logic seminar spring 2011

Intuitionistic Fuzzy Logic andIntuitionistic Fuzzy Set Theory

by G. Takeuti and S. Titani

JSL vol 49, 3, 1984

Valeria de Paiva

Stanford

April, 2011

Valeria de Paiva (Stanford) April, 2011 1 / 29

Introduction

Introduction

This paper is only 15 pages, 2 sections and 6 references. Hopefully itmakes sense for people who don’t like Set Theory...Takeuti and Titani have 3 papers in the subject:JSL1984,Archive for Mathematical Logic, 1992 "Fuzzy logic and fuzzy set theory","Heyting valued universes of intuitionistic set theory", 1981.Paper had a great impact:Cited by 163 in Scholar, 4 in Jstor.Work continuing...


Introduction

Introduction?...

1965 Zadeh: definition of fuzzy setsBasic idea: A notion of the membership relation which takes truth values inthe closed interval of real numbers [0, 1] .If an element really belongs to a set, the truth value is 1. If it really doesnot belong to the set then the truth value is 0, and all other reals are‘degrees’ of belonging.The logical operations implication and negation on [0, 1] used for Zadeh’sfuzzy sets look like Łukasiewciz’s logic, where p→ q = min(1, 1− p+ q),¬p = 1− p.For Łukasiewicz’s logic, P ∧ (P → Q)→ Q is not valid. Translating it tothe set version, it follows that the axiom of extensionality does not hold.Hay(1963) extended Łukasiewicz’s logic to a predicate logic and proved aweak completeness theorem: if P is valid then P + Pn is provable for eachpositive integer n. Hay also showed that one can (without losingconsistency) obtain completeness of the system by use of an additionalinfinitary rule.


Introduction

Introduction

On the other hand (...)The closed interval [0, 1] is a complete Heyting algebra (cHa).Grayson showed that for each cHa Ω the sheaf model V (Ω) over Ω is amodel of intuitionistic set theory ZFI .Ω-valued set theory is realized in V(Ω) as a model of ZFI .Denote the cHa [0, 1] by I, where:p→ q = r ∈ [0, 1]|p ∧ r ≤ q = 1 if p ≤ q, q if p ≥ qand ¬p = p→ 0 = 1ifp = 0, 0 if p > 0

Call the I-valued logic ’intuitionistic fuzzy logic’ (IF) and I-valued set theory’intuitionistic fuzzy set theory’ (ZFIF ).This paper develops IF and ZFIF .


Introduction

Summary

Section1 axiomatizes intuitionistic fuzzy logic IF in a language withpropositional variables, as suggested by R. Jensen.Section 1 also proves the consistency and strong completeness of the logicIF: if a sequent Γ⇒ ∆ is valid in every I-valued model then Γ⇒ ∆ isprovable, where Γ is a finite or infinite sequence of formulas and ∆ is asequence of at most one formula.Section 2 presents the intuitionistic fuzzy set theory ZFIF and develops itscalculus ZFIF.The axioms of ZFIF are the axioms of intuitionistic set theory ZFI , plusDependent Choice and Double Complement.


Introduction

Intuitionistic Set Theories?..

Apparently started in early 70s by Myhill and Friedman, IZF.Keep things as much as possible like in ZFC (Zermelo-Fraenkel withChoice), just change base logic to intuitionism. Must remove Choice andLEM, then check that LEM doesn’t come back...(nice article on SEP onconstructive and intuitionistic set theories).What is ZFI? Greyson [3] defines ZFI , considers set theories expressed ina first-order language with relation symbols ∈ , =, and governed by theusual axioms and rules of intuitionistic predicate logic.The usual schemas of Extensionality, Pairing, Unions, Power-set, Infinityand Separation:(Ext) ∀z(z ∈ x↔ z ∈ y)→ x = y(Pair) ∃z∀w(w ∈ z ↔ w = x ∨ w = y)(Union) ∃z∀w(w ∈ z ↔ ∃y ∈ xw ∈ y)(Power) ∃z∀w(w ∈ z ↔ ∀y ∈ wy ∈ x)(Inf) ∃x(∃y ∈ x ∧ ∀y ∈ x∃z ∈ xy ∈ z)(Sep) ∃z∀w(w ∈ z ↔ w ∈ x ∧ ϕ)


Introduction

Intuitionistic Set Theories?..

