Fuzzy Type Theory As Higher Order Fuzzy Logic - irafm

UNIVERSITY OF OSTRAVAInstitute for Research and Applications of Fuzzy ModelingFuzzy Type Theory As Higher Order FuzzyLogicVilém NovákResearch report No. 81Submitted/to appear:Proc. Int. Conference InTech’05, ThailandSupported by:Grant 201/04/1033 of the GA ČR, project MSM 6198898701 of the MŠMT ČRUniversity of OstravaInstitute for Research and Applications of Fuzzy Modeling30. dubna 22, 701 03 Ostrava 1, Czech Republictel.: +420-597 460 234 fax: +420-597 461 478e-mail: Vilem.Novak@osu.cz

2Fuzzy Type Theory As Higher Order Fuzzy LogicVilém Novák University of OstravaInstitute for Research and Applications of Fuzzy Modeling30. dubna 22, 701 03 Ostrava 1, Czech RepublicandInstitute of Information and Automation TheoryAcademy of Sciences of the Czech RepublicPod vodárenskou věží 4, 186 02 Praha 8, Czech RepublicEmail: Vilem.Novak@osu.czAbstractIn the paper, logical axioms, inference rules, semantics, and some specific properties including the completeness theorems offour kinds of fuzzy type theory are presented. This theory is a higher order fuzzy logic that can be used for precise formalization,for example, of the theory of computing with words, fuzzy IF-THEN rules, approximate reasoning, and others.I. INTRODUCTIONSuccessful development of the mathematical theory of fuzzy logic started in 1979 by the seminal paper of J. Pavelka [14].He developed a propositional fuzzy logic that has been extended to first order by V. Novák in [9] (see also [13]). It is specificfor this logic (called fuzzy logic with evaluated syntax and denoted by Ev Ł ) that it makes possible to consider axioms thatare not fully convincing. Consequently, both semantics as well as syntax of Ev Ł are evaluated. The concept of provability isgeneralized and the completeness theorem stating that the provability degree of a formula is equal to its truth degree is proved.After the seminal book of P. Hájek [7], mathematical fuzzy logic achieved a rapid development. Many formal systemscovering both propositional as well as predicate logic and differing in the structure of truth values appeared.To be able to capture semantics of natural language, with the goal to develop a formal theory of human reasoning (the, socalled, fuzzy logic in broader sense), fuzzy logic penetrated also into higher order, and a formal theory of fuzzy type theory(FTT) has been developed (see [10], [11]). This theory follows the development of classical type theory, as elaborated by A.Church [4] and L. Henkin [8], and later continued by P. Andrews in [1].Because of a large variety of possibilities, a discussion about, what is fuzzy logic, is now in progress. Based on the expressivepower and various experiences, several distinguished kinds of fuzzy logic took privilege over the other ones. Such logics areŁukasiewicz logic, Basic fuzzy logic (BL), and LΠ fuzzy logic. All these logics have propositional as well as predicate versions.It turns out that also fuzzy type theory can be developed in parallel ways. The original theory in [10] has been developed onthe basis of IMTL ∆ -algebra that is, a residuated lattice with prelinearity and double negation extended, moreover, by a specialunary operation of Baaz delta. In this paper we will present fuzzy type theories based on four different structures of truthvalues, namely IMTL ∆ , Łukasiewicz ∆ , BL ∆ and LΠ algebras. All these theories enjoy the generalized completeness property(i.e. completeness w.r.t. generalized models). It should be stressed that the fundamental connective in all of them is a fuzzyequality. Because of essential importance of this connective, the resulting theory is elegant and philosophically interesting.II. TRUTH VALUESAll above discussed fuzzy logics are based on extensions of residuated latticeL = 〈L, ∨, ∧, ⊗, →,0,1〉. (1)In this lattice, we define negation by ¬a = a → 0. Another important operation is biresiduation defined bya ↔ b = (a → b) ∧ (b → a).The residuated lattice (1) is an IMTL-algebra if it is prelinear, i.e.(a → b) ∨ (b → a) = 1, a,b ∈ L, (2)and the double negation ¬¬a = a holds true for all a ∈ L. It is a BL-algebra if it is prelinear (2) and divisible, i.e.a ⊗ (a → b) = a ∧ b, a,b ∈ L. (3)A Π-algebra is a BL-algebra fulfilling¬¬a ≤ (a → a ⊗ b) → b ⊗ ¬¬b, a,b ∈ L.

