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 ����� Vil´ em Nov´ ak and Irina Perfilieva (IRAFM, CZ) Future of MFL Future of MFL 7 / 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 ����� Vil´ em Nov´ ak and Irina Perfilieva (IRAFM, CZ) Future of MFL Future of MFL 7 / 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 ����� Vil´ em Nov´ ak and Irina Perfilieva (IRAFM, CZ) Future of MFL Future of MFL 7 / 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 ����� Vil´ em Nov´ ak and 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 ����� Vil´ em Nov´ ak and 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 . 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 ����� Vil´ em Nov´ ak and 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 . 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 ����� Vil´ em Nov´ ak and 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 ����� Vil´ em Nov´ ak and 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 ����� Vil´ em Nov´ ak and 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 ����� Vil´ em Nov´ ak and Irina Perfilieva (IRAFM, CZ) Future of MFL Future of MFL 9 / 53
Basic concepts Types Elementary types: o (truth values), ǫ (objects) Composed types: βα Formulas have types: A α ∈ Form α , A α ≡ B α ∈ Form o λ x α C β ∈ Form βα ∆ oo A o ∈ Form o ∆ ∆ Formulas of type o are propositions Interpretation of formulas A βα are functions M α − → M β ����� Vil´ em Nov´ ak and Irina Perfilieva (IRAFM, CZ) Future of MFL Future of MFL 10 / 53
Semantics of FTT Frame M = � ( M α , = α ) α ∈ Types , E ∆ � Fuzzy equality = α : M α × M α − → L [ x = α x ] = 1 (reflexivity) [ x = α y ] = [ y = α x ] (symmetry) [ x = α y ] ⊗ [ y = α z ] ≤ [ x = α z ] (transitivity) ����� Vil´ em Nov´ ak and 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 A o T ⊢ A o iff T | = A o . FTT has a lot of interesting properties and great explication power ����� Vil´ em Nov´ ak and Irina Perfilieva (IRAFM, CZ) Future of MFL Future of MFL 12 / 53
Natural language in FNL Standard � Lukasiewicz MV ∆ -algebra Two important classes of natural language expressions • Evaluative linguistic expressions • Intermediate quantifiers ����� Vil´ em Nov´ ak and Irina Perfilieva (IRAFM, CZ) Future of MFL Future of MFL 13 / 53
Natural language in FNL Standard � Lukasiewicz MV ∆ -algebra Two important classes of natural language expressions • Evaluative linguistic expressions • Intermediate quantifiers ����� Vil´ em Nov´ ak and 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 ����� Vil´ em Nov´ ak and 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. • 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 ����� Vil´ em Nov´ ak and 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. • 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 ����� Vil´ em Nov´ ak and Irina Perfilieva (IRAFM, CZ) Future of MFL Future of MFL 14 / 53
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 )ˆ ∧ ( ∃ u )ˆ ∧ ( ∃ u )ˆ Υ( ⊥ ∼ u ) ∧ ∧ Υ( † ∼ u ) ∧ ∧ Υ( ⊤ ∼ u ) ν ′ & ν ′ )( Hedge ν (EV10) NatHedge ¯ & & & ν ν ν & &( ∃ ν ν ν )( ∃ ν ν ν & ν & Hedge ν ν & ( ν ν ν 1 � ¯ ν ν ∧ ∧ ∧ ¯ ν ν � ν ν ν 2 )) ν ν (EV11) ( ∀ z )((Υ¯ ν ( LH z )) ∨ ∨ ∨ (Υ¯ ν ( MH z )) ∨ ∨ (Υ¯ ∨ ν ( RH z ))) ν ν ν ν ν ν ����� Vil´ em Nov´ ak and 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 w α o M p ( w α o ) = w : [0 , 1] − → M : w (0) = v L (left bound) w (0 . 5) = v S (central point) w (1) = v R (right bound) Set of contexts W = { w | w : [0 , 1] − → M } Crisp linear ordering ≤ w in each context w ����� Vil´ em Nov´ ak and 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 w α o M p ( w α o ) = w : [0 , 1] − → M : w (0) = v L (left bound) w (0 . 