Fuzzy Natural Logic: Theory and Applications

Vilem Novak

University of OstravaInstitute for Research and Applications of Fuzzy Modeling

NSC IT4InnovationsOstrava, Czech [email protected]

FSTA 2016, Liptovsky Jan, January 27, 2016

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

• 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, e.g.,

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

Vilem Novak (IRAFM, CZ) Fuzzy Natural Logic FSTA 2016 2 / 54

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Hypothesis (Lakoff)

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Hypothesis (Lakoff)

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FNL and semantics of natural language

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)

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

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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))

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Paradigm of FNL

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Paradigm of FNL

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Expected contribution of FNL

• Development of methods for construction of models of systems andprocesses on the basis of expert knowledge expressed in genuinenatural language

• Development of algorithms making computer to “understand” naturallanguage and behave accordingly

• Help to understand the principles of human thinking

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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

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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

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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

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Fuzzy Type Theory

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Fuzzy Type Theory

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Basic concepts

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

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Basic concepts

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Basic concepts

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Truth values for FNL

Standard Lukasiewicz MV∆-algebra

E = 〈[0, 1],∨,∧,⊗,∆,→, 0, 1〉

∨,∧ = minimum, maximum

a⊗ b = 0 ∨ (a + b − 1) ( Lukasiewicz conjunction)

a→ b = 1 ∧ (1− a + b) ( Lukasiewicz implication)

¬a = a→ 0 = 1− a ¬¬a = a

a↔ b = (a→ b) ∧ (b → a) ∆(a) =

{1 if a = 1,

0 otherwise.

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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)

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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

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Natural language in FNL

Two important classes of natural language expressions

• Evaluative linguistic expressions

• Intermediate quantifiers

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Natural language in FNL

Two important classes of natural language expressions

• Evaluative linguistic expressions

• Intermediate quantifiers

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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

FNL: Formal logical theory T Ev — semantics of evaluative linguisticexpressions

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Example

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Example

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Example

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Simple pure evaluative expressions

〈Hedge〉〈TE-adjective〉(i) TE-adjectives:

Gradable ajdectives (small, medium, big; weak, medium strong,strong)Evaluative adjectives (silly, normal, intelligent; good, average, bad)

(ii) Hedges:

• Narrowing: very, extremely, significantly• Widening: more or less, roughly, very roughly• Specifying: approximately, about, rather, precisely, typically

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Other evaluative expressions

(i) Compound evaluative expressionsroughly small or medium, small but not very (small)

(ii) Negative evaluative expressionsnot small, NOT very strong, NOT VERY strong

(iii) Fuzzy numbersaround two million, roughly one thousand

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Semantics of natural language in FNL

Fundamental concepts

(i) context/possible world

(ii) intension

(iii) extension

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Possible world

parameter w ; maximal consistent set of known facts

Evaluative linguistic expressions allow a special model:characterize a certain bounded part of an ordered scaleContext = Possible world

w = 〈vL, vS , vR〉, vL, vS , vR ∈ R

Example (Distance)

w = 〈10, 100, 400〉 (Slovakia)w = 〈100, 500, 3 000〉 (USA)

Set of possible worlds/contexts W

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Possible world

parameter w ; maximal consistent set of known facts

Evaluative linguistic expressions allow a special model:characterize a certain bounded part of an ordered scaleContext = Possible world

w = 〈vL, vS , vR〉, vL, vS , vR ∈ R

Example (Distance)

w = 〈10, 100, 400〉 (Slovakia)w = 〈100, 500, 3 000〉 (USA)

Set of possible worlds/contexts W

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Extension of a linguistic expression A

A class of elements in a given possible world/context having the propertydenoted by AMathematical model of extension (in a context w ∈W ):

Extw (A ) ⊂∼ U

Extension changes when changing the context

Example

Nearw1(150 km)= 0.4 (w1 =Czech Republic)Nearw2(150 km)= 0.95 (w2 =Russia)

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Extension of a linguistic expression A

A class of elements in a given possible world/context having the propertydenoted by AMathematical model of extension (in a context w ∈W ):

Extw (A ) ⊂∼ U

Extension changes when changing the context

Example

Nearw1(150 km)= 0.4 (w1 =Czech Republic)Nearw2(150 km)= 0.95 (w2 =Russia)

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Extensions of evaluative expressions

Context w = 〈0, 4, 10〉

Small Medium BigExtremely

0 4 10

VeSm? Small?Big ?

