Future of MFL: Fuzzy Natural Logic and Alternative Set Theory

VilémNovákand Irina Perfilieva

University of Ostrava Institute for Research and Applications of Fuzzy Modeling NSC IT4Innovations Ostrava, Czech Republic [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 the latter 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 logics Develop program solving special problems

IRAFM

VilémNovákand Irina Perfilieva (IRAFM, CZ) Future of MFL Future of MFL 2 / 53 Metamathematics of Mathematics?

The general properties of MFL are already known

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

IRAFM

VilémNovákand Irina Perfilieva (IRAFM, CZ) Future of MFL Future of MFL 2 / 53 Two of many possible directions

• Fuzzy Natural Logic • Fuzzy logic in the frame of the Alternative Set Theory

IRAFM

VilémNovákand Irina Perfilieva (IRAFM, CZ) Future of MFL Future of MFL 3 / 53 Two of many possible directions

• Fuzzy Natural Logic • Fuzzy logic in the frame of the Alternative Set Theory

IRAFM

VilémNovákand Irina Perfilieva (IRAFM, CZ) Future of MFL Future of MFL 3 / 53 The concept of natural logic George 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 natural language • to mesh with adequate linguistic descriptions of all natural languages

Natural logic is a collection of terms and rules that come with natural language and allow us to reason and argue in it Hypothesis (Lakoﬀ) Natural language employs a relatively small ﬁnite number of atomic predicates that are used in forming sentences. They are related to each other by meaning-postulates — axioms, e.g., REQUIRE(x, y) ⇒ PERMIT(x, y)

that do not vary from language to language. IRAFM

VilémNovákand Irina Perfilieva (IRAFM, CZ) Future of MFL Future of MFL 4 / 53 The concept of natural logic George 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 natural language • to mesh with adequate linguistic descriptions of all natural languages

that do not vary from language to language. IRAFM

VilémNovákand Irina Perfilieva (IRAFM, CZ) Future of MFL Future of MFL 4 / 53 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 theory Extension of Mathematical Fuzzy Logic; application of Fuzzy Type Theory

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

(Similar concept was initiated in 1995 by V. Nov´akunder the name Fuzzy logic broader sense (FLb))

IRAFM

VilémNovákand Irina Perfilieva (IRAFM, CZ) Future of MFL Future of MFL 5 / 53 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 theory Extension of Mathematical Fuzzy Logic; application of Fuzzy Type Theory

(Similar concept was initiated in 1995 by V. Nov´akunder the name Fuzzy logic broader sense (FLb))

IRAFM

VilémNovákand Irina Perfilieva (IRAFM, CZ) Future of MFL Future of MFL 5 / 53 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 theory Extension of Mathematical Fuzzy Logic; application of Fuzzy Type Theory

(Similar concept was initiated in 1995 by V. Nov´akunder the name Fuzzy logic broader sense (FLb))

IRAFM

VilémNovákand Irina Perfilieva (IRAFM, CZ) Future of MFL Future of MFL 5 / 53 FNL and semantics of natural language

Sources for the development of FNL • Results of classical linguistics • Logical analysis of concepts and semantics of natural language Transparent Intensional Logic (P. Tich´y,P. Materna) • Montague grammar (English as a Formal Language) • Results of Mathematical Fuzzy Logic

Fuzzy sets in linguistics: L. A. Zadeh The concept of a linguistic variable and its application to approximate reasoning (1975), Quantitative Fuzzy Semantics (1973), A computational approach to fuzzy quantiﬁers in natural languages (1983), Precisiated natural language (2004)

IRAFM

VilémNovákand Irina Perfilieva (IRAFM, CZ) Future of MFL Future of MFL 6 / 53 FNL and semantics of natural language

IRAFM

VilémNovákand Irina Perfilieva (IRAFM, CZ) Future of MFL Future of MFL 6 / 53 FNL and semantics of natural language