5) = v S (central point) w (1) = v R (right bound) Set of contexts W = { w | w : [0 , 1] − → M } Crisp linear ordering ≤ w in each context w ����� Vil´ em Nov´ ak and 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 ) = λ w λ x ( Aw ) x M ( Int ( A )) : W − → F ( w ([0 , 1]) Scheme of intension ����� Vil´ em Nov´ ak and 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 Int ( A ) = λ w λ x ( Aw ) x M ( Int ( A )) : W − → F ( w ([0 , 1]) Scheme of intension ֏ ֏ ֏ ����� Vil´ em Nov´ ak and 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 ) = [ v L ≈ w x ] MH ( a ) = [0 . 5 ∼ a ] , MH ( w x ) = [ v S ≈ w x ] RH ( a ) = [1 ∼ a ] , RH ( w x ) = [ v R ≈ w x ] ����� Vil´ em Nov´ ak and 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 1 LH RH MH v S v R v L LH ( a ) = [0 ∼ a ] , LH ( w x ) = [ v L ≈ w x ] MH ( a ) = [0 . 5 ∼ a ] , MH ( w x ) = [ v S ≈ w x ] RH ( a ) = [1 ∼ a ] , RH ( w x ) = [ v R ≈ w x ] ����� Vil´ em Nov´ ak and 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. ����� Vil´ em Nov´ ak and 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. ����� Vil´ em Nov´ ak and 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] ����� Vil´ em Nov´ ak and Irina Perfilieva (IRAFM, CZ) Future of MFL Future of MFL 20 / 53
Construction of extensions of evaluative expressions c MH RH LH b a,b,c a 1 2 v R v L v S a a Hedge Me Me Context ����� Vil´ em Nov´ ak and Irina Perfilieva (IRAFM, CZ) Future of MFL Future of MFL 21 / 53
Falakros/sorites paradox in evaluative expressions Theorem ( � linguistic hedge � small) • Zero is “(very) small” in each context ⊢ ( ∀ w )(( Sm ν ν ) w 0) ν • In each context there is p which surely is not “(very) small” ⊢ ( ∀ w )( ∃ p )(∆ ∆ ∆ ¬ ¬ ( Sm ν ¬ ν ) w p ) ν • In each context there is no n which is surely small and n + 1 surely is not small ⊢ ( ∀ w ) ¬ ¬ ¬ ( ∃ n )(∆ ∆ ∆( Sm ν ν ν ) w n & &∆ & ∆ ∆ ¬ ¬ ¬ ( Sm ν ν 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 ν ν ) w n ⇒ ⇒ At (( Sm ν ν ν ) w ( n + 1)) (At( A ) is measured by ( Sm ν ν ) w n ⇒ ⇒ ⇒ ( Sm ν ν ) w ( n + 1)) ν ν In each context there is no last surely small x and no first surely big x ����� Vil´ em Nov´ ak and 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 ����� Vil´ em Nov´ ak and 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 ����� Vil´ em Nov´ ak and 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 ����� Vil´ em Nov´ ak and 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 ����� Vil´ em Nov´ ak and Irina Perfilieva (IRAFM, CZ) Future of MFL Future of MFL 24 / 53
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 (( µ B ) z )) � �� � size of z is evaluated by Ev Ev — extension of a certain evaluative expression ( big, very big, small , etc.) ����� Vil´ em Nov´ ak and Irina Perfilieva (IRAFM, CZ) Future of MFL Future of MFL 25 / 53
Special intermediate quantifiers P: Almost all B are A := Q ∀ Bi Ex ( B , A ) ( ∃ z )((∆ ∆ ∆( z ⊆ B )& & &( ∀ x )( zx ⇒ ⇒ Ax )) ∧ ⇒ ∧ ( Bi Ex )(( µ B ) 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 ∀ ν ) ( B , A ) ¬ ( Sm ¯ ¬ ν ν ¬ G: Many B are not A := Q ∀ ν ) ( B , ¬ ¬ A ) ¬ ¬ ( Sm ¯ ν ¬ ¬ ν F: Few B are A := Q ∀ Sm Ve ( B , A ) H: Few B are not A := Q ∀ Sm Ve ( B , ¬ ¬ ¬ A ) ����� Vil´ em Nov´ ak and Irina Perfilieva (IRAFM, CZ) Future of MFL Future of MFL 26 / 53
Special intermediate quantifiers Classical quantifiers A: All B are A := Q ∀ ∆ ( B , A ) ≡ ( ∀ x )( Bx ⇒ ⇒ Ax ) , ⇒ Bi ∆ ∆ E: No B are A := Q ∀ ∆ ( B , ¬ ¬ ¬ A ) ≡ ( ∀ x )( Bx ⇒ ⇒ ⇒ ¬ ¬ ¬ Ax ) , Bi ∆ ∆ I: Some B are A := Q ∃ ∆ ( B , A ) ≡ ( ∃ x )( Bx ∧ ∧ Ax ) , ∧ Bi ∆ ∆ O: Some B are not A := Q ∃ ∆ ( B , ¬ ¬ ¬ A ) ≡ ( ∃ x )( Bx ∧ ∧ ∧ ¬ ¬ ¬ Ax ) . Bi ∆ ∆ ����� Vil´ em Nov´ ak and Irina Perfilieva (IRAFM, CZ) Future of MFL Future of MFL 27 / 53