0 4 10

Roughly

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Construction of extensions of evaluative expressions

Horizon

Each of the limit points vL, vS , vR is a starting point of a horizon runningfrom it in the sense of the ordering towards the next limit point

Horizon is determined by a special indiscernibility relation (fuzzy equality)

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Horizon

LH MH RH

vL vS vR

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Horizon

LH MH RH

vL vS vR

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

Hedges — shifts of horizon

ν : [0, 1] −→ [0, 1]

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

Hedges — shifts of horizon

ν : [0, 1] −→ [0, 1]

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Context

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Extensions of evaluative expressions

• Extremely small, Very small, Small, Very roughly small

• Medium

• Very roughly big, More or less big, Big, Rather big, Very big,Significantly big, Extremely big

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Intension of evaluative expression

Formalization of a property; it is invariant with respect to possibleworlds/contexts gives different truth values in various possible worlds

Mathematical model of intension

Int(A ) : W −→ F (U)

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Intension of simple evaluative expressions

Contexts Extensions

various contexts extensions

( ([0,1]))wFWIntension:

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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

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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

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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

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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

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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.)

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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)

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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).

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Nouns and objects

Simple noun phrase:

small house, very high tree, shallow sea

We evaluate one or more specific features of nouns:height, width, depth, beauty, etc.

Features have values (type α) taken from a set Mα = R

Mathematical model of the semantics of nouns

Objects = sets (vectors) of values of featuresNouns represent (fuzzy) sets of such objects

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Nouns and objects

Simple noun phrase:

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Nouns and objects

Simple noun phrase:

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Extension of “Very tall Swedes”

Extension of Swedes

Swedes

(=people)

Extension of Very tall

Features

Very tallSet of people

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Semantics of “Most Swedes are tall”

SwedesTall

Very big

0 0.5 1

(Most Swedes are tall)=

(=people)

Extension of Tall Extension of Very big (size)

Features

Set of people

Extension of Swedes

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Intension of “Most Swedes are tall”

Intension

λwωϕ · (∃zo(αϕ))((∆∆∆(z ⊆ Swede w) &&&

(∀hαϕ)(zh⇒⇒⇒ Bi(wv)(hv)) &&& BiVe((µ(Swede w)z))

Extension is a truth value

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Generalized (intermediate) syllogisms

Definition

• A syllogism 〈P1,P2,C 〉 — logical argument in which the conclusion Cis inferred from two premises — major P1 and minor P2.

• Intermediate syllogism: traditional syllogism where one or more of itsformulas are replaced by some containing intermediate quantifiers.

• We say that the syllogism is valid if

T IQ ` P1 &&& P2⇒⇒⇒ C ,

(equivalently, T IQ ` P1⇒⇒⇒ (P2⇒⇒⇒ C ))

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Definition

T IQ ` P1 &&& P2⇒⇒⇒ C ,

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Definition

T IQ ` P1 &&& P2⇒⇒⇒ C ,

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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 )

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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 )

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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)

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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)

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Square of opposition — basic relations

P1,P2 ∈ Formo are:

(i) contraries if T ` ¬¬¬(P1 &&& P2). By completeness:

M (P1)⊗M (P2) = 0

P1 and P2 cannot be both true but can be both false

(ii) sub-contraries if T ` P1∇∇∇P2. By completeness:

