Table of Contents Author Guidelines Submit a Manuscript
Applied Computational Intelligence and Soft Computing
Volume 2010, Article ID 907298, 7 pages
http://dx.doi.org/10.1155/2010/907298
Research Article

Consistency Degrees of Theories in Łukasiewicz Fuzzy and -Valued Propositional Logic Systems

Department of Mathematics, Quanzhou Normal University, Quanzhou, Fujian 362000, China

Received 7 October 2009; Revised 28 January 2010; Accepted 31 May 2010

Academic Editor: Etienne Kerre

Copyright © 2010 Jiancheng Zhang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

By means of theory of truth degrees of formulas, according to deduction theorems and completeness theorems, the new concepts of consistency degrees and polar index for general theories in Łukasewicz fuzzy and -valued propositional logic systems are introduced. Moreover, sufficient and necessary conditions for a theory to be consistent, inconsistent, and fully divergent are obtained. Finally, some important properties of truth degrees of formulas are proposed.

1. Introduction

Whether a theory (i.e., a set of formulas) is consistent or not (a theory is inconsistent if , otherwise it is consistent, where is a contradiction [1]) is one of the crucial questions in any logic system. The reason is that, in classical logic, a contradictory theory (i.e., a theory which is not consistent) turns into a useless theory in which everything is provable. Quite surprising is that the same result holds also in fuzzy and many-valued logic systems (it was proved in [1] that is inconsistent if and only if for each ). It is clear that theories containing a contradiction are inconsistent, but inconsistent theories need not contain a contradiction. For example, let , where is any atomic formula of , since is a theorem, then is an inconsistent theory not containing any contradiction. In [4], a systematic and exhaustive treatment of fuzzy logic () is proposed where both semantics and syntax are evaluated gradationally and even axioms there need not be fully true. Hence to propose a graded theory for expounding consistent degrees of fuzzy theories seems not easy to be accomplished as it is pointed out in [4] that “even the attempt to introduce some kind of degrees of consistency mostly does not work.” The concept of degree of inconsistency of a fuzzy theory has been introduced in [4] as follows: where is an inference sign (see [5]). It was proven that a fuzzy theory is consistent if and only if Incons. As was pointed out in [2], the truth valuation of a formula need not be zero in multivalued logic even though it looks like a contradiction. Hence, it seems that to define Incons by means of is not as good as to analyse the disparities of formulas of -conclusions.

How to measure the extent to which a theory is consistent is also one of the crucial questions in logic systems. For trying to grade the extent of consistency of different theories, many authors have proposed different methods in fuzzy (continuous-valued) logic systems and have obtained many good results [2, 3]. The concept of consistency degree of a fuzzy theory has been introduced in [2, 3] as follows: where , , there exists a deduction of A from of length , , , is the smallest integer at least as large as (for any real number ), and and were defined by Definitions 5 and 7.

The consistency degrees formula of theories of fuzzy propositional logic systems is obtained only in the condition of finite theories ( is a finite set of formulas) in [2]; it was pointed out in the conclusion of [2] that “How to define a reasonable concept of consistency degrees for infinite theories of Luk and how to extend the result obtained in the present paper to the more general fuzzy logic with graded syntax given in [5] would be attractive research topics.” In the condition of infinite theories, the consistency degrees formula is obtained by the definitions of divergence degree in the references [2, 3]. But the character of divergence degrees is different from the character of consistency degree. It is natural for us to find another measure to describe consistency degree. In this paper, from logical point of view and based on deduction theorems, completeness theorems, and the concept of truth degrees of formulas, we introduce a more natural and reasonable definition of consistency degrees of theories in -valued and fuzzy propositional logic systems.

2. Preliminaries

2.1. Logic Systems: and

Suppose that is a countable set, let and be unary and binary operations, respectively, and let be the free algebra of type generated by S, that is, is the smallest set containing and closed under the operation and . Elements of are called propositions or formulas and those of are called atomic propositions or atomic formulas. In fuzzy propositional logic (briefly, Luk), there are four axiom schemes as follows: , ), , . The axioms of -valued propositional logic (briefly, ) are those of Luk plus for any such that is not a divisor of . Where instead of , is (see [6]).

