Future of MFL: Fuzzy Natural Logic and Alternative …cintula/slides/fmfl_novak.pdfFuture of MFL: Fuzzy Natural Logic and Alternative Set Theory Vil em Nov ak and Irina Per lieva University

Future of MFL: Fuzzy Natural Logic andAlternative Set Theory

Vilem Novak and Irina Perfilieva

University of OstravaInstitute for Research and Applications of Fuzzy Modeling

NSC IT4InnovationsOstrava, Czech [email protected]

Future of MFL, Prague, June 16, 2016

Metamathematics of Mathematics?

The research in MFL has been mainly focused on its metamathematics

Repeated proclaim:

MFL should serve as a logic providing:

• Tools for the development of the mathematical model of vagueness

• Tools for various kinds of applications that have to cope with thelatter

This goal is not sufficiently carried out!

The general properties of MFL are already known

Let us focus on the inside of few specific fuzzy logicsDevelop program solving special problems

Vilem Novak and Irina Perfilieva (IRAFM, CZ) Future of MFL Future of MFL 2 / 53

Metamathematics of Mathematics?

The research in MFL has been mainly focused on its metamathematics

Repeated proclaim:

MFL should serve as a logic providing:

• Tools for the development of the mathematical model of vagueness

• Tools for various kinds of applications that have to cope with thelatter

This goal is not sufficiently carried out!

The general properties of MFL are already known

Let us focus on the inside of few specific fuzzy logicsDevelop program solving special problems


Two of many possible directions

• Fuzzy Natural Logic

• Fuzzy logic in the frame of the Alternative Set Theory


Two of many possible directions

• Fuzzy Natural Logic

• Fuzzy logic in the frame of the Alternative Set Theory


The concept of natural logicGeorge Lakoff: Linguistics and Natural Logic, Synthese 22, 1970

Goals

• to express all concepts capable of being expressed in natural language

• to characterize all the valid inferences that can be made in naturallanguage

• to mesh with adequate linguistic descriptions of all natural languages

Natural logic is a collection of terms and rules that come with naturallanguage and allow us to reason and argue in it

Hypothesis (Lakoff)

Natural language employs a relatively small finite number of atomicpredicates that are used in forming sentences. They are related to eachother by meaning-postulates — axioms, e.g.,

REQUIRE(x , y)⇒⇒⇒ PERMIT(x , y)that do not vary from language to language.



Goals





Hypothesis (Lakoff)





Goals





Hypothesis (Lakoff)




The concept of Fuzzy Natural Logic

Paradigm of FNL

(i) Extend the concept of natural logic to cope with vagueness

(ii) Develop FNL as a mathematical theoryExtension of Mathematical Fuzzy Logic;application of Fuzzy Type Theory

Fuzzy natural logic is a mathematical theory that provides models of termsand rules that come with natural language and allow us to reason andargue in it. At the same time, the theory copes with vagueness of naturallanguage semantics.

(Similar concept was initiated in 1995 by V. Novak under the nameFuzzy logic broader sense (FLb))



Paradigm of FNL







Paradigm of FNL






FNL and semantics of natural language

Sources for the development of FNL

• Results of classical linguistics

• Logical analysis of concepts and semantics of natural languageTransparent Intensional Logic (P. Tichy, P. Materna)

• Montague grammar(English as a Formal Language)

• Results of Mathematical Fuzzy Logic

Fuzzy sets in linguistics: L. A. ZadehThe concept of a linguistic variable and its application to approximatereasoning (1975), Quantitative Fuzzy Semantics (1973), A computationalapproach to fuzzy quantifiers in natural languages (1983), Precisiatednatural language (2004)


























Current constituents of FNL

• Theory of evaluative linguistic expressions

• Theory of fuzzy/linguistic IF-THEN rules and logical inference(Perception-based Logical Deduction)

• Theory of fuzzy generalized and intermediate quantifiers includinggeneralized Aristotle syllogisms and square of opposition

• Examples of formalization of special commonsense reasoning cases




















Higher-order fuzzy logic — Fuzzy Type Theory

Formal logical analysis of concepts and natural language expressionsrequires higher-order logic — type theory.

Why fuzzy type theory

• It is a constituent of Mathematical Fuzzy Logic, well established withsound mathematical properties.