3An MV-algebra is a BL-algebra fulfilling the law of double negation.Finally, ŁΠ-algebra is the algebraL = 〈L, ∨, ∧, ⊗, ⊙, Ł →, Π →,0,1〉 (4)where(i) 〈L, ∨, ∧, ⊗, Ł →,0,1〉 is an MV-algebra,(ii) 〈L, ∨, ∧, ⊙, Π →,0,1〉 is a Π-algebra,(iii) a ⊙ (b ⊖ c) = (a ⊙ b) ⊖ (a ⊙ c)holds for all a,b,c ∈ L wherea ⊖ b = a ⊗ Ł ¬ bŁ¬a = a Ł → 0.We will further define Π ¬ a = a Π → 0 and ∆a = Π ¬ Ł ¬ a. The algebra of truth values is complete if its lattice reduct is complete.A very important operation for the development of fuzzy type theory is the Baaz delta operation, which is a unary operationon L fulfilling the following properties:∆a ∨ ¬∆a = 1,∆a ≤ a,∆(a ∨ b) ≤ ∆a ∨ ∆b,∆a ≤ ∆∆a,∆(a → b) ≤ ∆a → ∆b, ∆1 = 1.For the details concerning properties of the above structures of truth values, see [5], [6], [7], [13].A. ExamplesTypical (standard) examples of the above introduced algebras have the support L = [0,1] so that the residuated lattice iscomplete and has the general formL = 〈[0,1], ∨, ∧, ⊗, →,0,1〉. (5)Example of IMTL-algebra is (5) whereThe Baaz delta operation isa ⊗ b =a → b ={a ∧ b, if a + b > 1,0 otherwise{1, if a ≤ b,¬a ∨ b otherwise(nilpotent minimum)(residuum)¬a = 1 − a (negation)a ↔ b = (¬a ∧ ¬b) ∨ (a ∧ b).∆(a) ={1 if a = 1,0 otherwise.(biresiduation)(Baaz delta)Example of BL-algebra is any residuated lattice (5) where ⊗ is a continuous t-norm. For the case when ⊗ is product orminimum, we get{{1 if a = 0, 0 if a = 0,¬a = a → 0 =¬¬a =0 otherwise, 1 otherwise.Łukasiewicz MV-algebra is a residuated lattice (5) wherea ⊗ b = 0 ∨ (a + b − 1),(Łukasiewicz conjunction)a → b = 1 ∧ (1 − a + b).(Łukasiewicz implication)Furthermore, ¬a = 1 − a, and a ↔ b = 1 − |a − b|.Example of ŁΠ-algebra isL = 〈L, ∨, ∧, ⊗, ·, Ł →, Π →,0,1〉

4wherea → Ł b = 1 ∧ (1 − a + b), a ⊗ b = 0 ∨ (a + b − 1),{1 a ≤ b,a → Π Łb =¬ a = 1 − a,Π¬ a =baotherwise,{1 a = 0,0 otherwise,a ⊖ b = 0 ∨ (a − b).All these examples are algebras considered in fuzzy logic theory. It has been proved that IMTL, BL and Łukasiewiczpropositional fuzzy logics are standard complete, i.e. their formulas are provable iff they are true in all the above standardalgebras.B. Fuzzy equality and extensionality of functionsImportant fuzzy relation is that of a fuzzy equality.=: M × M −→ Lwhich must fulfil the following properties:(i) reflexivity [m = . m] = 1,(ii) symmetry [m = . m ′ ] = [m ′ . = m],(iii) ⊗-transitivity [m = . m ′ ] ⊗ [m ′ . = m ′′ ] ≤ [m = . m ′′ ]for all m,m ′ ,m ′′ ∈ M where [m = . m ′ ] denotes a truth value of m = . m ′ .Example of a fuzzy equality on M = [u,v] ⊂ R with respect to IMTL ∆ -algebra is⎧1, if m = n,⎪⎨[m = ǫ n] =⎪⎩1v−u((v − m) ∧ (v − n))∨((m − u) ∧ (n − u)),m,n ∈ [u,v].Let F : M α −→ M β be a function and = α ,= β be fuzzy equalities in the respective domains M α and M β . Then F isextensional w.r.t = α and = β if there is a natural number q ≥ 1 such that[m = α m ′ ] q ≤ [F(m) = β F(m ′ )], m,m ′ ∈ M αwhere the power is taken with respect to ⊗. We say that F is strongly extensional if q = 1. It is weakly extensional ifThis equivalent to the condition[m = α m ′ ] = 1 implies that [F(m) = β F(m ′ )] = 1.