Here are two kinds of valuation sets: and . Define on and a unary operator and a binary operator as follows: . These two sets of valuation are just the standard -chain and the standard MV-algebra respectively.

A homomorphism (or ) of type from into the valuation set (or ), that is, , is called a valuation of . The set of all valuations will be denoted by .

propositional logic system is called the -valued propositional logic system if the set of valuation is . propositional logic system is called the Fuzzy propositional logic system if the set of valuation is . It has been proved that the semantics and syntax of and are in perfect harmony, that is, the standard completeness theorem holds in and respectively.

The deduction rule is Modus Ponens (briefly, MP) of which can be deduced from and . As it is well known, the following abbreviations introduced in and are popular [1]:

2.2. Truth Degrees of Formulas in

Definition 1 (see [7]). Suppose that are probability measure spaces; let then generates on a -algebra . There exists on a measure satisfying the following conditions.
is the set consisting of all -measurable sets.
For any measurable subset of is -measurable and is called the infinite product measure of and is called the infinite product of . can be abbreviated as if no confusion arises.

Suppose that is a fixed natural number, , and is an evenly distributed probability measure space, where that is, and . Let , and let be the infinite product of .

In the following, will be written as ; hence we have and can also be written as .

Let , then is decided by its restriction because generates the free algebra . Assume that , then an infinite-dimensional vector in is obtained. Conversely, let be any element of , then there exists a unique such that . Hence there exists a bijection defined by .

Definition 2 (see [8]). The abovementioned mapping is called the measured mapping of .

Definition 3 (see [8]). Suppose that , and define then is called the -valued truth degree of

Example 1. Let . Consider the truth degree . Notice that Let and then If then by there are a total of pairs such that If then by ; there are a total of pairs such that Thus,
Assume that is a formula generated by atomic formulas through connectives and . Substitute for in and keep the logic connectives in unchanged but explain them as the corresponding operators defined on the valuation lattice Then we get a function and call the truth degree function of .

Lemma 1 (see [8]). Let Then where if and only if and Obviously, and

2.3. Truth Degrees of Formulas in

Let be a formula obtained by connecting the atomic formulas with the connectives and , then determines a function , where is obtained by connecting the variables in with the operators and on in the same way as is constructed from . For example, if then . It is clear that and in general [9, 10], where is a McNaughton function [11].

Definition 4 (see [9, 10]). Let be a formula of and Then is called the truth degree of

Remark 1. It is easy to verify that if and only if is a (logical) theorem, and if and only if is a contradiction (see [9]).
Notice that can also be considered as a function of variables, where Hence if we are given two formulas and , we can choose an large enough such that both and can be written as and respectively.

2.4. Pseudometric of F(S) in and

Definition 5. A subset of is called a theory.
Let be a theory; . A deduction of from , in symbols, , is a finite sequence of formulas such that, for each is an axiom, or , or there are such that follows from and by MP. Equivalently, we say that is a conclusion of (or -conclusion). The set of all conclusions of is denoted by . By a proof of we will henceforth mean a deduction of from the empty set. We will also write in place of and call a theorem.

Definition 6 (see [9, 10]). Suppose that , then is called the resemblance degree between and .

Definition 7 (see [9, 10]). Suppose that , and define Then is a pseudometric on and is called the logic metric space.

Definition 8 (see [9, 10]). Suppose that is a theory; define Then is called the divergence degree of .   is said to be fully divergent if

2.5. Generalized Deduction Theorems

Deduction theorem is one of the most important theorems in classical mathematical logic, and it says that

The deduction theorem is no longer valid in fuzzy logic systems in general. Fortunately, it has been proved that there exist different weak forms of the deduction theorems (called Generalized deduction theorems) in different logic systems, respectively [1, 12].

Theorem 1. Suppose that is a theory, and , then
in if and only if
in , if and only if .

Remark 2. It is easy to verify that and are provably equivalent; hence by the definition of deduction, the generalized deduction theorem in and can be equivalently described as follows

3. Main Results

It is easy for the reader to check the following lemma.

Lemma 2. Suppose that and , then if and only if is a tautology in .