• Enables to include model of vagueness in the developed mathematicalmodels














Fuzzy Type Theory

Generalization of classical type theory

Syntax of FTT is an extended lambda calculus:

• more logical axioms

• many-valued semantics

Main fuzzy type theories

IMTL, Lukasiewicz, EQ-algebra


Fuzzy Type Theory








Fuzzy Type Theory








Basic concepts

Types

Elementary types: o (truth values), ε (objects)Composed types: βα

Formulas have types: Aα ∈ Formα,Aα ≡ Bα ∈ Formo

λxα Cβ ∈ Formβα

∆∆∆ooAo ∈ Formo

Formulas of type o are propositions

Interpretation of formulas Aβα are functions Mα −→ Mβ


Semantics of FTT

FrameM = 〈(Mα,=α)α∈Types ,E∆〉

Fuzzy equality =α: Mα ×Mα −→ L

[x =α x ] = 1 (reflexivity)

[x =α y ] = [y =α x ] (symmetry)

[x =α y ]⊗ [y =α z ] ≤ [x =α z ] (transitivity)


Generalized completeness (Henkin style)

Theorem

(a) A theory T of fuzzy type theory is consistent iff it hasa general model M .

(b) For every theory T of fuzzy type theory and a formula Ao

T ` Ao iff T |= Ao .

FTT has a lot of interesting properties and great explication power


Natural language in FNL

Standard Lukasiewicz MV∆-algebra

Two important classes of natural language expressions

• Evaluative linguistic expressions

• Intermediate quantifiers


Natural language in FNL

Standard Lukasiewicz MV∆-algebra

Two important classes of natural language expressions

• Evaluative linguistic expressions

• Intermediate quantifiers


Evaluative linguistic expressions

Example

very short, rather strong, more or less medium, roughly big, extremely big,very intelligent, significantly important, etc.

• Special expressions of natural language using which people evaluatephenomena and processes that they see around

• They are permanently used in any speech, description of any process,decision situation, characterization of surrounding objects; to bringnew information, people must to evaluate

We construct a special theory T Ev in the language of FTT formalizing 6general characteristics of the semantics of evaluative expressions



Example







Example






Axioms of T Ev

(EV1) (∃z)∆∆∆(¬¬¬z ≡ z)

(EV2) (⊥ ≡ w−1⊥w )∧∧∧ († ≡ w−1†w )∧∧∧ (> ≡ w−1>w )

(EV3) t ∼ t

(EV4) t ∼ u ≡ u ∼ t

(EV5) t ∼ u &&& u ∼ z · ⇒⇒⇒ t ∼ z

(EV6) ¬¬¬(⊥ ∼ †)

(EV7) ∆∆∆((t ⇒⇒⇒ u) &&&(u⇒⇒⇒ z))⇒⇒⇒ ·t ∼ z ⇒⇒⇒ t ∼ u

(EV8) t ≡ t ′&&& z ≡ z ′⇒⇒⇒ ·t ∼ z ⇒⇒⇒ t ′ ∼ z ′

(EV9) (∃u)Υ(⊥ ∼ u)∧∧∧ (∃u)Υ(† ∼ u)∧∧∧ (∃u)Υ(> ∼ u)

(EV10) NatHedge ννν&&&(∃ννν)(∃ννν ′)(Hedge ννν&&& Hedge ννν ′&&&(ννν1 ννν ∧∧∧ ννν ννν2))

(EV11) (∀z)((Υννν(LH z))∨∨∨ (Υννν(MH z))∨∨∨ (Υννν(RH z)))


Relevant characteristics — Context

(A) Nonempty, linearly ordered and bounded scale, threedistinguished limit points: left bound, right bound, and acentral point

Context wαo Mp(wαo) = w : [0, 1] −→ M:

w(0) = vL (left bound)

w(0.5) = vS (central point)

w(1) = vR (right bound)

Set of contexts W = w | w : [0, 1] −→ MCrisp linear ordering ≤w in each context w


Relevant characteristics — Context

(A) Nonempty, linearly ordered and bounded scale, threedistinguished limit points: left bound, right bound, and acentral point

Context wαo Mp(wαo) = w : [0, 1] −→ M:

w(0) = vL (left bound)

w(0.5) = vS (central point)

w(1) = vR (right bound)

Set of contexts W = w | w : [0, 1] −→ MCrisp linear ordering ≤w in each context w


