as follows:

(1) D( ||⊥⊥|| )=⊥⊥ || || (flipping over the horizontal axis)

(2) E(⊥⊥ || || )=|| ||⊥⊥ (flipping over the vertical axis)

(3) F( | | ||⊥⊥)= ||⊥⊥ || (flipping over the horizontal and vertical axis).

The unary operators I , K , G, H are defined in the following way:

(∀x∈M�) (I (x)= l(x)·x ·r(x)), (3)

(∀x∈M�) (K(x)= l′(x)·x ·r′(x)), (4)

(∀x∈M�) (G(x)=r(x)·x ·l(x)), (5)

(∀x∈M�) (H (x)=r′(x)·x ·l′(x)). (6)

All operators � ∈{I ,K,G,H } are involutions (7), and they keep the distance between the startingand ending symbols of the M -word unchanged (8):

� ◦� =1 (7)

(∀x∈M�) (δ(l(�(x)),r(�(x)))=δ(l(x),r(x)). (8)




For example,

(I ◦I )(x)= I (I (x))= I (lxxrx)= lxlxxrxrx ≡zip x,

so that for the operator I , property (7) is fulfilled. Property (8) follows from the fact that l(I (x))= l(x)and r(I (x))=r(x), hence, it directly follows that

δ(l(I (x)),r(I (x)))=δ(l(x),r(x)).

EXAMPLE 3Let the set of symbols � be the English alphabet �={a,...,z}.Let x=croatian≡zip croian and y= language≡zip le

The equivalence is the consequence of equal shells and equal λ- lengths.

D(x)=D(croatian)=xilzgrzm≡zip xilrzmE(x)=E(croatian)=naitaorc≡zip naiorcF(x)=F(croatian)=mzrgzlix≡zip mzrlix

D(xy)=D(x)D(y); E(xy)=E(y)E(x); F(xy)=F(y)F(x);E(croatianlanguage)=E(language)E(croatian)=egaugnalnaitaorc≡zip elnaiorc

I (x)= I (croatian)=CroatiaN ≡zip CroiaNK(x)=K(croatian)=xcroatianm≡zip xoianmG(x)=G(croatian)=ncroatianc≡zip ncroicH (x)=H (croatian)=mcroatianx≡zip mcroianx.

Two binary operations are defined on M� space. They are called serial (�) and parallel ( �)connection. In the theory of linear electrical networks, these operations correspond to the serial andparallel connection of electric twoports as explained in the Introduction.

A serial connection of M -words x and y is defined as

x�y= ii ·x ·tj ·y ·ss,

where i is the infimum of the starting symbols of x and y, t is the supremum of r(x) and l(y), j is theinfimum of r(x) and l(y), while s is the supremum of the ending symbols of x and y, that is

x�y= (lx ↓ ly)·(lx ↓ ly)·x ·(rx ↑ ly)·(rx ↓ ly)·y ·(rx ↑ry)·(rx ↑ry). (9)

A parallel � connection of M -words x and y is defined as

x �y=ss ·x ·jt ·y ·ii,

where s is the supremum of the starting symbols of x and y, j is the infimum of r(x) and l(y), t isthe supremum of r(x) and l(y), while i is the infimum of the ending symbols of x and y:

x �y= (lx ↑ ly)·(lx ↑ ly)·x ·(rx ↓ ly)·(rx ↑ ly)·y ·(rx ↓ry)·(rx ↓ry). (10)




All the above operations are associative and closed on M� , so that the set M� forms semigroups(M� , �) and (M� , �) with respect to the indicated operations. The neutral element in (M� , �) is ωα

and is called s-zero. Similarly, αω is a neutral element in the semigroup (M� , �) and is called p-zero:

(∀x∈M�) (x�ωα=x ∧ x �αω=x).

For example, to justify the equality x�ωα=x, it is easy to see that x and x�ωα= lxlxxωrxωαrxrx

have the same shell and their λ length values are

λ(x�ωα)=λ(lxlxxωrxωαrxrx)

=−δ(lx,lx)+δ(lx,lx)+λ(x)+δ(rx,rx)−δ(rx,ω)+δ(ω,α)−δ(α,rx)+δ(rx,rx)

=λ(x)−δ(α,rx)−δ(rx,ω)+δ(ω,α)=λ(x),

where the last equality follows from the fact that δ(α,rx)+δ(rx,ω)=δ(α,ω). These two operationscan also be called M-conjunction and M-disjunction, because they generalize conjunction and dis-junction in logical algebra, so that we retain the same notation ∧� and ∨� .