105 valid generalized Aristotle’s syllogisms Figure I Figure II Q 1 M is Y Q 1 Y is M Q 2 X is M Q 2 X is M Q 3 X is Y Q 3 X is Y Example (ATT-I) ⇒ All women ( M ) are well dressed ( Y ) ( ∀ x )( M x ⇒ ⇒ Y x ) ( Q ∀ Most people in the party ( X ) are women ( M ) Bi Ve x )( X , M ) ( Q ∀ Most people in the party ( X ) are well dressed ( Y ) Bi Ve x )( X , Y ) Example (ETO-II) ¬ ∧ No lazy people ( Y ) pass exam ( M ) ¬ ¬ ( ∃ x )( Yx ∧ ∧ Mx ) ( Q ∀ Most students ( X ) pass exam ( M ) Bi Ve x )( X , M ) Some students ( X ) are not lazy people ( Y ) ( ∃ x )( X ∧ ∧ ∧ ¬ ¬ Y ) ¬ ����� Vil´ em Nov´ ak and Irina Perfilieva (IRAFM, CZ) Future of MFL Future of MFL 28 / 53
105 valid generalized Aristotle’s syllogisms Figure I Figure II Q 1 M is Y Q 1 Y is M Q 2 X is M Q 2 X is M Q 3 X is Y Q 3 X is Y Example (ATT-I) ⇒ All women ( M ) are well dressed ( Y ) ( ∀ x )( M x ⇒ ⇒ Y x ) ( Q ∀ Most people in the party ( X ) are women ( M ) Bi Ve x )( X , M ) ( Q ∀ Most people in the party ( X ) are well dressed ( Y ) Bi Ve x )( X , Y ) Example (ETO-II) ¬ ∧ No lazy people ( Y ) pass exam ( M ) ¬ ¬ ( ∃ x )( Yx ∧ ∧ Mx ) ( Q ∀ Most students ( X ) pass exam ( M ) Bi Ve x )( X , M ) Some students ( X ) are not lazy people ( Y ) ( ∃ x )( X ∧ ∧ ∧ ¬ ¬ Y ) ¬ ����� Vil´ em Nov´ ak and Irina Perfilieva (IRAFM, CZ) Future of MFL Future of MFL 28 / 53
105 valid generalized Aristotle’s syllogisms Figure III Figure IV Q 1 M is Y Q 1 Y is M Q 2 M is X Q 2 M is X Q 3 X is Y Q 3 X is Y Example (PPI-III ) ( Q ∀ Almost all old people ( M ) are ill ( Y ) Bi Ex x )( Mx , Yx ) ( Q ∀ Almost all old people ( M ) have gray hair ( X ) Bi 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 ) ( Q ∀ Bi 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 ) ����� Vil´ em Nov´ ak and Irina Perfilieva (IRAFM, CZ) Future of MFL Future of MFL 29 / 53
105 valid generalized Aristotle’s syllogisms Figure III Figure IV Q 1 M is Y Q 1 Y is M Q 2 M is X Q 2 M is X Q 3 X is Y Q 3 X is Y Example (PPI-III ) ( Q ∀ Almost all old people ( M ) are ill ( Y ) Bi Ex x )( Mx , Yx ) ( Q ∀ Almost all old people ( M ) have gray hair ( X ) Bi 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 ) ( Q ∀ Bi 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 ) ����� Vil´ em Nov´ ak and Irina Perfilieva (IRAFM, CZ) Future of MFL Future of MFL 29 / 53
Square of opposition — basic relations P 1 , P 2 ∈ Form o are: (i) contraries if T ⊢ ¬ ¬ ¬ ( P 1 & & & P 2 ). P 1 and P 2 cannot be both true but can be both false (ii) sub-contraries if T ⊢ P 1 ∇ ∇ ∇ P 2 . weak sub-contraries if T ⊢ Υ( P 1 ∨ ∨ ∨ P 2 ) P 1 and P 2 cannot be both false but can be both true (iii) P 1 , P 2 ∈ Form o are contradictories if ¬ ∆ & ∆ ∆ ∇ ∆ T ⊢ ¬ ¬ (∆ ∆ P 1 & &∆ ∆ P 2 ) as well as T ⊢ ∆ ∆ P 1 ∇ ∇ ∆ ∆ P 2 . P 1 and P 2 cannot be both true as well as both false ⇒ (iv) The formula P 2 is a subaltern of P 1 in T if T ⊢ P 1 ⇒ ⇒ P 2 ( P 1 a superaltern of P 2 ) ����� Vil´ em Nov´ ak and Irina Perfilieva (IRAFM, CZ) Future of MFL Future of MFL 30 / 53
Generalized 5-square of opposition ����� Vil´ em Nov´ ak and 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) ����� Vil´ em Nov´ ak and 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 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) ����� Vil´ em Nov´ ak and 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 ����� Vil´ em Nov´ ak and 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 ����� Vil´ em Nov´ ak and Irina Perfilieva (IRAFM, CZ) Future of MFL Future of MFL 33 / 53