Greyson shows axiom of Regularity would bring back LEM so he introducesinstead ε-induction,

∀x[∀y ∈ xϕ(y)→ ϕ(x)]→ ∀xϕ(x)

From infinity and ε-induction follows the existence of a set ω of ’naturalnumbers’, a result of Powell.To complete the system ZFI Greyson adds the axiom of Replacement(Rep) ∀y ∈ x∃zϕ→ ∃w∀y ∈ x∃z ∈ wϕ


Introduction

Dependent Choice? Double Complement?

The axiom of dependent choices, denoted DC, is a weak form of the axiomof choice (AC) which is still sufficient to develop most of real analysis.The axiom of dependent choice implies the Axiom of countable choice, andis strictly stronger.The axiom of double complement is

∃u∀z(z ∈ u↔ ¬¬z ∈ x)

Yeah, related to Stable sets

x : ∀u(¬¬u ∈ x→ u ∈ x)

BUT Powell75 needs to be read to make sense of why ZFI plus DependentChoice plus Double Complement is the set theory we want to have.We will just assume it.


Introduction

Summary

In intuitionistic set theory, the natural numbers ω and the rationals Qbehave as those in classical set theory, but the real numbers R do notThe calculus in V I is almost classical. Only the concept of subset isnot classical. (for example there exists an infinite and Dedekind finitesubset of ω in V I)The calculus is almost classical in ZFIF in the sense that it is absolutefor Powell’s innermodel, which is a model of ZF constructed in ZFIF.Powell constructed the innermodel in [5] to prove the consistency ofZF relative to ZFI with the axiom of double complement.


Introduction

The intuitionistic fuzzy logic IF

The language of IF consists of:individual free variables: a0, a1, . . .;individual bound variables: x0, x1, . . .;individual constants: c0, ..., ci, . . . (i < u);propositional variables: z0, z1, , . . .

propositional constants:p0, , . . . pj , , . . . (j < i);predicate constants: R0(∗, . . . , ∗), . . . , Rk(∗, . . . , ∗), . . .logical symbols: ∧,∨,→,¬, ∃, ∀


Introduction


The proof system for IF consists of intuitionistic logic (Gentzen’s LJ) plusthe following axiom schemata:1.⇒ (A→ B) ∨ ((A→ B)→ B);2. (A→ B)→ B ⇒ (B → A) ∨B;3. (A ∧B)→ C ⇒ (A→ C) ∨ (B → C);4. A→ (B ∨ C)⇒ (A→ B) ∨ (A→ C);5. ∀x(C ∨A(x))⇒ C ∨ ∀xA(x), where x does not occur in C;6. ∀xA(x)→ C ⇒ ∃x(A(x)→ D) ∨ (D → C), where x not in D;Extra Inference Rule:.Γ⇒ A ∨ (C → z) ∨ (z → B) then Γ⇒ A ∨ (C → B)where z does not occur in the lower sequent.This inference rule (called density of rule TT) asserts that the truth valueset is dense.


Introduction


This section has 3 theorems.Theorem 1: The following (19) sequents are provable in IF, where in(6)-(9), (11), (12) x does not occur in C and in (10), (13) x does notoccur in D.Theorem 2 If Γ⇒ A is provable in IF, then Γ⇒ A is valid. (they provevalidity of the density inference rule)Theorem 3 (main theorem: strong completeness) Let IF be a system ofintuitionistic fuzzy logic with countably many constants. If a sequentΣ⇒ ∆ is valid then Σ⇒ ∆ is provable, where Σ may be infinite.


Introduction



Introduction


The proof of the theorem 3 in TT84 is long and not very enlightening.

