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Page 1: A note on the first-order logic of complete BL-chains

Math. Log. Quart. 54, No. 4, 435 – 446 (2008) / DOI 10.1002/malq.200710058

A note on the first-order logic of complete BL-chains

Petr Hajek∗1 and Franco Montagna∗∗2

1 Institute of Computer Science, Academy of Sciences of the Czech Republic,Pod vodarenskou vezı 2, Prague, 182 07 Czech Republic

2 Department of Mathematics and Computer Science, University of Siena,Pian dei Mantellini 44, 53100 Siena, Italy

Received 8 August 2007, accepted 3 September 2007Published online 1 July 2008

Key words First-order fuzzy logics, standard semantics, complete BL-chains.MSC (2000) 03B50, 03B52

In [10] it is claimed that the set of predicate tautologies of all complete BL-chains and the set of all standardtautologies (i. e., the set of predicate formulas valid in all standard BL-algebras) coincide. As noticed in [11],this claim is wrong. In this paper we show that a complete BL-chain B satisfies all standard BL-tautologies ifffor any transfinite sequence (ai : i ∈ I) of elements of B, the condition

∧i∈I

(a2

i

)= (

∧i∈I ai)

2 holds in B.

c© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

1 Introduction

In [10], the authors claim that the set of standard tautologies of BL∀ is equal to the set of formulas of BL∀ thatare valid in all complete BL-chains, and that, as a consequence, the predicate logic of all complete BL-chains isnot arithmetical. As noted in their corrigendum [11], the second claim (non-arithmeticity of standard tautologiesof BL∀) is true, but the first claim is false: the formula ∀x(ϕ(x) & ϕ(x)) → ((∀xϕ(x)) & (∀xϕ(x))) is a stan-dard tautology, but it can be invalidated in the complete BL-chain B consisting of the ordinal sum of the 3-ele-ment MV-chain M3 followed by the negative cone Z− of the ordered group Z of integers (see the next sectionfor the definitions): let P be a unary predicate letter, let M be a B-first-order structure with the set ω of naturalnumbers as domain such that PM (n) = −n, and let a0 < a1 < a3 denote the elements of M3 in increasing or-der. Then, recalling that in our ordinal sum the top element of M3 is identified with the top element of Z−,

‖∀x(P (x) & P (x))‖BM = ‖∀x(P (x))‖B

M = a1,

whereas

‖∀x(P (x)) & ∀x(P (x))‖BM = a2

1 = a0 < a1.

The problem of standard tautologies in many-valued logics constitutes an important aspect of fuzzy logic andhas involved many authors, cf. e. g. [3, 5, 7, 8, 9, 13]. Thus it makes sense to look for the largest class K of com-plete BL-chains whose predicate tautologies are precisely the standard tautologies of BL∀. In this note we willprove that K is precisely the class of all BL-chains B whose monoid operation ∗ satisfies, for every transfinitesequence (ai : i ∈ I) of elements of B, the condition

(RC)∧

i∈I a2i = (

∧i∈I ai)2,

where for every n > 1, an denotes a ∗ · · · ∗ a, n times. We will denote this condition by (RC) because it reflectssomehow the right continuity of ∗ with respect to the order topology.

∗ Corresponding author: e-mail: [email protected]∗∗ e-mail: [email protected]

c© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

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436 P. Hajek and F. Montagna: A note on the first-order logic of complete BL-chains

2 Preliminaries

Definition 2.1 A commutative and integral residuated lattice (cf. [14]) is an algebra A = (A, ∗,→,∨,∧, 1)such that (A, ∗, 1) is a commutative monoid, (A,∨,∧) is a lattice with top element 1, and → is a binary operationsatisfying the residuation property: x ∗ y ≤ z iff x ≤ y → z. A commutative and integral residuated lattice issaid to be bounded if it has a minimum m and its signature has an additional constant 0 that is interpreted as m,prelinear or representable if it satisfies the identity (x → y) ∨ (y → x) = 1 (equivalently, if it is isomorphic toa subdirect product of totally ordered commutative and integral residuated lattices also called chains), and divi-sible if it satisfies the equation x ∗ (x → y) = x ∧ y. A BL-algebra is a commutative, integral, bounded, divisi-ble, and prelinear residuated lattice. An MV-algebra is a BL-algebra satisfying −(−x) = x, where −x is an ab-breviation for x → 0. A product algebra is a BL-algebra satisfying the equation

−x ∨ ((x → (x ∗ y)) → y) = 1.

A Godel algebra is BL-algebra satisfying the equation x2 = x.A BL-algebra (an MV-algebra, a product algebra, a Godel algebra, respectively) is said to be standard if its

lattice reduct is ([0, 1],max,min). We note that all standard MV-algebras (product algebras, Godel algebras, re-spectively) are mutually isomorphic, whereas there are many non-isomorphic standard BL-algebras.

