Non-commutative Fuzzy Logic psMTL

An Alternative Proof for the Standard Completeness Theorem

Denisa Diaconescu

Faculty of Mathematics and Computer Science, University of Bucharest, Bucharest, Romania

Keywords:

Non-commutative Fuzzy Logic, psMTL Logic, Standard Semantics.

Abstract:

In (Jenei and Montagna, 2003) was proved that the non-commutative psMTL logic introduced in (H´ajek,

2003b) is the logic of left-continuous non-commutative t-norms or, equivalently, that the logic psMTL enjoys

standard completeness. In the present paper we provide an alternative proof for the standard completeness

theorem for the logic psMTL and we furthermore show that this result can be obtained also for ﬁnite theories.

1 INTRODUCTION

In general, a non-commutative logic is a logic

equipped with a non-commutative conjunction &, i.e.

the formula ϕ&ψ is not equivalent with the formula

ψ&ϕ. Motivations for considering non-commutative

logics appear in different areas, within and also out-

side mathematics. Some situations that give rise

to non-commutative conjunctions are the semantics

of parallel programming, quantum mechanics which

make use of non-commutative observables, Lambek

calculus (Lambek, 1958) modelling sentence struc-

tures using concatenation as a non-commutative op-

eration, logics based on formal matrix multiplication,

linear logics studied in (Girard, 1995), quantales that

have the non-commutative ”and” given by closure

of the product subspace (Rosenthal, 1990), (Mulvey

and Nawaz, 1995). On the other hand, in ordinary

language a commutative ”and” between clauses indi-

cates independence, while a non-commutative ”and”

indicates a pragmatic dependence as pointed out by

(Schmerling, 1975). Consider, for example, state-

ments from natural language such as ”he lost weight

and he got sick” vs. ”he got sick and he lost weight”.

T-norms were introduced by (Schweizer and

Sklar, 1960) in order to formulate properly the trian-

gle inequality in probabilistic metric spaces. Since

then, t-norms have been applied in various other

mathematical disciplines including game theory, the-

ory of non-additive measures and integrals, theory

of measure-free conditioning, theory of aggregation

operations, fuzzy set theory, fuzzy logic, fuzzy con-

trol, preference modeling and decision analysis, ar-

tiﬁcial intelligence. The role of t-norms in the the-

ory of many-valued logics is extremely important.

Many-valued t-norm based logics are non-classical

logical systems that use t-norms as truth-functions for

the conjunction connective and their residua as truth-

functions for the implicative connective. The seman-

tics of many-valued logics given by t-norms is usually

called standard semantics.

H´ajek introduced in his inﬂuential monograph

(H´ajek, 1998) a very general many-valued logic,

called BL logic (BL stands for ’Basic Logic’), with

the idea to formalize the many-valued logics induced

by continuous t-norms and their residua; this fact was

proved later in (Cignoli et al., 2000). On the other

hand, Esteva and Godo observed that the minimal

condition for a t-norm to have a residuum, and there-

fore to determine a logic, is left-continuity, not con-

tinuity. Therefore, in (Esteva and Godo, 2001) they

proposed a weaker logic, called MTL logic (MTL

stands for ’Monoidal T-norm based Logic’). In (Jenei

and Montagna, 2002) was proved that MTL is indeed

the logic of left-continuous t-norms and their residua.

The non-commutative counterparts of t-norms

were investigated by Flondor, Georgescu and

Iorgulescu in (Flondor et al., 2001) under the name of

pseudo-t-norms. Every continuous pseudo-t-norm is

commutative, but there are left-continuous pseudo-t-

norms which are not commutative. As a consequence,

the non-commutativecounterpart of the logic BL can-

not provide a fruitful logic, since the standard seman-

tics do not have a non-commutative nature; despite

this weakness, the non-commutativeBL logic was de-

veloped in (H´ajek, 2003a).

On the other hand, the non-commutative MTL

logic introduced in (H´ajek, 2003b) under the name of

350

Diaconescu D..

Non-commutative Fuzzy Logic psMTL - An Alternative Proof for the Standard Completeness Theorem.

DOI: 10.5220/0004151603500356

In Proceedings of the 4th International Joint Conference on Computational Intelligence (FCTA-2012), pages 350-356

ISBN: 978-989-8565-33-4

 2012 SCITEPRESS (Science and Technology Publications, Lda.)

psMTL logic (psMTL stands for ’pseudo Monoidal

T-norm based Logic’) enjoys standard completeness

with respect to left-continuous pseudo-t-norms, as

proved in (Jenei and Montagna, 2003).

