Fuzzy logic and self-referential reasoning: a ... - Semantic Scholar
Artif Intell Rev DOI 10.1007/s10462-011-9311-1
Fuzzy logic and self-referential reasoning: a comparative study with some new concepts Mohammad Reza Rajati · Hamid Khaloozadeh · Witold Pedrycz
© Springer Science+Business Media B.V. 2012
Abstract In this paper, we consider metacognition and self-reference realized in the framework of fuzzy computing. Different aspects of self-reference are discussed from the viewpoint of logic, linguistics, and artificial intelligence. Previous work in the area of fuzzy logic is also revisited in this context. It is shown that the reasoning based on the truth qualification principle requires some modifications so that it naturally gives rise to type-2 fuzzy sets, in order to capture the uncertainty about selection of a solution for a set of self-referential statements. Keywords Self-reference · Metacognition · Truth · Fuzzy logic · Liar’s paradox · Type-2 fuzzy sets · Veristic fuzzy sets
1 Introduction Self-reference and metacognition have been ongoing topics of interest, cf. (Bolander et al. 2006; Cox 2005; Crivelli 2004; Givant and McKenzie 1986; Seuren 2005; Smilek et al. 2007; Zadeh 1979). Logicians, mathematicians, psychologists, cognitive scientists, linguists, and philosophers have paid attention to this problem. Besides this long-term historical research
M. R. Rajati (B) Department of Electrical Engineering, University of Southern California, Los Angeles, CA, USA e-mail: [email protected] M. R. Rajati · H. Khaloozadeh Department of Electrical Engineering, K. N. Toosi University of Technology, Tehran, Iran H. Khaloozadeh e-mail: [email protected] W. Pedrycz Department of Electrical and Computer Engineering, University of Alberta, Edmonton, AB, Canada e-mail: [email protected] W. Pedrycz Systems Research Institute, Polish Academy of Sciences, Warsaw, Poland
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trail, the artificial intelligence community has focused on this issue since the inception of the area. Fuzzy logic, as a radically different approach to reasoning, has also contributed to provision of some understanding about self-referential reasoning. In this paper, we study the previous research on self-referential reasoning via fuzzy logic most of which is inspired by the original work of Zadeh, and present a modification to the reasoning procedure realized via truth qualification principle, which naturally gives rise to type-2 fuzzy sets. A thorough analytical study of the previously done works demonstrates that they cannot assign truth values to multiple self-referential statements in many cases. Some of the approaches also suffer from difficulty in reasoning about a large number of self-referential statements. It is worth noting that in this paper, we consider sets of sentences which only state some facts about the truth and falsehood of themselves or each other. The most well-known instance of such self-referential statements is the liar paradox, which indicates the falsehood of itself, and as we see later, has been the motivation of extensive research in many areas due to its paradoxical nature: p: This sentence is false. An example of a set of abovementioned self-referential statements would be: p1 := p2 is fairly false and ( p3 is very false or p3 is true) p2 := It is very true that ( p1 is fairly true if ( p3 is fairly false and p4 is fairly true)) p3 := ( p2 is very true if and only if p3 is very false) and p4 is fairly true p4 := It is very very true that ( p2 is very false if and only if ( p1 is true or p2 is very false or p3 is fairly true)) Although most of the works claim to accomplish the reasoning about the liar, they have difficulty even in the representation of multiple self-referential statements, and hence, fail to provide analysis tools to be generalized so that all self-referential statements containing utterances about the truth or falsehood of each other could be analyzed. Furthermore, they are not able to decide in many cases when the reasoning process yields more than one truth value for each statement. In contrast, the approach introduced by this paper first represents the set of multiple self-referential statements via the simple principles of fuzzy logic, and more specifically, translates them into multivariable mappings, and then shows that the possible truth values for each statement can be derived by finding the fixed points of such mappings. When there is more than a single truth value for each sentence, the reasoning process assigns their aggregation as type-2 truth values to each statement. We show that although the assignment of type-2 fuzzy sets removes some information about the original set of statements, it becomes necessary to complete the reasoning process. Actually, without such a modification to the truth qualification of self-referential statements, fuzzy logic is unable to decide about the ill-posed problem of self-referential reasoning. In order to provide the reader with an overall picture of the previous work on self-referential reasoning, we review some of the research in the areas of artificial intelligence, logic, mathematics, and philosophy, which we believe may contribute to a better comprehension of the works completed in the fuzzy logic community, and exhibit a potential to motivate further research on self-reference in the fuzzy logic community. The study is organized in the following manner. In Sect. 2, an overview of self-referential reasoning in artificial intelligence, logic, mathematics, and philosophy is presented. In Sect. 3, some comments are made about the research on self-reference in the area of fuzzy logic. In Sect. 4, we review some generic ideas of fuzzy logic that are sought as a prerequisite for further investigations presented in the paper. In Sect. 5, the ideas are related to the recently proposed general framework of fuzzy computation. In Sect. 6, some existing ideas on representation of self-reference via dynamic semantic maps are revisited. In Sect. 7, implicative
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representation of self-reference is considered. An attempt for the unification of the ideas is offered in Sects. 6 and 7. In Sect. 8, the computational verb approach to the evaluation of paradoxes is discussed. In Sect. 9, an enhanced method for truth qualification of self-referential statements is proposed, based on the conventions of fuzzy logic. In Sect. 10, some additional comments are made about the material presented in Sect. 9. Finally, some conclusions and future directions are discussed in Sect. 11.
2 Self-reference in artificial intelligence, logic, mathematics, and philosophy As we noted earlier, self-referential reasoning has attracted a considerable attention in artificial intelligence research. From the viewpoint of artificial intelligence, self-reference is a consequence of metacognitive reasoning (Cox 2005). An agent, who is able to make decisions for its future behavior, should be capable of realizing metacognition. It should decide which of different actions to perform under the current situation. It also must reason whether it has analyzed the decision adequately or more information and further analysis becomes necessary. After performing an action according to its deductions, it must ‘think’ about the outputs of its actions, and the strengths and weaknesses of its method of reasoning. Such reasoning procedures are called metacognition, since they involve ‘cognition about cognition’. Henceforth, a smart agent encounters reasoning about reasoning, before, during and after each of its actions (Cox 2005). In the sequel many subfields of artificial intelligence are involved in the research on metacognition. Model-based and case-based reasoning concern learning and reasoning about reasoning failure. Metacognition is also studied by the researchers in the area of decision theory and planning to enhance the choice of reasoning actions. Expert systems field contributes the facets of metaknowledge and rule control. Logical foundations are also provided for changing the beliefs, belief representation, and introspection (Bolander 2002). Metacognition involves self-reference (but as we discuss in this section, metacognition is more general), and when the truth of the beliefs of an agent is considered by it (or other agents), it may bring about logical inconsistency. Minsky (1961, 1965, 1968) believes that in order for a machine to describe its environment completely and describe its situation in its environment, it would have to possess an executable model of itself. From Minsky’s viewpoint, the agent should have a model of the environment E, and a model of itself M, in the environment. The model of the environment is denoted by E . It is used to process and answer inquiries about the environment. To answer inquiries about itself in the environment, the agent should have within E , a model of itself, denoted by M . Minsky notes that the agent must model its model of the environment as E in order to answer questions about its knowledge of its environment. M is agent’s knowledge of its knowledge and thinking methodology. The hierarchical introduction of higher order models could be continued, however it becomes heuristically less meaningful. An elaborate version of this model of metacognition could be found in the book Society of Mind (Minsky 1985). The model of Minsky for metacognitive agents could be generalized for multiagent systems. Agent i has a model of the environment, Ei , a model of itself in the environment, Mi , a model of any other agent, say j, Mij , a meta-model of the environment, Ei , and a meta-model of the environment perceived by agent j, Eij . Therefore, metacognition could be considered as more general than self-reference. Self-reference occurs when the agent reasons about itself, or reasons about other agents which directly or indirectly reason about it. By indirect self-reference, it is meant that agent r may reason about agent s, agent s may
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reason about agent t, and this process may continue until it reaches the situation in which an agent z is found which reasons about the first agent r . Therefore, agent r may be included in an ‘indirect’ self-referential reasoning procedure. In a multiagent system, when metacognition is about the truth of the cognitive processes of another agent (or the agent itself), it could result in logical inconsistency. McCarthy (1968) asserts that for an agent to adequately behave smartly, it must declaratively represent its knowledge. He furthermore emphasizes on agent introspection (McCarthy 1979), and having beliefs about mental states rather than about the environment. Accordingly, researchers working on the logical aspects of artificial intelligence have focused on reasoning about knowledge, belief, and internal states (Konolige 1985, 1988; Moore 1985, 1995). They have also tried to avoid the occurrence of logical inconsistency (Attardi and Simi 1991; Perlis 1985, 1988). From the viewpoint of logic, mathematics, and philosophy, self-reference is important mainly due to its aspect of paradox generation. Self-referential reasoning processes which convey deductions about the truth of propositions are ill-posed in many cases. They deviate from the fundamental principles of logic, mathematics, and philosophy. In terms of Aristotelian logic, self-reference could not be captured by the principle of bivalence, which asserts that every proposition is either true or false. In other words, many self-referential statements (at least seem to) contradict the well-known principle of excluded middle (Beal and Bueno 2002; Rieger 2001). Self-reference, the liar paradox, and similar paradoxes were addressed in the very old works of Epimenides, Eubulides of Miletus (Seuren 2005), and even Aristotle (Crivelli 2004). They noticed that the truth or falsehood of sentences like: “This very sentence is false” could not be evaluated easily, and results in a circular debate. Predictably, there is a controversial argument about the truth or validity of such a proposition (Barwise and Etchemendy 1987; Martin 1978, 1984; McGee 1991). The liar paradox was known for many Eastern philosophers and lead to thorough arguments among them. Among the solutions they provided, that of the renowned Iranian philosopher, mathematician, physicist, astronomer, and physician K. N. Toosi (or Tusi) is extremely interesting (Alwishah and Sanson 2009): “The liar is the result of a judgment that applies truth and falsity to something to which they in no way apply, and to apply them in any way is the misuse of a predicate. In other words, the fallacy is to suppose that the predicates “true” and “false” apply to a self-referential sentence like the liar.” Toosi’s argument is appealing since it tries to begin with the fact that truth predicates of a language could not be applied to the propositions of the same language (as we will see, to which the argument of Tarski is quite similar) and then implicitly tries to evaluate self-referential statements such as liar with a third truth value, neither true nor false (to which the solution of Kripke is quite similar, as we will see). From the viewpoint of linguistics and pure logic, self-referential statements call for an elaborate study of truth predicates. There are some formal theories about truth which share ideas between mathematics, philosophy, logic, and linguistics. Some famous examples are Kripke’s theory of truth and semantic theory of truth (Givant and McKenzie 1986; Kripke 1975). The semantic theory of truth has the following general principle for a given language: ‘ p’ is true if and only if p where ‘ p’ is a reference to the sentence (the sentence’s title), and p is just the sentence itself.