Definition 2.2 A lattice-ordered Abelian group (also called Abelian �-group) is an algebra

G = (G, +,−,∨,∧, 0),

where (G, +,−, 0) is an Abelian group, (G,∨,∧) is a lattice, and the identities

x + (y ∨ z) = (x + y) ∨ (x + z) and x + (y ∧ z) = (x + y) ∧ (x + z)

hold.The negative cone of an Abelian �-group G is the residuated lattice G− whose domain is {x ∈ G : x ≤ 0},

whose lattice operations and whose monoid operation are the restrictions to G− of the corresponding operationsin G, and whose residual → is defined by x → y = (y − x) ∧ 0.

Let (Li : i ∈ I) be an indexed family of commutative, integral, and totally ordered residuated lattices suchthat if i = j, then Li ∩ Lj = {1}. Further, assume that I is totally ordered by ≤. Then the ordinal sum

⊕i∈I Li

of the family (Li : i ∈ I) is defined as follows:(a) The domain of

⊕i∈I Li is

⋃i∈I Li.

(b) The operations are as follows:

x ∗ y =

⎧⎪⎨⎪⎩

x ∗i y if x, y ∈ Li (i ∈ I),x if x ∈ Li \ {1}, y ∈ Lj with i < j,

y if y ∈ Li \ {1}, x ∈ Lj with i < j,

x → y =

⎧⎪⎨⎪⎩

x →i y if x, y ∈ Li (i ∈ I),1 if x ∈ Li \ {1}, y ∈ Lj with i < j,

y if y ∈ Li \ {1}, x ∈ Lj with i < j,

x ∧ y =

⎧⎪⎨⎪⎩

x ∧i y if x, y ∈ Li (i ∈ I),x if x ∈ Li \ {1}, y ∈ Lj with i < j,

y if y ∈ Li \ {1}, x ∈ Lj with i < j,

x ∨ y ==

⎧⎪⎨⎪⎩

x ∨i y if x, y ∈ Li (i ∈ I),x if y ∈ Li \ {1}, x ∈ Lj with i < j,

y if x ∈ Li \ {1}, y ∈ Lj with i < j.

c© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim www.mlq-journal.org

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Math. Log. Quart. 54, No. 4 (2008) / www.mlq-journal.org 437

We quote the following result from [1].Theorem 2.3 Every BL-chain B is isomorphic to the ordinal sum of an indexed family (Bi : i ∈ I) of

commutative and integral residuated lattices, called the Wajsberg components of B, which are either MV-chainsor negative cones of Abelian totally ordered �-groups. Moreover, the totally ordered index set I has a minimum i0and the bottom element of B is the bottom of Bi0 . Conversely, each ordinal sum as above is a BL-chain. Final-ly, the Wajsberg components Bi of B are uniquely determined by B.

We will need the following result, which is a consequence of Mundici’s equivalence between MV-algebrasand Abelian �-groups with strong unit [12].

Theorem 2.4 MV-algebras are precisely the algebras (A,�,→,∨,∧, 0, 1) for which there are an Abelian�-group G and a positive element 1 of G such that

A = {x ∈ G : 0 ≤ x ≤ 1}, x � y = (x + y − 1) ∨ 0, x → y = (1 − x + y) ∧ 1,

where ∨ and ∧ are the restrictions to A of the lattice operations of G.

The following result is folklore.Theorem 2.5 Product chains are precisely the ordinal sums of the two-element MV-algebra followed by the

negative cone of an Abelian �-group.

Note also that in any BL-algebra meet and join are definable by

x ∧ y = x ∗ (x → y) and x ∨ y = ((x → y) → y) ∧ ((y → x) → x).

For this reason, the logic counterpart of BL-algebras, Hajek’s basic fuzzy logic BL, has only a conjunction &and an implication → as basic connectives.

Definition 2.6 Hajek’s basic fuzzy logic BL means the propositional logic whose language contains two bi-

nary connectives & and → and the constant 0, whose only rule is Modus PonensA A → B

B, and whose axiom

schemes are:(A1) (ϕ → ψ) → ((ψ → γ) → (ϕ → γ)),(A2) (ϕ & ψ) → ϕ,(A3) (ϕ & ψ) → (ψ & ϕ),(A4) (ϕ & (ϕ → ψ)) → (ψ & (ψ → ϕ)),(A5) (ϕ → (ψ → γ)) → ((ϕ & ψ) → γ),(A6) ((ϕ & ψ) → γ) → (ϕ → (ψ → γ)),(A7) ((ϕ → ψ) → γ) → (((ψ → ϕ) → γ) → γ),(A8) 0 → ϕ.

The connectives ∧ and ∨ and the constant 1 are defined by

ϕ ∧ ψ = ϕ & (ϕ → ψ), ϕ ∨ ψ = ((ϕ → ψ) → ψ) ∧ ((ψ → ϕ) → ϕ), and 1 = 0 → 0.