∆[m = α m ′ ] ≤ [F(m) = β F(m ′ )].It is easy to prove that each fuzzy equality = α (as a binary function) is strongly extensional w.r.t. itself and ↔. The ↔ is afuzzy equality on L strongly extensional w.r.t. itself; ∧ is strongly, and ∆ is weakly extensional w.r.t. ↔.A. SyntaxIII. SYNTAX AND SEMANTICS OF IMTL-FTTa) Basic syntactical elements: Let ǫ,o be distinct objects. Then Types is the smallest set of symbols fulfilling the conditions(i) ǫ,o ∈ Types,(ii) If α,β ∈ Types then αβ ∈ Types.Each formula in the fuzzy type theory is assigned a type α ∈ Types written as its subscript. Then we introduce primitivesymbols, that are variables x α ,..., special constants c α ,..., ι ǫ(oǫ) ,ι o(oo) (description operators), the symbol λ and variouskinds of brackets. Among special constant we put E (oα)α (equality), C (oo)o (conjunction) and D oo (Baaz delta).

5b) Formulas: the set of formulas is the smallest set Form fulfilling for all α,β ∈ Types the conditions(i) x α ∈ Form,(ii) c α ∈ Form,(iii) if B βα ∈ Form and A α ∈ Form then (B βα A α ) ∈ Form,(iv) if A β ∈ Form then λx α A β ∈ Form.Let us remark that formulas are also called lambda-terms, especially in fuzzy type theories developed for the applications incomputer science (cf. [2] and elsewhere).To simplify the notation, we will often write A ∈ Form α to stress that A is a formula of type α (without writing explicitlyits type as a subscript).The following are basic definitions of symbols of FTT:(a) Fuzzy equivalence/fuzzy equality≡ := λx α (λy α E (oα)α y α )x α .(b) Conjunction ∧ := λx o (λy o C (oo)o y o )x o .(c) Baaz delta ∆ := λx o D oo x o .(d) Representation of truth ⊤ := (λx o x o ≡ λx o x o )and falsity ⊥ := (λx o x o ≡ λx o ⊤).(e) Negation ¬ := λx o (⊥ ≡ x o ).(f) Implication ⇒ := λx o (λy o ((x o ∧ y o ) ≡ x o )).(g) Special connectives∨ := λx o (λy o (((x o ⇒ y o ) ⇒ y o ) ∧ ((y o ⇒ x o ) ⇒ x o ))),& := λx o (λy o (¬(x o ⇒ ¬y o ))).(h) General quantifier (∀x α )A o := (λx α A o ≡ λx α ⊤).(disjunction)(strong conjunction)B. SemanticsSemantics of FTT is a generalization of the semantics of first-order fuzzy logic. Let D be a set of objects and L be a setof truth values. A basic frame is a system of sets (M α ) α∈Typeswhere M ǫ = D is a set of objects, M o = L is a set of truthvalues and if γ = βα then M γ ⊆ M Mαβ. A frame is a systemM = 〈(M α ,= α ) α∈Types, L〉where L the algebra of truth values; namely either of IMTL ∆ , Łukasiewicz ∆ , BL ∆ or ŁΠ-algebra depending on the concretefuzzy type theory (see further explanation). Furthermore, = α is a fuzzy equality on M α and for α ≠ o,ǫ, each functionF ∈ M α is weakly extensional.A general model is a frame such that every formula A α , α ∈ Types, has interpretation in it (i.e. there is an element in thecorresponding set M α of the frame that interprets A α ).Because of lack of space, we will omit precise definition of interpretation of formulas. The reader may find it in [10]. Letus remark only that the formula of the form λx α A β is in M interpreted as a function assigning to every m ∈ M α an elementfrom M β that is obtained as an interpretation of A β in which all occurrences of x α are replaced