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Logician and philosopher Alfred Tarski developed the theory for formal languages such as formal logic. Here he restricted it in this way: that no language could contain its own truth predicate, that is, the expression “is true” could only apply to sentences in some other language. He called the latter an object language, the language being talked about. (It may, in turn, have a truth predicate that can be applied to sentences in still another language.) The reason for his restriction was that languages that contain their own truth predicate will contain paradoxical sentences like the liar: “This sentence is not true”. This statement, also known as the paradox of the liar, is deduced to be false, if considered to be true and vice versa, and this circular argument continues ad infinitum. As a result, Tarski held that the semantic theory could not be applied to natural languages, such as English, because they contain their own truth predicates (Givant and McKenzie 1986; Tarski 1944). On the other hand, Kripke’s theory of truth allows the existence of truth predicates in languages. Kripke builds a language in the following manner: Firstly, he begins with a subset of sentences in a natural language which do not contain expressions “is true” or “is false”. In the next step, he defines a truth predicate for such sentences. Then the definition of truth is generalized so that “is true” or “is false” could be applied to the sentences which are obtained in the preceding step. This process could be continued ad infinitum without any occurrence of liar-like inconsistencies. Note that Kripke’s theory implies the existence of a third truth value (say, “neither true nor false”) for sentences which are not built by his method. As a result, the liar is “neither true nor false”. Philosophical investigations about truth might be considered as the essence of philosophical challenges, and could be sought for in the early works of many schools of traditional and modern philosophy. One may refer to the works and scopes of Socrates, Plato, Aristotle, Avicenna, Aquinas, Kant, Kierkegaard, Nietzsche, James, Dewey, Belnap, and Habermas, among others. Treatment of such philosophical investigations is beyond the scope of this paper. Their relationship to the philosophical and logical aspects of the theory of fuzzy sets is an interesting topic which may be investigated in the future. The mathematical view of paradoxes of self-reference is highly relevant to that of mathematical and philosophical logic and linguistics. Perhaps Russell’s paradox is the most famous paradox of self-reference in mathematics, which is closely related to the liar paradox. Russell’s paradox is the result of accepting an axiom (due to Frege) in set theory, called the axiom of (unrestricted) comprehension: if Ψ is a predicate in the language of set theory, then there is a set that contains exactly those elements x such that Ψ (x). In other words, {x|Ψ (x)} is a set itself. Therefore assuming Ψ (x) to be x ∈ / x, one obtains Russell’s paradox: Let us consider = {x|x ∈ / x}
(1)
It is easily observed that every naïve set is a member of . However, investigating whether ∈ reveals the paradox. If it is true, then from the definition of , it is inferred immediately that ∈ / . If it is false, similarly it is inferred that ∈ . This circular and paradoxical debate quite resembles the argument on the truth or falseness of the liar. There have been several attempts to solve this paradox which implies that the very foundations of mathematics contain inconsistency. An example is the theory of types (Russell 1902) insisting that properties fall into different types, and that the type of a property is never the same as the entities to which it applies. Thus, the question never even arises as to whether a property applies to itself. Hence, the theory elaborates on illustrating the hierarchies of properties. A logical language that divides entities into such a hierarchy is said to employ the theory of types. The theory of types could be viewed as a mathematical counterpart of Tarski’s results. Another attempt was by Zermelo (1908) who formulated set theory to avoid the paradoxes of naïve
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set theory by replacing arbitrary set comprehension with weaker existence axioms, such as his axiom of separation (Aussonderung). Modifications to this axiomatic theory proposed in the 1920s by Abraham Fraenkel, Thoralf Skolem, and by Zermelo himself resulted in the axiomatic set theory called ZFC. This theory became widely accepted once Zermelo’s axiom of choice ceased to be controversial, and ZFC has remained the canonical axiomatic set theory until now (Kunen 1980). Some efforts have been performed to build a set theory in which a universal set exists which contains itself (Church 1974; Forster 1995; Oberschelp 1973; Quine 1937). Since our paper studies self-reference in the framework of fuzzy logic, it is appropriate to inquire that if fuzzy set theory is consistent with the existence of a universal set. It has been shown by White that infinite-valued predicate logic of Lukasiewicz is consistent with the axiom of comprehension and hence the existence of Russell’s set of all sets (White 1979).
3 Liar’s paradox in fuzzy logic Since the seminal work of Zadeh (1965), fuzzy logic has introduced itself as a dynamic area of research in many disciplines. From the early years of its formation, it has been subject to continuing arguments and philosophical, mathematical and technological investigations. For detailed discussions, see for example (Bˇehounek 2008; Cornelis et al. 2006; Entemann 2002; Hajek 1998; Novak 2006; Pelletier 2004; Trillas and Alsina 2001). Interestingly, there were a number of controversial yet sometimes not fully grounded and legitimized opinions (Elkan 1993, 1994a,b, 2001). The paradox of liar was also a subject investigated by researchers in the area of fuzzy logic. Zadeh has shown that the liar’s paradox could be captured by fuzzy logic (Zadeh 1979; Klir and Yuan 1996). As it will be seen in the sequel, following Zadeh’s notion, some researchers have developed methods for self-referential reasoning via fuzzy logic. On the other hand, Hajek, Paris, and Shepherdson (Hajek et al. 2000) have attempted at extension of Tarski’s results to the case of an infinite valued truth predicate. The crux of their approach is the result that a logical system with crisp Peano arithmetic and an infinite valued truth predicate can be consistent. Although their approach is merely logically inclined, it seems that it has been inspired by the approach of Zadeh. As they stress, if the arithmetic is kept to be two-valued, and the unary truth predicate Tr(.) is allowed to be multi-valued, the liar need not be contradictory as long as it is considered to possess the truth value of 1/2 in the context of predicate Lukasiewicz logic. Yet another logically grounded treatment of the liar is the approach of Zhang and Zhang (2004), which introduces a novel fuzzy logic. In their proposed fuzzy logic, they aim to model the internal states of an agent, or interactive relationships among agents as “bipolar” facts. The truth of the “negative” pole of each fact could be any value in [−1, 0] (or −V with the notations of this paper) and the truth of the positive pole could be any value in [0, 1] (or V). They start from the heuristic rule “(self-negation, self-assertion) ⇒ (self-consciousness, self-assurance)” in which the antecedent and the conclusion are bipolar facts about an agent. It is a self-negative statement, therefore we have the following fact for the liar: (self-negation, self-assertion) = (−1,0). Then, by the so called bipolar modus ponens ([(ϕ − , ϕ + ) ∧ ((ϕ − , ϕ + ) ⇒ (Ψ − , Ψ + ))] ⇒ (Ψ − , Ψ + )), it is concluded that the liar should be considered as an honest entity which quite reasonably talks about its mental state: “Sorry! I am wrong”, with the bipolar truth value (−1, 1). However, such a treatment is obviously difficult to be extended to a set of multiple self-referential statements, since there are no ways to extract such heuristic rules for an arbitrary set of metacognitive agents having beliefs
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about truth or falsehood of themselves and/or other agents. We do not compare this line of research with other methods of self-referential reasoning in this study, due to the fact that it could not be extended to multiple self-referential statements. In Kehagias and Vezerides (2006); Vezerides and Kehagias (2003), an algorithmic way is developed for converting a set of self-referential statements to a system of nonlinear equations and finding consistent truth value assignments to them. Although Vezerides and Kehagias have been successful in representation of a quite general class of self-referential statements, their approach does not determine how to ‘select’ a particular set of consistent truth value assignments, when there are more than one solution to the corresponding system of nonlinear equations. 4 Preliminaries In this section, we concisely present some preliminaries with intent to make the study selfcontained. In particular, we recall the notions such as of type-2 fuzzy sets, linguistic variables, linguistic truth variables, fuzzy logic, and possibility theory. 4.1 Fuzzy logic According to Zadeh,1 in fuzzy logic (Klir and Yuan 1995) for the expression “x is A”, the numeric degree of truth Tr(“x is A”) is defined as the degree of compatibility of x with the soft constraint A. Assume for example that: p := John is young Given that John is 25, the truth of p, Tr( p) is μ young (25). 4.2 Type-2 fuzzy sets Type-2 fuzzy sets are a special category of L-fuzzy sets introduced firstly by Goguen (1967). They were formulated in the primary form by Zadeh (1975a) to model the uncertainty about the membership values of a fuzzy set. Definition 1 Assume that U is the universe of discourse. A type-2 fuzzy subset of U, A˜ is defined by the membership values: μ A˜ : U → FV μ A˜ (u) = ϕu
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wherein FV is the family of all fuzzy subsets of the unit interval V = [0, 1] and ϕu denotes a member of the family whose membership function is defined by: μϕu : Gu → V
(3)
In (3), Gu is a subset of V and x ∈ Gu is equivalent to μϕu (x) = 0, i.e. Gu is the support of ϕu . In other words, any function which is defined from Gu to V−{0} can serve as the membership function, depending on the fuzzy set of interest. Actually, a fuzzy set ϕu is assigned as the membership of each u instead of a unique membership value, to stress that the membership value itself is uncertain, and can be called 1 It should be noted that fuzzy logic is treated formally as a legitimate logical system obeying a set of axioms
(Cintula and Hajek 2010). For the purposes of this paper, the above broad interpretation of fuzzy logic, which is inspired by possibility theory, is sufficient.