Luckily three years later Takano in the TSUKUBA J . MATH came out with"Another proof of the completeness of the Intuitionistic Fuzzy Logic",which is short and sweet.

Takano reminds us that in 1969 Horn had introduced and proved (weak)complete a system, H, which was LJ with two extra axioms:∀x(C ∨A(x))⇒ C ∨ ∀xA(x), where x does not occur in C;⇒ (A→ B) ∨ (B → A)He also reminds us that Ono had characterized the system H by means ofKripke modelsNow since the first axiom above is number 5 in the system TT and sincethe second axiom above (sometimes called D for Dummett) is provable inLJ from the first two extra axioms in TT, Takano proves the following:


Introduction


Theorem [Takano, Takeuti and Titani] Call TT the system described byTakeuti and Titani in JSL84 and TT− the system without the density rule.Suppose that the language concerned is countable. The following properties(a)-(d) of a sequent Σ⇒ A are equivalent, where Σ may be infinite:(a) Σ⇒ A is valid.(b) Σ⇒ A is provable in H.(c) Σ⇒ A is provable in TT−.(d) Σ⇒ A is provable in TT.The proof is simple, algebraic and connects to the structure of the cHa[0,1] as "a countable linearly ordered structure with the distinct maximaland minimal elements".Takano also mentions that he doesn’t know how to do a syntactical proofof (d)→(c). But we now have an easy proof of TT’s theorem 3 and aconnection to linearly ordered kripke models...


Introduction

Takano’s easy proof:

Want to show (a) implies (b) where (a) Σ⇒ A is valid.(b) Σ⇒ A is provable in H.Method: Assume Σ⇒ A is unprovable in H and then construct a model〈A, [[.]]〉 in which Σ⇒ A is not valid. Assume (WLOG) infinitely manyvariables that do not occur in Σ⇒ A and that ∆ = A.Let Let A and F be the sets of all terms and all formulas, respectively.Four propositions.


Introduction


Prop1. There exists a set G of formulas which satisfiesthe followingconditions (l)-(3): (1) Σ ⊆ G and A 6∈ G. (2) If G satisfies a disjunctionB1 ∨B2 ∨Bv, then Bi ∈ G for some i. (3) If B(t) ∈ G for every t in A,then ∀xB(x) is an element of G. (inductively build up the sets from Σ andA).Prop2. 〈F/,≤〉 is a countable linearly ordered structure with the distinctmaximal element |A→ A| and the minimal element ¬(A→ A).Prop3. The following properties hold in 〈F/,≤〉:|B ∧ C| = min(|B|, |C|), blah, etc... (describe the Heyting algebra...)Prop4.(Horn69) If 〈L,≤〉 is a countable linearly ordered structure with thedistinct maximal and minimal elements, then there exists a monomorphismon〈L,≤〉 to 〈[0, 1],≤〉, which preserves the maximal and the minimalelements as well as all existing supremums and infimums in 〈L,≤〉. Hencethere exists such a monomorphism on 〈L,≤〉 to 〈[0, 1],≤〉.


Introduction


Almost fits in one slide, but not quite..

Put 2 and 4 together to construct a monomorphism h, and put[[B]] = h(|B|) for every B in F and we obtain a model 〈A, [[.]]〉 by prop3.This gives a contradiction, as for every B we have[[B]] = 1↔ |B| = 1↔ B ∈ G. So in this model B ∈ Σ impliesB ∈ G ↔ [[B]] = 1, while A is not in G, so [[A]] 6= 1, hence Σ⇒ A is notvalid.


Introduction


Horn, 1969: It is known that the theorems of the intuitionist predicatecalculus are exactly those formulas which are valid in every Heyting algebra.The simplest kind of Heyting algebra is a linearly ordered set. This paperconcerns the question of determining all formulas which are valid in everylinearly ordered Heyting algebra.Baaz and Zach, 2000 [Use Avron’s hypersequents to prove cutelimination and "density elimination" of IF] The logic is known to beaxiomatizable, but no deduction system amenable to proof-theoretic, andhence, computational treatment, has been known.Ciabattoni and Metcalfe, 2005 Density elimination by substitutions isintroduced as a uniform method for removing applications of theTakeuti-Titani density rule from proofs in first- order hypersequent calculi.