by the corresponding m. Forexample, interpretation of λx α A o is a fuzzy set on M α determined by the property represented by the formula A o .IV. LOGICAL AXIOMS AND INFERENCE RULES OF IMTL-FTTIn this section, we will overview logical axioms of IMTL-FTT that have been in details presented in [10]. Logical axiomsof the other three FTT will be obtained as a modification of them and presented in the next section.The logical axioms of IMTL-FTT are divided into several groups.Fundamental equality axioms(FT I 1) ∆(x α ≡ y α ) ⇒ (f βα x α ≡ f βα y α )(FT I 2 1 ) (∀x α )(f βα x α ≡ g βα x α ) ⇒ (f βα ≡ g βα )(FT I 2 2 ) (f βα ≡ g βα ) ⇒ (f βα x α ≡ g βα x α )(FT I 3) (λx α B β )A α ≡ C βwhere C β is obtained from B β by replacing all free occurrences of x α in it by A α , provided that A α is substitutableto B β for x α (lambda conversion).(FT I 4) (x ǫ ≡ y ǫ ) ⇒ ((y ǫ ≡ z ǫ ) ⇒ (x ǫ ≡ z ǫ ))Truth structure axioms(FT I 5) (x o ≡ y o ) ≡ ((x o ⇒ y o ) ∧ (y o ⇒ x o ))

6(FT I 6) (A o ≡ ⊤) ≡ A o(FT I 7) (A o ⇒ B o ) ⇒ ((B o ⇒ C o ) ⇒ (A o ⇒ C o ))(FT I 8) (A o ⇒ (B o ⇒ C o )) ≡ (B o ⇒ (A o ⇒ C o ))(FT I 9) ((A o ⇒ B o ) ⇒ C o ) ⇒ (((B o ⇒ A o ) ⇒ C o ) ⇒ C o )(FT I 10) (¬B o ⇒ ¬A o ) ≡ (A o ⇒ B o )(FT I 11) A o ∧ B o ≡ B o ∧ A o(FT I 12) A o ∧ B o ⇒ A o(FT I 13) (A o ∧ B o ) ∧ C o ≡ A o ∧ (B o ∧ C o )(FT I 14) (g oo (∆x o ) ∧ g oo (¬∆x o )) ≡ (∀y o )g oo (∆y o )(FT I 15) ∆(A o ∧ B o ) ≡ ∆A o ∧ ∆B oQuantifier axiom(FT I 16) (∀x α )(A o ⇒ B o ) ⇒ (A o ⇒ (∀x α )B o ) where x α is not free in A o .Axioms of descriptions(FT I 17) ι α(oα) (E (oα)α y α ) ≡ y α ,α = o,ǫc) Inference rules and provability.: There are two inference rules in all the discussed fuzzy type theories.(R)Let A α ≡ A ′ α and B ∈ Form o . Then, infer B ′ whereB ′ comes from B by replacing one occurrence of A α ,which is not preceded by λ, by A ′ α.(N) Let A o ∈ Form o . Then infer ∆A o from A o .V. FORMAL THEORIES IN IMTL-FTTA theory T over FTT is set of formulas of type o (truth value). The provability in the theory T is defined as usual; T ⊢ A omeans that A o is provable in T .The following theorems summarize some of the important properties of IMTL-FTT. For the precise proofs of them see [10].Theorem 1(a) If ⊢ A o then ⊢ (∀x α )A o (Rule of generalization).(b) ⊢ (∀x α )B o ⇒ B o,xα [A α ] (Substitution axioms).(c) If T ⊢ A o and T ⊢ A o ⇒ B o then T ⊢ B o (Rule of modus ponens).It can be proved that IMTL-FTT includes formal system of the first-order IMTL-logic.Theorem 2 (Deduction theorem)Let T be a theory, A o ∈ Form o a formula. Thenholds for every formula B o ∈ Form o .Theorem 3 (Equality theorem)Theorem 4 (Description operator theorem)T ∪ {A o } ⊢ B o iff T ⊢ ∆A o ⇒ B o⊢ ∆(x β ≡ y β ) ⇒ ∆((f αβ ≡ g αβ ) ⇒ (f αβ x β ≡ g αβ y β ))holds for every type γ ∈ Types.⊢ ι γ(oγ) (E (oγ)γ y γ ) ≡ y γNote that interpretation of p oα is a fuzzy set. It can be demonstrated that interpretation of ι α(oα) p oα is a typical element(prototype) of p oα , namely, an arbitrary but fixed element from its kernel (if this fuzzy set is subnormal then there is no suchelement).Let T be a theory of IMTL-FTT.(i) T is contradictory ifT ⊢ ⊥.

