Next Article in Journal
A New Methodology for Improving Service Quality Measurement: Delphi-FUCOM-SERVQUAL Model
Next Article in Special Issue
The Logic of Pseudo-Uninorms and Their Residua
Previous Article in Journal
Optimal Allocation of Virtual Machines in Cloud Computing
 
 
Order Article Reprints
Font Type:
Arial Georgia Verdana
Font Size:
Aa Aa Aa
Line Spacing:
Column Width:
Background:
Article

Logics for Finite UL and IUL-Algebras Are Substructural Fuzzy Logics

Faculty of Science, Zhejiang Sci-Tech University, Hangzhou 310018, China
Symmetry 2018, 10(12), 755; https://doi.org/10.3390/sym10120755
Received: 12 November 2018 / Revised: 2 December 2018 / Accepted: 12 December 2018 / Published: 15 December 2018
(This article belongs to the Special Issue Mathematical Fuzzy Logic and Fuzzy Set Theory)

Abstract

:
Semilinear substructural logics UL ω and IUL ω are logics for finite UL and IUL -algebras, respectively. In this paper, the standard completeness of UL ω and IUL ω is proven by the method developed by Jenei, Montagna, Esteva, Gispert, Godo, and Wang. This shows that UL ω and IUL ω are substructural fuzzy logics.

1. Introduction

In [1], we constructed three semilinear substructural logics UL ω , IUL ω , and HpsUL ω * by adding one simple axiom:
( FIN ) ( A e ) ( A A e )
to Metcalfe and Montagna’s uninorm logic UL , involutive uninorm logic IUL [2], and a suitable extension HpsUL * [3] of Metcalfe, Olivetti, and Gabbay’s pseudo-uninorm logic HpsUL [4], respectively. Especially, we show that UL ω and IUL ω are complete with respect to finite UL and IUL -algebras, respectively. That is, they are logics for finite UL and IUL-algebras, respectively.
In this paper, we prove that UL ω and IUL ω are standard complete by Wang’s constructions in [5] and [6], which are some generalizations of the Jenei and Montagna-style approach for proving standard completeness for monoidal t-norm-based logic MTL [7] and the proof of the standard completeness for IMTL given by Esteva, Gispert, Godo, and Montagna in [8]. These constructions have been extended by Yang in [9,10,11,12].
Substructural logics are logics that lack some of the three basic structural rules of contraction, weakening, and exchange. For a survey, see [13]. Substructural fuzzy logics are substructural logics that are complete with respect to algebras whose lattice reduct is the real unit interval [ 0 , 1 ] , i.e., logics that are standard complete [2]. Our result in this paper thus shows that UL ω and IUL ω are substructural fuzzy logics.
As pointed out in [6], our construction in Lemma 5 also presents a method to construct uninorms and involutive uninorms. Then, the standard completeness for UL ω and IUL ω gives a characterization of uninorms and involutive uninorms and their residua constructed by finite UL and IUL -algebras from our constructions. That is, the identity ( Fin ) holds in all these standard UL and IUL -algebras; see Definition 5. These new classes of uninorms and involutive uninorms may be used in the theory of evaluation as the aggregation operators or combining functions [14,15].
We have proven that HpsUL * is standard complete in [16]. However, we are unable to prove whether HpsUL ω * is standard complete or complete with respect to finite HpsUL * -algebras and left them as open problems. In addition, we have proven that IUL is standard complete in [17], which is a longstanding open problem in the circle of fuzzy logic. Unfortunately, such a great work has not been accepted by our community since 2015, although one referee thought that the central ideas in our proof are reasonable and could find no significant flaws in the reasoning. The referee also said that he would also not be confident that the proof is correct if the proof were his own and he had spent many months laboring over it.