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the fuzzy valued membership grade of A at u. μϕu can be called the secondary membership function of A at u. When for all u ∈ U, μϕu is a constant and Gu is an interval, especially when μϕu = 1, the type-2 fuzzy set is called an interval-valued fuzzy set. 4.3 Linguistic variables and linguistic truth variables For a better treatment of the concept of truth in fuzzy logic, it is instructive to recall the definition and concept of a linguistic variable (Zadeh 1975a,b,c). Definition 2 A linguistic variable is formally denoted by a quintuple (X , T (X ), U, G, M). X is the name or the label of the variable. T (X ) is a set containing the names of linguistic values related to X. G is a syntactic rule for generating the names X ∈ T (X ). M is a semantic rule (function) assigning a meaning M(X ) to each X . This meaning is a fuzzy subset of the universe of discourse U. Definition 3 A linguistic variable X is called structured if T (X ) and M could be characterized algorithmically. Thereby, the elements of T (X ) and their corresponding meaning assigned by M could be derived in an algorithmic fashion. This issue makes the manipulation of structured linguistic variables easier when the number of elements of the set T (X ) is infinite. Truth variables play a key role in everyday deductions, since it is of interest to characterize the degree of truth of statements by expressions like very true, true, fairly true, more or less true, absolutely false, and false. Correspondingly, truth, as a linguistic variable and the linguistic truth values such as ‘true’, ‘very true’, ‘false’, and ‘absolutely false’ have found an important role in approximate reasoning. Essentially, a linguistic truth variable is a structured linguistic variable, and its term set could be derived via algorithms. The term set of the linguistic variable “truth” contains linguistic truth values which could be derived in an algorithmic fashion. Furthermore, a semantic rule M could be found to assign meanings (and hence fuzzy sets) to each truth value. This view of fuzzy truth values is essentially possibilistic, and represents an uncertainty in the value of a numeric truth value. Hence, a linguistic truth value could be seen as a possibility distribution, and is not actually a genuine truth value leading to a genuine logical system. On the other hand, treating truth values as genuine truth values leads to a fuzzy linguistic logic (Dubois et al. 2007). As a result, we could define logical operations on genuine fuzzy truth values. Definition 4 Let τ1 and τ2 be linguistic truth values. The following operations are defined (Zadeh 1975b): The negation of τ1 , ¬τ1 is defined as: ¬τ1 = 1 τ1 μ¬τ1 (u) = μτ1 (1 − u)
(4)
in which denotes the extended subtraction operation. The conjunction τ1 ∧ τ2 and disjunction τ1 ∨ τ2 of τ1 and τ2 , are respectively defined as: τ1 ∧ τ2 = min (τ1 , τ2) μτ1 ∧τ2 (z) = sup min μτ1 (x), μτ2 (y) z=min(x,y)
τ1 ∨ τ2 = max (τ1 , τ2) min μτ1 (x), μτ2 (y) μτ1 ∨τ2 (z) = sup z=max(x,y)
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in which τ1 ∧ τ2 and τ1 ∨ τ2 are determined by fuzzy extensions of min and max, respectively. As noticed by Zadeh, such definitions are semantically different from the conventional complement “ ”, union “∪”, and intersection “∩” encountered in fuzzy set theory. More precisely, when τ1 and τ2 are linguistic truth values of two statements p1 and p2 , τ1 ∨ τ2 and τ1 ∧ τ2 are the truth values of “ p1 or p2 ” and “ p1 and p2 ” respectively. Moreover, ¬τ1 is the truth value of “not p1 ”. On the other hand, A1 ∩ A2 and A1 ∪ A2 are the ‘meanings’ of “A1 and A2 ” and “A1 or A2 ”, for two arbitrary fuzzy sets A1 and A2 . Similarly, A1 is the meaning of “not A1 ”. Essentially, a truth value as a genuine truth value is pertinent to type-2 fuzzy logic. In type-2 fuzzy sets, we assign a type-1 fuzzy membership value to each element, instead of a numeric value. Therefore, the truth functionality (Eqs. 4, 5) applies: it is equivalent to Zadeh’s fuzzy logical notion of truth of a statement. On the other hand, when dealing with truth as a linguistic variable, we must handle it like possibility distributions, and it represents uncertainty about numeric truth values. As possibility theory is the meaning-oriented notion of information processing, this view is equivalent to Zadeh’s notion of meaning of a statement. We must apply conventional complement, union, and intersection of possibility distributions, and truth functionality is not permitted. For an elaborate discussion on this issue, see Dubois et al. (2007); Haack (1979). 4.4 Possibility theory Possibility theory is initiated with the concept of a fuzzy restriction (Zadeh 1975d). If x is a variable which takes values from the universe of discourse U , and A is a fuzzy set in U , the sentence “x is A” is considered as introducing a fuzzy restriction on x where such an ‘elastic’ restriction is characterized by the membership function of A. This leads us to consider μ A (u) as the degree of possibility of x=u. Given a fuzzy set A in U , the sentence “x is A” provides a possibility distribution function assigned to x, denoted by πx : ∀u ∈ U πx (u) = μ A (u)
(6)
Zadeh establishes a rather simple, but extremely useful mathematical basis to extend the theory of possibility. The development of this theory is well-known in the fuzzy computing community, so we avoid repeating the material and refer the interested readers to the Zadeh’s original work (Zadeh 1978), as a contribution which is highly illuminating. Also, there are other approaches to the theory of possibility, which are axiomatic and based on Dempster–Shafer theory of evidence (Klir and Yuan 1995). Despite the fact that they are helpful, the approach of Zadeh is very capable in the ‘meaning-oriented’ possibility, and as we denote in this paper, it is very well-behaved for interpretation of self-referential sentences. So while trying to extend it, we will adhere to the essential features of the existing approach. In his paper, Zadeh proposes three principles for analyzing the information content of sentences, namely truth qualification principle, probability qualification principle and possibility qualification principle. As our paper concerns truth and possibility qualification principles, we reconsider them briefly. 4.5 Truth qualification and possibility qualification principles Truth qualification principle tries to judge the truth of sentences which speak about the truth of other sentences. If:
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x is A → πx (u) = μ A (u) then x is A isv τ → πx (u) = μτ (μ A (u)) in which τ is a linguistic truth value (a veristic constraint) such as: true very true, more or less true, and false. “→” shows that the statement induces a possibility distribution function. The truth qualification principle has been applied to the medical diagnosis problem by the aid of relational equations (Sanchez 1996). Recently, there have been some studies on the truth values (Gera and Dombi 2007, 2008; Vychodil 2006; Walker and Walker 2005) and the truth qualification principle (Own 2009; Zadeh 2008). It is evident that the possibility qualification principle qualifies sentences speaking about the possibilities of other sentences and events. If: x is A → πx (u) = μ A (u) then x is A is α-possible → π˜ x (u) = ϕu Gu = [min(α, μ A (u)), min(1, 1 − μ A (u) + α)] ⊆ V Here α-possible is a linguistic possibility value (α ∈ [0, 1]), and π˜ x represents a type-2 possibility distribution. This type-2 fuzzy set is constructed to show the uncertainty caused by weakening the proposition by a linguistic possibility value. Zadeh, proposed that μϕu (x) = a ∈ [0, 1] ∀x ∈ Gu , i.e. an interval-valued fuzzy set. Replacing α = 1 for ‘completely possible’ Gu reduces to: Gu = [μ A (u), 1]. Other versions of the possibility qualification principle could be envisioned as well in which we assign type-2 fuzzy sets as the possibility distribution of statements.
5 Fuzzy truth values, generalized theory of uncertainty and self-reference There are some studies which consider self-referential reasoning in the scope of fuzzy logic. As we noted, Zadeh’s work on the resolution of the paradox of the liar via truth qualification principle is the first work in this field (Klir and Yuan 1996; Zadeh 1979). It follows his notion of information processing as manipulation of fuzzy constraints (Zadeh 1975d). This view on information processing was fertilized with the establishment of possibility theory, especially via the qualification principles we reviewed briefly (Zadeh 1978) and is now more fruitful because of Zadeh’s recent works on development of a Generalized Theory of Uncertainty (GTU) (Zadeh 2005, 2006, 2008). Generalized Theory of Uncertainty tries to model information, whatever its form, as a generalized constraint. A generalized constraint is represented as: X isr R where X is a constraint variable, R is a constraining relation and r is an indexing variable denoting the semantics of the constraint. Different kinds of constraints are distinguishable on information; possibilistic (r = blank); probabilistic (r = p); veristic (r = v); usuality (r = u); random set (r = rs); fuzzy graph (r = fg); bimodal (r = bm) and group (r = g). Such constraints could be qualified, combined, and propagated, leading to the Generalized Constraint Language.