M (P1)⊕M (P2) = 1

P1 and P2 cannot be both false but can be both true

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Square of opposition — basic relations

(i) P1,P2 ∈ Formo are contradictories if both

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

By completeness:M (∆∆∆P1)⊗M (∆∆∆P2) = 0 M (∆∆∆P1)⊕M (∆∆∆P2) = 1

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

(ii) The formula P2 is a subaltern of P1 in T if T ` P1⇒⇒⇒ P2. Bycompleteness:

M (P1) ≤M (P2)

(P1 a superaltern of P2)

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Generalized 5-square of opposition

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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)

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Focus is negated

Example

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Focus is negated

Example

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Focus is negated

Example

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Focus is negated

Example

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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

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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

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Negation in FNL

not unhappy 6≡ happyThis is not violation of double negation

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Negation in FNL

not unhappy 6≡ happyThis is not violation of double negation

unhappy is antonym of happy

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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 Am THEN Y is Bm

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

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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 Am THEN Y is Bm

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

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Perception-based logical deduction

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

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Perception-based logical deduction

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

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Perception-based logical deduction (PbLD)

• It imitates human way of reasoning

• It realizes algorithms based on an expert knowledge specified innatural language

• Linguistic description can be learned from datapresence of an expert is not necessary

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Applications of FNL

• Fuzzy/linguistic control (real applications)

• Decision making

• Classification and evaluation

• Time series forecasting (forecasting of trend or trend-cycle)

• Mining information from time series in the form of natural languageexpressions

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Applications of FNL

• Decision making

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Applications of FNL

• Decision making

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Applications of FNL

• Decision making

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Applications of FNL

• Decision making

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Conclusions

• The concept of fuzzy natural logic was presented

• Formalization of the semantics of two classes of natural languageexpressions

• Human reasoning: perception-based logic deduction (PbLD),intermediate syllogisms, generalized square of opposition

• FNL includes a mathematical model of the vagueness phenomenon.

Model of the Sorites/Falakros paradox appearing in the semanticsof evaluative expressions

• Many open problems:

• How surface structures can be transformed into logical formulasrepresenting their meaning

• Formalization of negation• Formalization of hedging• Formalization of presupposition and its consequence• Formalization of the meaning of verbs

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Conclusions

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Conclusions

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Conclusions

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Conclusions

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Thank you for your attention

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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 and for every n, if the latter is smallthen it is almost true that n + 1 is also small

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

At(A) – “it is almost true that A”, measured by the implication(Smννν)w n⇒⇒⇒ (Smννν)w (n + 1)

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Theorem (〈linguistic hedge〉 big)

• In each context there is p which surely is not big` (∀w)(∃p)(∆∆∆¬¬¬(Bi ννν)w p)

• In each context there is no n which is surely bigand n − 1 surely is not big

• In each context and for every n, if the latter is bigthen it is almost true that n − 1 is also big

` (∀w)(∀n)((Bi ννν)w n⇒⇒⇒ At((Bi ννν)w (n − 1))

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Vagueness of evaluative expressions

In each context there is no last surely small xand no first surely big x

Theorem

TEv ` (∀w)¬¬¬(∃x)(∀y)(∆∆∆(Smννν)wx &&&(x <w y ⇒⇒⇒∆∆∆¬¬¬(Smννν)wy)),

TEv ` (∀w)¬¬¬(∃x)(∀y)(∆∆∆(Bi ννν)wx &&&(y <w x ⇒⇒⇒∆∆∆¬¬¬(Bi ννν)wy))

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

Dvorak, A. and Holcapek, M., L-fuzzy Quantifiers of the Type 〈1〉 Determined byMeasures, Fuzzy Sets and Systems, 160(2009), 3425-3452

V. Novak, On fuzzy type theory, Fuzzy Sets and Systems 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, Fuzzy Sets and Systems 159(2008), 2939-2969.

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

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

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