Lemma 3. In and Luk, if is a logical theorem, then .

Lemma 4. Let in and Luk, then the following properties hold.
If is a theorem, then .
one has  .

Now we are ready to define the concept of consistency degree for finite and infinite theories. Let us first take an analysis on the inconsistency of a theory in and Suppose that is a theory and is inconsistent, then the contradiction is a conclusion of that is to say, holds. It follows from Remark 2 that there exits a finite string of formulas and such that holds. By standard completeness theorem, is a tautology; then in and respectively. Conversely, if there are a finite sequence of formulas and such that , by Lemma 2 and Remark 1, is a theorem; hence by the generalized deduction theorem, holds, and so is inconsistent. From the above analysis, in order to judge whether a given theory is inconsistent or not, it suffices to calculate of all possible formulas of the form where and . Moreover, the larger of such formulas is, the more closer that is to be inconsistent (see Theorem 6). Note that there may be an infinite number of such formulas for a given theory even though is finite; hence it is natural and reasonable for us using the supremum of of all formulas with the form where and to measure the inconsistency of .

Definition 9. Suppose that is a theory, and is a contradiction. Define where

Then is called the first polar index of and is called the second polar index of in and respectively.

Definition 10. Suppose that is a theory and define where Consist is called the consistency degree of in and respectively.

Remark 3. Since, for all and are logically equivalent, the definition of does not depend on the order of 's.
It is easy to verify that, for all hence if then .

Theorem 2. Suppose that is a theory, then, in ,

Proof. For every , if , then it is easy to check that . If , then holds by axioms. Since , thus, for every , holds; then by Lemma 3, . For the same reason, for any and any , we have and . Then .

Theorem 3. Suppose that , then,
in .  
In .

Proof. By Definition 9, Theorem 2, and Lemma 4, it is easy to check the theorem above. The proof is left to the reader.

Theorem 4. Suppose that is a theory, and is a contradiction. Then

Proof. For all since thus then . Conversely, for all , it follows from Remark 2 that and such that , by axioms; thus by MP. It follows from Lemma 3 that ; then . This completes the proof.

Theorem 5. Suppose that is a theory. Then the following are given.
(i) is consistent in and if and only if in and respectively.
(ii) is inconsistent in and if and only if in and respectively.

Proof. Since is consistent or inconsistent, and , it suffices to prove
Assume that is consistent, that is, For any given and , then it follows from generalized deduction theorems that is not a theorem; then, by Lemma 2 and Remark 1, ; thus . Conversely, if , then, for any and , it follows from Definition 9 that ; hence is not a theorem by Lemma 2 and Remark 1. Thus by generalized deduction theorems; then, is consistent.

Theorem 6. Suppose that is a theory, then, in and   
is consistent if and only if ,
is inconsistent if and only if ,
   is consistent and full divergent if and only if .

Proof. In by Theorem 5, is consistent if and only if ; hence holds.
If is inconsistent, then by Theorem 5, thus and Consist. Conversely, if Consist, then ; hence and is inconsistent by Theorem 5.
If is consistent and full divergent, then and by Theorem 5. For all , since and , it follows from generalized deduction theorems that there exist and , such that and . Since and holds, hence by hypothetical syllogism. By axioms, , so by Lemma 3 we have ; thus and ; then Consist. Conversely, if Consist, then ; thus and . For any , by axioms. If is a theorem, since , where , thus by Definitions 9 and 8, then , so is consistent and full divergent.
The proof of is analogous to that of and so it is omitted.

Definition 11. Suppose that is a theory, and define . It can be verified that and .

Theorem 7. In and if is consistent, or is inconsistent but , then if and only if .

Proof. Suppose that . First of all, we prove that , if is consistent, and by Theorem 5; hence and , so , and then . If is inconsistent but , it follows from Theorems 5 and 6 that and , hence and ; it follows from Theorem 6 that is consistent and ; thus we have , and then . Secondly, we prove that . Hence there exists such that . By definition, there exist and such that and, for any and , ; thus , where and . Thus then holds.
Conversely, suppose that ; since there exists such that , it follows from Definition 11 that , and for all such that , thus , and . Then . This completes the proof.