Introduction

Intuitionistic fuzzy set theory ZFIF

The logic of ZFIF is the intuitionistic fuzzy logic IF whose constants arepredicate constants ∈ , = and EThe axioms of ZFIF are those of intuitionistic set theory ZFI together withthe axiom of Dependent Choices (DC) and Double Complement (¬¬)


Introduction


Equality Axioms: u = u; u = v ⇒ v = u; u = v ∧ v = w ⇒ u = w;u = v ∧A(u)→ A(v);(Eu ∨ Ev → u = v)→ u = vOther Axioms:Extensionality. ∀z(z ∈ x↔ z ∈ y) ∧ (Ex↔ Ey)→ x = y.∈-strictness. x ∈ u→ Ex ∧ Eu.∈-induction. ∀x(∀y ∈ x(ϕ(y)→ ϕ(x))→ ∀xϕ(x).Separation. ∀x∃u∀z(z ∈ u↔ z ∈ x ∧ ϕ(z)).Collection. ∀x[∀y ∈ x∃zϕ(y, z)→ ∃u∀y ∈ x∃z ∈ u(ϕ(y, z)].Pair. ∀x, y∃u∀z[z ∈ u(z = x) ∨ (z = y)].Union. ∀x∃uz[z ∈ u∃y ∈ x(z ∈ y)].Power. ∀x∃u∀z[z ∈ u∀y ∈ z(y ∈ x)].Infinity.∃x[∃y(y ∈ x) ∧ ∀y ∈ x∃z ∈ x(y ∈ z)].DepChoice. ∀x ∈ u∃y ∈ uϕ(x, y)→ ∀z ∈ u∃f [f : ω → u ∧ f(ω) =z ∧ ∀x ∈ ωϕ(f(x), f(x+ 1))]DNeg ∃u∀z[¬¬z ∈ x↔ z ∈ u].


Introduction


Grayson proved the following three facts about ZFI :(I) The principle of ω-induction:

ϕ(0) ∧ ∀y[ϕ(y)→ ϕ(y + 1)]⇒ ∀y ∈ ωϕ(y)

and the trichotomy of elements of ω:

∀x, y ∈ ω(x ∈ y ∨ x = y ∨ y ∈ x)

The arithmetic sum, product and exponent are given by the usual recursivedefinitions. The set of integers Z and its order and arithmetic areconstructed as usual, and similarly for the rationals Q.(II) The real numbers R (Dedekind reals) are defined as pairs of subsets ofQ, <L, U>, such that 5 properties hold.(III) DC implies the axiom of Countable Choice:CAC: ∀x ∈ ω∃y ∈ uϕ(x, y)→ ∃f [f : ω → u ∧ ∀x ∈ ωϕ(x, f(x))],CAC implies the fact that every Dedekind real number is a Cauchy real (alimit of a Cauchy sequence of rational numbers), i.e.∀x ∈ R∃f : ω → Q∀n(|x− f(n)| < 1/n)


Introduction

Sheaf models...

From Fact III they say it follows that:Theorem 2.1 Every Dedekind real number is Cauchy real in ZFIF .

Provide a definition of a model of ZFIF , V I , following Grayson:Set V I

0 = ∅.Assuming that V I

α has been constructed and each element of V Iα is a pair

〈|u|, Eu〉, where |u| is the characteristic function, D(u)→ I, and Eu ∈ Iis the truth value of "there exists u", define:V Iα+1 = 〈|u|, Eu〉|D(u) ⊆ V I

α , |u| : D(u)→ I, ∀t ∈ D(u)(|u|(t) ≤Eu ∧ Et), Eu ∈ I;if β is a limit ordinal, V I

β =⋃α<β

V Iα ; and V

I = ∪α∈OnV Iα .