Relevant characteristics —Intension

(B) Function from the set of contexts into a set of fuzzy sets

Int(A ) = λw λx (Aw)x M (Int(A )) : W −→ F (w([0, 1])

Scheme of intension


Relevant characteristics —Intension

(B) Function from the set of contexts into a set of fuzzy sets

Int(A ) = λw λx (Aw)x M (Int(A )) : W −→ F (w([0, 1])

Scheme of intension

֏

֏

֏

various contexts extensions

( ([0,1]))wFWIntension:


Horizon

(C) Each of the limit points is a starting point of some horizonrunning from it in the sense of the ordering of the scale towardsthe next limit point (the horizon vanishes beyond)

Three horizons

LH(a) = [0 ∼ a], LH(w x) = [vL ≈w x ]

MH(a) = [0.5 ∼ a], MH(w x) = [vS ≈w x ]

RH(a) = [1 ∼ a], RH(w x) = [vR ≈w x ]


Horizon

(C) Each of the limit points is a starting point of some horizonrunning from it in the sense of the ordering of the scale towardsthe next limit point (the horizon vanishes beyond)

Three horizons

1

LH MH RH

vL vS vR

LH(a) = [0 ∼ a], LH(w x) = [vL ≈w x ]

MH(a) = [0.5 ∼ a], MH(w x) = [vS ≈w x ]

RH(a) = [1 ∼ a], RH(w x) = [vR ≈w x ]


Relevant characteristics — horizon

(D) Each horizon is represented by a special fuzzy setdetermined by a reasoning analogous to that leading to thesorites paradox.

Sorites paradox

One grain does not make a heap. Adding one grain to what is not yet aheap does not make a heap. Consequently, there are no heaps.


Relevant characteristics — horizon

(D) Each horizon is represented by a special fuzzy setdetermined by a reasoning analogous to that leading to thesorites paradox.

Sorites paradox

One grain does not make a heap. Adding one grain to what is not yet aheap does not make a heap. Consequently, there are no heaps.


Construction of extensions of evaluative expressions

Extension of evaluative expression is delineated by shifting of the horizonusing linguistic hedge

Hedges — shifts of horizon

ν : [0, 1] −→ [0, 1]


Construction of extensions of evaluative expressions

a

b

c

a,b,c

vL

vS

vR

1

Mea2

Mea

LHMH

RH

Context

Hedge


Falakros/sorites paradox in evaluative expressions

Theorem (〈linguistic hedge〉 small)

• Zero is “(very) small” in each context` (∀w)((Smννν)w 0)

• In each context there is p which surely is not “(very) small”` (∀w)(∃p)(∆∆∆¬¬¬(Smννν)w p)

• In each context there is no n which is surely smalland n + 1 surely is not small

` (∀w)¬¬¬(∃n)(∆∆∆(Smννν)w n &&& ∆∆∆¬¬¬(Smννν)w(n + 1))

• In each context: if n is small then it is almost true that n + 1 is alsosmall

` (∀w)(∀n)((Smννν)w n⇒⇒⇒ At((Smννν)w (n + 1))

(At(A) is measured by (Smννν)w n⇒⇒⇒ (Smννν)w (n + 1))In each context there is no last surely small xand no first surely big x


Intermediate quantifiers

P. L. Peterson, Intermediate Quantifiers. Logic, linguistics, and Aristoteliansemantics, Ashgate, Aldershot 2000.

Quantifiers in natural language

Words (expressions) that precede and modify nouns; tell us how many orhow much. They specify quantity of specimens in the domain of discoursehaving a certain property.

Example

All, Most, Almost all, Few, Many, Some, NoMost women in the party are well dressedFew students passed exam

Intermediate quantifiers form an important subclass of generalized(possibly fuzzy) quantifiers


Intermediate quantifiers


Quantifiers in natural language

Words (expressions) that precede and modify nouns; tell us how many orhow much. They specify quantity of specimens in the domain of discoursehaving a certain property.

Example

All, Most, Almost all, Few, Many, Some, NoMost women in the party are well dressedFew students passed exam

Intermediate quantifiers form an important subclass of generalized(possibly fuzzy) quantifiers


Semantics of intermediate quantifiers

Main idea

They are classical quantifiers ∀ and ∃ taken over a smaller class ofelements. Its size is determined using an appropriate evaluative expression.