EXAMPLE 4Let the set of symbols be the English alphabet, �={a,...,z}.Let x=croatian≡zip croian and y= language≡zip le. Then the serial and parallel connections aregiven by

x�y= (c↓ l) (c↓ l)·croatian ·(n↑ l)·(n↓ l)·language ·(n↑e)·(n↑e)

=cccroatiannllanguagenn=CcroatiaNLanguageN ≡zip croian,

x �y= (c↑ l) (c↑ l)·croatian ·(n↓ l)·(n↑ l)·language ·(n↓e)·(n↓e)

= llcroatianlnlanguageee=LcroatianlnlanguagEe≡zip lroianle

y�x=ClanguagECroatiaNn≡zip croian=x�y ,

y �x=LlanguagececroatianE ≡zip lecroiee≡zip lroianle=x �y.

Note that these operations are commutative.

4 M -system as many-valued logic system

In the M -system theory, the truth value of an M -word x is completely determined by its shell q(x).As the shell of each M -word can be one of n2 atoms made from symbols from an alphabet with nelements, it obviously follows that the M -system will represent a many-valued logic system.

4.1 AND and OR logic functions

Even words from a 2-generator system (�={a,b}) will have four possible truth values, and to thoseextremes we can join the classical concept of truth and falsity: ba= and ab=⊥. The other twovalues (aa and bb) may be declared as a kind of ambiguity or uncertainty (see [9], [10]).

Since the serial interconnection is represented by the logic function AND, and the parallel inter-connection by the function OR, both symbols are emphasized in the caption of Table 2. The first




TABLE 2. Truth values of the AND (∧�) and OR (∨�) logical functions for 2-generator atoms

TABLE 3. AND (∧�) and OR (∨�) logical functions for the only two classical Boolean values

∧� ba abba ba abab ab ab (abbaab)

∨� ba abba ba (baabba) baab ba ab

symbol indicates the algorithm (defined in (9) and (10)) used to obtain the elements in the table, whilethe second symbol indicates the corresponding logic function. In this way, for example, algorithm(9) for x=aa and y=bb, x∧� y gives the following:

x�y= (lx ↓ ly)·(lx ↓ ly)·x ·(rx ↑ ly)·(rx ↓ ly)·y ·(rx ↑ry)·(rx ↑ry)

aa�bb= (a↓b)·(a↓b)·aa·(a↑b)·(a↓b)·bb·(a↑b)·(a↑b)

=a·a·aa·b·a·bb·b·b=aaaababbbb

=zip ab

q(ab)=ab

as presented in Table 2, as well. Analogously, one can obtain/calculate any other entry in the table,or any other entry in the OR table (∨�), using rule (10). Using the unary operator K (4) with anyof the two functions, we can obtain all other functions. In the remainder of the article, we will useonly the symbols of logic algebra, having in mind that their realizations are made by the unary andbinary functions of the M-system.

Classical tables for the logical operations AND and OR can be obtained using only atoms declaredas true and false (ba and ab), as shown in Table 3.

This illustrates a well-known fact that only true AND (∧�) true give true (ba), while the rest isfalse and only false OR (∨�) false provide false (ab). All other combinations give true. In Table 3,the complete M -word obtained by the AND and OR operation is written as well (in parentheses),not only its shell. In fact, it is worth noting that the M -system is also qualitatively different from theclassical one — two falsities in the AND function have double weight (i.e. the obtained expressioncannot be compressed), just like two truths in the OR function.

Note that Table 2 has a block structure with two different 2×2 blocks, where one of them appearsseveral times:

∧� :[

X2 Y2

Y2 Y2

]∨� :

[V2 V2

V2 U2

](11)

for every 2-generator system, and even for (as we will see soon) groups of atoms.

The same procedure can be done for the n-generator M -system, giving thereby results which aretheoretically founded but in a useful algorithmic form (applicable for computer programs).




TABLE 4. AND logic function for 3-generator atoms

TABLE 5. OR logic function for 3-generator atoms

Let us take, for example, a 3-generator M -system with �={a,b,c} in which, as in the 2-generatorcase, Table 4 for the AND (∧�) function is obtained. Let us derive some conclusions from it.

In Table 4, only the shells of M -words obtained by ∧� applied to atoms are provided. Of course,some of the results which are obtained cannot be compressed (as was also shown in Table 3): cc∧�

ba=bacc, cc∧� bb=bbcc, cb∧� aa=aabb, cc∧� ab=aacc, bb∧� aa=aabb, bc∧� bc=bccbbc,bc∧� aa=accabc, aa∧� bc=aabc, ab∧� ab=abbaab, ab∧� ac=abbaac and ac∧� ac=accaac.