BL-chains set up a fundamental tool for the semantics of first-order fuzzy logic. In this case, we have a lan-guage with individual variables, constant symbols, function symbols, predicate symbols, BL-connectives, andpropositional constants, and with the quantifiers ∃ and ∀. Models for predicate BL logic BL∀ are pairs (B,M),where B is a BL-chain and M is a pair (M,M ), where M is a non-empty set and M is a function which associ-ates to every constant symbol c an element cM of M , to every n-ary function symbol f a function fM from Mn

into M , and to every n-ary predicate symbol P a function PM from Mn into B. Such a pair is also said to be aB-first-order structure. A valuation is a function v from the set V of variables into M . For each valuation v andeach variable x, we denote by Vv,x the set of all valuations which differ from v at most on x. Then given a mo-del (B,M) and a valuation v, we define for every term t its value t(M ,v) as in classical model theory. Moreover,we define for each formula ϕ its truth value ‖ϕ‖B

M ,v in the model (B,M) according to the valuation v by induc-tion as follows:

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438 P. Hajek and F. Montagna: A note on the first-order logic of complete BL-chains

(1) If ϕ is an atomic formula of the form P (t1, . . . , tn), then

‖ϕ‖BM ,v = PM (tM ,v

1 , . . . , tM ,vn ).

Moreover, ‖0‖BM ,v = 0.

(2) If ϕ = ψ & ψ (ϕ = γ → ψ, respectively), then ‖ϕ‖BM ,v is defined iff ‖ψ‖B

M ,v, ‖γ‖BM ,v are defined, and in

this case, ‖ϕ‖BM ,v = ‖ψ‖B

M ,v ∗ ‖γ‖BM ,v (‖ψ‖B

M ,v → ‖γ‖BM ,v , respectively).

(3) If ϕ = ∀xψ, then ‖ϕ‖BM ,v is defined iff

(a) for all v′ ∈ Vv,x, ‖ψ‖BM ,v′ is defined;

(b)∧

v′∈Vv,x‖ψ‖B

M ,v′ exists in B.

In this case,

‖ϕ‖BM ,v =

∧v′∈Vv,x

‖ψ‖BM ,v′ .

(4) If ϕ = ∃xψ, then ‖ϕ‖BM ,v is defined iff

(a) for all v′ ∈ Vv,x, ‖ψ‖BM ,v′ is defined;

(b)∨

v′∈Vv,x‖ψ‖B

M ,v′ exists in B.

In this case,

‖ϕ‖BM ,v =

∨v′∈Vv,x

‖ψ‖BM ,v′ .

If the free variables in ϕ are among x1, . . . , xn and v is a valuation with v(xi) = ci for i = 1, . . . , n, then wewrite ‖ϕ(c1, . . . , cn)‖B

M instead of ‖ϕ‖BM ,v . If ϕ is a sentence, then we write ‖ϕ‖B

M instead of ‖ϕ‖BM ,v . A mo-

del (B,M) is said to be safe if ‖ϕ‖BM is defined for every sentence ϕ.

Let Γ be a set of sentences, ψ be a sentence, and K be a class of BL-chains. We say that ϕ is a semantic con-sequence of Γ in K and write Γ �K ψ iff we have that for every B ∈ K and for every B-first-order structure Msuch that (B,M) is safe, if ‖γ‖B

M = 1 for all γ ∈ Γ, then ‖ψ‖BM = 1. We say that ψ is valid in K, or that ψ is a

K-first-order tautology, iff ∅ �K ψ, and that ψ is valid in a BL-chain B if it is valid in K = {B}. Finally, a sen-tence ψ is said to be a standard tautology of BL∀ if ψ is valid in the class of all standard BL-algebras. For anyclass K of BL-chains, the set of first-order K-tautologies is denoted by TAUT(K)∀.

The first-order logic of all BL-chains, that is, TAUT(K)∀ when K is the class of all BL-chains, has beenaxiomatized by Hajek in [3] as follows.

Definition 2.7 The logic BL∀ is the logic in the predicate language described above, whose rules are ModusPonens and

ϕ

∀xϕ, and whose axiom schemes are the following:

(0) all axiom schemes of BL;(1) the scheme ∀xψ → ψ(x/t), t substitutable for x in ψ;(2) the scheme ψ(x/t) → ∃xψ, t substitutable for x in ψ;(3) the scheme ∀x(ϕ → ψ) → (ϕ → ∀xψ), where x is not free in ϕ;(4) the scheme ∀x(ψ → ϕ) → (∃xψ → ϕ), where x is not free in ϕ;(5) the scheme ∀x(ψ ∨ ϕ) → ((∀xψ) ∨ ϕ), where x is not free in ϕ.Then in [3] the following is shown.Theorem 2.8 Let K be the class of all BL-chains, Γ be any set of sentences, and ϕ be any sentence. The fol-

lowing are equivalent:(1) Γ �K ϕ.(2) ϕ is derivable from Γ in BL∀.We also quote the following result from [9].Theorem 2.9 The class of all standard tautologies of BL∀ is not arithmetical.

c© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim www.mlq-journal.org

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Math. Log. Quart. 54, No. 4 (2008) / www.mlq-journal.org 439

3 Main result

We start from the following lemmas.Lemma 3.1 Let B denote a complete BL-chain satisfying the condition (RC) mentioned in the introduction.

Then the following hold:(a) For every non-empty subset X of B which entirely consists of idempotents, inf(X) and sup(X) are idem-

potent.(b) If c ∈ B is not an idempotent, then there are idempotents a, b such that a < c < b and the set

[a, b] = {x ∈ B : a ≤ x ≤ b}

is the domain of either an MV-algebra or of a product algebra, denoted by B[a,b], with respect to the operations

x ∗[a,b] y = x ∗ y and x →[a,b] y = min{x → y, b}.