The goal of the present paper is to provide an al-

ternative method to prove the standard completeness

theorem for the logic psMTL and to show that this

result holds even for ﬁnite theories, i.e. we prove

the ﬁnite strong standard completeness theorem for

psMTL logic.

This paper is organized as follows: in Section

2 we recall the deﬁnitions and the basic properties

on psMTL logic; Section 3 is the innovative part of

this paper, where we give an alternative proof for

the ﬁnite strong standard completeness theorem for

psMTL logic; Section 4 contains some concluding re-

marks.

2 THE LOGIC PSMTL

H´ajek introduced the non-commutative logic psMTL

in (H´ajek, 2003b). In this section we recall this log-

ical system in a somewhat different formulation but

equivalent with that of H´ajek, following the line of

(Diaconescu, 2012).

The fact that the logic psMTL has a non-

commutative conjunction & leads to the existence

of two implications. Therefore, the language of the

propositional calculus psMTL consists of denumer-

able many propositional variables, whose set is de-

noted by Var, the primitive connectives →, , &, ∧,

∨ and the constant

0. From this primitive connectives,

further connectives and constants are deﬁned by:

¬ϕ is ϕ →

∼ ϕ is ϕ

1 is 0 → 0,

ϕ ↔ ψ is (ϕ → ψ)&(ψ → ϕ),

ϕ ! ψ is (ϕ ψ)&(ψ ϕ).

The formulas are deﬁned by structural induction as

usual and we denote by Form

psMTL

the set of all for-

mulas of the logic psMTL. For any formula ϕ of

psMTL, we deﬁne the formula ϕ

•

obtained by inter-

changing the two implications → and and revers-

ing the arguments of &. Notice that (ϕ

•

)

•

= ϕ. The

axioms of psMTL are:

I. a formula which has one of the following forms is

an axiom:

(A1) (ψ → χ) → ((ϕ → ψ) → (ϕ → χ))

(A2) (ϕ&ψ) → ϕ

(A3) (ϕ∧ ψ) → ϕ

(A4) (ϕ∧ ψ) → (ψ∧ ϕ)

(A5) ((ϕ → ψ)&ϕ) → (ϕ∧ ψ)

(A6a) (ϕ → (ψ → χ)) → ((ϕ&ψ) → χ)

(A6b) ((ϕ&ψ) → χ) → (ϕ → (ψ → χ))

(A7) ((ϕ → ψ) → χ) → (((ψ → ϕ) → χ) → χ)

(A8a) (ϕ∨ψ) → (((ϕ ψ) → ψ) ∧((ψ ϕ) → ϕ))

(A8b) (((ϕ ψ) → ψ) ∧((ψ ϕ) → ϕ)) → (ϕ∨ψ)

(A9)

0 → ϕ

II. if ϕ is an axiom of the form (A1), (A2), (A5),

(A6a), (A6b), (A7), (A8a) or (A8b), then ϕ

•

is also

an axiom.

The deduction rules of psMTL are:

(MP1)

ϕ ϕ → ψ

(Impl1)

ϕ → ψ

ϕ ψ

(MP2)

ϕ ϕ ψ

(Impl2)

ϕ ψ

ϕ → ψ

A theory over psMTL logic is any set of formu-

las. Provability in a theory T is deﬁned in the ob-

vious way, using all the deduction rules. We note

by T ⊢ ϕ the fact that ϕ is a T-theorem and by

Theor

psMTL

(T) the set of all T-theorems (we use the

notation Theor

psMTL

when T is the empty set).

Proposition 2.1. For any formula ϕ of psMTL and

any theory T, T ⊢ ϕ iff T ⊢ ϕ

•

The algebraic models for psMTL logic were intro-

duced by Flondor, Georgescu and Iorgulescu:

Deﬁnition 2.1. (Flondor et al., 2001). A psMTL-

algebra is a structure A = (A, ∨,∧,⊙,→, , 0,1)

such that the following conditions are fulﬁlled:

(i) (A, ∨,∧,0,1) is a bounded lattice,

(ii) (A, ⊙,1) is a monoid,

(iii) x ⊙ y ≤ z iff x ≤ y → z iff y ≤ x z,

(iv) (x → y) ∨ (y → x) = (x y) ∨ (y x) = 1.