IRAFM

VilémNovákand Irina Perfilieva (IRAFM, CZ) Future of MFL Future of MFL 6 / 53 FNL and semantics of natural language

IRAFM

VilémNovákand Irina Perfilieva (IRAFM, CZ) Future of MFL Future of MFL 6 / 53 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 including generalized Aristotle syllogisms and square of opposition • Examples of formalization of special commonsense reasoning cases

IRAFM

VilémNovákand Irina Perfilieva (IRAFM, CZ) Future of MFL Future of MFL 7 / 53 Current constituents of FNL

IRAFM

VilémNovákand Irina Perfilieva (IRAFM, CZ) Future of MFL Future of MFL 7 / 53 Current constituents of FNL

IRAFM

VilémNovákand Irina Perfilieva (IRAFM, CZ) Future of MFL Future of MFL 7 / 53 Current constituents of FNL

IRAFM

VilémNovákand Irina Perfilieva (IRAFM, CZ) Future of MFL Future of MFL 7 / 53 Higher-order fuzzy logic — Fuzzy Type Theory

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

Why fuzzy type theory • It is a constituent of Mathematical Fuzzy Logic, well established with sound mathematical properties. • Enables to include model of vagueness in the developed mathematical models

IRAFM

VilémNovákand Irina Perfilieva (IRAFM, CZ) Future of MFL Future of MFL 8 / 53 Higher-order fuzzy logic — Fuzzy Type Theory

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

IRAFM

VilémNovákand Irina Perfilieva (IRAFM, CZ) Future of MFL Future of MFL 8 / 53 Higher-order fuzzy logic — Fuzzy Type Theory

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

IRAFM

VilémNovákand Irina Perfilieva (IRAFM, CZ) Future of MFL Future of MFL 8 / 53 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

IRAFM

VilémNovákand Irina Perfilieva (IRAFM, CZ) Future of MFL Future of MFL 9 / 53 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

IRAFM

VilémNovákand Irina Perfilieva (IRAFM, CZ) Future of MFL Future of MFL 9 / 53 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

IRAFM

VilémNovákand Irina Perfilieva (IRAFM, CZ) Future of MFL Future of MFL 9 / 53 Basic concepts

Types Elementary types: o (truth values), (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

IRAFM

VilémNovákand Irina Perfilieva (IRAFM, CZ) Future of MFL Future of MFL 10 / 53 Semantics of FTT

Frame

M = h(Mα, =α)α∈Types , E∆i

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)

IRAFM

VilémNovákand Irina Perfilieva (IRAFM, CZ) Future of MFL Future of MFL 11 / 53 Generalized completeness (Henkin style)

Theorem (a) A theory T of fuzzy type theory is consistent iﬀ it has a 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

IRAFM

VilémNovákand Irina Perfilieva (IRAFM, CZ) Future of MFL Future of MFL 12 / 53 Natural language in FNL

StandardLukasiewicz MV∆-algebra

Two important classes of natural language expressions • Evaluative linguistic expressions • Intermediate quantiﬁers

IRAFM

VilémNovákand Irina Perfilieva (IRAFM, CZ) Future of MFL Future of MFL 13 / 53 Natural language in FNL

StandardLukasiewicz MV∆-algebra

Two important classes of natural language expressions • Evaluative linguistic expressions • Intermediate quantiﬁers

IRAFM

VilémNovákand Irina Perfilieva (IRAFM, CZ) Future of MFL Future of MFL 13 / 53 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 evaluate phenomena and processes that they see around • They are permanently used in any speech, description of any process, decision situation, characterization of surrounding objects; to bring new information, people must to evaluate

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

IRAFM

VilémNovákand Irina Perfilieva (IRAFM, CZ) Future of MFL Future of MFL 14 / 53 Evaluative linguistic expressions

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

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

IRAFM

VilémNovákand Irina Perfilieva (IRAFM, CZ) Future of MFL Future of MFL 14 / 53 Evaluative linguistic expressions