Moreover, B[a,b] is Archimedean, that is, if a < c < d < b, then there is n such that dn < c.(c) If c ∈ B is an idempotent, then there are a ≤ c and b ≥ c such that [a, b] entirely consists of idempotents

and(c1) for all h < a there is h′ such that h ≤ h′ < a and h′ is not an idempotent;(c2) for all k > b there is k′ such that b < k′ ≤ k and k′ is not an idempotent; thus if a < b, then B[a,b] de-

fined as in (b) is a Godel algebra.

P r o o f. (a) is an easy consequence of the completeness of B and of (RC). We prove (b). By Theorem 2.3,every x ∈ B belongs to a (unique) interval J such that either J has a minimum a and (J ∪ {1}, ∗,→, a, 1) is anMV-algebra or J has no minimum and (J ∪ {1}, ∗,→, 1) is the negative cone of an Abelian �-group (thereforewe have that J ∪ {1} is the Wajsberg component of B which x belongs to). In any case, by (RC),

a = inf{xn : n ∈ ω} = inf(J)

must be an idempotent (since infn xn = infn x2n = (infn xn)2). Therefore if (J ∪ {1}, ∗,→, 1) is the negativecone of an Abelian �-group, then (J ∪ {1, a}, ∗,→, 1) is a product algebra.

Now suppose that J has more than one element (this is the case if x is not an idempotent). Then there is anelement b which is the smallest element strictly bigger than all elements of J and this b is an idempotent. Indeed,let

Z = {x : (∀y ∈ J)(y < x)} and b = inf(Z).

Since Z is an union of Wajsberg components it is closed under ∗ and by (RC) we have

b2 = (inf(Z))2 = inf{x2 : x ∈ Z} = inf(Z) = b.

Thus b is an idempotent and b /∈ J .Clearly a < x < b. Moreover, for x ∈ [a, b),

b ∗ x = 1 ∗ x = x and b → x = x.

Thus letting, for x, y ∈ [a, b],

x ∗[a,b] y = x ∗ y and x →[a,b] y = min{x → y, b},

the algebra ([a, b], ∗[a,b],→[a,b], a, b) is either an MV-algebra or a product algebra; it will be denoted by B[a,b].We prove that B[a,b] is Archimedean: let c, d with a < c < d < b be given. Then since inf{dn : n ∈ ω} is anidempotent and there are no idempotents between a and b (as B[a,b] is either an MV-chain or a product chain),we must have inf{dn : n ∈ ω} = a. Thus there is n such that dn < c.

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440 P. Hajek and F. Montagna: A note on the first-order logic of complete BL-chains

(c) Simply let a be the infimum of all h ≤ c such that [h, c] entirely consists of idempotents and b be thesupremum of all k ≥ c such that [c, k] entirely consists of idempotents, and use (a) to conclude that a and b arealso idempotents. Note that it is possible that a = b = c.

Definition 3.2 A BL-chain B is called weakly saturated1) iff the conditions (b) and (c) of Lemma 3.1 hold.In this case, a subalgebra B[a,b] as in Lemma 3.1(b) is called an MV-component of B if B[a,b] is an MV-alge-bra and a product component if B[a,b] is a product algebra. Moreover, an algebra B[a,b] as in Lemma 3.1(c) withmore than one element will be called a Godel component of B. We also say that B[a,b] is a component to meanthat B[a,b] is either an MV-component or a product component, or a Godel component of B.

Note that a non-idempotent element belongs to either an MV-component or to a product component, whereasfor an idempotent element x there are many possibilities: either x is an endpoint of a product component or ofan MV-component, or belongs to a Godel component, or is the infimum of a set of endpoints of MV- or productcomponents.

Lemma 3.3 Let B be a weakly saturated BL-chain and a ∈ B \ {1} be an idempotent element which doesnot belong to a Godel component or to an MV-component or to a product component. Then a is the limit pointof a set of idempotents which are endpoints of MV- or product components of B.

P r o o f. Since a is not in a Godel component, then there is c0 such that a < c0 < 1 and c0 is not an idempo-tent. By Lemma 3.1, (b), there exist a0 < c0 < b0 such that B[a0,b0] is a product component or an MV-compo-nent of B. Since a is not an endpoint of that component, it must be the case that a < a0. Thus there is c1 suchthat c1 is not an idempotent and a < c1 < a0. Then we obtain idempotent elements a1, b1 such that

a < a1 < c1 < b1 < a0

and B[a1,b1] is a product component or an MV-component of B. Iterating the construction we get a sequence

(B[an,bn] : n ∈ ω)

of components such that

b0 > a0 > · · · > bn > an > · · · > a.

If inf{an : n ∈ ω} = a, we are done. Otherwise, there is aω such that a < aω < an for every n. Now let cω bea non-idempotent element such that a < cω < aω, and we restart the procedure from cω, thus getting a compo-nent B[aω+1,bω+1] with a < aω+1 < cω < bω, etc. By a cardinality argument, the procedure must end, thus get-ting a transfinite decreasing sequence (aγ : γ < α), α an infinite ordinal, converging to a.