Condition (iii) is called the adjointness property,

while (iv) is known as the prelinearity condition. We

say that a psMTL-algebra is a chain if the underlying

lattice order is linear.

For any psMTL-algebra A, an A-evaluation of

propositional variables is a mapping e : Var → A.

Any A-evaluation of propositional variables e can be

uniquely extended to an A-evaluation of all formulas

(denoted also by e) using the operations on A as truth

functions, i.e. e(

0) = 0, e(ϕ&ψ) = e(ϕ) ⊙ e(ψ), and

e(ϕ◦ ψ) = e(ϕ) ◦ e(ψ), where ◦ ∈ {∨,∧, →, }.

If T is a theory of the logic psMTL, A is a psMTL-

algebra and e is an A-evaluation such that e(ψ) = 1,

for any ψ ∈ T, then we call e an A-model of T.

K¨uhr introduced a particular class of psMTL-

algebras, namely representable psMTL-algebras. The

variety of psMTL-algebras do not enjoy the subdirect

Non-commutativeFuzzyLogicpsMTL-AnAlternativeProoffortheStandardCompletenessTheorem

351

representation property (i.e. each algebra is a subal-

gebra of a direct product of chains), but it was proved

that a psMTL-algebra is subdirectly representable if

and only if it is a representable psMTL-algebra.

Deﬁnition 2.2. (K¨uhr, 2003). A representable

psMTL-algebra (psMTL

-algebra, for short) is a

psMTL-algebra in which the following identities are

valid:

(R1) (y → x) ∨ (z ((x → y) ⊙ z)) = 1,

(R2) (y x) ∨ (z → (z⊙ (x y))) = 1.

Conditions (R1) and (R2) are also known as K

uhr’s

identities.

H´ajek reﬂected this problem in the logical frame-

work by introducing in (H´ajek, 2003b) the logic

psMTL

as an extension of psMTL by the axioms:

(A10) (ϕ → ψ) ∨ (χ ((ψ → ϕ)&χ))

(A10

•

) (ϕ ψ) ∨(χ → (χ&(ψ ϕ)))

Axioms (A10) and (A10

•

) are just the logical reﬂec-

tion of K¨uhr’s identities. Due to the properties of

psMTL

-algebras, a chain completeness theorem for

psMTL

was proved by H´ajek:

Theorem 2.1. (H

ajek, 2003b). Let T be a theory over

psMTL

and ϕ be a formula. T proves ϕ iff for each

psMTL

-chain L and each L-model e of T, e(ϕ) = 1.

The non-commutative generalizations of t-norms

were studied by Flondor, Georgescu and Iorgulescu:

Deﬁnition 2.3. (Flondor et al., 2001). A pseudo-t-

norm is a binary relation ⊗ on the real unit inter-

val [0,1] that is associative, non-decreasing in both

arguments and satisfying x ⊗ 1 = 1 ⊗ x = x, for all

x ∈ [0,1].

A pseudo-t-norm ⊗ is left-continuous if

i∈I

⊗

b) = (

i∈I

) ⊗ b and

i∈I

(b ⊗ a

) = b ⊗ (

i∈I

Moreover, any left-continuous pseudo-t-norm ⊗ has

a left residuum → and a right residuum given by

a → b = sup{c | c⊗ a ≤ b},

a b = sup{c | a⊗ c ≤ b}.

The structure ([0,1],max,min,⊗,→, , 0,1) is a

psMTL

-chain, called a standard psMTL-algebra.

Notice that every standard psMTL-algebra is repre-

sentable.

It was proved in (Flondor et al., 2001) that any

continuous pseudo-t-norm is commutative, but there

are left-continuous pseudo-t-norms which are not

commutative as shown by the following example:

Example 2.1. (Flondor et al., 2001).

Let 0 < a

< a

< b

< 1 and T : [0,1]× [0,1] → [0,1]

deﬁned by

T(x,y) =



, a

< x ≤ a

, a

< y ≤ b

min(x,y), otherwise

Then T is a left-continuous pseudo-t-norm which is

not commutative.

3 STANDARD COMPLETENESS

This section is the main contribution of this paper.

In (Jenei and Montagna, 2003) it was proved that

the logic psMTL

is complete with respect to left-

continuous pseudo-t-norms and their residua, by gen-

eralizing to the non-commutative case the proof for

the standard completeness theorem for MTL logic

from (Jenei and Montagna, 2002).