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

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

IRAFM

VilémNovákand Irina Perfilieva (IRAFM, CZ) Future of MFL Future of MFL 14 / 53 Axioms of T Ev (EV1)( ∃z)∆∆∆(¬z ≡ z) −1 −1 −1 (EV2)( ⊥ ≡ w ⊥w ) ∧ († ≡ w †w ) ∧ (> ≡ w >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 ≡ t0 & z ≡ z0 ⇒ ·t ∼ z ⇒ t0 ∼ z0 (EV9)( ∃u)Υ(ˆ ⊥ ∼ u) ∧ (∃u)Υ(ˆ † ∼ u) ∧ (∃u)Υ(ˆ > ∼ u) (EV10) NatHedge ν¯ &&(∃ν)(∃ν 0)(Hedge ν & Hedge ν 0 & (ν 1 ν¯ ∧ ν¯ ν 2))

(EV11)( ∀z)((Υν¯(LH z)) ∨ (Υν¯(MH z)) ∨ (Υν¯(RH z))) IRAFM

VilémNovákand Irina Perfilieva (IRAFM, CZ) Future of MFL Future of MFL 15 / 53 Relevant characteristics — Context

(A) Nonempty, linearly ordered and bounded scale, three distinguished limit points: left bound, right bound, and a central 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] −→ M} Crisp linear ordering ≤w in each context w

IRAFM

VilémNovákand Irina Perfilieva (IRAFM, CZ) Future of MFL Future of MFL 16 / 53 Relevant characteristics — Context

(A) Nonempty, linearly ordered and bounded scale, three distinguished limit points: left bound, right bound, and a central 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] −→ M} Crisp linear ordering ≤w in each context w

IRAFM

VilémNovákand Irina Perfilieva (IRAFM, CZ) Future of MFL Future of MFL 16 / 53 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

IRAFM

VilémNovákand Irina Perfilieva (IRAFM, CZ) Future of MFL Future of MFL 17 / 53 Relevant characteristics —Intension

(B) Function from the set of contexts into a set of fuzzy sets Intension : W F (w ([0,1])) Int(A ) = λw λx (Aw)x M (Int(A )) : W −→ F (w([0, 1]) Scheme of intensionvarious contexts extensions

IRAFM

VilémNovákand Irina Perfilieva (IRAFM, CZ) Future of MFL Future of MFL 17 / 53 Horizon (C) Each of the limit points is a starting point of some horizon running from it in the sense of the ordering of the scale towards the 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] IRAFM

VilémNovákand Irina Perfilieva (IRAFM, CZ) Future of MFL Future of MFL 18 / 53 Horizon (C) Each of the limit points is a starting point of some horizon running from it in the sense of the ordering of the scale towards the next limit point (the horizon vanishes beyond)

Three horizons

LH RH MH

v v v L S R

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] IRAFM

VilémNovákand Irina Perfilieva (IRAFM, CZ) Future of MFL Future of MFL 18 / 53 Relevant characteristics — horizon

(D) Each horizon is represented by a special fuzzy set determined by a reasoning analogous to that leading to the sorites paradox.

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

IRAFM

VilémNovákand Irina Perfilieva (IRAFM, CZ) Future of MFL Future of MFL 19 / 53 Relevant characteristics — horizon

(D) Each horizon is represented by a special fuzzy set determined by a reasoning analogous to that leading to the sorites paradox.