We also quote the following.Proposition 3.4

(a) Every totally ordered Archimedean group G can be embedded into the ordered group R of reals by a com-plete embedding.

(b) Every countable totally ordered Archimedean group G can be embedded into a countable densely totallyordered subgroup G′ of R by a complete embedding (i. e., one which preserves existing suprema and infima).

P r o o f.

(a) This is just Holder’s theorem, cf. [2].(b) Let X be the set of strictly positive elements of G. If X has a minimum, then it is readily seen that G is

isomorphic to the ordered group Z of integers, and then it suffices to take G′ = Q (the totally ordered group ofrationals). Otherwise, G is already densely ordered: if a < b, then b − a > 0 and there is c with 0 < c < b − a.Thus a < a + c < b. It follows that G is densely ordered. Moreover, by (a), G can be embedded in R by a com-plete embedding. Thus we may take G′ = G.

1) Compare with the notion of a saturated BL-chain in [4, 6]; being saturated implies being weakly saturated but not conversely.

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Math. Log. Quart. 54, No. 4 (2008) / www.mlq-journal.org 441

Corollary 3.5 Let B be a countable and weakly saturated BL-algebra. Then the following hold:(a) Every MV- (product, respectively) component B[a,b] of B can be embedded into a countable and dense

subalgebra of the standard MV-algebra (product algebra, respectively) on [0, 1] by a complete embedding.(b) Every countable Godel component B[a,b] of B embeds into a countable and densely ordered subalgebra

of the standard Godel algebra on [0, 1] by a complete embedding.

P r o o f.

(a) It follows from Lemma 3.1(b), from Proposition 3.4, and from Theorems 2.4 and 2.5.(b) (Cf. [3, Lemma 5.3.1]) We shall say that x ∈ [a, b] has a predecessor iff there is y < x in [a, b] such that

there is no z with y < z < x. Let X be the set of elements of B[a,b] having a predecessor and let Y = [a, b] \ Xbe the set of the remaining elements of B[a,b]. Let

Z = {(y, 1) : y ∈ Y } ∪ {(x, q) : x ∈ X, q ∈ Q ∩ (0, 1]}.

For (x, q), (y, r) ∈ Z, define (x, q) � (y, r) iff either x < y or x = y and q ≤ r. Then (Z,�) is a totally anddensely ordered set, therefore one can obtain a Godel chain C from it with

(x, q) ∗ (y, r) = min{(x, q), (y, r)}

(where min is meant with respect to �) and with the residuum of ∗ as implication. Moreover, one can easily seethat the map x �−→ (x, 1) is a complete embedding of B[a,b] into C.

Lemma 3.6 Every countable and weakly saturated BL-chain B embeds into a countable, weakly saturated,and densely ordered BL-chain B′ by a complete embedding.

P r o o f. B′ is obtained from B by replacing each non-dense component by a densely ordered one accordingto Corollary 3.5. Clearly in this way we obtain a weakly saturated and countable BL-chain, thus we only have toverify density. Let x < y ∈ B′. Let us verify that there is z with x < z < y. Suppose first that x is not an idem-potent. Then x belongs to a densely ordered MV-component or product component B′

[a,b] of B′. If y ∈ B′[a,b],

then the existence of such a z follows from the density of B′[a,b], otherwise we may take z = b. The argument is

similar if y is not an idempotent. Finally, if x and y are both idempotent, then either there is a non-idempotent zsuch that x < z < y and we are finished, or x and y belong to a Godel component B′

[a,b], and the claim followsagain from the density of B′

[a,b].

The MacNeille completion of a BL-chain B is the algebra B+ = (B+, ∗+,→+, ∅, B) defined as follows:(1) Let, for X ⊆ B, K(X) denote the smallest subset Y of B such that

(1a) X ⊆ Y ;(1b) if y ∈ Y and z ≤ y, then z ∈ Y ;(1c) if sup(Y ) exists in B, then sup(Y ) ∈ Y .

Then the domain of B+ consists of all sets of the form K(X), X ⊆ B.(2) For I, J ∈ B+, we define

I ∗+ J = K(I ∗ J), I ∧+ J = I ∩ J, I ∨+ J = I ∪ H,

I →+ J = {z ∈ B : z ∗ I ⊆ J}.

Moreover, define 0+ = ∅ and 1+ = B.Note that B+ is a commutative, integral, and bounded residuated lattice, which is totally and completely or-

dered with respect to inclusion, and the map x �−→ {z ∈ B : z ≤ x} is a complete embedding of B into B+.

Lemma 3.7 If B is a weakly saturated densely ordered BL-chain, B+ is also a weakly saturated BL-chain.If in addition B is countable, then B+ is isomorphic to a standard BL-algebra.

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442 P. Hajek and F. Montagna: A note on the first-order logic of complete BL-chains

P r o o f. We know that B+ is a commutative, integral, bounded, and totally ordered residuated lattice; we ve-rify the divisibility condition. Thus suppose I ⊂ J and let us verify that I = J ∗+ (J →+ I). The inclusion

I ⊇ J ∗+ (J →+ I)

holds in any residuated lattice, so let us verify that

I ⊆ J ∗+ (J →+ I).