Remark 4. From Theorem 7, the definition of consistency degree is reasonable. If the consistency degree is higher, the distance between every conclusion of and contradiction should be longer.

Example 2. In , calculate and Consist for , and .

Solution
Since and hence . By Theorem 3, , and
By [1], ; then it follows from Lemma 3 that . Let . Then . Hence if and only if , and . Since there are a total of -tuples and among them only imply that , hence ; thus ,. We have . It is easy to check that ; then and .

Example 3. In Luk, calculate and Consist for , and .

Solution
Since , then , and Consist
Since by [1], then . Let and . Lebesgue measure . In , we have Thus Thus . It is easy to check that ; then and .

Remark 5. We compared the consistency degree concepts in [2, 3] with the definitions of this paper. From [2, 3], we see that the concept of consistency degree and of a theory based on the concept of the divergence degree, moreover, though [2] has given the concept of consistency degree based on the condition of finite theories while the concept of consistency degree of is clearly is based on the truth degrees of formulas and deduction theorems. From the logical point of view, the definition of and is more natural and reasonable. It is a more theoretical idea to understand while comparing to the facilitating calculation in this paper. Therefore, it is important and essential to give out the definition and characterization of the consistency degree in this paper.

4. Concluding Remarks

Based on deduction theorems, completeness theorems, and by means of the theory of truth degrees of formulas, the present paper defines a new polar index of a theory and a new concept of consistency degrees of theories in fuzzy and -valued propositional logic system, which is more natural and reasonable from the logical point of view. Moreover, some important properties of truth degree of formula are proposed. Finally, sufficient and necessary conditions for a theory to be consistent, inconsistent, and fully divergent are obtained.

Acknowledgment

This project was supported by the Natural Science Foundation of Fujian Province of China (no. 2006J0221).

References

  1. P. Hájek, Metamathematics of Fuzzy Logic, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1998.
  2. G.-J. Wang and W.-X. Zhang, “Consistency degrees of finite theories in Łukasiewicz prepositional fuzzy logic,” Fuzzy Sets and Systems, vol. 149, no. 2, pp. 275–284, 2005. View at Publisher · View at Google Scholar · View at Scopus
  3. X.-N. Zhou and G.-J. Wang, “Consistency degrees of theories in some systems of propositional fuzzy logic,” Fuzzy Sets and Systems, vol. 152, no. 2, pp. 321–331, 2005. View at Publisher · View at Google Scholar · View at Scopus
  4. S. Gottwald and V. Novák, “On the consistency of fuzzy theories,” in Proceedings of the 7th International Fuzzy Systems Association World Congress, pp. 168–171, Academi, Prague, Czech Republic, 1997.
  5. V. Novák, I. Perfilieva, and J. Močkoř, Mathematical Principles of Fuzzy Logic, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1999.
  6. R. Tuziak, “An axiomatization of the finite-valued Łukasiewicz calculus,” Studia Logica, vol. 47, no. 1, pp. 49–55, 1988. View at Publisher · View at Google Scholar · View at Scopus
  7. P.R. Halmos, Measure Theory, Springer, New York, NY, USA, 1974.
  8. G. J. Wang, “Theory of truth degrees of formulas in Łukasiewicz n-valued propositional logic and a limit theovem,” Science China Information Sciences E, vol. 35, no. 6, pp. 561–569, 2005 (Chinese). View at Google Scholar
  9. G.-J. Wang and Y. Leung, “Integrated semantics and logic metric spaces,” Fuzzy Sets and Systems, vol. 136, no. 1, pp. 71–91, 2003. View at Publisher · View at Google Scholar · View at Scopus
  10. G. J. Wang, Theory of Non-Classical Mathematical Logic and Approximate Reasoning, Science in China Press, Beijing, China, 2000.
  11. R. Cignoli, I. M. L. D'ottaviano, and D. Mundici, Algebraic Foundations of Many-Valued Reasouing, Kluwer Academic Publishers, Dordrecht, The Netherlands, 2000.
  12. S. Gottwald, A Treatise on Many-Valued Logics, Studies in Logic and Computation, Research Studies Press, Baldock, UK, 2001.