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The centerpiece of this paper is the concept of veristic fuzzy sets and veristic constraints, so we follow Zadeh’s recent notation. When we want to accommodate a veristic constraint τ on a statement p j we have the following statement pi : pi := p j isv τ in which τ is a truth value. Taking a glance at the liar sentence: p := p isv not true It could be concluded that this sentence puts a constraint on its truth. Thus its truth could be evaluated via the truth qualification principle. The proposition is reformulated as: p := The truth value (constraint on verity) of p isv τ in which τ is equal to ‘not true’ or false. Actually, this statement assigns a linguistic truth value τ to the verity distribution2 of the statement p. Considering the membership function of the constraint on the truth of p (i.e. the verity distribution associated with p) as v p (u), it is easily observed that by the truth qualification principle, we have: 1 (7) 2 Remark It should be noted that the truth qualification principle is not about assigning linguistic truth values to numeric truth or possibility values. Therefore, we did not use the truth qualification principle to derive a numeric truth value v = 1/2 for the liar. However, some authors (Kosko 1996) prefer to view the truth qualification procedure for the liar as yielding a numeric truth value for it. v p (u) = 1 − v p (u) ⇒ v p (u) =
It is worth emphasizing again that v p (u) represents a verity distribution, and therefore, should be treated by the calculi of possibility theory. Actually, Zadeh accepts the fixed points of the membership function μτ (u) as the value of the membership function v p (u) associated with the verity distribution of the self-referential statement, for all u ∈ [0, 1]. 5.1 Semantics imposed by Zadeh’s method of self-referential reasoning It is by Zadeh that the truth value represented by the membership function μ(u) = 1 could be defined as the truth value unknown (Zadeh 1975a). He then tries to relax the ‘ignorance’ imposed by the term ‘unknown’, via defining the set of truth values ‘α − unknown’. μ(u) = α is representative of these truth values. One may conclude that the reasoning procedure in (7) yields the truth value 1/2-unknown for the liar sentence. However, it should be noted that this type of conclusion is not very useful. Actually, when we are seeking the truth space, we demonstrate that, when a truth value is unknown, it can be each member of the truth space. It is just the same as this trivial semantic interpretation: “when a truth value is unknown, it is equally possible to be any member of V = [0, 1]”. On the other hand, consider the following reasoning process: p := John is young q := The truth value of p isv unknown q models complete ignorance about the truth of p. Truth qualification yields: π Age(J ohn) (u) = 1. Therefore, John’s age may be any arbitrary value in the universe of discourse. By the notion of fuzzy logic presented in Sect. 4.1., the truth value of q, T r (q), is the compatibility 2 A verity distribution is essentially a possibility distribution on the space of numeric truth values V = [0, 1].
Therefore, as a linguistic value is analogous to a possibility distribution, a linguistic truth value is analogous to a verity distribution.
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of the truth of p with unknown which is always 1, whatever the truth value of p is. For self-referential statements, both of p and q are the same. Because the sentence itself contains information about its truth, the result of the reasoning process needs more effort to be interpreted. For example, for the liar sentence, referring to the definition of fuzzy logic in Sect. 4.1., supposing the truth value of p be any value x in V = [0, 1], it is deduced that the truth value of p is the degree of compatibility of x with v p , i.e. v p (x) = 1/2. Then there is no way but to assert that p is half true, because any assumption will yield that: Tr( p) = 1/2. It is possible to interpret Zadeh’s solution of the liar paradox more easily. Concerning the concept of fuzzy logic presented in Sect. 4.1., it is easy to show via the truth qualification principle that when the truth value of a statement p is known to be Tr( p), the truth value of the statement q : p is τ is μτ (T r ( p)). Therefore, assuming the numeric truth value of liar to be v, one can easily infer that: v =1−v ⇒v =
1 2
(8)
This could be a justification for dealing with the numeric truth value of the liar, as Kosko suggests (Kosko 1996). Another helpful idea is the viewpoint of two different kinds of truth values we presented in Sect. 4.3.: truth values as members of the term set of the linguistic variable TRUE, and truth values as genuine logical entities which are pertinent to type-2 fuzzy logic, i.e. they can be employed as fuzzy membership values (secondary membership grades) in type-2 fuzzy sets. Obviously, here we deal with the first notion of fuzzy truth values which is possibilistic. Therefore, the result v p (u) = 1/2 says that, the verity (analogous to the possibility) that the numeric truth value of the liar is u, assuming that u is any member of the unit interval, is 1/2. However, one may prefer the first interpretation of Zadeh’s solution to the liar, since despite its intricacy, it perfectly illustrates the special condition of the liar in which the subject and object are the same, and both are the statement itself: It starts with an unknown verity distribution function for the liar, and shows that after the truth qualification procedure, assuming any numeric truth value for the liar leads to the conclusion that its truth value is 1/2. Henceforward, in this paper, we assume verity distribution functions for self-referential statements. Another important fact reveals itself here: essentially the truth qualification principle manipulates the ‘meaning’ of the statement which it tries to qualify, which comes from its possibilistic nature. As we noted in Sect. 4.3., ‘not true’ could be treated in two ways: when one refers to the meaning of ‘not true’, μnot tr ue (u) = 1 − μtr ue (u). In contrast, when ‘true’ is the linguistic truth value of a statement p, the truth value of ‘not p’ is characterized by μtr ue (1 − u). The nature of truth qualification principle is meaning-oriented and it is quite compatible with the semantics of natural languages. Consequently, when Zadeh applies the truth qualification principle to the liar which states a fact about its truth value, he preserves the ‘meaning-oriented’ nature of the truth qualification principle. Therefore, he does not interpret the liar as: v p (u) = v p (1 − u)
(9)
which implies an infinite number of linguistic truth values for the liar. This interpretation is not semantically valid. Accordingly, when connectives are present in the statements, one must use conventional connectives of fuzzy sets rather than logical operations on fuzzy truth values. This fact will be used in Sect. 9.
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6 Representation of self-reference via dynamic semantic maps Inspired by sequential reasoning about the liar, Grim views the liar sentence as a dynamical system (Grim 1993). He asserts that the oscillating nature of reasoning about the liar, whose truth results in its falsehood and vice versa, might be modeled by a simple dynamical system: tn+1 = 1 − tn
(10)
He concludes that this bivalent map, which oscillates between 0 and 1 (true and false) is also valid when the initial value of tn is assumed to be a member of [0, 1]. The behavior of the map is still oscillatory although it can take values in the interval of [0, 1]. This discussion leads to interpretation of self-referential reasoning as a dynamical system. For example, in Grim’s paper, the following proposition is called the fuzzy logistic: p := It isv very false that p isv true iff it isv false Utilizing the fuzzy equivalents of ‘very false’, ‘true’ and ‘false’ and generalizing the previous discussion, the following map is obtained: tn+1 = (1 − (1 − abs(tn − (1 − tn ))))2
(11)
Grim’s approach is extendable to multiple sentences, each of which is speaking about the truth of itself or the others. They also yield dynamical mappings whose state variables are the truth values of the sentences. The fuzzy inconsistent dualist is a sound example: p1 := p1 isv true iff p2 isv true p2 := p2 isv true iff p1 isv very false The corresponding dynamical system is: tn+1 = 1 − abs(tn − rn ) rn+1 = 1 − abs(rn − (1 − tn )2 )
(12)
In the rest of his article, Grim inspects many sets of self-referential sentences and derives their resultant dynamical systems. He shows that the so-called semantic systems may be chaotic or have strange attractors. The main goal of his studies is to illustrate that fuzzy logic does not remedy the problem of self-reference and the dynamical systems which model the reasoning process may not converge. Grim does not address the work of Zadeh in his article; instead, he reaches the ‘dynamical system’ approach via generalization of the cyclic reasoning about the liar. Furthermore, he considers numeric truth values as the states of the dynamic systems pertinent to self-referential statements, in contrast to truth qualification approach which essentially deals with verity distributions (although, as we mentioned in Sect. 5.1., truth qualification procedure can somehow support considering numeric truth values for the liar and other self-referential statements). However, about the liar (and as it could be seen, all other self-referential sentences), the work of Grim, verifies Zadeh’s insight indirectly. The abovementioned dynamical systems may be chaotic or oscillatory unless they are initiated from one of their fixed points. Hence the best solution is to accept to start the reasoning process from the fixed points of the related dynamical system. They are proved to exist under some rather weak assumptions as we see in Sect. 9. This is the same as accepting the fixed point itself as the solution, since the system states do not change in this case. Consider the dynamical map generated by the liar sentence: tn+1 = 1 − tn t0 = 1/2 ⇒ tn = 1/2
∀n>0
(13)
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It is obvious that starting from the fixed point of the system, it would not exhibit oscillatory behavior. This is analogous to the approach of Zadeh in accepting the fixed point of false as the membership function of the truth value of the liar.
7 Implicative interpretation of self-reference In another attribution to self-referential reasoning in the framework of fuzzy logic, Chen (1999) revisits the liar. The work is apparently influenced by Zadeh’s method for resolution of the liar. Chen’s paper only considers the liar and does not refer to other forms of selfreference, except in a section in which indirect self-reference in the inconsistent dualist is considered: p1 := p2 isv true p2 := p1 isv false However, Chen views these two sentences as a special form of the liar. The article employs the cyclic reasoning caused by the liar to construct an alternative interpretation of it: q1 := If p isv true, then p isv false q2 := If p isv false, then p isv true This is obviously another aspect of the paradoxical nature of the liar. It generates q1 , q2 which are inconsistent, at least in classical logic. Again, assuming that v p (u) is the membership function of the truth value of p, statements q1 , q2 are translated to the following: vq1 (u) = I (v p (u), 1 − v p (u)) vq2 (u) = I (1 − v p (u), v p (u))
(14)
in which I is a fuzzy implication function:3 I : [0, 1] × [0, 1] → [0, 1] I (0, 0) = I (0, 1) = I (1, 1) = 1 I (1, 0) = 0
(15)
To reflect the paradoxical inherence of p, q1 and q2 should be equally true, thereby it is intuitively realized that: I (v p (u), 1 − v p (u)) = I (1 − v p (u), v p (u))
(16)
It is clear that independent of selecting a particular implication function, v p (u)=1/2 is a solution to (16), due to its symmetric nature. From this viewpoint, the paradox of the liar reveals itself when someone tends to decide whether the liar’s sentence is true or false. Generalizing this argument, one might be inclined to demonstrate whether the self-referential sentence: p := ( p is τ ) is α or β, where α, β and τ denote truth values. We have the following reasoning procedure: q1 := If p isv α then p isv τ isv α q2 := If p isv β then p isv τ isv β vq1 (u) = I (μα (v p (u)), μα (μτ (v p (u)))) vq2 (u) = I (μβ (v p (u)), μβ (μτ (v p (u))))
(17)
3 Implications are also shown by the symbol ‘⇒’ in fuzzy logic. We have used this notation in this paper as
well. The main reason for using I (., .) in this section is its suitability for presenting Chen’s approach.