7Otherwise it is consistent.(ii) T is maximal consistent if each its extension T ′ , T ′ ⊃ T is inconsistent.(iii) T is complete ifT ⊢ A o ⇒ B o or T ⊢ B o ⇒ A o .holds for every two formulas A o ,B o .(iv) T is extensionally complete if for every closed formula of the form A βα ≡ B βα , T ⊬ A βα ≡ B βα implies that there isa closed formula C α such that T ⊬ A βα C α ≡ B βα C α .Theorem 5Every consistent theory T can be extended to a maximal consistent theory T which is complete. To every consistent theory Tthere is an extensionally complete consistent theory T which is an extension of T .Theorem 6 (Completeness)(a) A theory T of fuzzy type theory is consistent iff it has a general model M.(b) For every theory T and a formula A o ,T ⊢ A o iff T |= A o .VI. OTHER FUZZY TYPE THEORIESIn this section, we will present systems of logical axioms of the other three fuzzy type theories, namely Ł-FTT (Łukasiewiczfuzzy type theory), BL-FTT and ŁΠ-FTT. Their syntax and semantics is essentially the same. They differ in some of thedefinitions and by logical axioms that are obtained as modification of the logical axioms of IMTL-FTT.A. Łukasiewicz-fuzzy type theoryThis is another important kind of fuzzy type theory that differs from IMTL-FTT by the definition of disjunction:∨ := λx o (λy o (x o ⇒ y o ) ⇒ y o ).Logical axioms of Ł-FTT are (FT I 1)–(FT I 17) and, further, the axiom(FT Ł 18) (A o ∨ B o ) ≡ (B o ∨ A o ).There is also simpler alternative which uses Rose-Rosser implication axioms for characterization of the structure of truthvalues. Then, axioms (FT I 7)–(FT I 10) should be replaced by the following axioms.Implication axioms(FT Ł 7) A o ⇒ (B o ⇒ A o )(FT Ł 8) (A o ⇒ B o ) ⇒ ((B o ⇒ C o ) ⇒ (A o ⇒ C o ))(FT Ł 9) (¬B o ⇒ ¬A o ) ≡ (A o ⇒ B o )(FT Ł 10) (A o ∨ B o ) ≡ (B o ∨ A o )Completeness of Ł-FTT formulated in Theorem 6 is provable.B. BL-fuzzy type theoryRecall that BL stands for basic fuzzy logic developed by P. Hájek in [7]. Since the law of double negation does not hold inBL-algebra, we must introduce a new special constant S o(oo) interpreted by the operation of supremum. Moreover, we mustintroduce the following definition of the disjunction connective:∨ := λx o λy o (S (oo)o y o )x o .Obviously this is a formula of type (oo)o. In the same spirit as above, we will write x o ∨ y o instead of (∨x o )y o .Axioms of BL-FTT are (FT I 1)–(FT I 17) and, moreover,(FT BL 18) (A o ∧ B o ) ⇒ A&(A o ⇒ B o )(FT BL 19) ⊢ B o,xα [A α ] ⇒ (∃x α )B o .(FT BL 20) (∀x α )(A o ⇒ B o ) ⇒ ((∃x α )A o ⇒ B o )(FT BL 21) (∀x α )(A o ∨ B o ) ⇒ ((∀x α )A o ∨ B o )On the basis of Theorem 5, it is possible to prove the following lemma:Lemma 1Let T be an extensionally complete theory of BL-FTT. Then it is Henkin: if T ⊬ (∀x α )A o holds for a closed formula (∀x α )A othen there is a closed formula C α such that T ⊬ A o,xα [C α ].This lemma enables to prove completeness of BL-FTT in the form of Theorem 6. However, the item (a) holds w.r.t. safegeneral models, that is, models in which all the necessary suprema and infima exist (cf. [7]). The reason is that completion ofthe structure of truth values is not generally possible.