A similar table is also constructed for the OR (∨�) function on 3-generator atoms (Table 5), and itcan be compared with Table 4.

To be able to better observe the structure of tables with AND and OR, numbers from 1 to 9 areassigned to atoms (Table 6), and the tables are divided by lines into sub-matrices.5

∧� :⎡⎣ X3 Y3 Z3

Y3 Y3 Z3

Z3 Z3 Z3

⎤⎦ ∨� :

⎡⎣ W3 W3 W3

W3 V3 V3

W3 V3 U3

⎤⎦, (12)

where matrices X3,Y3,U3,V3 are like X2,Y2,U2,V2 from (11) but with one extra column and row whichcorrespond to the additional alphabet symbol. So, tables for a 3-generator system are obtained byextending the sub-matrices from the 2-generator system and by adding one new block-row andblock-column (the last row and column for ∧� and the first row and column for ∨�) consisting of

5This will also be done with the function of implication and equivalence, because we want to highlight the internalstructure of many-valued logic spaces.




TABLE 6. Table of inner structures for AND and OR functions

one new block-matrix (Z3 for ∧� and W3 for ∨�). The tables for n-generator systems also have thesame form, obtained recursively from the tables for systems of a smaller dimension.

Applications of the M -system using computers can be realized by mapping individual M -atoms towords or sentences or wrapping words/sentences with the corresponding shell. Truth tables for anylogical function of classical Boolean algebra can be checked in a strictly formal way with any wordof natural language, which can be assigned by a certain logical value. Function τ assigns to eachshell of an M -word (i.e. to atoms from the �0 base) its truth value. For example, for 3-generatorM� , atoms are from the set {aa,ab,ac,ba,bb,bc,ca,cb,cc}, with truthfulness according to an agreedtruth table. The truth value of complex propositions will then be obtained by logical operations fromM -structures.

Thus, the truth value τ of a conjunction of two false atomic propositions is τ (ac∧� ac)=τ (accaac)=τ (ac). Analogously, τ of a disjunction of two true propositions is τ (ca∨� ca)=τ (caacca)=τ (ca).Shells obtained by these operations give ac falsity and ca truth, as expected.

It should be noted that the conjunction of two false propositions or the disjunction of two truepropositions is ‘more false’ or ‘more true’ than just one such proposition. The quality of truth is stilljust a shell ac, that is, ca, but the quantity in both cases is higher — the resulting words (accaac,respectively caacca) cannot be compressed.

In logical algebra, which often explains logical results via turned on/turned off bulbs, it can besaid, that it can be seen if the bulb glows from an M -word because it is connected to only one, or twoor more switches. An M -system, beside the quality, also introduces quantity to logical operations.

EXAMPLE 5Let x=robin, y=chicken, z=penguin, w=cow be animal species (see [15],[14]), where the firstthree come from bird species and the last one from mammals. By the evidence of their belonging tothe species of birds, we could assign ca to x, ba to y, aa to z, while for w, we can assign the biggestuntruth ac from the 3-generator M -system. The truths of certain compound statements, accordingto Table 4 and Table 5, are:

τ (robinANDchickenORcow)=τ (robin�chicken �cow)=τ (ca∧� ba∨� ac)=τ (ba∨� ac)=ba= (medium truth)

If the variable is wrapped by the associated logical shell, then it is written by capital letters, becausethis ensures the number of symbols in the M-word is even (note that this does not affect the shell of




the word, i.e. it’s truth value):

τ (penguinORchickenANDcow)=τ (aPENGUINa �bCHICKENa�aCOW c)=τ (bPENCKENGUANa∧� aCOWc)=τ (aNEPbaUGNCWANEKc)=ac=⊥

4.2 Implication and equivalence

In M -system theory, the binary operations � and �, i.e. s- and p- connection (addition), correspond,as shown, to the conjunction (AND) and disjunction (OR) from classical algebra. The complemen-tarity operator K in M -system theory corresponds to negation from logical algebra. Therefore, theimplication from logical algebra, in M -system theory, corresponds to the binary operations K(x)�y

and K(x) �y which are called s-implication (symbol:s−→), and p-implication (symbol:

p−→ ):

(∀x,y∈M�) xs−→y=K(x)�y , x

p−→y=K(x) � y. (13)

It makes sense for these M� operations to be called implications, because they follow the logicalrule of origin (p−→q is logically equivalent to ¬p∨q) and are valid in the rules of propositionallogic (see Example 6).