For H ∈ B+ and each component B[a,b] of B, let us say that H belongs to B[a,b] iff a ∈ H and b /∈ H . More-over, for H,K ∈ B+, let us say that H and K belong to the same component if there exists a component B[a,b]

such that H and K both belong to B[a,b]. Note that the set of elements of B+ belonging to a component B[a,b]

is the MacNeille completion of B[a,b], therefore, recalling the fact that B[a,b] is a densely ordered subalgebra ofthe standard MV- (or product, or Godel) algebra on [0, 1], this MacNeille completion is isomorphic to the stan-dard MV- (or product, or Godel) algebra on [0, 1]. Thus if I and J belong to the same component, then the con-dition I ⊆ J ∗+ (J →+ I) holds. Now suppose that I and J do not belong to the same component. Then thereexists y ∈ J \ I such that for all x ∈ I we have x ∗ y = y → x = x. Indeed if not, then for each y ∈ J \ I therewould be x ∈ I such that x and y belong to the same MV- or product component B[a,b], and this would implythat I and J belong both to B[a,b], which has been excluded before. Now let y ∈ J \ I be such an element, andlet x ∈ I . Then for all z ∈ J we have z ∗ x ≤ x, therefore x ∈ J → I . Hence x = y ∗ x ∈ J ∗+ (J →+ I), anddivisibility is proved.

Now MacNeille completion does not introduce new components, thus B+ is also weakly saturated. If more-over B is countable, then assume that its domain is the unit interval of rationals and its order is the standard or-der of rationals. Then replace each component B[a,b] of B by MacNeille completion and extend the operationsin the obvious way. It is easy to see that you get a standard BL-algebra (recall Mostert-Shields representation ofcontinuous t-norms).

Definition 3.8 Let (B,M) and (C,N) be two models of BL∀. We say that (B,M) is an elementarysubmodel of (C,N) if B is a subalgebra of C, M is a subset of N , and for every formula ϕ(x1, . . . , xn)with free variables x1, . . . , xn and for all c1, . . . , cn ∈ M we have: ‖ϕ(c1, . . . , cn)‖C

N = ‖ϕ(c1, . . . , cn)‖BM .

We will prove an analogue of the Lowenheim-Skolem theorem for models of BL∀ (and more generally formodels for any predicate fuzzy logic). To this purpose we note that in some sense a model (B,M) of BL∀ canbe viewed as a classical model. More precisely, we associate to (B,M) (where up to isomorphism we can safe-ly assume that the domains of B and of M are disjoint) a model C(B,M) of classical logic as follows:

(1) The domain of C(B,M) is the union of the domains of B and of M .(2) C(B,M) has four binary operations ∗C , →C , ∨C , and ∧C such that if x, y ∈ B, then

x ∗C y = x ∗ y, x →C y = x → y, x ∨C y = x ∨ y, x ∧C y = x ∧ y;

otherwise

x ∗C y = x →C y = x ∨C y = x ∧C y = 0.

Moreover, C(B,M) has a constant 0C , interpreted as the bottom element 0 of B.(3) C(B,M) has two unary predicates, B(x) and M(x), interpreted as the domain of B and the domain of M .

Moreover, C(B,M) has a binary predicate = for equality.(4) For every constant c of the language, C(B,M) has a constant cC interpreted as c, and for every n-ary func-

tion symbol f of the language, C(B,M) has a function fC such that

fC(c1, . . . , cn) =

{fM (c1, . . . , cn) if c1, . . . , cn ∈ M,

0 otherwise.

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Math. Log. Quart. 54, No. 4 (2008) / www.mlq-journal.org 443

(5) For any formula ϕ(x1, . . . , xn) in the free variables shown, there exists an n-ary operation fϕ(x1, . . . , xn)in C(B,M) defined by

fϕ(c1, . . . , cn) =

{‖ϕ(c1, . . . , cn)‖B

M if c1, . . . , cn ∈ M,

0 otherwise.

Easy to see, the clauses defining models of BL∀ can be translated in classical first-order logic. In other wordswe can write a set Γ of classical first-order axioms such that:

(a) Every model C(B,M) defined as above satisfies Γ.(b) If C satisfies Γ, then

(b1) letting

BC = {c ∈ C : C � B(c)},

BC is the universe of a BL-chain BC with bottom element 0C and binary operations ∗C , →C , ∨C , and ∧C ;(b2) letting

MC = {c ∈ C : C � M(c)},

MC is the universe of a BC-structure MC such that

fϕ(c1, . . . , cn) = ‖ϕ(c1, . . . , cn)‖BC

MC

for every formula ϕ(x1, . . . , xn) and for every c1, . . . , cn ∈ M .For instance, the axiom of Γ describing the clause corresponding to ∀ is

∀�x∀y(M(�x ) → ((f∀zϕ(z)(�x ) = y) ↔ Θ(�x, y))),

where ∀�x is an abbreviation for ∀x1 . . .∀xn, M(�x ) is an abbreviation for M(x1) & · · · & M(xn), �x is an ab-breviation for x1, . . . , xn, and Θ(�x, y) denotes the formula

(B(y) & ∀u(B(u) → (u ≤ y ↔ ∀v(M(v) → u ≤ fϕ(�x, v))))).