8C. ŁΠ-fuzzy type theoryIn this case, we must consider two fuzzy equalities, namely Ł ≡ and Π ≡. Further, we will consider a new special constantP (oo)o interpreted by the special product ⊙ (in the case that the support of truth values is L = [0,1] then ⊙ is the ordinaryproduct of reals).Let τ,ν ∈ {Ł,Π}. The following are definitions of special symbols:D. Logical axioms := λx o (λy o P oo y o )x oττ⇒ := λx o λy o · x o ∧ y o ≡ xo ,ττ¬ := λx o (x o ⇒ ⊥)& := λx o λy o · Ł¬(x oŁ⇒ Ł ¬y o )∆ := λx o · Π¬ Ł ¬x o∨ := λx o λy o · (x oŁ⇒ y o ) Ł ⇒ y o ,G⇒ := λx o λy o · ∆(x oŁ⇒ y o ) ∨ y o ,(∀x α )A o := (λx α A oŁ≡ λx α ⊤),The following are logical axioms of ŁΠ-FTT where again τ,ν ∈ {Ł,Π}.Fundamental equality axioms(FT ŁΠ1) ∆(x ατ≡ yα ) Ł ⇒ (f βα x αν≡ fβα y α ),where τ ≠ ν,(FT2 ŁΠ1) (∀x α )(f βα x ατ≡ gβα x α ) Ł ⇒ (f βατ≡ gβα )(FT2 ŁΠ2) (f βατ≡ gβα ) Ł ⇒ (f βα x ατ≡ gβα x α )(FT ŁΠ3) (λx α B β )A ατ≡ Cβwhere C β is obtained from B β by replacing all free occurrences of x α in it by A α , provided that A α is substitutableto B β for x α (lambda conversion).(FT ŁΠ4) (x ǫτ≡ yǫ ) τ ⇒ ((y ǫτ≡ zǫ ) τ ⇒ (x ǫτ≡ zǫ ))Truth structure axiomsτ(FT ŁΠ5) (x o ≡ yo ) ≡ Ł τ τ((x o ⇒ yo ) ∧ (y o ⇒ xo ))τ Ł(FT ŁΠ6) (A o ≡ ⊤) ≡ A oτ(FT ŁΠ7) (A o ⇒ Bo ) ⇒ τ τ((B o ⇒ Co ) ⇒ τ τ(A o ⇒ Co ))τ τ(FT ŁΠ8) (A o ⇒ (Bo ⇒ Co )) ≡ Ł τ τ(B o ⇒ (Ao ⇒ Co ))τ(FT ŁΠ9) ((A o ⇒ Bo ) ⇒ τ C o ) ⇒ τ τ(((B o ⇒ Ao ) ⇒ τ C o ) ⇒ τ C o )(FT ŁΠ10) ( ¬ Ł ŁB o ⇒ ¬A Ł o ) ≡ Ł (A oŁ⇒ B o )Ł(FT ŁΠ11) A o ∧ B o ≡ B o ∧ A o(FT ŁΠ12) A o ∧ B oŁ⇒ A oŁ(FT ŁΠ13) (A o ∧ B o ) ∧ C o ≡ A o ∧ (B o ∧ C o )(FT ŁΠ14) Π ¬ Π ¬ A oΠ⇒ ((AoΠ⇒ Ao B o ) Π ⇒ ( Π ¬ Π ¬B o B o ))(FT ŁΠ15) A o (A oΠ⇒ Bo ) Ł ≡ A o ∧ B oQuantifier axiom(FT ŁΠ16) (∀x α )(A oŁ⇒ B o ) Ł ⇒ (A oŁ⇒ (∀x α )B o )where x α is not free in A oAxioms of descriptions(FT ŁΠ17) ι α(oα) (E (oα)α y α ) Ł ≡ y α ,α = o,ǫThe inference rule (R) from Section IV is formulated only for τ = Ł. Similarly as in the case of BL-fuzzy type theory,completeness Theorem 6 holds w.r.t. safe general models.VII. CONCLUSIONIn this paper, we have formed four kinds of fuzzy type theories as fuzzy logics of higher order. These logics are necessarywhen we want to model natural human reasoning which is specific by the use of natural language. Because of the complexity