2. HpsUL ω * , UL ω , IUL ω and Algebras Involved

The Hilbert system HpsUL is the logic of bounded representable residuated lattices, which is based on a countable propositional language with formulas built inductively as usual from a set of propositional variables, binary connectives , , , , , and constants e , f , , , with definable connectives:
¬ φ : = φ f ,
φ ψ : = ( φ ψ ) ( ψ φ ) ,
λ χ ( φ ) : = ( χ φ χ ) e ,
ρ χ ( φ ) : = ( χ χ φ ) e .
Definition 1.
HpsUL consists of the following axioms and rules [4]:
( A 1 ) φ φ ( A 2 ) ( φ ψ ) ( ( χ φ ) ( χ ψ ) ) ( A 3 ) φ ( ( φ ψ ) ψ ) ( A 4 ) ( φ ( ψ χ ) ) ( ψ ( φ χ ) ) ( A 5 ) ψ ( φ φ ψ ) ( A 6 ) ( ψ ( φ χ ) ) ( φ ψ χ ) ( A 7 ) ( ψ ψ ( ψ φ ) ) ( ψ φ ) ( A 8 ) ( φ t ) ( ψ t ) φ ψ ( A 9 ) φ ψ ψ ( A 10 ) φ ψ φ ( A 11 ) ( χ φ ) ( χ ψ ) ( χ φ ψ ) ( A 12 ) φ φ ψ ( A 13 ) ψ φ ψ ( A 14 ) ( φ χ ) ( ψ χ ) ( φ ψ χ ) ( A 15 ) e ( A 16 ) φ ( e φ ) ( A 17 ) φ ( A 18 ) φ ( PRL ) ( λ χ ( φ ψ φ ) ) ( ρ χ ( φ ψ ψ ) ) ( MP ) φ , φ ψ ψ ( ADJ U ) φ φ e ( PN ) φ ψ φ ψ ( PN ) φ ψ ψ φ
Definition 2
([2,3]). A logic is a schematic extension (extension for short) of HpsUL if it results from HpsUL by adding axioms in the same language. In particular,
  • HpsUL * is HpsUL plus (WCM) ( φ e ) ( φ e ) ;
  • UL is HpsUL plus φ ψ ψ φ ;
  • IUL is UL plus ¬ ¬ φ φ .
Definition 3.
New extensions of HpsUL are defined as follows.
  • HpsUL ω * is HpsUL * plus ( FIN ) ( φ e ) ( φ φ e ) ;
  • UL ω and IUL ω are UL and IUL plus ( FIN ) , respectively.
Let L { HpsUL * , UL , IUL , HpsUL ω * , UL ω , IUL ω } in the remainder of this section. A proof in L of a formula φ from a set Γ of formulas is defined as usual. We write Γ L φ if such a proof exists.
Definition 4
([4]). An HpsUL -algebra is a bounded residuated lattice A = A , , , · , , , e , f , , with universe A, binary operations , , · , , , and constants e , f , , such that:
(i) 
A , , , , is a bounded lattice with top element and bottom element ⊥;
(ii) 
A , · , e is a monoid;
(iii) 
x , y , z A , x · y z iff x y z iff y x z ;
(iv) 
x , y , u , v A , ( λ u ( x y x ) ) ( ρ v ( x y y ) ) = e , where, for any a , b A , λ a ( b ) : = ( a b · a ) e , ρ a ( b ) : = ( a a · b ) e .
We use the convention that · binds stronger than other binary operations, and we shall often omit·; we will thus write x y instead of x · y , for example. Suitable classes of algebras of extensions of HpsUL are defined as follows.