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Upon considering that v p1 is equal to v p2 we have: I (μα (v p (u)), μα (μτ (v p (u)))) = I (μβ (v p (u)), μβ (μτ (v p (u))))
(18)
This leads to definition of α–β–τ legitimacy. When we are to reason about the relationship between assigning a truth value of α or β to a self-referential proposition and assigning a truth value of τ to it, it makes sense to try to find a v p (u) such that (18) holds. When a solution v p (u) exists, the self-referential expression is called to be α–β–τ legitimate, i.e. its problematic meaning could be captured in the landscape of fuzzy logic. The solution(s) v p (u) is (are) therefore called the I -truth value(s) of the liar paradox. In this sense, the liar paradox is true-false-false legitimate. Chen tries to find conditions for μτ , μα , μβ , which make (18) have a solution. As stated previously, Chen’s paper devotes a section to indirect self-reference. It considers the inconsistent dualist as a special form of the liar. So it utilizes a similar method, based on implication, to interpret it: p1 := p2 isv true p2 := p1 isv false For p1 we have the inference: q1 := If p1 isv true, then p1 isv false q2 := If p1 isv false, then p1 isv true We could also construct similar sentences for p2 . The following equation is then concluded: I (v p1 (u), 1 − v p1 (u)) = I (1 − v p1 (u), v p1 (u))
(19)
In a general setting, the following pair of sentences could be considered: p1 := p2 isv γ p2 := p1 isv δ in which γ , δ are truth values. Inspecting if p1 and p2 are α or β we can assert that: q1 := If p1 isv α, then p2 isv γ is α, then p1 isv δ isv γ isv α Thereby: vq1 (u) = I (μα (v p1 (u)), μα (μγ (μδ (v p1 (u)))))
(20)
In a similar way: q2 := If p1 isv β, then p2 isv γ is β, then p1 isv δ isv γ isv β whose truth is: vq2 (u) = I (μβ (v p1 (u)), μβ (μγ (μδ (v p1 (u)))))
(21)
As the past, the truth values of both of the fuzzy implications are argued to be identical: I (μα (v p1 (u)), μα (μγ (μδ (v p1 (u))))) = I (μβ (v p1 (u)), μβ (μγ (μδ (v p1 (u)))))
(22)
We have also the following for v p2 : I (μα (v p2 (u)), μα (μδ (μγ (v p2 (u))))) = I (μβ (v p2 (u)), μβ (μγ (μδ (v p2 (u)))))
(23)
Again, the self-referential set of sentences is α–β–γ –δ legitimate if there is at least one solution pair (v p1 , v p2 ) to (22) and (23). It is somewhat difficult to conclude about the existence
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of solutions to this problem generally. However, using Brouwer’s fixed point theorem (see Sect. 9), Chen tries to show that there are solutions for v p1 (u) and v p2 (u), in some special cases such as when α and δ denote true and both β and γ denote τ , and μτ is a continuous isomorphism on V = [0, 1]. Unfortunately, Chen’s approach is not easy to extend for other self-referential sentences. To illustrate this issue, let us consider the following set of statements: p1 := p2 isv γ p2 := p3 isv δ p3 := p1 isv η We are going to decide if p2 is α or β, using Chen’s approach: If p2 is α, then p3 is δ is α, then p1 is η is δ is α, then p2 is γ is η is δ is α. On the other hand, if p2 is β, then p3 is δ is β, then p1 is η is δ is β, then p2 is γ is η is δ is β. At last we have: I (μα (v p2 (u)), μα (μδ (μη (μγ (v p2 (u)))))) = I (μβ (v p2 (u)), μβ (μδ (μη (μγ (v p2 (u))))))
(24)
This procedure becomes more complex and less algorithmic when the number of sentences becomes large, and the logical functions of each sentence go more intricate. For example, this procedure is tedious for the following set of propositions (The Greek letters represent truth values): p1 p2 p3 p4
:= := := :=
p2 isv γ if and only if ( p3 isv γ or p4 isv ε) p1 isv ζ or p3 isv κ p1 isv η p2 isv ω or p4 isv λ
Moreover, finding conditions on the truth values to force the equations to have solutions (making them legitimate in some sense) becomes harder. Thus, taking Chen’s approach as a general reasoning tool is impractical. Furthermore, Chen does not propose a method for ‘optimal’ selection of implication functions (and also other logical functions). He does not offer any criterion for selection of a solution, when more than one solution is found for the truth values. Another problem to his approach is the possibility of the definition of higher orders of legitimacy for any set of self-referential statements. By deriving the same equations as (22), we could judge whether a member of a set of self-referential statements is e.g. α1 -α2 -α3 -α4 -τ legitimate, but this requires generating four sentences q1 , q2 , q3 , q4 for each statement, and try to find a v pi (u) such that it makes vq1 (u) = vq2 (u) = vq3 (u) = vq4 (u). It is debatable that which order of legitimacy is ‘adequate’ for resolution of a set of self-referential statements. Obviously, reasoning about higher order legitimacy of the statements suffers from more severe versions of the abovementioned problems, and this may justify neglecting it. It is also worth noting that assigning numeric truth values to self-referential statements via this approach is imaginable, following our previous discussion on assigning a numeric truth value to the liar, suggested by Kosko (cf. Sect. 5.1.) Chen’s approach could be a tool for generation of dynamical semantic maps, such as those considered by Grim. However, again the abovementioned problems make it an impractical one.
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8 Computational verb systems and liar’s paradox In another treatment of the paradox of the liar, computational verb systems are employed by Yang (2001). Computational verb systems are proposed to capture the dynamical semantics imposed by the use of verbs in natural languages. A computational verb α is defined by the following: (1)
A set Eα which is called the evolving system in the context of computational verbs:
Eα = {T, X s , X p , Aα , Bα }
(2)
where T ⊂ , X s ⊂ 3 and X p ⊂ k symbolize time, space and cognition and associate these features with a verb. Aα is a function called the inner system and models self-reflection in the brain: Aα = T × X s × X p → T × X s × X p
(3)
(25)
(26)
The outer system Bα is a model of the existing world and is defined as: Bα = T × X s × X p → T × X s × X p
(27)
Here, T ⊂ , X s ⊂ 3 and X p ⊂ k are the ‘observed’ time, space and cognition. To involve the context in which a verb sentence is asserted, and to evaluate its truth or falsehood, a transformation is made which is called a BE-transformation. It transforms the verb sentence into the following so-called BE-proposition: S: [Subject V er b Object] BE word
(28)
in which word could be an adjective. Now, the numeric truth value of the above statement could be evaluated. To capture this concept mathematically, the canonical BE-transformation is defined as a mapping which transforms a point of the time-space-cognition P to a numeric truth value between 0 and 1 inclusive. Essentially, if the inner system of a verb is known, the BE-transformation I word (V er b, P) could be illustrated as: I word (V er b, P) : × T × X s × X p → [0, 1]
(29)
wherein V er b ∈ is a verb in the set of all of the verbs of the language of discourse and P is a point in the time-space-cognition space. If the outer system of the Verb is known, we have: O (V er b, P) : × T × X s × X p → [0, 1] (30) word wherein Verb∈ is a verb in the set of all of the verbs of the language of discourse and P is a point in the time-space-cognition space. To obtain an interpretation of the paradox of the liar which could be captured by computational verb sentences, Yang starts from Chen’s interpretation which is mentioned in Sect. 7. Then he concludes that Chen’s interpretation can be unified into a single verb statement: p is what p is not Consequently, a general form of the liar is extracted as a verb sentence:
S : [Subject V er b what Subject NOT V er b]
(31)
Yang also offers some examples like the ones below: I say what I do not say
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I do not feel what I feel Then, a BE-transformation is applied to such sentences: S : [Subject V er b] BE word [Subject NOT V er b] BE word
(32)
Now, regarding the context in which the BE-transformation of Eq. (32) is performed, a verb sentence might be paradoxical or non-paradoxical. Yang uses a function : [0, 1]×[0, 1] → [0, 1] to show the paradoxical value of the verb sentence. Therefore, the paradoxical value of a verb sentence could vary between 1 (completely paradoxical) and 0 (non-paradoxical). Let us consider the paradox of the liar, which is transformed to “This sentence is what this sentence is not”. In order to find a reasonable paradoxical value for the liar, let v1 be the numeric truth value of the first part (i.e. the assertion that the sentence is true) and v2 be the numeric truth value of the second part (i.e. the assertion that the sentence is not true). Then Yang concludes that: (1, 0) = (1, 0) = 1 and (0, 0) = (1, 1) = 0. It is obvious that an infinite number of candidates for are available. Yang chooses the following: 1 (v1 − 0.5)(v2 − 0.5) < 0 (v1 , v2 ) = (33) 0 (v1 − 0.5)(v2 − 0.5) > 0 The function (33) returns a Boolean paradoxical value. To obtain a function which yields a continuum of truth values between 0 and 1, Yang proposes the following paradoxical value function: (v1 , v2 ) =
1 − exp(abs((v1 − 0.5) − (v2 − 0.5))) 1 − exp(1)
(34)
Yang tries to classify the paradoxes into three categories. First, he defines the so-called paradoxical strength of a verb sentence: word∈ P∈χ word (V er b, P), word (not V er b, P) Ψ (S) = (35) word∈ P∈χ 1 in which S is the verb sentence of interest, is collection of all possible words which can be used to derive BE-transformations of S.χ is a discrete subset of the time-space-cognition space. Actually it includes all contexts that words can impose. For a continuous χ, obviously the summation is converted to a Riemannian integral: word∈ χ word (V er b, P), word (not V er b, P) d P (36) Ψ (S) = word∈ χ d P A Linguistic Classification is then done according to the paradoxical strength. If Ψ (S) is near 1, it indicates a strong verb paradox. If it is near 0, it indicates no verb paradox. If it is between 0 and 1, it indicates a weak verb paradox. As the previous sections, here we try to analytically compare Yang’s method with the other methods. The main difference is that the approach does not tend to solve the paradox with fuzzy logic. Rather it utilizes fuzzy logic to sort the degree with which a sentence is paradoxical. The method claims to include context in the reasoning process. The main problem with such an approach is that it is not justifiable to see the liar paradox similar to the sentence (31). In classical logic, sentences like “This sentence is what this sentence is not” are simply false and this will not result in a circular reasoning continuing ad infinitum.