In this section we provide an alternative proof for

the standard completeness theorem for psMTL logic.

Moreover, we prove a strong standard completeness

theorem with respect to ﬁnite theories, namely:

Theorem 3.1 (Finite Strong Standard Completeness).

Let T be a ﬁnite theory over psMTL

and ϕ be a

formula. T proves ϕ iff for each standard psMTL-

algebra L and each L-model e of T, e(ϕ) = 1.

This theorem is proved along this section. The

idea behind the proof is to adapt to the non-

commutative case the proof given by (Horˇcik, 2007)

for the standard completeness theorem for MTL logic.

Let us consider a ﬁnite theory T and a formula

ϕ over the logic psMTL

such that T 0 ϕ. Thus, by

Theorem 2.1, there is a psMTL

-chain

L = (L,⋆

,→

,≤,0,1)

and an L-model e

of T such that e

(ϕ) < 1. We will

show that L can be embedded into a standard psMTL-

algebra. As in (Horˇcik, 2007), we deﬁne the follow-

ing set:

G = {e

(ψ) | ψ is a subformula of χ ∈ T ∪ {ϕ}}.

Let S be the submonoid of L generated by the set

G, i.e. S = (S,⋆,≤,0,1), where ⋆ denotes the restric-

tion of ⋆

to S. Observe that S is ﬁnitely generated

since G is ﬁnite.

Lemma 3.1. The monoid S is countable and inversely

well ordered, i.e. each subset of S has a maximum.

Proof. Let g

,... ,g

be the generators of S and M

a subset of S. For any element of m ∈ M we have

|m|

= k

, for each i ∈ {1,...,n}, where |m|

repre-

sents the number of occurrences of g

in m. There-

fore we can assign to each element of M an n-tuple

,... ,k

) ∈ N

. Let us denote by m

,...,k

)

the ele-

ment from M associated with the n-tuple (k

,... ,k

Thus there is a subset H ⊆ N

such that (k

,... ,k

) ∈

H iff m

,...,k

)

∈ M. Moreover, if (k

,... ,k

) ≤

,... ,t

), we have m

,...,k

)

≥ m

,...,t

)

since ⋆ is

IJCCI2012-InternationalJointConferenceonComputationalIntelligence

352

order-preserving. Since H has only ﬁnitely many

minimal elements, one of them must correspond to the

maximum of M, thus S is inversely well ordered.

Due to the previous lemma, we can introduce the

following residua on S:

a → b = max{z ∈ S | z⋆ a ≤ b},

a b = max{z ∈ S | a⋆ z ≤ b}.

Therefore we can prove the following result:

Proposition 3.1. The enriched monoid

S = (S,⋆,→, ,≤, 0,1)

is a psMTL

-chain and there exists an evaluation e

of T such that e

(ϕ) < 1.

Proof. We know that S is an integral totally ordered

monoid and that ≤ is compatible with ⋆. The fact that

a⋆ b ≤ c iff a ≤ b → c iff b ≤ a c follows immedi-

ately from the deﬁnitions of → and . Hence S is an

psMTL

-chain.

Let us deﬁne an evaluation e

(p) = e

(p), for

every propositional variable p appearing in any χ ∈

T ∪ {ϕ} and arbitrarily, otherwise. Let χ ∈ T ∪ {ϕ}.

We show by induction on the complexity of χ that

(ψ) = e

(ψ), for each subformula ψ of χ. Al-

most all cases are trivial, therefore we treat only the

case when ψ = ψ

. By deﬁnition we have

(ψ) = e

(ψ

) e

(ψ

) = max{z ∈ S | e

(ψ

)⋆z ≤

(ψ

)}. Since ψ is a subformula of χ, we have

(ψ) ∈ S. Consequently, e

(ψ

)⋆ e

(ψ) = e

(ψ

)∗

(ψ) ≤ e

(ψ

). Thus e

(ψ) ≤ e

(ψ). Now sup-

pose there is an element z

′

∈ S such that z

′

> e

(ψ)

and e

(ψ

) ⋆ z

′

≤ e

(ψ

). Since z

′

∈ L, we get z

′

≤

(ψ

) e

(ψ

) = e

(ψ) (contradiction). Hence

(ψ) = e

(ψ).

Since e

(τ) = e

(τ) = 1, for any τ ∈ T, e

is an

S-model of T. Moreover, e

(ϕ) = e

(ϕ) < 1.