Definition 5
([1,3,4]). Let A = A , , , · , , , e , f , , be an HpsUL -algebra. For L , an extension of HpsUL , A is an L -algebra if all axioms of L are valid in A . An L -chain is an L -algebra that is linearly ordered. In particular:
  • A is an HpsUL * -algebra if the weak commutativity (Wcm) holds: x y e iff y x e for all x , y A ;
  • A is a UL -algebra if x y = y x for all x , y A ;
  • A is an IUL -algebra if it is a UL -algebra such that ¬ ¬ x = x for all x A ;
  • A is an HpsUL ω * -algebra ( UL ω or IUL ω -algebra) if it is an HpsUL * -algebra ( UL or IUL -algebra) such that the following identity ( Fin ) holds: x e = x 2 e for all x A .
Definition 6
([4]). Let A = A , , , · , , , e , f , , be an L -algebra. (i) An A -valuation v is a homomorphism from the term algebra determined by formulas in L to A ; (ii) A formula φ is valid in A if v ( φ ) e holds for any A -valuation v; (iii) The relation of semantic consequence Γ A φ holds if each A -evaluation that validates all formulae in a theory Γ validates φ as well.
Theorem 1
([4,18]). Γ L φ iff Γ A φ for every L -chain A , i.e., L is a semilinear substructural logic.
Proposition 1.
Let A = A , , , · , , , e , f , , be an HpsUL * -algebra. Then, A is an HpsUL ω * -algebra if and only if x y e iff x y 2 e for all x , y A .
Proof. 
For the proof of the necessity part, see Lemma 2.4(i) of [1]. For the sufficiency part, assume that x y e iff x y 2 e for all x , y A . Suppose that x e > x 2 e . Then, x 2 ( x e ) > e by Definition 4 (iii), and hence, ( x e ) x 2 > e by (Wcm). Therefore, ( x e ) x > e by the assumption. Then, x e > x e , a contradiction, and thus, x e x 2 e . x 2 e x e is proven by a similar way. Hence, x e = x 2 e for all x A , i.e., A is an HpsUL ω * -algebra. □
Lemma 1.
Let A be an HpsUL ω * -chain and s , t , u A . Then:
(1) 
s t u = s implies s t = s and s u = s ;
(2) 
s t u = u implies s u = u and t u = u ;
(3) 
s t = e implies s = t = e .
Proof. 
Only (1) is proven as follows; for the others, see [1]. If t u e , then t u t e and u t u e by Proposition 1 and (Wcm). Thus, s t u t s and s t u t u s t . Hence, s t s and s s t . Therefore, s t = s . The case of t u > e is proven in the same way. □
Clearly, Lemma 1 holds for all UL ω and IUL ω -chains.