The p-implication (see Table 7) is false if and only if the antecedent in the implication is true andthe consequent is false,6 as it is known in logical algebra:

(∀x,y∈M�) ((xp−→y)∈⊥� ⇐⇒ (x∈ � ∧y∈⊥�)). (14)

The s-implication in the M -system gives a dual logical table compared to the p-implication table,as can be seen from Table 8.

EXAMPLE 6(Two rules of inference)∀x,y∈M� holds:

Modus ponendo ponensLogical formula: τ ((x ∧� (x

p−→y))p−→y)= �

Note: This law ensures, for example, that if something is chicken and chicken is a bird, then that isalso a bird:τ ((CHICKEN ∧� (CHICKEN

p−→BIRD))p−→BIRD)

=τ ((CHICKEN ∧� xCICKENmBRD)p−→BIRD)

=τ ((CICKeCICKENmBRDN )p−→BIRD)

=τ (xCICKeCICKENmBRDNmBRD)= �

6Remark: So far, due to commutativity, the order of variables in AND and OR logic functions has been arbitrary. However,since the implication is not commutative, logical tables for s- and p- implications should be written and interpreted correctly.The columns of the logic table correspond to the antecedent and the rows to the consequence. This, for example, means that

bcp−→cc=ba is obtained from Table 7 as p-implications of the sixth column and the third row.




TABLE 7. p-implication (p−→) for 3-generator atoms

TABLE 8. Table of inner structures for p-implication and s-implication

Dual modus ponendo ponensLogical formula: τ ((x ∨� (x

s−→y))s−→y)=⊥�

Note: This law is the dual of the previous law. Note that the p- and s- marks, and also ∧� and ∨� ,are interchanged:τ ((CHICKEN ∨� (CHICKEN

s−→BIRD))s−→BIRD)

=τ ((CHICKEN ∨� BcICKEnRDM )s−→BIRD)

=τ ((CICKENBcICKEnRDM )s−→BIRD)

=τ (BcICKENBcICKEnRDnRDN )=⊥�

We can notice that each sub-matrix X , Y , Z has the same structure

p−→:⎡⎣X Y Z

X Y YX X X

⎤⎦ s−→:

⎡⎣X ′ Y ′ Z ′

X ′ Y ′ Y ′

X ′ X ′ X ′

⎤⎦ (15)

for every set of atoms (domain) and similarity with the main matrix. At the same time, it is easyto notice the dual (D(·)) structure of the (sub)matrices with respect to the p- and s- implication:X =D(X ′), Y =D(Y ′) i Z =D(Z ′). Using these properties, one can easily generate matrices of ahigher order.




TABLE 9. Table of inner structures for p-equivalence and its dual s-equivalence

Here, p-equivalence (symbolp⇐⇒) corresponds to the standard equivalence in classical logic, while

s-equivalence (symbols⇐⇒) corresponds to the exclusive disjunction in classical logic.

(∀x,y∈M�) xp⇐⇒y = (K(x) � y) � (K(y) � x) (16)

(∀x,y∈M�) xs⇐⇒y = (K(x) � y) � (K(y) � x). (17)

Both operations are commutative and associative.As can be seen from Table 9, s-equivalence generates a dual matrix compared to the matrix of

p-implication. We can also notice that each matrix X , Y , Z has the same structure:

p⇐⇒:⎡⎣X Y Z

Y Y YZ Y X

⎤⎦ s⇐⇒:

⎡⎣Z ′ Y ′ X ′

Y ′ Y ′ Y ′

X ′ Y ′ Z ′

⎤⎦ (18)

for any group of atoms (domain) and is similar to the main matrix. It is also easy to notice the dualnature of the sub-matrices with respect to p and s equivalence: X =D(Z ′), Y =D(Y ′) i Z =D(X ′).Higher-order matrices can easily be generated.

5 Partition of logical space

While in a 2-generator model the atom ba is associated with the truth, and to falsity ab, in a 3-generator system, the truth can be ca, and falsity is the atom ac. However, ba can remain truth, butslightly smaller. Of course, in that case aa would also be truth, but it is the smallest one. Accordingly,cc, bc and ac are untruths, listed in the order from the smallest to the largest. Atoms with the symbolb at the end are uncertainties in the 3-generator system, also with a certain intensity with respect tothe first symbols of atoms. In the 2-generator system these atoms could be untruths, depending onthe observed domain. It is important to emphasize that the atom can have a different interpretation,depending on the domain in which it is located. This is consistent with the Bochvar and Lukasiewiczlogical view (see [5, 16]).