Therefore we have the psMTL

-chain S which is

countable and inversely well ordered, and the evalu-

ation e

on S such that e

(ϕ) < 1. The next step is

to build a new psMTL

-chain S

′

order-isomorphic to

[0,1] in which S can be embedded.

In order to deﬁne such a psMTL

-chain, we will

use a similar construction as in the original proof of

the standard completeness theorem from (Jenei and

Montagna, 2003). We deﬁne the new universe by:

′

= {(a,x) | a ∈ S \{0}, x ∈ (0,1]} ∪{(0,1)}.

The order ≤

′

on S

′

is the lexicographic order, i.e.

(a,x) ≤

′

(b,y) iff a < b or (a = b and x ≤ y).

Let I be the set of all idempotents of S, i.e x⋆ x = x.

We deﬁne the following monoidal operation on S

′

(a,x)⋆

′

(b,y) =







(a⋆ b,1), a⋆ b < a∧ b

(a,xy), a = b, a ∈ I

min{(a,x),(b, y)}, otherwise

where xy stands for the usual product of reals. Note

that in any case, the ﬁrst coordonate always equals

a⋆ b.

The proof of the following result follows closely

the proof of Lemma 6 from (Horˇcik, 2007), but dif-

ferences do appear since we are dealing with a non-

commutativity operation ⋆

′

Lemma 3.2. The structure

′

= (S

′

,⋆

′

,≤

′

,(0,1), (1,1))

is a totally ordered integral monoid, where (1,1) is

the neutral element and the top element as well, (0,1)

is the bottom element and ⋆

′

is monotone with respect

to ≤

′

on both arguments.

Proof. It is obvious that (1,1) is the neutral element

and the top element and that (0, 1) is the bottom ele-

ment. Let us prove that ⋆

′

is associative, i.e.

(a,x) ⋆

′

((b,y) ⋆

′

(c,z)) = ((a,x) ⋆

′

(b,y)) ⋆

′

(c,z).

We denote the left-hand side (right-hand side, respec-

tively) of the above equation by L (R, respectively).

Let P(a,b) denote the following property of a,b ∈ S

′

P(a,b) : a = b and a ∈ I.

Clearly, if P(a,b) holds, then P(b,a) holds as well.

We must analyse several cases:

1. Suppose that none of P(a,b), P(b,c), P(a,b ⋆ c),

P(a⋆ b,c) is valid. We distinguish two cases:

(i) a⋆b⋆ c = a∧ b∧ c. Then both L and R are equal

to min{(a,x),(b, y),(c,z)} and therefore associa-

tivity of ⋆

′

holds.

(ii) a ⋆ b ⋆ c 6= a∧ b ∧ c. We claim that in this case

both L and R are equal to (a⋆ b⋆ c,1). Let us con-

sider the case of L, since for R we have a similar

proof. If b⋆ c = b∧ c, then a⋆ (b⋆ c) 6= a∧(b⋆ c),

therefore L = (a ⋆ b⋆ c,1). If b ⋆ c 6= b ∧ c, then

(b,y) ⋆

′

(c,z) = (b ⋆ c,1) and a ⋆ b⋆ c 6= a. Thus

either a⋆ b ⋆ c 6= b ⋆ c and then L = (a ⋆ b ⋆ c,1),

or a⋆ b⋆ c = b⋆ c and then L = (a,x)∧ (b⋆c, 1) =

(b⋆ c,1) = (a⋆ b⋆ c,1).

2. Suppose that P(b,c) holds. Then b = c, b is idem-

potent and L = (a,x) ⋆

′

(b,yz). We obtain that

L =











(b,xyz), if a = b

(a⋆ b,1), if a ⋆ b < a ∧ b

(a,x), if a⋆ b = a∧ b,a < b

(b,yz), if a⋆ b = a∧ b,a > b

For R we have the following cases:

Non-commutativeFuzzyLogicpsMTL-AnAlternativeProoffortheStandardCompletenessTheorem

353

− if a = b, then R = (b,xy) ⋆

′

(c,z) = (b⋆ c, xyz) =

(b,xyz);

− if a⋆b < a∧b, then (a⋆b)⋆b= a⋆b = (a⋆b)∧b.

Moreover a⋆ b < a ∧b ≤ b. Thus R = (a ⋆ b,1) ⋆

′

(b,z) = (a⋆ b,1);

− if a⋆b = a∧b and a < b, then R = (a,x)⋆

′

(b,z) =

(a,x);

− if a⋆b = a∧b and b < a, then R = (b,y)⋆

′

(b,z) =

(b,yz).