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

IRAFM

VilémNovákand Irina Perfilieva (IRAFM, CZ) Future of MFL Future of MFL 19 / 53 Construction of extensions of evaluative expressions

Extension of evaluative expression is delineated by shifting of the horizon using linguistic hedge

Hedges — shifts of horizon ν : [0, 1] −→ [0, 1]

IRAFM

VilémNovákand Irina Perfilieva (IRAFM, CZ) Future of MFL Future of MFL 20 / 53 Construction of extensions of evaluative expressions

c MH LH RH b a,b,c a 1 2 v a v v Hedge L Me S aMe R

Context

IRAFM

VilémNovákand Irina Perfilieva (IRAFM, CZ) Future of MFL Future of MFL 21 / 53 Falakros/sorites paradox in evaluative expressions

Theorem (hlinguistic hedgei small) • Zero is “(very) small” in each context ` (∀w)((Sm ν)w 0) • In each context there is p which surely is not “(very) small” ` (∀w)(∃p)(∆∆¬(Sm ν)w p) • In each context there is no n which is surely small and n + 1 surely is not small ` (∀w)¬(∃n)(∆∆∆(Sm ν)w n &&∆∆¬(Sm ν)w(n + 1)) • In each context: if n is small then it is almost true that n + 1 is also small ` (∀w)(∀n)((Sm ν)w n ⇒ At((Sm ν)w (n + 1))

(At(A) is measured by( Sm ν)w n ⇒ (Sm ν)w (n + 1)) In each context there is no last surely small x

and no ﬁrst surely big x IRAFM

VilémNovákand Irina Perfilieva (IRAFM, CZ) Future of MFL Future of MFL 22 / 53 Intermediate quantifiers P. L. Peterson, Intermediate Quantifiers. Logic, linguistics, and Aristotelian semantics, Ashgate, Aldershot 2000. Quantifiers in natural language Words (expressions) that precede and modify nouns; tell us how many or how much. They specify quantity of specimens in the domain of discourse having a certain property.

Example All, Most, Almost all, Few, Many, Some, No Most women in the party are well dressed Few students passed exam

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

VilémNovákand Irina Perfilieva (IRAFM, CZ) Future of MFL Future of MFL 23 / 53 Intermediate quantifiers P. L. Peterson, Intermediate Quantifiers. Logic, linguistics, and Aristotelian semantics, Ashgate, Aldershot 2000. Quantifiers in natural language Words (expressions) that precede and modify nouns; tell us how many or how much. They specify quantity of specimens in the domain of discourse having a certain property.

Example All, Most, Almost all, Few, Many, Some, No Most women in the party are well dressed Few students passed exam

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

VilémNovákand Irina Perfilieva (IRAFM, CZ) Future of MFL Future of MFL 23 / 53 Semantics of intermediate quantifiers

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

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

IRAFM

VilémNovákand Irina Perfilieva (IRAFM, CZ) Future of MFL Future of MFL 24 / 53 Semantics of intermediate quantifiers

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

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

IRAFM

VilémNovákand Irina Perfilieva (IRAFM, CZ) Future of MFL Future of MFL 24 / 53 Formal theory of intermediate quantifiers

T IQ = T Ev+ 4 special axioms “hQuantiﬁeri B’s are A”

∀ (QEv x)(B, A) := (∃z)((∆∆∆(z ⊆ B) & | {z } “the greatest” part of B’s (∀x)(z x ⇒ Ax)) ∧ | {z } each of B’s has A Ev((µB)z)) | {z } size of z is evaluated by Ev

Ev — extension of a certain evaluative expression (big, very big, small, etc.)

IRAFM

VilémNovákand Irina Perfilieva (IRAFM, CZ) Future of MFL Future of MFL 25 / 53 Special intermediate quantifiers

∀ P: Almost all B are A := QBi Ex (B, A) (∃z)((∆∆∆(z ⊆ B)&&&(∀x)(zx ⇒ Ax)) ∧ (Bi Ex)((µB)z)) ∀ B: Almost all B are not A := QBi Ex (B,¬A) ∀ T: Most B are A := QBi Ve (B, A) ∀ D: Most B are not A := QBi Ve (B,¬A) ∀ K: Many B are A := Q¬(Sm ν¯)(B, A) ∀ G: Many B are not A := Q¬(Sm ν¯)(B,¬A) ∀ F: Few B are A := QSm Ve (B, A) ∀ H: Few B are not A := QSm Ve (B,¬A)