In this way, the symbols of the shells of M -words carry information about the level of truth.The set �0 of all atoms can be partitioned into subsets where atoms with the same first symbol areelements of the same subset. Each of these subsets will present one logical domain. The first symbolcarries information about the hierarchical logic domain, level of truth, while the other (second)carries information on the level of untruth within that domain. If the untrue is larger, the second




symbol of the atom will be farther away from the starting symbol of the alphabet. The heaviestuntruth will have the second symbol equal to the ω-symbol of the alphabet, in any domain. Thelargest truth, of course, will have the first symbol equal to the ω symbol of the alphabet, and allother truths will have first symbols which are lexicographically smaller.

However, the described partition of logical space in the substructures that carry all the featuresof the above structures is not the only way for the logical partition of a set of shells derived froman n-generator system. With the following partition, each set of M -words can be divided into fivedisjoint subsets, i.e. every M -word x is an element of exactly one of the following sets:

(1) � ={x∈M� : (lx = (lx ↑ l′x) ∧ rx = (rx ↓r′x)) ∧ (lx >� rx)}

(2) A� ={x∈M� : (lx = (lx ↓ l′x) ∧ rx = (rx ↓r′x)) ∧ ¬(lx = l′x ∨ rx =r′

x)}(3) 0� ={x∈M� : (lx = l′x ∧ rx =r′

x)}(4) Z� ={x∈M� : (lx = (lx ↑ l′x) ∧ rx = (rx ↑r′

x)) ∧ ¬(lx = l′x ∨ rx =r′x)}

(5) ⊥� ={x∈M� : (lx = (lx ↓ l′x) ∧ rx = (rx ↑r′x)) ∧ (lx <� rx)}.

The set of the above subsets of M�-space is called the logical partition of M�-space and will bedenoted by L� for each alphabet �:

L� ={ �,A�,0�,Z�,⊥�}. (19)

EXAMPLE 7For alphabet �={1,2,3,4,5}, with standard ordering by the relation <, the partition of base �0 hasthe following form:

(1) � ={53,52,51,43,42,41,32,31},(2) A� ={11,12,21,22},(3) 0� ={33},(4) Z� ={55,54,45,44},(5) ⊥� ={13,14,15,23,24,25,34,35}.

The relation of equivalence generated by the partition L� is denoted by ∼� and it is called the logicalequivalence relation of M -words over the alphabet �:

(∀x∈M�) x∼� y⇔ (∃X ∈L�)(x,y∈X ).

The projection of the set M� on L� is denoted by τ and is called the truthful quality of an M -word:

(∀x∈M�)(∀Y ∈L�) (τ (x)=Y ⇔ x∈Y ).

If the cardinality of the alphabet � is even, then the alphabet centre does not have a joined alphabetelement, so the 0� set is an empty set. In that case, Cayley7 tables for ∧� and ∨� operations aregiven in Table 10.

The described set L� together with the operator K (negation) represents a Boolean algebra, i.e. acomplemented distributive lattice ({ � , A� , Z� , ⊥�}, K , �, �) (see [6], [8]).

7British mathematician (1821–1895).




TABLE 10. Cayley tables for ({ �,A�,Z�,⊥�},∧�) and ({ �,A�,Z�,⊥�},∨�)

∧� � A� Z� ⊥�

� � A� Z� ⊥�

A� A� A� ⊥� ⊥�

Z� Z� ⊥� Z� ⊥�

⊥� ⊥� ⊥� ⊥� ⊥�

∨� � A� Z� ⊥�

� � � � �

A� � A� � A�

Z� � � Z� Z�

⊥� � A� Z� ⊥�

TABLE 11. Dunn/Belnap’s B4 logic

∧ T B N FT T B N FB B B F FN N F N FF F F F F

∨ T B N FT T T T TB T B T BN T T N NF T B N F

On the other hand, classical two-valued Boolean algebra ({⊥, },¬,∧,∨) which has only twotruth values (true and false) is isomorphic to the M-structure ({ �,⊥�},K,∧�,∨�).

The given logical partition L� includes Dunn/Belnap’s-logic (B4)8 (Table 11) which combinesKleene’s three-valued K3 and Priest’s three-valued logic P3 (see [1, 11, 17]). A set of values A�

corresponds to Dunn/Belnap’s variable B (overdetermined truth value), while Z� corresponds to N(underdetermined truth value). Of course, � and ⊥� in the table are denoted by T (true) and F(false).

The result obtained after applying logical functions on M-words appears in one of the partitionsets (Table 10), which are made by the partition of logical space as described above. If the partitionis reduced to only two sets (instead of four, with 0 exception), then we have classical logicalalgebra, with only two Boolean values. This means that although the n-generator M -system offersthe possibility of many-valued logic, it is up to us how many logical variables or logical domainswe use.