Then it is known that V I is a model of ZFI , and V is embedded in V I

with the mapping : V → V I , given by D(u) = t|t ∈ u, u(t) = 1 fort ∈ u, Eu = 1.(Need to read ’Heyting valued universes of intuitionistic set theory’...)


Introduction

Theorems in section 2..

Let V be a model of ZFC in which we construct a model V I of ZFIF .We denote the sets of natural numbers, rationals and reals in V by ω, Qand R, respectively.


Introduction


THEOREM 2.5. V I is a model of ZFIF .Proof: Since it is clear that 〈V I , I, [[.]]〉 is a model of IF, it suffices to showthat [[DC]] = [[¬¬]] = 1, which they show by showing that [[DC]] = 1and that [[¬¬]] = 1 separately.


Introduction


The following theorem shows that the real numbers in V I are classical.Theorem 2.6[[R = R]] = 1.THEN they can do usual definitions like:Definition For u ∈ R and ε ∈ Q+, a neighborhood B(u, ε) of u is definedby B(u, ε) = x ∈ R such that |x− u| < ε.and many others like:A is an open set if all its elements have open neighborhoods contained inA, or:A is open if ∀x ∈ A∃ε ∈ Q+ such that (B(x, ε) ⊆ A),The closure of A, A = x ∈ R|∃f : ω → A(limn→∞, f(n) = x), etc..


Introduction

Traditional analysis theorems in V I

in page 15 we have: A formula A is said to be decidable if A ∨ ¬A,i.e.,¬¬A↔ A.All traditional development of continuous and differentiable functions sothat we obtain the classical forms of Rolle’s theorem, the mean valuetheorem, and Taylor’s theorem.ROLLE’S THEOREM. Let a, b ∈ R, f : [a, b]→ R continuous, and fdifferentiable on (a, b). Let f(a) = f(b) = 0. Then there is c ∈ (a, b) suchthat f ′(c) = 0.MEAN VALUE THEOREM. Let a, b ∈ R andf : [a, b]→ R and f be adifferentiable function on the interval (a, b). Then there is c ∈ (a, b) suchthat f(b)− f(a) = f ′(c)(b− a).TAYLOR’S THEOREM. Let U be a closed interval of R, a, b ∈ U , and letf : U → R be n+ 1 times differentiable on U . Then there exists c ∈ Rwith min(a, b) < c < max(a, b) such that

f(b) = Σnk=0f

(k)(a)(b− a)k + f (n+1)(c)(b− c)n(b− a).


Introduction

Traditional analysis theorems in V I

FinallyTheorem 2.10 If g : ω → R and α ∈ R, then the formulalimn→∞g(n) = α is decidable.Integrals are also decidable and the functions ex, logx, sinx and cosx canbe defined via their powerseries and integral, so:The calculus without using the concept of subset is classical in V I , in thesense that for each sentence A in the calculus we have A↔ ¬¬A↔ AV

It follows that the calculus without using the concept of subset is classicalin ZFIF , in the sense that for each A in the calculus, A↔ AS .but what does this mean?...


Introduction

Open Problems

If a sentence A holds in V , then AS is valid in V I .Let T be the set AS |A holds in V .Then, is every valid formula in V I provable from T in ZFIF ?


Introduction

What happened since?..

About the logic we saw already:Takano, 87, simplified proof,CSL2000 (Baaz and Zach: Hypersequent and the Proof Theory ofIntuitionistic Fuzzy Logic), syntactic cut-elimination plus density eliminationCiabattoni and Metcalfe: syntactic density elimination...About the set theory?....A natural interpretation of fuzzy sets and fuzzy relations M. Shimoda2002? 18 citFuzzy sets Zadeh 1965 Cited by 27153.a different notion of "intuitionistic fuzzy sets" by Atanassov and co-workerslong controversy’Krassimir Atanassov has published about 370 papers in journals, 140conference reports, 20 monographs.’ says wikipedia...


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