Classical logic: No substantiation why and how the range of quantificationshould be made smaller


Semantics of intermediate quantifiers

Main idea

They are classical quantifiers ∀ and ∃ taken over a smaller class ofelements. Its size is determined using an appropriate evaluative expression.

Classical logic: No substantiation why and how the range of quantificationshould be made smaller


Formal theory of intermediate quantifiers

T IQ = T Ev+ 4 special axioms

“〈Quantifier〉 B’s are A”

(Q∀Ev x)(B,A) := (∃z)((∆∆∆(z ⊆ B)︸︷︷︸“the greatest” part of B’s

&&&

(∀x)(z x ⇒⇒⇒ Ax))︸︷︷︸each of B’s has A

∧∧∧

Ev((µB)z))︸︷︷︸size of z is evaluated by Ev

Ev — extension of a certain evaluative expression

(big, very big, small, etc.)


Special intermediate quantifiers

P: Almost all B are A := Q∀Bi Ex(B,A)

(∃z)((∆∆∆(z ⊆ B) &&&(∀x)(zx ⇒⇒⇒ Ax))∧∧∧ (Bi Ex)((µB)z))

B: Almost all B are not A := Q∀Bi Ex(B,¬¬¬A)

T: Most B are A := Q∀Bi Ve(B,A)

D: Most B are not A := Q∀Bi Ve(B,¬¬¬A)

K: Many B are A := Q∀¬¬¬(Sm ννν)(B,A)

G: Many B are not A := Q∀¬¬¬(Sm ννν)(B,¬¬¬A)

F: Few B are A := Q∀SmVe(B,A)

H: Few B are not A := Q∀SmVe(B,¬¬¬A)


Special intermediate quantifiers

Classical quantifiers

A: All B are A := Q∀Bi∆∆∆(B,A) ≡ (∀x)(Bx ⇒⇒⇒ Ax),

E: No B are A := Q∀Bi∆∆∆(B,¬¬¬A) ≡ (∀x)(Bx ⇒⇒⇒¬¬¬Ax),

I: Some B are A := Q∃Bi∆∆∆(B,A) ≡ (∃x)(Bx ∧∧∧ Ax),

O: Some B are not A := Q∃Bi∆∆∆(B,¬¬¬A) ≡ (∃x)(Bx ∧∧∧¬¬¬Ax).


105 valid generalized Aristotle’s syllogisms

Figure I

Q1 M is Y

Q2 X is M

Q3 X is Y

Figure II

Q1 Y is M

Q2 X is M

Q3 X is Y

Example (ATT-I)

All women (M) are well dressed (Y ) (∀x)(M x ⇒⇒⇒ Y x)Most people in the party (X ) are women (M) (Q∀Bi Vex)(X ,M)

Most people in the party (X ) are well dressed (Y ) (Q∀Bi Vex)(X ,Y )

Example (ETO-II)

No lazy people (Y ) pass exam (M) ¬¬¬(∃x)(Yx ∧∧∧Mx)Most students (X ) pass exam (M) (Q∀Bi Vex)(X ,M)

Some students (X ) are not lazy people (Y ) (∃x)(X ∧∧∧¬¬¬Y )


105 valid generalized Aristotle’s syllogisms

Figure I

Q1 M is Y

Q2 X is M

Q3 X is Y

Figure II

Q1 Y is M

Q2 X is M

Q3 X is Y

Example (ATT-I)

All women (M) are well dressed (Y ) (∀x)(M x ⇒⇒⇒ Y x)Most people in the party (X ) are women (M) (Q∀Bi Vex)(X ,M)

Most people in the party (X ) are well dressed (Y ) (Q∀Bi Vex)(X ,Y )

Example (ETO-II)

No lazy people (Y ) pass exam (M) ¬¬¬(∃x)(Yx ∧∧∧Mx)Most students (X ) pass exam (M) (Q∀Bi Vex)(X ,M)

Some students (X ) are not lazy people (Y ) (∃x)(X ∧∧∧¬¬¬Y )


105 valid generalized Aristotle’s syllogismsFigure III

Q1 M is Y

Q2 M is X

Q3 X is Y

Figure IV

Q1 Y is M

Q2 M is X

Q3 X is Y

Example (PPI-III )