From a philosophical and linguistic point of view, this means that managing the truthfulness ofwords and/or sentences can have a dynamic character — in the case of the expansion of context,the number of logical variables can be extended, or within the same domain, more ambiguities canbe created (with different intensities). If one, for example, decides to work only with three logicalvalues, but not in the same domain, but rather in the whole logical space, then besides two classicalvalues (truth- ca and false - ac), for the third, uncertainty value, one can choose between cb, bb andab. One of the resulting tables, Table 12, shows the same properties as the original table from whichit came. For three more sub-matrices of other logical domains also have these properties. The sametables are shown using the symbols of logical algebra ( , | and ⊥) to better highlight the symmetryof these basic logic functions.

With the choice of four, instead of three logical variables, e.g. with the M-shells: da, ca, ac, ad,(‘da’: all Ss are Ps, ‘ac’: no Ss are Ps, ‘ca’: some Ss are Ps, ‘ad’: not all Ss are Ps) we can investi-gate the Aristotle’s famous square (Square of Opposition) AEIO contraries and contradictories, butalso mappings to Necessary (A), Impossible (E), Possible (I) and Non necessary (O) modality inpropositional modal logic or even universal logic (see [2, 3, 13]).

Modal logic introduces the device of possible worlds. There is a great deal of controversy overwhat possible worlds are, but it is important that they provide a way of understanding possibility

8Table 2 is already the table of Dunn/Belnap’s four-valued logic, with bb=B and aa=N .




TABLE 12. AND and OR truth tables for three Boolean values on the set of 3- generator atoms

∧� ca bb acca ca bb acbb bb bb acac ac ac ac (accaac)

∧ | ⊥ | ⊥| | | ⊥⊥ ⊥ ⊥ ⊥

∨� ca bb acca ca (caacca) ca cabb ca bb bbac ca bb ac

∨ | ⊥ | | |⊥ | ⊥

and necessity: a sentence that is necessarily true is one that is true in every possible world, and asentence that is possibly true is one that is true in at least one possible world.

The existence of hypothetical worlds in the M-system can be assumed in any tables alreadymentioned (e.g. Tables 2 and 7). This particularly holds for the Table 13, which was generated withfour elements of alphabet �={a,b,c,d} (i.e. sixteen M-atoms) which has led to an investigationin the direction of extending Dunn/Belnap’s B4 logic, where significant results have so far beenachieved (see [18, 19]). With each logical value in Table (13) B4 truth-degrees {T ,B,N ,F} a alsowritten from the implemented logical partition, e.g. ’dc:N ’ means that the dc - logical variable in theM-system, has N meaning in B4. Of course, the table is created according to the already describedAND structures (11) and (12), respectively.

The wn sets of sorted M-atoms, where n is a cardinal number of the set �, make possible worldswhich may enter into mutual accessibility relations. The elements of these sets for Table 13 (�={a,b,c,d};n=4) are: w1 = {da, db, dc, dd}, w2 = {ca, cb, cc, cd}, w3 = {ba, bb, bc, bd}, w4 = {aa,ab, ac, ad}.

Verifications of the accessibility relations through the logical function f can be performed withinthe possible worlds in the following way:

• for necessity:

∀jε{1,...,n},∃i | f :wi ×wj →wi,

where i ε{1,...,n};• for possibility:

∃jε{1,...,n}| f :wi ×wj →wk ,

where i,k ε {1,...,n} and k �= i.

It is easy to see, from Table 13, that the logical function AND represents modal necessity for theelements belonging to the set w4 (i.e. the logical relation maps elements of w4 with any set/worldinto the elements of the same set w4).

This is not the case with the set/world w2 because it gives the same set only in relation to w1 anditself, i.e. not in all accessibility relations. Therefore, this is the modal possibility.

Modal logic can be considered completely analogously over the elements of the set. In this way,the logical value ad in Table 13 will remain the same (unchanged) in interaction with an arbitrary