3. Suppose that P(a, b) holds. The proof followssim-

ilarly with the case 2.

4. Suppose that P(a,b⋆ c) holds and none of P(a,b),

P(b,c) is valid. Then a = b⋆ c which implies a ≤

b,c. Moreover a < b, since a = b implies P(a,b).

From a < b, we obtain a = a⋆a < a⋆b, but a⋆b ≤ a,

thus a⋆ b = a. Similar we get that a = b⋆a = a⋆ c =

c⋆ a. Therefore we have:

R = (a,x) ⋆

′

(c,z) =



(a,xz), if a = c

(a,x), if a < c

If a = c, then L = (a,x) ⋆

′

((b,y) ⋆

′

(a,z)) = (a,x) ⋆

′

(a,z) = (a,xz). On the other hand, if a < c, then

a < b∧ c since a < b. As b⋆ c = a < b∧ c, we get L

= (a,x) ⋆

′

(b⋆ c,1) = (a, x).

5. Suppose that P(a ⋆ b,c) is valid and none of

P(a,b), P(b,c) is valid. Similarly with the previous

case, we can show that L = R.

Finally, let us prove that ⋆

′

is monotone, i.e.

(a,x) ≤

′

(b,y) implies (a,x) ⋆

′

(c,z) ≤

′

(b,y) ⋆

′

(c,z)

and (c, z) ⋆

′

(a,x) ≤ (c, z) ⋆

′

(b,y). We prove only the

ﬁrst implication, since the other has a similar proof.

We distinguish several cases:

1. Suppose that none of P(a,c), P(b,c) holds. We

have the following cases:

− a⋆c = a∧c and b⋆c= b∧c. We have that (a,x)⋆

′

(c,z) = min{(a,x),(c,z)} ≤

′

min{(b,y),(c, z)} =

(b,y) ⋆

′

(c,z).

− a ⋆ c = a∧ c and b⋆ c < b∧ c. Notice that a⋆ c ≤

b⋆ c implies (a,x) ⋆

′

(c,z) = min{(a,x),(c,z)} ≤

′

(b⋆ c,1) = (b, y) ⋆

′

(c,z).

− a⋆c < a∧c and b⋆c = b∧c. We have that a⋆c <

a ∧ c ≤ b ∧ c = b ⋆ c. Therefore (a, x) ⋆

′

(c,z) =

(a⋆c,1) and the ﬁrst component of (b,y)⋆

′

(c,z) is

b⋆ c = b∧ c > a⋆ c. Thus (a,x) ⋆

′

(c,z) ≤

′

(b,y) ⋆

′

(c,z).

− a ⋆ c < a∧ c and b⋆ c < b∧ c. Notice that a⋆ c ≤

b ⋆ c and (a, x) ⋆

′

(c,z) = (a ⋆ c,1) ≤

′

(b ⋆ c, 1) =

(b,y) ⋆

′

(c,z).

2. Suppose that P(a, c) holds. Then a ⋆ b = a, since

a ≤ b and a is idempotent. We have (a,x) ⋆

′

(c,z) =

(a,xz). Moreover,

(b,y) ⋆

′

(c,z) =



(a,yz), if a = b

(a,z), if a < b

If a = b, then x ≤ y and (a,xz) ≤

′

(a,yz) since the

usual product of reals is monotone. If a < b, then

(a,xz) ≤

′

(a,z) since xz ≤ z.

3. Suppose that P(b,c) holds. Moreover, suppose

that P(a,c) is not valid. Then b = c and a < b. Thus

(b,y) ⋆

′

(c,z) = (b,yz) and

(a,x) ⋆

′

(c,z) =



(a⋆ b,1), if a ⋆ b < a

(a,x), if a⋆ b = a

Since a⋆b≤ b, we get (a,x)⋆

′

(c,z) ≤

′

(b,y)⋆

′

(c,z).

Moreover, we can endow S

′

with two additional

operations such that it becomes a psMTL

-chain:

Lemma 3.3. The structure

′

= (S

′

,⋆

′

,→

′

,≤

′

,(0,1), (1,1))

is a psMTL

-chain, where

(a,x) →

′

(b,y) = max{(c,z) | (c,z)⋆

′

(a,x) ≤

′

(b,y)},

(a,x)

′

(b,y) = max{(c,z) | (a,x) ⋆

′

(c,z) ≤

′

(b,y)}.