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Since this approach relies on interpretation of self-referential statements by Chen’s method, it suffers from the same limitations as his approach, and is not amenable to be generalized to multiple statements. Multiple statements should be translated to a single statement of the form (31) and then using BE-transformations, their paradoxical value : [0, 1]m → [0, 1], in which m is the number of multiple statements, is determined. This approach is not quite algorithmic and comprehensible.
9 Self-reference needs extended truth qualification In this section, we present a unified framework for manipulation of self-referential expressions via fuzzy logic as follows: For simplicity, assume that we have sentences in which every sentence only speaks about the truth of itself and other sentences in the set. First, we start with a rather general form of such sentences. Assume that: Si j := H j1 p j isv τ ij1 •H j2 p j isv τ ij2 • · · · •H jm( j) p j isv τ ijm( j) (37) Here, Si j is an expression which contains m( j) statements about the truth of statement p j . Definitely, each τ ijk , k = 1, . . ., m( j), is a truth value. The sign • represents any of the logical connectives ∧, ∨, ⇒, ⇔ . H represents a set of hedges. Each sentence in the set of self-referential statements is of the form: pi := Hi (Si1 •Si2 • · · · •Siq(i) )
(38)
Hi stands for a set of hedges. It should be noted that the aforementioned form of self-referential statements is not the most general case, albeit it can cover a vast range of self-referential statements. It is presented merely for obtaining a better understanding of the problem. For example, the following sentence could not be interpreted in the form of Eq. (38): p1 := p2 isv true and it isv fairly false that ( p3 isv very false and p3 isv fairly false) However, it is quite straightforward to derive its logical interpretation:
v p1 (u) = v p2 (u)
1 − (1 − v p3 (u))2 1 − v p3 (u)
(39)
wherein we have used the common definitions for fairly and very and algebraic product for the logical connective ‘and’. It is worth noting that Kehagias and Vezerides have previously attempted at building a language which can transform the members of a quite general class containing sets of selfreferential statements to nonlinear equations (Kehagias and Vezerides 2006; Vezerides and Kehagias 2003). One may be interested in the exact algorithmic way of converting a set of self-referential statements to a system of nonlinear equations. As a general rule, one must build a language whose atomic terms are statements of the form p j isv τ ij1 and derive a suitable algorithm. However, it seems that such a conversion is easily performed by an expert, although building formal languages is quite possible by extending the approach of Vezerides and Kehagias or by means of the results concerning the concept of fuzzy formulas (Shen et al. 1988). As a comprehensive example, consider the following set of self-referential statements:
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p1 := p2 isv true and ( p3 isv very false or p3 isv fairly true) p2 := It isv very true that ( p2 isv very true iff ( p3 isv fairly false and p4 isv fairly true)) p3 := ( p1 isv very true iff p3 isv very false) and p4 isv fairly true p4 :=It isv very very true that ( p1 isv false iff ( p1 isv true or p2 isv fairly false or p3 isv false)) Its resultant map is thereby: f : R4 → R4 f (v p1 (u), v( p2 (u), v p3 (u), v p4 (u)) = ( f1 , f2 , f3 , f4 ) 2 , v (u)) (u)) f 1 = v p2 (u) × max((1 − v p3 p3
f 2 = (1 − abs(v p2 (u)2 − ( 1 − v p3 (u) × v p4 (u))))2 f 3 = (1 − abs(v 2p1 (u) − (1 − v p3 (u))2 )) × v p4 (u)
f 4 = (1 − abs(1 − v p1 (u) − max(v p1 (u), 1 − v p2 (u), 1 − v p3 (u))))4
(40)
In the logical interpretation of the statements, the conventional functions for hedges ‘very’ and ‘fairly’, maximum for ‘or’ and algebraic product for ‘and’ are used arbitrarily. One has many other choices for them. We discuss this fact in this section more thoroughly. Following Zadeh’s approach in accepting the fixed point of the mapping generated by the liar as the solution, the fixed point(s) of the mapping generated by the set of equations (we derived such a mapping in (38)) could be considered as the solution(s) of self-referential sentences. By Brouwer’s fixed point theorem, it is obvious that for the existence of a solution it is sufficient that the t-norms, t-conorms, c-norms and hedge functions are continuous on V = [0, 1]. It is rather a mild condition and we could always use many well-known and continuous functions in the reasoning procedure: Theorem 1 (Brouwer’s Fixed Point Theorem). Let g(x) : D → D be a continuous surjective function, and D ⊂ Rm be closed and connected. The equation g(x) = x has at least one solution x0 ∈ D which is called a fixed point of g(x) (Rudin 1976). In such an approach, the maps may have many (and maybe an infinite number of) fixed points. Also, there might be different mappings for the sentences due to various choices of t-norms, t-conorms and c-norms, and hence different sets of fixed points. An important question arises here: Which solution to select? One might be inclined to define a measure to select the solution, or make use of the opinion of an ‘expert’. But it seems that it is somewhat meaningless to select a solution and refuse others. Also, it is difficult to find an objective function to evaluate the solutions. These barriers lead us to model the uncertainty in the selection of proper solutions via fuzzy logic tools. Type-2 fuzzy sets are efficient tools in modeling uncertainty in membership functions of fuzzy sets. Verity distributions are represented by veristic fuzzy sets (fuzzy truth values). Therefore, we could use a ‘set of solutions’ as the membership values of the veristic fuzzy set. This approach is quite consistent with the conventional methods of fuzzy logic. Essentially, this is an extension to the truth qualification procedure. Furthermore, it is yet another step to enrich the ability of fuzzy logic such that it can consistently assign truth values to sets of self-referential expressions. For non self-referential sentences, it is easy to apply the truth qualification principle and other laws of fuzzy logic to obtain the possibility distributions. For self-referential sentences, the output of the qualification process may be a type-2 fuzzy set. Assume that F = {V 1 , . . ., V 2 , . . ., V k , . . .} is the set of all fixed points of the mapping f generated by a set of self-referential statements. Thereby each V i could be shown as:
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T j j j V j = v p1 (u), v p2 (u), . . . , v pn (u) F may be a countable or uncountable, finite or infinite set. The truth value of each sentence becomes: v pi (u) = Gui = v 1pi (u),v 2pi (u),…,v kpi (u), . . . (41) This is a multi-valued or infinite valued membership function for each veristic fuzzy set. This approach is plausible and consistent with the methods of reasoning in fuzzy logic, but it is not flawless. Its main problem is that it does not always yield one kind of type-2 fuzzy set. In some cases, v pi (u) is a uni-valued (thus type-1) or multi-valued membership function and in the other occasions, it becomes an interval valued fuzzy set. It may be sometimes more convenient to interpret the verity distributions as interval valued fuzzy sets, especially in case they happen to be used in the design of a knowledge based system, or are exploited in another reasoning procedure. In this case, we suggest the following verity distribution functions: v pi (u) = inf Gui , sup Gui (42) This approach is similar to the method of considering uncertainty bounds in membership functions of rule-based fuzzy decision making and control (Wu and Mendel 2002). The next task is assigning a membership function to Gui to make the set valued membership functions be fuzzy set valued ones: v pi (u) = iu
(43)
iu is a fuzzy subset of Gui . To accomplish this task, Mendel’s proposals for modeling uncertainties via type-2 fuzzy sets are helpful (Mendel 2003). He believes that asking persons to assign anything other than a uniform weighting to the set valued memberships would be difficult. Thereby, we are guided to choose: μiu (x) = c x ∈ Gui , u ∈ U
(44)
c is a constant, and especially it may be taken 1. Mendel also suggests to model uncertainties existing among a group of people by means of aggregation of each person’s equally weighted Gui ’s. If some of the experts are more trustworthy, their opinion might be weighted more than the others. Thus, instead of assigning a predetermined μ to the members of Gui , we could refer to a group of experts in the area of fuzzy reasoning and logic and build a type-2 fuzzy set according to Mendel’s prescriptions. It should be noted that discussion about proper choices for μ is less important than the previous issues on taking the uncertainties about selection of the solutions into account. To present an illustrative yet simple example, let us consider the so-called truth-teller: p : p isv true. The truth-teller is known to be logically problematic, and some authors (Goldstein 1992) consider it as a counterpart of the liar. This statement seems to produce no paradox. However, it could be assumed to be both true and false! To capture its truth value via the proposed method, we have the following equation by truth qualification principle: v p (u) = v p (u)
(45)
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The set of admissible solutions to the above truth value assignment problem is [0, 1], i.e. any number in the unit interval could be assigned as a truth value to the truth-teller. To be more precise, as the solution v p (u) is a fuzzy set over the unit interval V, not only is any point c in [0, 1] a solution (v p (u) = c), but also the membership function of any fuzzy set v p over V satisfies Eq. (45). But it seems that the reasoning procedure is incomplete: Finally, what is the truth value of the truth-teller statement? The provisional method discussed above proposes that all of the admissible truth values are aggregated into an interval-valued fuzzy set, and assigned to the truth-teller as its truth value: ∀u ∈ V v p (u) = Gu = [0, 1]
(46)
In this problem, the fuzziness in the truth of the self-referential statement is induced by existence of more than one fixed point for the mapping which represents the self-referential statement. Accepting a set of truth values rather than one truth value is quite in accord with the conventions of fuzzy logic. This problem illustrates the strength of Zadeh’s method of treating self-reference. Although the truth-teller is not paradoxical like the liar, any numeric truth value could be ascribed to it in the realms of multiple-valued and two-valued logic. Applying Zadeh’s method reveals the problematic nature of this statement, and depicts that any verity distribution could be ascribed to the truth-teller.