9of the latter, first-order (fuzzy) logic is not sufficient. However, fuzzy type theory is extremely powerful formal system whichcan be used in arbitrary mathematical model.The reason why we confined just to IMTL-FTT, Ł-FTT, BL-FTT and ŁΠ-FTT lays in their potential for various kinds ofapplications. The IMTL-FTT should be taken as fundamental fuzzy type theory so that the other ones are obtained as specificextensions of it. The most powerful is ŁΠ-FTT which, on the other hand, is the most complicated. Therefore, if the productconnective is unnecessary, it is better to confine to Ł-FTT only. If our main task is the fuzzy approximation theory thenBL-FTT is convenient. Its drawback, however, is the lack of the law of double negation that is very important when modelingnatural human reasoning. Note that ŁΠ-FTT contains all the other three fuzzy type theories. Let us also remark that a slightlydifferent formal system of ŁΠ-FTT that is not based on λ-notation has been developed in [3] with the goal to establish formalbackground for fuzzy mathematics.It is probable that we can develop more kinds of FTT than the above four ones. This should be done, however, only whenthere is essential substantiation for them.ACKNOWLEDGMENTThe research was supported by the grant 201/04/1033 of the GA ČR and partially also by the project MSM 6198898701 ofthe MŠMT ČR.REFERENCES[1] Andrews, P., An Introduction to Mathematical Logic and Type Theory: To Truth Through Proof. Kluwer, Dordrecht, 2002.[2] Barnedregt, H., Lambda calculi with types, in Abramsky, S., Gabbay, D. M., Maibaum, T. S. E. (Eds.), Handbook of Logic in Computer Science, Vol.2, Clarendon Press, Oxford 1992, 117–310.[3] Běhounek, L., Cintula, P., Fuzzy class theory, Fuzzy Sets and Systems 154(2005), 34–55.[4] Church, A., A formulation of the simple theory of types. J. Symb. Logic 5(1940), 56–68.[5] Cintula, P., The ŁΠ and ŁΠ 1 Propositional and Predicate Logics. Fuzzy Sets and Systems 124(2001), 289–302.2[6] Esteva, F. and L. Godo, Monoidal t-norm based logic: towards a logic for left-continuous t-norms. Fuzzy Sets and Systems 124(2001), 271–288.[7] Hájek, P., Metamathematics of fuzzy logic. Kluwer, Dordrecht, 1998.[8] Henkin, L., Completeness in the theory of types. J. Symb. Logic 15(1950), 81–91.[9] Novák, V.: On the syntactico–semantical completeness of first–order fuzzy logic. Part I. Syntax and Semantics, Part II. Main Results. Kybernetika26(1990), 47–66, 134–154.[10] Novák, V.: On fuzzy type theory. Fuzzy Sets and Systems 149(2005), 235–273.[11] Novák, V.: From Fuzzy Type Theory to Fuzzy Intensional Logic. Proc. Third Conf. EUSFLAT 2003, University of Applied Sciences at Zittau/Goerlitz,Germany 2003, 619–623.[12] Novák, V.: Descriptions in Full Fuzzy Type Theory. Neural Network World 13(2003), 5, 559–569.[13] Novák, V., Perfilieva I. and J. Močkoř, Mathematical Principles of Fuzzy Logic. Kluwer, Boston/Dordrecht 1999.[14] Pavelka, J., On fuzzy logic I, II, III. Zeit. Math. Logic. Grundl. Math. 25(1979), 45–52; 119–134; 447–464.