Moreover, the mapping ψ : S → S

′

deﬁned by ψ(x) =

(x,1) is an embedding of psMTL-algebras.

Proof. By Lemma 3.2, it is enough to show that ⋆

′

has

a left and a right residuum. Therefore we must show

that each set of the form M

= {(c,z) | (c,z)⋆

′

(a,x) ≤

′

(b,y)} or M

= {(c, z) | (a,x) ⋆

′

(c,z) ≤

′

(b,y)} has

a maximum. Let us consider the case of M

. Since

S is inversely well ordered by Lemma 3.1, π

)

has a maximum c

, where π

is the projection on

the ﬁrst component. Thus there is an element of the

form (c

,z) ∈ M

. If c

⋆ a < b, then (c

,1) is the

maximum of M

. Thus suppose that c

⋆ a = b. We

distinguish several cases:

1. Suppose c

⋆ a < c

∧ a. Then, for any z, we

have (c

,z) ⋆

′

(a,x) = (c

⋆ a,1). Thus (c

,1)

must be the maximum of M

2. Suppose that P(c

,a) holds. Then we have

,z) ⋆

′

(a,x) = (c

,zx). If x ≤ y, then (c

,1)

is the maximum of M

. If x > y, then the maximum

of M

is (c

,y/x).

3. Suppose that c

⋆ a = c

∧ a. Moreover, let us

assume that P(c

,a) is not valid. Then (c

,z) ⋆

′

(a,x) = min{(c

,z),(a,x)}. The have several

cases:

· if a = c

, then min{(c

,z),(a,x)} = (c

,x∧z)

and the maximum of M

is either (c

,1) when

x ≤ y, or (c

,y) otherwise;

IJCCI2012-InternationalJointConferenceonComputationalIntelligence

354

· if a < c

, then min{(c

,z),(a,x)} = (a,x) and

,1) is the maximum of M

;

· if a > c

, then min{(c

,z),(a,x)} = (c

,z)

and (c

,y) is the maximum of M

It follows immediately from the deﬁnitions of →

′

and

′

that the following hold:

(a,x) ⋆

′

(b,y) ≤

′

(c,z) iff (a,x) ≤

′

(b,y) →

′

(c,z) iff

(b,y) ≤

′

(a,x) (c,z).

Finally, it can be easily veriﬁed that the mapping ψ

satisﬁes the following equalities:

ψ(a) ⋆

′

ψ(b) = (a,1) ⋆

′

(b,1) = (a⋆ b,1) = ψ(a ⋆ b),

ψ(a) →

′

ψ(b) = max{(c,z)|(c,z)⋆

′

(a,1) ≤

′

(b,1)} =

(a → b,1) = ψ(a → b),

ψ(a)

′

ψ(b) = max{(c,z)|(a,1)⋆

′

(c,z) ≤

′

(b,1)} =

(a b,1) = ψ(a b).

Therefore the proof is completed.

The remaining step is to show that S

′

is order-

isomorphic to [0,1]. As shown in (Hrbacek and Jech,

1999), any totally ordered set X is order-isomorphic

to [0,1] if it satisﬁes the following properties: X is

complete, X has a maximum and a minimum and X

has a countable subset D which is dense in X, i.e. for

each x,y ∈ X such that x < y, there is z ∈ D such that

x < z < y.

Thus it is enough to prove that S

′

satisﬁes all the

above mentioned conditions. Clearly, S

′

has a maxi-

mum and a minimum. Further, the subset

{(a,x) | a ∈ S, x ∈ Q ∩ [0,1]}

of S

′

is countable and dense in S

′

. Finally, given X ⊆

′

, X 6=

0, let Z = π

(X). Then Z ⊆ S and Z 6=

0, hence

Z has a maximum a

, since S is inversely well ordered

by Lemma 3.1. Now let

α = sup{x ∈ (0, 1] | (a

,x) ∈ X}.

Then (a

,α) = sup(X). In conclusion S

′

is complete

and we have the following result:

Lemma 3.4. The set S

′

is order-isomorphic to [0,1],

i.e. there is a bijection Φ : S

′

→ [0, 1] such that

(a,x) ≤

′

(b,y) implies Φ(a,x) ≤ Φ(b,y).