It follows that if C and D satisfy Γ and C is an elementary substructure of D, then (BC ,MC) is an elemen-tary susbtructure of (BD,MD). Thus the Lowenheim-Skolem theorem for classical logic gives us:

Lemma 3.9 Every model (B,M) of BL∀ with a countable language has an elementary submodel (B′,M ′)such that both B′ and M ′ are countable.

Lemma 3.10 Let B be a complete BL-chain satisfying (RC) and M be a B-first-order structure. Let (C,N)be a countable elementary substructure of (B,M). Then there exists a structure (D,N) such that C is a subal-gebra of D, D is a countable and weakly saturated subalgebra of B, and (C,N) is an elementary substructureof (D,N).

P r o o f. By (RC) and Lemma 3.1, B consists of components and of suprema/infima of endpoints of compo-nents. D is obtained from C as follows: let B[a,b] be a component of B which intersects C. Then if b /∈ C andthe set {x ∈ C : x > b} has no minimum, then we add b to C. Moreover, if [a, b] ∩ C has no minimum and ei-ther the set {x ∈ C : x < a} has no maximum, or it has a maximum which is not an idempotent, then we add ato C. Note that if {x ∈ C : x > b} has a minimum m, then m must be an idempotent, for otherwise we wouldhave b < m2 < m, which gives a contradiction. Note also that [a, b] ∩ C can fail to have a minimum only if itis a Godel or a product component, because MV-components B[a,b] are Archimedean and if for some c ∈ C wehave a ≤ c < b, then there is n with cn = a and a ∈ C. Thus since a, b are idempotent, we have that if a /∈ C,then for all x ∈ C we have that

a ∗ x ∈ {a, x} and a → x ∈ {x, 1}.

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444 P. Hajek and F. Montagna: A note on the first-order logic of complete BL-chains

Similarly if b /∈ C, then for all x ∈ C we have that

b ∗ x ∈ {b, x} and b → x ∈ {x, 1}.

Thus adding such endpoints we still have a subalgebra D of B, which is weakly saturated because every com-ponent of D has a maximum (either b or min{x ∈ C : x > b}) and a minimum (either a or min([a, b] ∩ C),or max{x ∈ C : x < a} if such a maximum is an idempotent), and countable since only countably many com-ponents of B can intersect C. Now N is a D-first-order structure, since it is a C-first-order structure and C isa subalgebra of D. For any closed formula ϕ (possibly with parameters from N ), we prove by induction on ϕthat ‖ϕ‖C

N = ‖ϕ‖DN . For atomic formulas the claim is obvious and the induction steps corresponding to connec-

tives follow from the fact that C is a subalgebra of D.Now suppose ϕ = ∃xψ(x). Then

‖ϕ‖DN =

∨Dc∈N ‖ψ(c)‖D

N =∨D

c∈N ‖ψ(c)‖CN ≤ ∨C

c∈N ‖ψ(c)‖CN = ‖ϕ‖C

N ;

the superscripts D and C in∨D

c∈N and in∨C

c∈N mean that the supremum is computed in D (in C, respective-ly). Moreover, ‖ϕ‖D

N can only differ from ‖ϕ‖CN if ‖ϕ‖D

N is an endpoint of a component B[a,b] of B that is notin C. Distinguish two cases:

(a) ‖ϕ‖DN = a and ‖ϕ‖C

N = d > a. Then [a, b] ∩ C has no minimum (otherwise we would not have added ato C). It follows that there is e ∈ C such that a < e < d, and since

∨Dc∈N ‖ψ(c)‖D

N = a,

for all c ∈ N we have ‖ψ(c)‖DN < e. But by the induction hypothesis, ‖ψ(c)‖D

N = ‖ψ(c)‖CN < e, which contra-

dicts∨C

c∈N ‖ψ(c)‖CN = ‖ϕ‖C

N = d.

(b) ‖ϕ‖DN = b and ‖ϕ‖C

N = d > b. Then {x ∈ C : x > b} has no minimum, and therefore there exists e ∈ C

such that b < e < d, and since∨D

c∈N ‖ψ(c)‖DN = b, ‖ψ(c)‖D

N < e for all c ∈ N . But by the induction hypothe-sis, ‖ψ(c)‖D

N = ‖ψ(c)‖CN < e, which contradicts

∨Cc∈N ‖ψ(c)‖C

N = ‖ϕ‖CN = d.

Finally suppose ϕ = ∀xψ(x). Then

‖ϕ‖DN =

∧Dc∈N ‖ψ(c)‖D

N =∧D

c∈N ‖ψ(c)‖CN ≥ ∧C

c∈N ‖ψ(c)‖CN = ‖ϕ‖C

N .