3. Wang’s Construction and Standard Completeness

In this section, let L ω { UL ω , IUL ω } , A = A , , , · , , , e , f , , be a finite or countable L ω -chain and s , t , u be arbitrary elements of A.
Definition 7
([3,5]). Let A be an UL ω -chain. For each s A , t is the immediate predecessor of s in A if: (i) t A , t < s ; (ii) u A , u < s implies u t . For each s A , let s denote the immediate predecessor of s in A if it exists, otherwise take s = s .
Let X = { ( s , 1 ) : s A } { ( s , q ) : s A , s > s , q Q ( 0 , 1 ) } ; we define:
( s , q ) ( t , r ) iff either s < S t , or s = t and q r and,
I 1 : = { ( s , t ) : s , t A , s t = s t , s > s t } I 2 : = { ( s , t ) : s , t A , s t = t s , t > s t } I 3 : = { ( s , t ) : s , t A , s t = t = s , s > s t } I 4 : = { ( s , t ) : s , t A , ( s t t and s t s ) or    ( s t = s t = s ) or ( s t = s t = t ) } .
Now define, for ( s , q ) , ( t , r ) X :
( s , q ) ( t , r ) = ( s , q ) ( s , t ) I 1 , ( t , r ) ( s , t ) I 2 , ( s , q ) X ( t , r ) ( s , t ) I 3 , ( s t , 1 ) ( s , t ) I 4 ,
where by X and X are meant min X and max X with respect to X , respectively. We will omit the index if it does not cause confusion.
Lemma 2.
Let A be an UL ω -chain. Then, ( s , q ) ( t , r ) ( e , 1 ) iff ( s , q ) ( t , r ) ( t , r ) ( e , 1 ) for all ( s , q ) , ( t , r ) in X.
Proof. 
Let ( s , q ) ( t , r ) ( e , 1 ) . Since ( s , q ) ( t , r ) = ( s t , ) for some { q , r , 1 } by Definition 7, then s t e . Thus, s t t e by (Fin). Hence, ( s , q ) ( t , r ) ( t , r ) ( e , 1 ) . The sufficiency part of the lemma is proven in the same way. □
Definition 8
([6,8]). Let A be an IUL ω -algebra. Let:
I * : = { ( s , t ) : s , t A , s < s , t < t , t = ¬ s } ,
I * * : = { ( s , t ) : s , t A , s s = s s = s = t } .
( s , q ) , ( t , r ) X , define:
( s , q ) ( t , r ) = ( s , q ) ( t , 1 ) ( s , 1 ) ( t , r ) i f ( s , t ) I * , q + r 1 , ( s , q r ) ( s , 1 ) i f ( s , t ) I * * , ( s , q ) ( t , r ) o t h e r w i s e .
Lemma 3.
Let A be an IUL ω -chain and s , t A . (i) If s t s , s t e , s t e , then s t t e ; (ii) if s t = s t and s t e , then s t t e ; (iii) ( s , q ) ( t , r ) ( s , q ) ( t , r ) .
Proof. 
(i) If s t e , then s t t e by Proposition 1, and thus, s t t s t t e . If t e , then s t t t e by s t e . Thus, let s t > e and t > e in the following.
t e by t > e . t e by s t s . Then, t > e . Thus, s t s . Hence, s t > s by s t s . s t e by Lemma 1(3) and t > e . Therefore, s t < e by s t e . Then, s t < e < t . Thus, s t < t . Hence, s < e .
Suppose that s t t . Then, s s t s t e . Thus, s t e by Proposition 1, a contradiction, and, hence s t > t . Therefore, s t t . s t t by s < e . Then, s t = t .
Suppose that s t s , then s t t s t > e . Thus, s t > e by Proposition 1, a contradiction, and hence, s t < s .
Therefore, s t s . s t s by t > e . Then, s t = s . Then, s s t = s by s t = t . Thus, s s = s by Lemma 1(1).
Suppose that s s = s , then s t = s s t s , a contradiction with s t > s , and hence, s s < s by s s s . Then, s s s .
Thus, s = s s s s s . Hence, s s = s . Then, ( s s ) t = s t = s and s ( s t ) = s t = t . Thus, s = t by ( s s ) t = s ( s t ) , a contradiction with s < e < t . Thus, the case of s t > e and t > e does not exist. This completes the proof of (i).
(ii) It follows from s t e that s t t e by Proposition 1. Then, s t t = s t t s t t e by s t = s t , and thus, s t t e .
(iii) See Proposition 3.7 (2) of [6]. □
Lemma 4.
Let A be a finite IUL ω -chain. Then, ( s , q ) ( t , r ) ( e , 1 ) if and only if ( s , q ) ( t , r ) ( t , r ) ( e , 1 ) for all ( s , q ) , ( t , r ) in X.
Proof. 