Let us deﬁne the following operations on [0,1]:

a⊙ b = Φ(Φ

−1

(a) ⋆

′

−1

(b)),

a →

⊙

b = Φ(Φ

−1

(a) →

′

−1

(b)),

⊙

b = Φ(Φ

−1

(a)

′

−1

(b)).

Then the structure

[0,1]

⊙

= ([0,1], ⊙,→

⊙

,≤,0,1)

is a standard psMTL-algebra and we have an [0,1]

⊙

model of T such that Φ(ψ(e

(ϕ))) < 1. Thus the

proof of Theorem 3.1 is ﬁnished.

4 CONCLUSIONS

In this paper we proved the ﬁnite strong standard

completeness theorem for the non-commutative logic

psMTL. For proving this result, we used a construc-

tion of standard psMTL-algebras which is interesting

on its own. Our proof can also be seen as an alterna-

tive proof for the standard completeness theorem for

psMTL logic given in (Jenei and Montagna, 2003).

As a future research, it would be interesting to

extend our proof for the ﬁnite strong standard com-

pleteness theorem for psMTL for some schematic ex-

tensions of psMTL logic. We mention that in (Dia-

conescu, 2012), the proof for the standard complete-

ness theorem for psMTL logic given in (Jenei and

Montagna, 2003) was extended for the extensions

psSMTL and psIMTL of psMTL logic.

REFERENCES

Cignoli, R., Esteva, F., Godo, L., and Torrens, A. (2000).

Basic fuzzy logic is the logic of continuous t-norms

and their residua. Soft Computing, 4:106–112.

Diaconescu, D. (2012). On the non-commutative proposi-

tional logic psmtl and its extensions. submitted.

Esteva, F. and Godo, L. (2001). Monoidal t-norm based

logic: Towards a logic for left-continuous t-norms.

Fuzzy Sets and Systems, 124(3):271–288.

Flondor, P., Georgescu, G., and Iorgulescu, A. (2001).

Pseudo-t-norms and pseudo-bl algebras. Soft Com-

puting, 5:355–371.

Girard, J.-Y. (1995). Linear logic: its syntax and seman-

tics. In Girard, J.-Y., Lafont, Y., and Regner, L., edi-

tors, Advances in Linear Logic,, volume 222 of Lon-

don Math. Soc. Lecture Notes, pages 1–42. Cambridge

Univ. Press.

H´ajek, P. (1998). Metamathematics of fuzzy logic. Trans.

in Logic, 4.

H´ajek, P. (2003a). Fuzzy logics with non-commutative con-

junctions. J. Logic. Comput., 13:469–479.

H´ajek, P. (2003b). Observations on non-commutative fuzzy

logic. Soft Computing, 8:38–43.

Horˇcik, R. (2007). Alternative proof of standard complete-

ness theorem for mtl. Soft Computing, 11(2):123–129.

Hrbacek, K. and Jech, T. (1999). Introduction to set the-

ory. Monographs and textbooks in pure and applied

mathematics. Dekker, New York.

Jenei, S. and Montagna, F. (2002). A proof of standard

completeness for esteva and godo’s logic mtl. Studia

Logica, 70(2):183–192.

Jenei, S. and Montagna, F. (2003). A proof of standard

completeness for non-commutative monoidal t-norm

logic. Neural Network World, 13:481–488.

K¨uhr, J. (2003). Pseudo-bl-algebras and drl-monoids. Math.

Bohem., 128:199–208.

Non-commutativeFuzzyLogicpsMTL-AnAlternativeProoffortheStandardCompletenessTheorem

355

Lambek, J. (1958). The mathematics of sentence structure.

Amer. Math. Montly, 65:154–169.

Mulvey, C. and Nawaz, M. (1995). Quantales: quantale

sets. In H¨ohle, U. and Klement, E., editors, Non-

Classical Logics and Their Applications to Fuzzy Sub-

sets, pages 219–234. Kluwer Acad. Publ., Dordrecht,

Boston, London.

Rosenthal, K. (1990). Quantales and their applications. In

Pitman Research Notes in Mathematics, volume 234.

Longman Scientiﬁc and Technical, Essex.

Schmerling, S. (1975). Asymmetric conjunction and rules

of conversation. In Cole, P. and Morgan, J., editors,

Speech Acts of Syntax and Semantics, volume 3, pages

211–231. Academic Press, New York.

Schweizer, B. and Sklar, A. (1960). Statistical metrics. Pac.

J. Math., 10:313–334.

IJCCI2012-InternationalJointConferenceonComputationalIntelligence

356