Negation in FNL not unhappy �≡ happy This is not violation of double negation ����� Vil´ em Nov´ ak and Irina Perfilieva (IRAFM, CZ) Future of MFL Future of MFL 34 / 53
Negation in FNL not unhappy �≡ happy This is not violation of double negation unhappy is antonym of happy ����� Vil´ em Nov´ ak and 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 A 1 THEN Y is B 1 IF X is A 2 THEN Y is B 2 . . . . . . . . . . . . . . . . . . . . . . . . IF X is A m THEN Y is B m Text describing one’s behavior in some (decision) situation ����� Vil´ em Nov´ ak and 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: R 1 := IF Distance is very small THEN Acceleration is very big R 2 := IF Distance is small THEN Brake is big R 3 := 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 ����� Vil´ em Nov´ ak and 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: R 1 := IF Distance is very small THEN Acceleration is very big R 2 := IF Distance is small THEN Brake is big R 3 := 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 ����� Vil´ em Nov´ ak and 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. Vopˇ enka Different understanding to infinity Vopˇ enka: “Mathematics uses infinity whenever it faces vagueness!” ����� Vil´ em Nov´ ak and Irina Perfilieva (IRAFM, CZ) Future of MFL Future of MFL 37 / 53
Source of Natural Infinity Example Consider the number 10 120 Imagine counting each second 10 12 atoms Counting 10 120 atoms by counting each second 10 12 of them would take 10 100 years!! Visible universe has about 10 80 atoms and exists less than 10 11 years! ����� Vil´ em Nov´ ak and Irina Perfilieva (IRAFM, CZ) Future of MFL Future of MFL 38 / 53
Source of Natural Infinity Example Consider the number 10 120 Imagine counting each second 10 12 atoms Counting 10 120 atoms by counting each second 10 12 of them would take 10 100 years!! Visible universe has about 10 80 atoms and exists less than 10 11 years! ����� Vil´ em Nov´ ak and Irina Perfilieva (IRAFM, CZ) Future of MFL Future of MFL 38 / 53
Source of Natural Infinity Example Consider the number 10 120 Imagine counting each second 10 12 atoms Counting 10 120 atoms by counting each second 10 12 of them would take 10 100 years!! Visible universe has about 10 80 atoms and exists less than 10 11 years! ����� Vil´ em Nov´ ak and Irina Perfilieva (IRAFM, CZ) Future of MFL Future of MFL 38 / 53
Source of Natural Infinity Example Consider the number 10 120 Imagine counting each second 10 12 atoms Counting 10 120 atoms by counting each second 10 12 of them would take 10 100 years!! Visible universe has about 10 80 atoms and exists less than 10 11 years! ����� Vil´ em Nov´ ak and 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!? ����� Vil´ em Nov´ ak and 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! ����� Vil´ em Nov´ ak and 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! ����� Vil´ em Nov´ ak and 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! ����� Vil´ em Nov´ ak and 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! ����� Vil´ em Nov´ ak and 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! ����� Vil´ em Nov´ ak and 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 ����� Vil´ em Nov´ ak and Irina Perfilieva (IRAFM, CZ) Future of MFL Future of MFL 41 / 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 ����� Vil´ em Nov´ ak and Irina Perfilieva (IRAFM, CZ) Future of MFL Future of MFL 41 / 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 ����� Vil´ em Nov´ ak and 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 ����� Vil´ em Nov´ ak and 