Let ( s , q ) ( t , r ) ( e , 1 ) . There are three cases to be considered.
Case 1. ( s , t ) I * and q + r 1 . Then, ( s , q ) ( t , r ) = ( s , q ) ( t , 1 ) ( s , 1 ) ( t , r ) ( e , 1 ) . Thus, s t e , s t e . Then, s t t e by Proposition 1. If ( s , q ) ( t , r ) = ( s , 1 ) ( t , r ) , then ( s , q ) ( t , r ) ( t , r ) = ( ( s , 1 ) ( t , r ) ) ( t , r ) ( ( s , 1 ) ( t , r ) ) ( t , r ) ( s t t , 1 ) ( e , 1 ) by Lemma 3(iii). Let ( s , q ) ( t , r ) = ( s , q ) ( t , 1 ) in the following. If ( s , q ) ( t , 1 ) = ( s , q ) , then ( s , q ) ( t , r ) ( t , r ) = ( s , q ) ( t , r ) ( e , 1 ) . Otherwise, s t s or s t = s t . Then, s t t e by Lemmas 3(i) and 3(ii). Thus, ( s , q ) ( t , r ) ( t , r ) = ( ( s , q ) ( t , 1 ) ) ( t , r ) ( ( s , q ) ( t , 1 ) ) ( t , r ) ( s t t , 1 ) ( e , 1 ) .
Case 2. ( s , t ) I * * , then s s = s s = s = t and ( s , q ) ( t , r ) = ( s , q r ) ( s , 1 ) ( e , 1 ) . Thus, s s e . Hence, s s s e by Proposition 1 and (Wcm). Therefore, ( s , q ) ( t , r ) ( t , r ) = ( ( s , q r ) ( s , 1 ) ) ( s , r ) ( ( s , q r ) ( s , 1 ) ) ( s , r ) ( s s s , 1 ) ( e , 1 ) .
Case 3. ( s , q ) ( t , r ) = ( s , q ) ( t , r ) ( e , 1 ) , then s t e . Thus, s t t e by Proposition 1. Hence, by Lemma 3(iii), ( s , q ) ( t , r ) ( t , r ) ( s , q ) ( t , r ) ( t , r ) ( s t t , 1 ) ( e , 1 ) .
By a similar procedure, we prove that ( s , q ) ( t , r ) ( e , 1 ) if ( s , q ) ( t , r ) ( t , r ) ( e , 1 ) . □
Lemma 5.
Let A be an HpsUL ω * -chain, X, and the binary operation ∘ on X be as in Definition 7. The following conditions hold:
(a) 
X is densely ordered and has a maximum X = ( , 1 ) and a minimum X = ( , 1 ) .
(b) 
X , , X , e X is a linearly-ordered monoid, where e X = ( e , 1 ) .
(c) 
∘ is left-continuous with respect to the order topology on X , X .
(d) 
There is a map Φ from A into X such that Φ is an embedding of the structure A , , , · , e , , into X , X , X , , e X , X , X , and for all s , t A , Φ ( s t ) is the residuum of Φ ( s ) and Φ ( t ) in X , X , X , , e X , X , X , respectively.
(e) 
( s , q ) , ( t , r ) X , ( s , q ) ( t , r ) ( e , 1 ) iff ( s , q ) ( t , r ) ( t , r ) ( e , 1 ) .
Proof. 
Claim (e) has been proven by Lemma 2. As pointed out in [3], the associativity of ∘ is mainly dependent on Lemma 1(1)∼(2). Other claims are proven in the same way as that of Theorem 4.5 in [3]. □
Lemma 6.
Every countable UL ω * -chain can be embedded into a standard UL ω * -algebra.
Proof. 
Let X , A , etc., be as in Definition 7. We can assume, without loss of generality, that X = Q [ 0 , 1 ] . Now, define for α , β [ 0 , 1 ] , α * β = sup { x y : x , y X , x α , y β } . The proof of the weak commutativity, the monotonicity, associativity, left-continuity, etc., of * is the same as that of Theorem 4.6 in [3]. The neutral element of * is e X in Q [ 0 , 1 ] . By the left-continuity of *, the following property holds.
(P) α , β , γ [ 0 , 1 ] , α * β * γ = sup { x y z : x , y , z X , x α , y β , z γ } .
We prove that α * β e X iff α * β * β e X for any α , β in [ 0 , 1 ] . Given α * β e X , then x y e X for all x , y X , x α , y β . Let x , y , z X , x α , y β , z β . Then, x y e X , x z e X . Thus, x y y e X , x z z e X by Lemma 5(e). Hence, x y z max { x y y , x z z } e X . Therefore, α * β * β e X by (P). The sufficient part of the claim is proven in a similar way. □
By Lemma 1, Definition 8, Lemma 4, we can prove the claims similar to Lemma 5 and 6 for IUL ω -algebras. As a consequence of these lemmas, and extending Theorem 3.3 of [7] in the obvious way, we obtain the following standard completeness.
Theorem 2.
UL ω and IUL ω are complete with respect to the class of standard algebras involved.