TABLE 13. ‘Sweet SIXTEEN ’ of the M-system

AND da:T db:T dc:N dd:N ca:T cb:T cc:N cd:Nda:T da:T db:T dc:N dd:N ca:T cb:T cc:N cd:Ndb:T db:T db:T dc:N dd:N cb:T cb:T cc:N cd:Ndc:N dc:N dc:N dc:N dd:N cc:N cc:N cc:N cd:Ndd:N dd:N dd:N dd:N dd:N cd:N cd:N cd:N cd:Nca:T ca:T cb:T cc:N cd:N ca:T cb:T cc:N cd:Ncb:T cb:T cb:T cc:N cd:N cb:T cb:T cc:N cd:Ncc:N cc:N cc:N cc:N cd:N cc:N cc:N cc:N cd:Ncd:N cd:N cd:N cd:N cd:N cd:N cd:N cd:N cd:Nba:B ba:B bb:B bc:F bd:F ba:B bb:B bc:F bd:Fbb:B bb:B bb:B bc:F bd:F bb:B bb:B bc:F bd:Fbc:F bc:F bc:F bc:F bd:F bc:F bc:F bc:F bd:Fbd:F bd:F bd:F bd:F bd:F bd:F bd:F bd:F bd:Faa:B aa:B ab:B ac:F ad:F aa:B ab:B ac:F ad:Fab:B ab:B ab:B ac:F ad:F ab:B ab:B ac:F ad:Fac:F ac:F ac:F ac:F ad:F ac:F ac:F ac:F ad:Fad:F ad:F ad:F ad:F ad:F ad:F ad:F ad:F ad:F

For reasons of space, the table is vertically divided into: this left part,

AND ba:B bb:B bc:F bd:F aa:B ab:B ac:F ad:Fda:T ba:B bb:B bc:F bd:F aa:B ab:B ac:F ad:Fdb:T bb:B bb:B bc:F bd:F ab:B ab:B ac:F ad:Fdc:N bc:F bc:F bc:F bd:F ac:F ac:F ac:F ad:Fdd:N bd:F bd:F bd:F bd:F ad:F ad:F ad:F ad:Fca:T ba:B bb:B bc:F bd:F aa:B ab:B ac:F ad:Fcb:T bb:B bb:B bc:F bd:F ab:B ab:B ac:F ad:Fcc:N bc:F bc:F bc:F bd:F ac:F ac:F ac:F ad:Fcd:N bd:F bd:F bd:F bd:F ad:F ad:F ad:F ad:Fba:B ba:B bb:B bc:F bd:F aa:B ab:B ac:F ad:Fbb:B bb:B bb:B bc:F bd:F ab:B ab:B ac:F ad:Fbc:F bc:F bc:F bc:F bd:F ac:F ac:F ac:F ad:Fbd:F bd:F bd:F bd:F bd:F ad:F ad:F ad:F ad:Faa:B aa:B ab:B ac:F ad:F aa:B ab:B ac:F ad:Fab:B ab:B ab:B ac:F ad:F ab:B ab:B ac:F ad:Fac:F ac:F ac:F ac:F ad:F ac:F ac:F ac:F ad:Fad:F ad:F ad:F ad:F ad:F ad:F ad:F ad:F ad:F

…and its right part.

element (because in the AND function it represents ‘maximal falsity’), while da will hold only in its‘own’ set, in the world w1. In this case, the function AND represents modal necessity for the ‘ad’element, while the same function represents possibility for the element ‘da’. In the future researchwe shall verify reflexive, transitive and symmetric relations for elements and their worlds in thisand the other tables.




5.1 The M-system as an analyser of syllogisms

The M -system can conclude on the truthfulness of syllogisms regardless of the natural language inwhich these statements are written. The only condition is that the words/expressions are properlysubstituted into logical formulas / tautologies.

In the next example, the classical check of transitivity of an implication for statements written inthree languages is given:

EXAMPLE 8Let us observe the following three statements written in three languages:

Croatian: SVI LJUDI SU SMRTNIEnglish: ALL MEN ARE MORTALFrench: TOUS LES HOMMES SONT MORTELS

Croatian: SOKRAT JE CVJEKEnglish: SOCRATES IS A MANFrench: SOCRATE EST UN HOMME

Croatian: SOKRAT JE SMRTANEnglish: SOCRATES IS MORTALFrench: SOCRATE EST MORTEL

If processed using the M -system, they have the following form:

Croatian:τ (((SOKRAT

p−→COVJEK) ∧� (COVJEKp−→SMRTAN ))

p−→p−→(SOKRAT

p−→SMRTAN ))

=τ ((hSKRATgCVEKG ∧� xCVEmTAN )p−→sKRATGmTANG)

=τ (hSKRATgCVEKgCVEmTANp−→sKRATGmTANG)

=τ (SKRATgCVEKgCVEmTANmhSKRATGmTANG)= �

English:

τ (((SOCRATESp−→MAN ) ∧� (MAN

p−→MORTALS))p−→

p−→(SOCRATESp−→MORTALS))

=τ ((mSCRATEShANH ∧� nAnTASM )p−→mSCRATESHmTASH )

=τ (mSCRATEShANhAnTASMp−→mSCRATESHmTASH )

=τ (nhSCRATEShANhAnTASmhSCRATESHmTASH )= �

French:τ (((SOCRATE

p−→HOMME) ∧� (HOMMEp−→MORTELS))

p−→p−→(SOCRATE

p−→MORTELS))

=τ ((hSCRATEhOE ∧� sHOEmTES)p−→mSCRATEmTES)

=τ (hSCRATEhOEhOEmTESp−→mSCRATEmTES)

=τ (SCRATEhOEhOEmTESHSCRATEmTESH )= �




Note that, in the evaluation of tautology, the parity of the number of symbols of the linguisticword is not important, i.e. it is possible to translate it directly into an M-word without the need towrap it with the shell of an M-word. Furthermore, the evaluation with respect to the T ,A,Z ,F domainof truthfulness is not important either.