Once again, ‖ϕ‖DN can only differ from ‖ϕ‖C

N if ‖ϕ‖DN is an endpoint of a component B[a,b] of B which is not

in C. Now distinguish two cases:(a) ‖ϕ‖D

N = a and ‖ϕ‖CN = d < a. Then [a, b] ∩ C has no minimum (otherwise we would not have added a

to C). If {x ∈ C : x < a} has no maximum, then there is e ∈ C such that d < e < a. Moreover,

∧Dc∈N ‖ψ(c)‖D

N = a

implies that ‖ψ(c)‖DN ≥ a > e for all c ∈ N , and since by the induction hypothesis ‖ψ(c)‖D

N = ‖ψ(c)‖CN we get

a contradiction. Finally, suppose that {x ∈ C : x < a} has a maximum m which is not an idempotent. Then itmust be the case that m = d (if d < m, then there would be c ∈ N such that ‖ψ(c)‖D

N = ‖ψ(c)‖CN < m, which

gives a contradiction). Note that d = m belongs either to an MV-component or to a product component B[a′,b′]

of B and a′ < m < b′. Moreover, for every c ∈ N we have ‖ψ(c) & ψ(c)‖DN ≥ a2 = a > d, therefore letting

η = ∀z(∀xψ(x) → (ψ(z) & ψ(z))),

we have ‖η‖CN = 1. Since (C,N) is an elementary substructure of (B,M), we get ‖η‖B

M = 1. By the same rea-son,

‖∀xψ(x)‖BM =

∧Bc∈M ‖ψ(c)‖B

M = d = m.

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Math. Log. Quart. 54, No. 4 (2008) / www.mlq-journal.org 445

Then there are two cases: if there is no z ∈ B such that m < z < b′, there exists c ∈ N with ‖ψ(c)‖BM = m and

since ‖η‖BM = 1 we get

‖∀xψ(x)‖BM ≤ m2 < m,

which is a contradiction. If there is z ∈ B such that m < z < b′, then there must be c ∈ N with ‖ψ(c)‖BM < z,

therefore, using again ‖η‖BM = 1 we get

m = ‖∀xψ(x)‖BM < z2,

therefore there is c1 ∈ M such that ‖ψ(c1)‖BM < z2. By the same argument we get c2 ∈ M such that

m ≤ ‖ψ(c1)‖BM < z4,

and iterating we get m ≤ z2n for every n, contradicting the fact that B[a′,b′] is Archimedean.

(b) ‖ϕ‖DN = b and ‖ϕ‖C

N = d < b. Then {x ∈ C : x > b} has no minimum, and therefore there exists e ∈ C

such that b < e < d, and since∧D

c∈N ‖ψ(c)‖DN = b, there is c ∈ N such that ‖ψ(c)‖D

N < e. But by the inductionhypothesis, ‖ψ(c)‖D

N = ‖ψ(c)‖CN < e, and we obtain a contradiction with

∧Cc∈N ‖ψ(c)‖C

N = d.

Theorem 3.11 Let C be a complete BL-chain. The following are equivalent:(1) C satisfies the condition (RC).(2) The formula ∀x(ϕ(x) & ϕ(x)) → ((∀xϕ(x)) & (∀xϕ(x))) is valid in C.(3) Every standard tautology of BL∀ is valid in C.

P r o o f.(1) ⇒ (2): Trivial.(2) ⇒ (3): We prove that every sentence ϕ of BL∀ that is not valid in C can be invalidated in some standard

BL-algebra. Let M be a C-first-order structure with ‖ϕ‖CM < 1. By Lemma 3.9, we can obtain a countable ele-

mentary submodel (D,N) of (C,M). Then by Lemma 3.10, we get a weakly saturated subalgebra E of C suchthat (D,N) is an elementary substructure of (E,N). Next we apply Lemma 3.6, thus obtaining a countable anddensely ordered BL-chain F in which E embeds by a complete embedding. Finally by Lemma 3.7 we embed Finto a standard BL-chain H , again by a complete embedding. The completeness of the embeddings makes surethat (E,N) is an elementary substructure of (H,N). Thus ϕ is invalidated in a standard BL-algebra.

(3) ⇒ (1): We argue contrapositively. Suppose that the condition (RC) is not satisfied. Take a transfinite se-quence (ai : i ∈ I) in C such that

∧i∈I a2

i > (∧

i∈I ai)2 and a set D = {ci : i ∈ I} bijective to {ai : i ∈ I}.Let P be a unary predicate letter, and consider a first-order fuzzy structure M on C such that PM (ci) = ai forall i ∈ I . Then ‖∀x(P (x) & P (x))‖C

M =∧

i∈I a2i > (

∧i∈I ai)2 = ‖(∀xP (x)) & (∀xP (x))‖C

M .Hence the formula ∀x(P (x) & P (x)) → ((∀xP (x)) & (∀xP (x))), which is a standard tautology, is not va-

lid in C.

Corollary 3.12 Let K be the class of complete BL-chains satisfying the condition (RC). Then TAUT(K)∀equals the set of all standard BL∀-tautologies, and therefore by [5], it is not arithmetical.

Acknowledgements The work of the first author was partly supported by grant A100300503 of the Grant Agency of theAcademy of Sciences of the Czech Republic and partly by the Institutional Research Plan AV0Z10300504.

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