Almost all old people (M) are ill (Y ) (Q∀Bi Exx)(Mx ,Yx)Almost all old people (M) have gray hair (X ) (Q∀Bi Exx)(Mx ,Xx)

Some people with gray (X ) hair are ill (Y ) (∃x)(Xx ∧∧∧ Yx)

Example (TAI-IV )

Most shares with downward trend (Y ) are from car industry (M)(Q∀Bi Vex)(Yx ,Mx)

All shares of car industry (M) are important (X ) (∀x)(Mx ,Xx)

Some important shares (X ) have downward trend (Y ) (∃x)(Xx ∧∧∧ Yx)


105 valid generalized Aristotle’s syllogismsFigure III

Q1 M is Y

Q2 M is X

Q3 X is Y

Figure IV

Q1 Y is M

Q2 M is X

Q3 X is Y

Example (PPI-III )

Almost all old people (M) are ill (Y ) (Q∀Bi Exx)(Mx ,Yx)Almost all old people (M) have gray hair (X ) (Q∀Bi Exx)(Mx ,Xx)

Some people with gray (X ) hair are ill (Y ) (∃x)(Xx ∧∧∧ Yx)

Example (TAI-IV )

Most shares with downward trend (Y ) are from car industry (M)(Q∀Bi Vex)(Yx ,Mx)

All shares of car industry (M) are important (X ) (∀x)(Mx ,Xx)

Some important shares (X ) have downward trend (Y ) (∃x)(Xx ∧∧∧ Yx)


Square of opposition — basic relations

P1,P2 ∈ Formo are:

(i) contraries if T ` ¬¬¬(P1 &&& P2).P1 and P2 cannot be both true but can be both false

(ii) sub-contraries if T ` P1∇∇∇P2.weak sub-contraries if T ` Υ(P1 ∨∨∨ P2)P1 and P2 cannot be both false but can be both true

(iii) P1,P2 ∈ Formo are contradictories if

T ` ¬¬¬(∆∆∆P1 &&& ∆∆∆P2) as well as T `∆∆∆P1∇∇∇∆∆∆P2.

P1 and P2 cannot be both true as well as both false

(iv) The formula P2 is a subaltern of P1 in T if T ` P1⇒⇒⇒ P2

(P1 a superaltern of P2)


Generalized 5-square of opposition


Negation in FNLTopic—focus articulation:

Each sentence is divided into topic (known information) and focus (newinformation)

Focus is negated

Example

It is not true, that:

(i) JOHN reads Jane’s paperSomebody else reads it

(ii) John READS JANE’S PAPERJohn does something else

(iii) John reads JANE’S PAPERJohn reads some other paper

(iv) John reads Jane’s PAPERJohn reads Jane’s book (lying on the same table)


Negation in FNLTopic—focus articulation:

Each sentence is divided into topic (known information) and focus (newinformation)

Focus is negated

Example

It is not true, that:

(i) JOHN reads Jane’s paperSomebody else reads it

(ii) John READS JANE’S PAPERJohn does something else

(iii) John reads JANE’S PAPERJohn reads some other paper

(iv) John reads Jane’s PAPERJohn reads Jane’s book (lying on the same table)


Negation in FNL

The law of double negation is in NL fulfilled (and so, in FNL as well)

We do not think (Th) that John did not read this paper (R(J, p))

¬¬¬Th(¬¬¬R(J, p)) ≡ Th(R(J, p))

However:It is not clear (Cl) whether John did not read Jane’s paper (R(J, p))

¬¬¬Cl(¬¬¬R(J, p)) ≡ Cl(R(J, p))?

Consequence: We suspect John to read Jane’s paper


Negation in FNL

The law of double negation is in NL fulfilled (and so, in FNL as well)

We do not think (Th) that John did not read this paper (R(J, p))

¬¬¬Th(¬¬¬R(J, p)) ≡ Th(R(J, p))

However:It is not clear (Cl) whether John did not read Jane’s paper (R(J, p))

¬¬¬Cl(¬¬¬R(J, p)) ≡ Cl(R(J, p))?