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 ) } 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 ����� Vil´ em Nov´ ak and 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 ����� Vil´ em Nov´ ak and 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 ����� Vil´ em Nov´ ak and 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 ����� Vil´ em Nov´ ak and 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. ����� Vil´ em Nov´ ak and 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. ����� Vil´ em Nov´ ak and Irina Perfilieva (IRAFM, CZ) Future of MFL Future of MFL 44 / 53
Indiscernibility relation � x ≡ y iff � x , y � ∈ { R n | n ∈ FN } sequence of still sharper criteria R n Indiscernibility of rational numbers | x − y | < 1 x . = y iff n ; n ∈ FN x . x ∈ IS iff = 0 infinitely small 1 x ∈ IL iff x ∈ IS infinitely large Figure X is a figure if x ∈ X and y ≡ x implies y ∈ X ����� Vil´ em Nov´ ak and Irina Perfilieva (IRAFM, CZ) Future of MFL Future of MFL 45 / 53
Indiscernibility relation � x ≡ y iff � x , y � ∈ { R n | n ∈ FN } sequence of still sharper criteria R n Indiscernibility of rational numbers | x − y | < 1 x . = y iff n ; n ∈ FN x . x ∈ IS iff = 0 infinitely small 1 x ∈ IL iff x ∈ IS infinitely large Figure X is a figure if x ∈ X and y ≡ x implies y ∈ X ����� Vil´ em Nov´ ak and 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 ����� Vil´ em Nov´ ak and 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 ����� Vil´ em Nov´ ak and 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 ����� Vil´ em Nov´ ak and Irina Perfilieva (IRAFM, CZ) Future of MFL Future of MFL 46 / 53
Fuzzy approach Indiscernibility ≡ sharpened to � ≡ = � { R β | β ≤ γ } The intensity of our “effort” to discern the objects x and y — a number α of � x , y � ∈ R β , β ∈ α Degree of equality ν = α x ≡ ν y iff γ . Theorem (a) x ≡ 1 x. (b) x ≡ ν y implies y ≡ ν x. (c) x ≡ ν 1 y and y ≡ ν 2 z implies x ≡ ν 3 z where ν 1 ⊗ ν 2 ⋖ ν 3 ����� Vil´ em Nov´ ak and Irina Perfilieva (IRAFM, CZ) Future of MFL Future of MFL 47 / 53
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 from elements of the kernel Y . X F A fuzzy set approximating the real class X ����� Vil´ em Nov´ ak and Irina Perfilieva (IRAFM, CZ) Future of MFL Future of MFL 48 / 53
Fuzzy sets Theorem (a) X F 1 ( x ) ⊗ X F 2 ( x ) ⋖ ( X 1 ∩ X 2 ) F ( x ) ⋖ X F 1 ( x ) ∧ X F 2 ( x ) (b) X F 1 ( x ) ∨ X F 2 ( x ) ⋖ ( X 1 ∪ X 2 ) F ( x ) ⋖ X F 1 ( x ) ⊕ X F 2 ( x ) (c) ( X ) F ( x ) = ( V − X ) F ( x ) . = 1 − X F ( x ) ����� Vil´ em Nov´ ak and 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 ����� Vil´ em Nov´ ak and Irina Perfilieva (IRAFM, CZ) Future of MFL Future of MFL 50 / 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 ����� Vil´ em Nov´ ak and Irina Perfilieva (IRAFM, CZ) Future of MFL Future of MFL 50 / 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 ����� Vil´ em Nov´ ak and Irina Perfilieva (IRAFM, CZ) Future of MFL Future of MFL 50 / 53
References Vopˇ enka, P., Mathematics in the Alternative Set Theory. Teubner, Leipzig 1979. Vopˇ enka, P., Fundamentals of the Mathematics In the Alternative Set Theory. Alfa, Bratislava 1989 (in Slovak). Vopˇ enka, P., Calculus Infinitesimalis. Pars Prima. Introduction to differential calculus of functions of one variable. KANINA 2010 (in Czech). Vopˇ enka, P., Calculus Infinitesimalis. Pars Secunda. Integral of real functions of one variable. KANINA 2011 (in Czech). Many papers in Commentationes Mathematicae Universitatis Carolinae (CMUC) ����� Vil´ em Nov´ ak and Irina Perfilieva (IRAFM, CZ) Future of MFL Future of MFL 51 / 53
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