IRAFM

VilémNovákand Irina Perfilieva (IRAFM, CZ) Future of MFL Future of MFL 26 / 53 Special intermediate quantifiers

Classical quantiﬁers

∀ A: All B are A := QBi∆(B, A) ≡ (∀x)(Bx ⇒ Ax), ∀ E: No B are A := QBi∆(B,¬A) ≡ (∀x)(Bx ⇒ ¬Ax), ∃ I: Some B are A := QBi∆(B, A) ≡ (∃x)(Bx ∧ Ax), ∃ O: Some B are not A := QBi∆(B,¬A) ≡ (∃x)(Bx ∧ ¬Ax).

IRAFM

VilémNovákand Irina Perfilieva (IRAFM, CZ) Future of MFL Future of MFL 27 / 53 105 valid generalized Aristotle’s syllogisms Figure I Figure II

Q1 M is Y Q1 Y is M

Q2 X is M Q2 X is M Q3 X is Y 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)(QBi Ve x)(X , M) ∀ Most people in the party (X ) are well dressed (Y )(QBi Ve x)(X , Y )

Example (ETO-II) No lazy people (Y ) pass exam (M) ¬(∃x)(Yx ∧ Mx) ∀ Most students (X ) pass exam (M)(QBi Ve x)(X , M)

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

VilémNovákand Irina Perfilieva (IRAFM, CZ) Future of MFL Future of MFL 28 / 53 105 valid generalized Aristotle’s syllogisms Figure I Figure II

Q1 M is Y Q1 Y is M

Q2 X is M Q2 X is M Q3 X is Y Q3 X is Y

Example (ETO-II) No lazy people (Y ) pass exam (M) ¬(∃x)(Yx ∧ Mx) ∀ Most students (X ) pass exam (M)(QBi Ve x)(X , M)

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

VilémNovákand Irina Perfilieva (IRAFM, CZ) Future of MFL Future of MFL 28 / 53 105 valid generalized Aristotle’s syllogisms Figure III Figure IV

Q1 M is Y Q1 Y is M

Q2 M is X Q2 M is X Q3 X is Y Q3 X is Y

Example (PPI-III) ∀ Almost all old people (M) are ill (Y )(QBi Ex x)(Mx, Yx) ∀ Almost all old people (M) have gray hair (X )(QBi Ex x)(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) ∀ (QBi Ve x)(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)IRAFM

VilémNovákand Irina Perfilieva (IRAFM, CZ) Future of MFL Future of MFL 29 / 53 105 valid generalized Aristotle’s syllogisms Figure III Figure IV

Q1 M is Y Q1 Y is M

Q2 M is X Q2 M is X Q3 X is Y Q3 X is Y

Example (TAI-IV) Most shares with downward trend (Y ) are from car industry (M) ∀ (QBi Ve x)(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)IRAFM

VilémNovákand Irina Perfilieva (IRAFM, CZ) Future of MFL Future of MFL 29 / 53 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)

IRAFM

VilémNovákand Irina Perfilieva (IRAFM, CZ) Future of MFL Future of MFL 30 / 53 Generalized 5-square of opposition

IRAFM

VilémNovákand Irina Perfilieva (IRAFM, CZ) Future of MFL Future of MFL 31 / 53 Negation in FNL Topic—focus articulation: Each sentence is divided into topic (known information) and focus (new information)

Focus is negated

Example It is not true, that: (i) JOHN reads Jane’s paper Somebody else reads it (ii) John READS JANE’S PAPER John does something else (iii) John reads JANE’S PAPER John reads some other paper (iv) John reads Jane’s PAPER John reads Jane’s book (lying on the same table) IRAFM

VilémNovákand Irina Perfilieva (IRAFM, CZ) Future of MFL Future of MFL 32 / 53 Negation in FNL Topic—focus articulation: Each sentence is divided into topic (known information) and focus (new information)