Consequence: We suspect John to read Jane’s paper


Negation in FNL

not unhappy 6≡ happyThis is not violation of double negation


Negation in FNL

not unhappy 6≡ happyThis is not violation of double negation

unhappy is antonym of happy


Fuzzy/linguistic IF-THEN rules and logical inference

Fuzzy/Linguistic IF-THEN rule

IF X is A THEN Y is B

Conditional sentence of natural language

Example: IF X is small THEN Y is extremely strong

Linguistic description

IF X is A1 THEN Y is B1

IF X is A2 THEN Y is B2

. . . . . . . . . . . . . . . . . . . . . . . .IF X is Am THEN Y is Bm

Text describing one’s behavior in some (decision) situation


Perception-based logical deduction

Imitates human way of reasoning

Crossing strategy when approaching intersection:

R1 := IF Distance is very small THEN Acceleration is very big

R2 := IF Distance is small THEN Brake is big

R3 := IF Distance is medium or big THEN Brake is zero

• Gives general rules for driver’s behavior independently on the concreteplace

• People are able to follow them in an arbitrary signalized intersection

We can “distinguish” between the rules despite the fact that their meaningis vague


Perception-based logical deduction

Imitates human way of reasoning

Crossing strategy when approaching intersection:

R1 := IF Distance is very small THEN Acceleration is very big

R2 := IF Distance is small THEN Brake is big

R3 := IF Distance is medium or big THEN Brake is zero

• Gives general rules for driver’s behavior independently on the concreteplace

• People are able to follow them in an arbitrary signalized intersection

We can “distinguish” between the rules despite the fact that their meaningis vague


MFL and AST?

Suggestion:

Switch the development of MFL to a new mathematical frame based onthe Alternative Set Theory by P. Vopenka

Different understanding to infinity

Vopenka: “Mathematics uses infinity whenever it faces vagueness!”


Source of Natural Infinity

Example

Consider the number 10120

Imagine counting each second 1012 atoms

Counting 10120 atoms by counting each second 1012 of them wouldtake 10100 years!!

Visible universe has about 1080 atoms and exists less than 1011 years!



Example







Example







Example






God’s view

Ha, Ha, why are you boringme with such ridiculously smallnumbers! Where is INFINITY!?


God’s view


We cannot imagine even half of this way!Something is wrong!


God’s view


We cannot imagine even half of this way!Something is wrong!


Big numbers behave as infinite

Basic property

α ≈ α + 1

10001 people slept “whole night” on 10000 beds!



Basic property

α ≈ α + 1




Basic property

α ≈ α + 1



Alternative Set Theory

• An attempt at developing a new mathematics on the basis of criticismof classical one

• Take useful principles and replace others by more natural ones

• Make mathematics closer to human perception of reality












Fundamental concepts

Actualizability ↔ Potentiality

Class

Actualized grouping of objects

X = x | ϕ(x)

Set

Sharp class

• All its elements can be put on a list

• There exists ≤ according to which a set has the first and the lastelements

From classical point of view every set is classically finite



Actualizability ↔ Potentiality

Class

Actualized grouping of objects

X = x | ϕ(x)

Set

Sharp class

• All its elements can be put on a list

• There exists ≤ according to which a set has the first and the lastelements

From classical point of view every set is classically finite



Is every part Y ⊆ x necessarily a set?

Finite set

defined very sharply; no part of it can be unsharp; transparent

Fin(x) iff (∀Y ⊆ x)Set(Y )

Horizon

• threshold terminating our view of the world,

• the world continues beyond the horizon,

• part of the world before it is determined nonsharply,

• not fixed, moving around the world




Finite set



Horizon








Finite set



Horizon






Semisets

A class X is a semiset if there is a set a such that X ⊆ a

Theorem

Let a be an infinite set and ≤ its linear ordering. Put

X = x ∈ a | y ∈ a | y ≤ x is a finite set.

Then X is a proper semiset.


Semisets

A class X is a semiset if there is a set a such that X ⊆ a

Theorem

Let a be an infinite set and ≤ its linear ordering. Put

X = x ∈ a | y ∈ a | y ≤ x is a finite set.

Then X is a proper semiset.


Indiscernibility relation

x ≡ y iff 〈x , y〉 ∈⋂Rn | n ∈ FN

sequence of still sharper criteria Rn

Indiscernibility of rational numbers

x.

= y iff | x − y |< 1

n; n ∈ FN

x ∈ IS iff x.