Finally, we check the case when a tautological rule is rendered false on purpose. Let us assume thattrue premises yield the conclusion: NOT ‘SOCRATES IS MORTAL’. The change from ‘SOCRATESIS MORTAL’ to NOT ‘SOCRATES IS MORTAL’ is carried out by a simple exchange of the p-implication to its dual s-implication:

τ (((SOCRATESp−→MAN ) ∧� (MAN

p−→MORTALS))s−→

s−→(SOCRATESp−→MORTALS))

=τ ((mSCRATEShANH ∧� nAnTASM )s−→mSCRATESHmTASH )

=τ (mSCRATEShANhAnTASMs−→mSCRATESHmTASH )

=τ (mhSCRATEShANhAnTASMNmhSCRATESHmTASHN )=⊥�

which, after the usage of the M-system, yields the falsity (⊥�) for English and French, but truth( �) for Croatian.

If the tautological conclusion is changed on purpose, such that the premises which are true yield thenew truth: ‘SOCRATES IS NOT HUMAN/MORTAL’:τ (((SOCRATES

p−→MAN ) ∧� (MANp−→MORTALS))

p−→p−→(SOCRATES

s−→MORTALS))

=τ ((mSCRATEShANH ∧� nAnTASM )p−→hSCRATEsTAS)

=τ (mSCRATEShANhAnTASMp−→hSCRATEsTAS)

=τ (nhSCRATEShANhAnTASmhSCRATEsTASN )=Z�

which, after the usage of the M-system, yields the falsity ⊥� for Croatian and French, but Z�

(undefined) for English. This confirms the natural foundation of the syllogisms, but also of theM -system.

6 Future works

M -system theory can be used for the automated testing of the semantic truth of sentences in linguis-tics. The procedure for testing truthfulness is independent of the used language and it is applicablewith the classical two-valued or more valued set of different possible truth values. The testingprocedure is very suitable for automation.

Difficulties arise on how to order the truth-degrees in a meaningful way. Although in the articlewe have presented some solutions, we note that these do not represent the only reasonable orderings.Shramko and Wansing [18] suggest partial orderings: information order (≤i), truth order (≤t) andfalsity order (≤f ). In Dunn/Belnap’s B4 (FOUR2), the orderings are dependent (i.e. a increase intruth-content means a decrease in falsity content), while in the extension (SIXTEEN3) they becomeindependent.

Our first step will be to compare the results of other authors to many-valued logic with the possiblesolutions provided by the M-system. One direction of the research will be to study the M-system as amanyfold lattice and to compare it with the existing systems (e.g. threefold lattice [19]). The second




study will consider logical states with the same shell but different M-words (e.g. logical AND is asymmetric operation: cd ∧� bd = bd ∧� cd = bd, but their M-words are not: cd ∧� bd =bbcddbbd,bd ∧� cd =bddccd, because they can not be compressed.)

Further, we believe that M-system can be used to represent linguistic quantifiers (one, some, most,every, etc.), their generalizations (ever, any, at all, etc…) and negative environments (download-entailing) because they are directly related to the hierarchy of logic domains (the highest one includesall the lower ones, see Table 8). Furthermore, we plan to consider the use of M-words in orderto represent the semantics of tenses, moods and modals. The semantics-pragmatics interfaces forimplicatures will be most interesting and rewarding (see [4, 7]).

Acknowledgements

The authors would like to thank all colleagues who have contributed to this article, especially SreckoKovac, Domagoj Matijevic, Berislav Zarnic and Darko Zubrinic as well as the anonymous reviewersfor their valuable comments and suggestions which were helpful in improving the article.

Funding

This work has been supported in part by the Croatian Science Foundation under the projects IP-2014-09-9540 and UIP 11-2013-3624.

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Received 24 November 2016


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Theory of M-system - FSB Online of M-system MARIO ESSERT ∗, Faculty of Mechanical Engineering and Naval Architecture, University of Zagreb, Ivana Luci