Focus is negated

VilémNovákand Irina Perfilieva (IRAFM, CZ) Future of MFL Future of MFL 32 / 53 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

IRAFM

VilémNovákand Irina Perfilieva (IRAFM, CZ) Future of MFL Future of MFL 33 / 53 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

IRAFM

VilémNovákand Irina Perfilieva (IRAFM, CZ) Future of MFL Future of MFL 33 / 53 Negation in FNL

not unhappy 6≡ happy This is not violation of double negation

IRAFM

VilémNovákand Irina Perfilieva (IRAFM, CZ) Future of MFL Future of MFL 34 / 53 Negation in FNL

not unhappy 6≡ happy This is not violation of double negation

unhappy is antonym of happy

IRAFM

VilémNovákand Irina Perfilieva (IRAFM, CZ) Future of MFL Future of MFL 34 / 53 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

IRAFM

VilémNovákand Irina Perfilieva (IRAFM, CZ) Future of MFL Future of MFL 35 / 53 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 concrete place • People are able to follow them in an arbitrary signalized intersection We can “distinguish” between the rules despite the fact that their meaning is vague

IRAFM

VilémNovákand Irina Perfilieva (IRAFM, CZ) Future of MFL Future of MFL 36 / 53 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

IRAFM

VilémNovákand Irina Perfilieva (IRAFM, CZ) Future of MFL Future of MFL 36 / 53 MFL and AST?

Suggestion: Switch the development of MFL to a new mathematical frame based on the Alternative Set Theory by P. Vopˇenka

Diﬀerent understanding to inﬁnity

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

IRAFM

VilémNovákand Irina Perfilieva (IRAFM, CZ) Future of MFL Future of MFL 37 / 53 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 would take 10100 years!! Visible universe has about 1080 atoms and exists less than 1011 years!

IRAFM

VilémNovákand Irina Perfilieva (IRAFM, CZ) Future of MFL Future of MFL 38 / 53 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 would take 10100 years!! Visible universe has about 1080 atoms and exists less than 1011 years!

IRAFM

VilémNovákand Irina Perfilieva (IRAFM, CZ) Future of MFL Future of MFL 38 / 53 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 would take 10100 years!! Visible universe has about 1080 atoms and exists less than 1011 years!

IRAFM

VilémNovákand Irina Perfilieva (IRAFM, CZ) Future of MFL Future of MFL 38 / 53 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 would take 10100 years!! Visible universe has about 1080 atoms and exists less than 1011 years!

IRAFM

VilémNovákand Irina Perfilieva (IRAFM, CZ) Future of MFL Future of MFL 38 / 53 God’s view

Ha, Ha, why are you boring me with such ridiculously small numbers! Where is INFINITY!?

IRAFM

VilémNovákand Irina Perfilieva (IRAFM, CZ) Future of MFL Future of MFL 39 / 53 God’s view

Ha, Ha, why are you boring me with such ridiculously small numbers! Where is INFINITY!?

We cannot imagine even half of this way!

Something is wrong! IRAFM

VilémNovákand Irina Perfilieva (IRAFM, CZ) Future of MFL Future of MFL 39 / 53 God’s view

Ha, Ha, why are you boring me with such ridiculously small numbers! Where is INFINITY!?

We cannot imagine even half of this way!

Something is wrong! IRAFM

VilémNovákand Irina Perfilieva (IRAFM, CZ) Future of MFL Future of MFL 39 / 53 Big numbers behave as infinite

Basic property

α ≈ α + 1

10001 people slept “whole night” on 10000 beds!

IRAFM

VilémNovákand Irina Perfilieva (IRAFM, CZ) Future of MFL Future of MFL 40 / 53 Big numbers behave as infinite

Basic property

α ≈ α + 1

10001 people slept “whole night” on 10000 beds!