= 0 infinitely small

x ∈ IL iff1

x∈ IS infinitely large

Figure

X is a figure if x ∈ X and y ≡ x implies y ∈ X


Indiscernibility relation

x ≡ y iff 〈x , y〉 ∈⋂Rn | n ∈ FN

sequence of still sharper criteria Rn

Indiscernibility of rational numbers

x.

= y iff | x − y |< 1

n; n ∈ FN

x ∈ IS iff x.

= 0 infinitely small

x ∈ IL iff1

x∈ IS infinitely large

Figure

X is a figure if x ∈ X and y ≡ x implies y ∈ X


Some types of classes

Countable X ≈ FN

Uncountable X ≈ α, infinite, not countable

Real there is ≡ such that X is a figure

Imaginary not real, e.g., Ω is imaginaryImaginary classes are rare

Prolongation axiom

To each countable function F there exists a set function f such that

F ⊆ f



Countable X ≈ FN




Prolongation axiom


F ⊆ f



Countable X ≈ FN




Prolongation axiom


F ⊆ f


Fuzzy approachIndiscernibility ≡ sharpened to

≡ =⋂Rβ | β ≤ γ

The intensity of our “effort” to discern the objects x and y — a number αof

〈x , y〉 ∈ Rβ, β ∈ α

Degree of equality

x ≡ν y iff ν =α

γ.

Theorem

(a) x ≡1 x .

(b) x ≡ν y implies y ≡ν x .

(c) x ≡ν1 y and y ≡ν2 z implies x ≡ν3 z where ν1 ⊗ ν2 l ν3


Fuzzy sets

Membership degree of x in X

X F (x) =∨ν | (∃y ∈ Y )(x ≡ν y)

Measure of the greatest intensity of our “effort” to discern x fromelements of the kernel Y .

X F

A fuzzy set approximating the real class X


Fuzzy sets

Theorem

(a)X F

1 (x)⊗ X F2 (x) l (X1 ∩ X2)F (x) l X F

1 (x) ∧ X F2 (x)

(b)X F

1 (x) ∨ X F2 (x) l (X1 ∪ X2)F (x) l X F

1 (x)⊕ X F2 (x)

(c)(X )F (x) = (V − X )F (x)

.= 1− X F (x)


Conclusions

• MFL should focus on the detailed development of some well selectedlogics.

• MFL should help to develop the concept of Fuzzy Natural LogicOpen problems:• How surface structures can be transformed into logical formulas

representing their meaning• Formalization of negation• Formalization of hedging• Formalization of presupposition and its consequence• Formalization of the meaning of verbs; meaning of sentences

• MFL could be developed on the basis of ASTIntroducing degrees into AST


Conclusions






Conclusions






References

Vopenka, P., Mathematics in the Alternative Set Theory. Teubner,Leipzig 1979.

Vopenka, P., Fundamentals of the Mathematics In the Alternative SetTheory. Alfa, Bratislava 1989 (in Slovak).

Vopenka, P., Calculus Infinitesimalis. Pars Prima. Introduction todifferential calculus of functions of one variable. KANINA 2010 (inCzech).

Vopenka, P., Calculus Infinitesimalis. Pars Secunda. Integral of realfunctions of one variable. KANINA 2011 (in Czech).

Many papers in Commentationes Mathematicae Universitatis Carolinae(CMUC)


References

Holcapek, M., Monadic L-fuzzy quantifiers of the type 〈1n, 1〉, FSS 159(2008),1811-1835.

V. Novak, On fuzzy type theory, FSS, 149(2005), 235–273.

V. Novak, EQ-algebra-based fuzzy type theory and its extensions, Logic Journal ofthe IGPL 19(2011), 512-542

V. Novak, A comprehensive theory of trichotomous evaluative linguisticexpressions, FSS 159(2008), 2939-2969.

V. Novak, A formal theory of intermediate quantifiers, FSS 159(2008) 1229–1246

P. Murinova, V. Novak, A formal theory of generalized intermediate syllogisms,FSS 186(2012), 47-80

V. Novak, The Alternative Mathematical Model of Linguistic Semantics andPragmatics, Plenum, New York, 1992.



Thank you for your attention


Documents

Future of MFL: Fuzzy Natural Logic and Alternative …cintula/slides/fmfl_novak.pdfFuture of MFL: Fuzzy Natural Logic and Alternative Set Theory Vil em Nov ak and Irina Per lieva University