10 Remarks on extended truth qualification process It is notable that we can take two approaches in modeling the uncertainty about choosing the logical functions and representations of hedges: The first approach is reasoning with different sets of logical operators and including the fixed points obtained by each set of interest in the resulting type-2 fuzzy set. The second approach is considering the semantics of logical operators. It is reasonable to choose a unique set of operators for interpreting logical functions, because of different semantics of the operators. Thus one may reason that using different functions for negation, hedges, t-norms and t-conorms, illustrate different strengths and meanings and yields different versions of a sentence. However, this discussion is a complement to our argument on aggregation of the type-1 truth values of the sentences into type-2 fuzzy sets. It does not affect the generality of our argument at all. It is worth noting that our extended truth qualification method removes some ‘information’ from the reasoning process. We illustrate this with a simple example: Assume that V 1 = {v 1p1 , v 1p2 , v 1p3 }, V 2 = {v 2p1 , v 2p2 , v 2p3 }, V 3 = {v 3p1 , v 3p2 , v 3p3 } are three sets of solutions found for the equations corresponding to three self-referential statements, p1 , p2 , and p3 . When these solutions are combined (via constructing type-2 verity distributions), we assign a ‘set of truth values’ to each sentence, and therefore we have a set-valued (and therefore type-2) fuzzy set as the verity distribution of each statement: v p1 (u) = W 1 = v 1p1 , v 2p1 , v 3p1 (47) v p2 (u) = W 2 = v 1p2 , v 2p2 , v 3p2 v p3 (u) = W 3 = v 1p3 , v 2p3 , v 3p3 If one is interested in finding the original set of solutions from (47), one should begin a try and error search to extract V 1 , V 2 , V 3 from other invalid selections like {v 1p1 , v 2p2 , v 2p3 } which do
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not satisfy the original equations, i.e. test all three permutations of members of W 1 , W 2 , and W 3 in the equations to find V 1 , V 2 , and V 3 . The ‘vanished’ information is arbitrarily paid to make a valid reasoning process resulting in type-2 fuzzy sets which is in agreement with the conventions of fuzzy logic about modeling the uncertainties about membership functions with type-2 fuzzy sets. In case of using Eq. (42), the original set of truth values could not be recovered at all. Our example, truth-teller, is an appropriate illustration of the fuzziness induced by the existence of more than one solution to the truth qualification. The proposed reasoning method is simply a trade-off: some ambiguity is presented to the truth values of the sentences to make the reasoning process have a unique solution of the from of a type-2 fuzzy set. This ambiguity and the abovementioned vanished information about the truth values compensate the problem with finding an ‘optimal’ set of truth values for the statements. Henceforth, it is quite reasonable if one assumes the proposed reasoning process as a rational ‘suggestion’. This is quite extendable to some other reasoning methods encountered in fuzzy logic, for example possibility qualification principle (See Sect. 4.4). Zadeh (1978) noted that possibility qualification should be considered as speculative, especially until when the nature of possibility qualification is clarified very well. Interestingly, possibility qualification principle states that a possibility qualification procedure induces a type-2 fuzzy set, due to the weakening effect of a linguistic possibility assigned to a soft constraint characterized by a type-1 fuzzy set. To present how the proposed provisional method assigns the verity distributions to a set of multiple self-referential statements, we consider some examples: The following two sets of self-referential statements are considered to be analogous to the truth-teller (Goldstein 1992): p1 : p2 isv true. p2 : p1 isv true. The corresponding mapping is: f : R2 → R2 f (v p1 (u), v p2 (u)) = ( f 1 , f 2 ) f 1 = v p2 (u) f 2 = v p1 (u)
(48)
g : R2 → R2 g(vq1 (u), vq2 (u)) = (g1 , g2 ) g1 = 1 − vq2 (u) g2 = 1 − vq1 (u)
(49)
and q1 : q2 isv false. q2 : q1 isv false. Its associated mapping is:
The solution for the fixed point of the mappings f and g leads to the following equations: v p1 (u) = v p2 (u) vq1 (u) + vq2 (u) = 1
(50)
It is apparent that there is an infinite number of solutions for both equations. We aggregate all the solutions in a type-2 verity distribution to capture the uncertainty due to multiple solutions for the equations:
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v p1 (u) = v p2 (u) = [0, 1] vq1 (u) = vq2 (u) = [0, 1]
(51)
As a more complicated example, consider the following set of self-referential statements: p1 : It isv very false that p1 ⇔ ( p1 ⇔ p2 ). p2 : It isv very false that p1 ⇔ ¬ p3 . p3 : It isv very false that p3 ⇔ ( p1 ⇔ p3 ). Its corresponding mapping is: f : R3 → R3 f (v p1 (u), v p2 (u), v p3 (u)) = ( f 1 , f 2 , f 3 ) f 1 = abs(v p1 (u) − (1 − abs(v p1 (u) − v p2 (u))))2 f 2 = abs(v p1 (u) − (1 − v p3 (u)))2 f 3 = abs(v p3 (u) − (1 − abs(v p1 (u) − v p3 (u))))2
(52)
which admits at least the following fixed points: [0.29969, 0.19997, 0.35284]T , [1, 0, 0]T , [0.29295, 0.12715, 0.35047]T . It is somewhat difficult to asses whether the mapping has other fixed points or not. However, since our approach does not concern the method of finding the fixed points, we assume that these are the only fixed points of the mapping. Our method suggests that we consider the following (set-valued) verity distributions for the statements: v p1 (u) = {0.29295, 0.29969, 1} v p2 (u) = {0, 0.12715, 0.19997} v p3 (u) = {0, 0.35047, 0.35284}
(53)
We prefer to assign crisp sets as the truth value of each statement, since there is no reason to prefer a particular solution to the others, and assign a fuzzy set to each statement as its truth value. It should be noted that by such a truth value assignment, we do not mean to assign inconsistent truth values such as the following to the aforementioned set of self-referential statements. v p1 (u) = 1 v p2 (u) = 0.12715 v p3 (u) = 0
(54)
Instead, we try to show that the problem imposes an ambiguity in selecting a set of consistent truth values for a particular set of statements. In this way, some information on the possible sets of consistent truth values is missed, which makes our suggestion a provisional one. However, no other reasonable method could be envisioned to aggregate all possible consistent truth value assignments.
11 Conclusions and future works We have considered different methods of self-referential reasoning by fuzzy logic tools in this paper. We showed that the previous studies are not adequate both in representing multiple self-referential statements and in deciding the set of truth values to be accepted, unless the truth qualification process is modified for sets of self-referential statements resulting in mappings with multiple fixed points. It appeared to be reasonable not to simply ‘choose’ a set of truth values for each statement, but to aggregate all correct truth values for a statement in a
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type-2 fuzzy set. Although this process reduces some amount of information about multiple self-referential statements, it completes the reasoning process for them based on the previous approach of Zadeh in solving the paradox of the liar via fuzzy logic. Self-referential reasoning by fuzzy logic has many potential applications. It could be helpful in truth maintenance in knowledge-based systems and conflict management in multiagent systems. It could prevent the occurrence of logical inconsistency in decision-making and uncertainty management. In this paper, we considered only statements which speak about the truth and falsehood of each other. However, in more realistic cases, the sentences might be the internal beliefs of many agents or game players, which bring about an inconsistent system of beliefs. Hence, each agent may have other statements in the set of its beielfs together with judgments on the truth or falsehood of its beliefs or others’ beliefs. A simple example is: Agent1: Agent2 is young. What Agent2 believes isv false. Agent2: Agent1 is old. What Agent1 believes isv very true. The next step in developing methods for handling metacognitive reasoning is finding solutions for such problems, which seem to be quite reachable following the approach of this paper. A somewhat similar problem of inconsistency detection is considered recently in the framework of fuzzy logic and classical AI, however, without any direct attribution to the paradoxes arising in cases in which the agents are reasoning about the truth of the beliefs of each other (Patching et al. 2006). It would be interesting to study the implications of existence of a universal set in fuzzy set theory and fuzzy logic. In this way, one may refer to the previous studies on axiomization of fuzzy set theory (Hajek and Hanikova 2003), and more generally, the studies on mathematical fuzzy set theory and mathematical fuzzy logic (Gottwald 2005, 2006).