IRAFM

VilémNovákand Irina Perfilieva (IRAFM, CZ) Future of MFL Future of MFL 40 / 53 Big numbers behave as infinite

Basic property

α ≈ α + 1

10001 people slept “whole night” on 10000 beds!

IRAFM

VilémNovákand Irina Perfilieva (IRAFM, CZ) Future of MFL Future of MFL 40 / 53 Alternative Set Theory

• An attempt at developing a new mathematics on the basis of criticism of classical one • Take useful principles and replace others by more natural ones • Make mathematics closer to human perception of reality

IRAFM

VilémNovákand Irina Perfilieva (IRAFM, CZ) Future of MFL Future of MFL 41 / 53 Alternative Set Theory

IRAFM

VilémNovákand Irina Perfilieva (IRAFM, CZ) Future of MFL Future of MFL 41 / 53 Alternative Set Theory

IRAFM

VilémNovákand Irina Perfilieva (IRAFM, CZ) Future of MFL Future of MFL 41 / 53 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 last elements From classical point of view every set is classically ﬁnite

IRAFM

VilémNovákand Irina Perfilieva (IRAFM, CZ) Future of MFL Future of MFL 42 / 53 Fundamental concepts

Actualizability ↔ Potentiality Class Actualized grouping of objects

X = {x | ϕ(x)}

IRAFM

VilémNovákand Irina Perfilieva (IRAFM, CZ) Future of MFL Future of MFL 42 / 53 Fundamental concepts

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

IRAFM

VilémNovákand Irina Perfilieva (IRAFM, CZ) Future of MFL Future of MFL 43 / 53 Fundamental concepts

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

IRAFM

VilémNovákand Irina Perfilieva (IRAFM, CZ) Future of MFL Future of MFL 43 / 53 Fundamental concepts

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

IRAFM

VilémNovákand Irina Perfilieva (IRAFM, CZ) Future of MFL Future of MFL 43 / 53 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.

IRAFM

VilémNovákand Irina Perfilieva (IRAFM, CZ) Future of MFL Future of MFL 44 / 53 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.

IRAFM

VilémNovákand Irina Perfilieva (IRAFM, CZ) Future of MFL Future of MFL 44 / 53 Indiscernibility relation

\ x ≡ y iﬀ hx, yi ∈ {Rn | n ∈ FN}

sequence of still sharper criteria Rn Indiscernibility of rational numbers

. 1 x = y iff | x − y |< ; n ∈ n FN . x ∈ IS iff x = 0 infinitely small 1 x ∈ iff ∈ infinitely large IL x IS

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

IRAFM

VilémNovákand Irina Perfilieva (IRAFM, CZ) Future of MFL Future of MFL 45 / 53 Indiscernibility relation

\ x ≡ y iﬀ hx, yi ∈ {Rn | n ∈ FN}

sequence of still sharper criteria Rn Indiscernibility of rational numbers

. 1 x = y iff | x − y |< ; n ∈ n FN . x ∈ IS iff x = 0 infinitely small 1 x ∈ iff ∈ infinitely large IL x IS

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

IRAFM

VilémNovákand Irina Perfilieva (IRAFM, CZ) Future of MFL Future of MFL 45 / 53 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 imaginary Imaginary classes are rare

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

F ⊆ f

IRAFM

VilémNovákand Irina Perfilieva (IRAFM, CZ) Future of MFL Future of MFL 46 / 53 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 imaginary Imaginary classes are rare

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

F ⊆ f

IRAFM

VilémNovákand Irina Perfilieva (IRAFM, CZ) Future of MFL Future of MFL 46 / 53 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 imaginary Imaginary classes are rare

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

F ⊆ f

IRAFM

VilémNovákand Irina Perfilieva (IRAFM, CZ) Future of MFL Future of MFL 46 / 53 Fuzzy approach Indiscernibility ≡ sharpened to \ ≡b = {Rβ | β ≤ γ} The intensity of our “effort” to discern the objects x and y — a number α of hx, yi ∈ Rβ, β ∈ α

Degree of equality α x ≡ y iﬀ ν = . ν γ

Theorem

(a) x ≡1 x.