4. Concluding Remarks

Roughly speaking, the methodological significance of Jenei and Montagna’s proof is that it does not require a complete understanding of the structure of the MTL -algebras by embedding a countable MTL -algebra into a dense one. It is indeed different from the proof of the BL ’s standard completeness given by Hajek, Cignoli, Esteva, Godo, Torrens et al. in [19,20]. The validation of the structure X in Definitions 7, 8 and Lemmas 5, 6 is dependent on Lemma 1(1), which claims that s t u = s implies s t = s . However, we are unable to prove the condition that s t u = t implies s t = t in HpsUL ω * . It seems that we need to introduce some stronger axioms into HpsUL ω * to guarantee its completeness with respect to finite (or standard) HpsUL * -algebras.

Funding

This research was funded by the National Foundation of Natural Sciences of China (Grant Nos. 61379018, 61662044, 11571013 and 11471286).

Conflicts of Interest

The author declares no conflict of interest.

References

  1. Wang, S.M. Semilinear substructural logics with the finite embeddability property. Algebra Univ. 2018. [Google Scholar] [CrossRef]
  2. Metcalfe, G.; Montagna, F. Substructural fuzzy logics. J. Symb. Log. 2007, 7, 834–864. [Google Scholar] [CrossRef]
  3. Wang, S.M. Logics for residuated pseudo-uninorms and their residua. Fuzzy Sets Syst. 2013, 218, 24–31. [Google Scholar] [CrossRef]
  4. Metcalfe, G.; Olivetti, N.; Gabbay, D. Proof Theory for Fuzzy Logics; Springer Series in Applied Logic; Springer: New York, NY, USA, 2009; Volume 36, ISBN 9781402094095. [Google Scholar]
  5. Wang, S.M. Uninorm logic with the n-potency axiom. Fuzzy Sets Syst. 2012, 205, 116–126. [Google Scholar] [CrossRef]
  6. Wang, S.M. Involutive uninorm logic with the n-potency axiom. Fuzzy Sets Syst. 2013. [Google Scholar] [CrossRef]
  7. Jenei, S.; Montagna, F. A proof of standard completeness for Esteva and Godo’s logic MTL. Stud. Logica 2002, 70, 183–192. [Google Scholar] [CrossRef]
  8. Esteva, F.; Gispert, J.; Godo, L.; Montagna, F. On the standard and rational completeness of some axiomatic extensions of the monoidal t-norm logic. Stud. Logica 2002, 71, 199–226. [Google Scholar] [CrossRef]
  9. Yang, E. Weakening-free, non-associative fuzzy logics: Micanorm-based logics. Fuzzy Sets Syst. 2015, 276, 43–58. [Google Scholar] [CrossRef]
  10. Yang, E. Basic substructural core fuzzy logics and their extensions: Mianorm-based logics. Fuzzy Sets Syst. 2016. [Google Scholar] [CrossRef]
  11. Yang, E. Involutive basic substructural core fuzzy logics: Involutive mianorm-based logics. Fuzzy Sets Syst. 2017. [Google Scholar] [CrossRef]
  12. Yang, E. Substructural fuzzy-relevance logic. Notre Dame J. Formal Log. 2015, 56, 471–491. [Google Scholar] [CrossRef]
  13. Nikolaos, G.; Jipsen, P.; Kowalski, T.; Ono, H. Residuated Lattices: An Algebraic Glimpse at Substructural Logics; Elsevier: Amsterdam, The Netherlands, 2007; ISBN 978-0-444-52141-5. [Google Scholar]
  14. Dombi, J. Basic concepts for the theory of evaluation: the aggregative operator. Eur. J. Oper. Res. 1982, 10, 282–293. [Google Scholar] [CrossRef]
  15. Yager, R.; Rybalov, A. Uninorm aggregationoperators. Fuzzy Sets Syst. 1996, 80, 111–120. [Google Scholar] [CrossRef]
  16. Wang, S.M. The logic of pseudo-uninorms and their residua. arXiv, 2017; arXiv:1707.05441. [Google Scholar]
  17. Wang, S.M. Density Elimination for Semilinear Substructural Logics. arXiv, 2015; arXiv:1509.03472. [Google Scholar]
  18. Cintula, P.; Noguera, C. Implicational (semilinear) logics I: A new hierarchy. Arch. Math. Log. 2010, 49, 417–446. [Google Scholar] [CrossRef]
  19. Cignoli, R.; Esteva, F.; Godo, L.; Torrens, A. Basic fuzzy logic is the logic of continuous t-norms and their residua. Soft Comput. 2000, 4, 106–112. [Google Scholar] [CrossRef]
  20. Hájek, P. Basic fuzzy logic and BL-algebras. Soft Comput. 1998, 2, 124–128. [Google Scholar] [CrossRef][Green Version]

Share and Cite

MDPI and ACS Style

Wang, S. Logics for Finite UL and IUL-Algebras Are Substructural Fuzzy Logics. Symmetry 2018, 10, 755. https://doi.org/10.3390/sym10120755

AMA Style

Wang S. Logics for Finite UL and IUL-Algebras Are Substructural Fuzzy Logics. Symmetry. 2018; 10(12):755. https://doi.org/10.3390/sym10120755

Chicago/Turabian Style

Wang, Sanmin. 2018. "Logics for Finite UL and IUL-Algebras Are Substructural Fuzzy Logics" Symmetry 10, no. 12: 755. https://doi.org/10.3390/sym10120755

Note that from the first issue of 2016, MDPI journals use article numbers instead of page numbers. See further details here.

Article Metrics

Back to TopTop