(b) x ≡ν y implies y ≡ν x.

VilémNovákand Irina Perfilieva (IRAFM, CZ) Future of MFL Future of MFL 47 / 53 Fuzzy sets

Membership degree of x in X

F _ X (x) = {ν | (∃y ∈ Y )(x ≡ν y)}

Measure of the greatest intensity of our “eﬀort” to discern x from elements of the kernel Y . X F A fuzzy set approximating the real class X

IRAFM

VilémNovákand Irina Perfilieva (IRAFM, CZ) Future of MFL Future of MFL 48 / 53 Fuzzy sets

Theorem

(a) F F F F F X1 (x) ⊗ X2 (x) l (X1 ∩ X2) (x) l X1 (x) ∧ X2 (x) (b) F F F F F X1 (x) ∨ X2 (x) l (X1 ∪ X2) (x) l X1 (x) ⊕ X2 (x) (c) . (X )F (x) = (V − X )F (x) = 1 − X F (x)

IRAFM

VilémNovákand Irina Perfilieva (IRAFM, CZ) Future of MFL Future of MFL 49 / 53 Conclusions

• MFL should focus on the detailed development of some well selected logics. • MFL should help to develop the concept of Fuzzy Natural Logic Open 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 AST Introducing degrees into AST

IRAFM

VilémNovákand Irina Perfilieva (IRAFM, CZ) Future of MFL Future of MFL 50 / 53 Conclusions

IRAFM

VilémNovákand Irina Perfilieva (IRAFM, CZ) Future of MFL Future of MFL 50 / 53 Conclusions

IRAFM

VilémNovákand Irina Perfilieva (IRAFM, CZ) Future of MFL Future of MFL 50 / 53 References

Vopˇenka, P., Mathematics in the Alternative Set Theory. Teubner, Leipzig 1979. Vopˇenka, P., Fundamentals of the Mathematics In the Alternative Set Theory. Alfa, Bratislava 1989 (in Slovak). Vopˇenka, P., Calculus Infinitesimalis. Pars Prima. Introduction to differential calculus of functions of one variable. KANINA 2010 (in Czech). Vopˇenka, P., Calculus Infinitesimalis. Pars Secunda. Integral of real functions of one variable. KANINA 2011 (in Czech).

Many papers in Commentationes Mathematicae Universitatis Carolinae (CMUC)

IRAFM

VilémNovákand Irina Perfilieva (IRAFM, CZ) Future of MFL Future of MFL 51 / 53 References

Holˇcapek, M., Monadic L-fuzzy quantiﬁers of the type h1n, 1i, FSS 159(2008), 1811-1835. V. Nov´ak, On fuzzy type theory, FSS, 149(2005), 235–273.

V. Nov´ak, EQ-algebra-based fuzzy type theory and its extensions, Logic Journal of the IGPL 19(2011), 512-542

V. Nov´ak, A comprehensive theory of trichotomous evaluative linguistic expressions, FSS 159(2008), 2939-2969.

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

P. Murinov´a,V. Nov´ak, A formal theory of generalized intermediate syllogisms, FSS 186(2012), 47-80

V. Nov´ak,The Alternative Mathematical Model of Linguistic Semantics and Pragmatics, Plenum, New York, 1992.

P. L. Peterson, Intermediate Quantiﬁers. Logic, linguistics, and Aristotelian semantics, Ashgate, Aldershot 2000. IRAFM

VilémNovákand Irina Perfilieva (IRAFM, CZ) Future of MFL Future of MFL 52 / 53 Thank you for your attention

IRAFM

VilémNovákand Irina Perfilieva (IRAFM, CZ) Future of MFL Future of MFL 53 / 53