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Fuzzy logic and self-referential reasoning: a comparative study with some new concepts Mohammad Reza Rajati · Hamid Khaloozadeh · Witold Pedrycz
© Springer Science+Business Media B.V. 2012
Abstract In this paper, we consider metacognition and self-reference realized in the framework of fuzzy computing. Different aspects of self-reference are discussed from the viewpoint of logic, linguistics, and artificial intelligence. Previous work in the area of fuzzy logic is also revisited in this context. It is shown that the reasoning based on the truth qualification principle requires some modifications so that it naturally gives rise to type-2 fuzzy sets, in order to capture the uncertainty about selection of a solution for a set of self-referential statements. Keywords Self-reference · Metacognition · Truth · Fuzzy logic · Liar’s paradox · Type-2 fuzzy sets · Veristic fuzzy sets
1 Introduction Self-reference and metacognition have been ongoing topics of interest, cf. (Bolander et al. 2006; Cox 2005; Crivelli 2004; Givant and McKenzie 1986; Seuren 2005; Smilek et al. 2007; Zadeh 1979). Logicians, mathematicians, psychologists, cognitive scientists, linguists, and philosophers have paid attention to this problem. Besides this long-term historical research
M. R. Rajati (B) Department of Electrical Engineering, University of Southern California, Los Angeles, CA, USA e-mail: [email protected] M. R. Rajati · H. Khaloozadeh Department of Electrical Engineering, K. N. Toosi University of Technology, Tehran, Iran H. Khaloozadeh e-mail: [email protected] W. Pedrycz Department of Electrical and Computer Engineering, University of Alberta, Edmonton, AB, Canada e-mail: [email protected] W. Pedrycz Systems Research Institute, Polish Academy of Sciences, Warsaw, Poland
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trail, the artificial intelligence community has focused on this issue since the inception of the area. Fuzzy logic, as a radically different approach to reasoning, has also contributed to provision of some understanding about self-referential reasoning. In this paper, we study the previous research on self-referential reasoning via fuzzy logic most of which is inspired by the original work of Zadeh, and present a modification to the reasoning procedure realized via truth qualification principle, which naturally gives rise to type-2 fuzzy sets. A thorough analytical study of the previously done works demonstrates that they cannot assign truth values to multiple self-referential statements in many cases. Some of the approaches also suffer from difficulty in reasoning about a large number of self-referential statements. It is worth noting that in this paper, we consider sets of sentences which only state some facts about the truth and falsehood of themselves or each other. The most well-known instance of such self-referential statements is the liar paradox, which indicates the falsehood of itself, and as we see later, has been the motivation of extensive research in many areas due to its paradoxical nature: p: This sentence is false. An example of a set of abovementioned self-referential statements would be: p1 := p2 is fairly false and ( p3 is very false or p3 is true) p2 := It is very true that ( p1 is fairly true if ( p3 is fairly false and p4 is fairly true)) p3 := ( p2 is very true if and only if p3 is very false) and p4 is fairly true p4 := It is very very true that ( p2 is very false if and only if ( p1 is true or p2 is very false or p3 is fairly true)) Although most of the works claim to accomplish the reasoning about the liar, they have difficulty even in the representation of multiple self-referential statements, and hence, fail to provide analysis tools to be generalized so that all self-referential statements containing utterances about the truth or falsehood of each other could be analyzed. Furthermore, they are not able to decide in many cases when the reasoning process yields more than one truth value for each statement. In contrast, the approach introduced by this paper first represents the set of multiple self-referential statements via the simple principles of fuzzy logic, and more specifically, translates them into multivariable mappings, and then shows that the possible truth values for each statement can be derived by finding the fixed points of such mappings. When there is more than a single truth value for each sentence, the reasoning process assigns their aggregation as type-2 truth values to each statement. We show that although the assignment of type-2 fuzzy sets removes some information about the original set of statements, it becomes necessary to complete the reasoning process. Actually, without such a modification to the truth qualification of self-referential statements, fuzzy logic is unable to decide about the ill-posed problem of self-referential reasoning. In order to provide the reader with an overall picture of the previous work on self-referential reasoning, we review some of the research in the areas of artificial intelligence, logic, mathematics, and philosophy, which we believe may contribute to a better comprehension of the works completed in the fuzzy logic community, and exhibit a potential to motivate further research on self-reference in the fuzzy logic community. The study is organized in the following manner. In Sect. 2, an overview of self-referential reasoning in artificial intelligence, logic, mathematics, and philosophy is presented. In Sect. 3, some comments are made about the research on self-reference in the area of fuzzy logic. In Sect. 4, we review some generic ideas of fuzzy logic that are sought as a prerequisite for further investigations presented in the paper. In Sect. 5, the ideas are related to the recently proposed general framework of fuzzy computation. In Sect. 6, some existing ideas on representation of self-reference via dynamic semantic maps are revisited. In Sect. 7, implicative
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representation of self-reference is considered. An attempt for the unification of the ideas is offered in Sects. 6 and 7. In Sect. 8, the computational verb approach to the evaluation of paradoxes is discussed. In Sect. 9, an enhanced method for truth qualification of self-referential statements is proposed, based on the conventions of fuzzy logic. In Sect. 10, some additional comments are made about the material presented in Sect. 9. Finally, some conclusions and future directions are discussed in Sect. 11.
2 Self-reference in artificial intelligence, logic, mathematics, and philosophy As we noted earlier, self-referential reasoning has attracted a considerable attention in artificial intelligence research. From the viewpoint of artificial intelligence, self-reference is a consequence of metacognitive reasoning (Cox 2005). An agent, who is able to make decisions for its future behavior, should be capable of realizing metacognition. It should decide which of different actions to perform under the current situation. It also must reason whether it has analyzed the decision adequately or more information and further analysis becomes necessary. After performing an action according to its deductions, it must ‘think’ about the outputs of its actions, and the strengths and weaknesses of its method of reasoning. Such reasoning procedures are called metacognition, since they involve ‘cognition about cognition’. Henceforth, a smart agent encounters reasoning about reasoning, before, during and after each of its actions (Cox 2005). In the sequel many subfields of artificial intelligence are involved in the research on metacognition. Model-based and case-based reasoning concern learning and reasoning about reasoning failure. Metacognition is also studied by the researchers in the area of decision theory and planning to enhance the choice of reasoning actions. Expert systems field contributes the facets of metaknowledge and rule control. Logical foundations are also provided for changing the beliefs, belief representation, and introspection (Bolander 2002). Metacognition involves self-reference (but as we discuss in this section, metacognition is more general), and when the truth of the beliefs of an agent is considered by it (or other agents), it may bring about logical inconsistency. Minsky (1961, 1965, 1968) believes that in order for a machine to describe its environment completely and describe its situation in its environment, it would have to possess an executable model of itself. From Minsky’s viewpoint, the agent should have a model of the environment E, and a model of itself M, in the environment. The model of the environment is denoted by E . It is used to process and answer inquiries about the environment. To answer inquiries about itself in the environment, the agent should have within E , a model of itself, denoted by M . Minsky notes that the agent must model its model of the environment as E in order to answer questions about its knowledge of its environment. M is agent’s knowledge of its knowledge and thinking methodology. The hierarchical introduction of higher order models could be continued, however it becomes heuristically less meaningful. An elaborate version of this model of metacognition could be found in the book Society of Mind (Minsky 1985). The model of Minsky for metacognitive agents could be generalized for multiagent systems. Agent i has a model of the environment, Ei , a model of itself in the environment, Mi , a model of any other agent, say j, Mij , a meta-model of the environment, Ei , and a meta-model of the environment perceived by agent j, Eij . Therefore, metacognition could be considered as more general than self-reference. Self-reference occurs when the agent reasons about itself, or reasons about other agents which directly or indirectly reason about it. By indirect self-reference, it is meant that agent r may reason about agent s, agent s may
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reason about agent t, and this process may continue until it reaches the situation in which an agent z is found which reasons about the first agent r . Therefore, agent r may be included in an ‘indirect’ self-referential reasoning procedure. In a multiagent system, when metacognition is about the truth of the cognitive processes of another agent (or the agent itself), it could result in logical inconsistency. McCarthy (1968) asserts that for an agent to adequately behave smartly, it must declaratively represent its knowledge. He furthermore emphasizes on agent introspection (McCarthy 1979), and having beliefs about mental states rather than about the environment. Accordingly, researchers working on the logical aspects of artificial intelligence have focused on reasoning about knowledge, belief, and internal states (Konolige 1985, 1988; Moore 1985, 1995). They have also tried to avoid the occurrence of logical inconsistency (Attardi and Simi 1991; Perlis 1985, 1988). From the viewpoint of logic, mathematics, and philosophy, self-reference is important mainly due to its aspect of paradox generation. Self-referential reasoning processes which convey deductions about the truth of propositions are ill-posed in many cases. They deviate from the fundamental principles of logic, mathematics, and philosophy. In terms of Aristotelian logic, self-reference could not be captured by the principle of bivalence, which asserts that every proposition is either true or false. In other words, many self-referential statements (at least seem to) contradict the well-known principle of excluded middle (Beal and Bueno 2002; Rieger 2001). Self-reference, the liar paradox, and similar paradoxes were addressed in the very old works of Epimenides, Eubulides of Miletus (Seuren 2005), and even Aristotle (Crivelli 2004). They noticed that the truth or falsehood of sentences like: “This very sentence is false” could not be evaluated easily, and results in a circular debate. Predictably, there is a controversial argument about the truth or validity of such a proposition (Barwise and Etchemendy 1987; Martin 1978, 1984; McGee 1991). The liar paradox was known for many Eastern philosophers and lead to thorough arguments among them. Among the solutions they provided, that of the renowned Iranian philosopher, mathematician, physicist, astronomer, and physician K. N. Toosi (or Tusi) is extremely interesting (Alwishah and Sanson 2009): “The liar is the result of a judgment that applies truth and falsity to something to which they in no way apply, and to apply them in any way is the misuse of a predicate. In other words, the fallacy is to suppose that the predicates “true” and “false” apply to a self-referential sentence like the liar.” Toosi’s argument is appealing since it tries to begin with the fact that truth predicates of a language could not be applied to the propositions of the same language (as we will see, to which the argument of Tarski is quite similar) and then implicitly tries to evaluate self-referential statements such as liar with a third truth value, neither true nor false (to which the solution of Kripke is quite similar, as we will see). From the viewpoint of linguistics and pure logic, self-referential statements call for an elaborate study of truth predicates. There are some formal theories about truth which share ideas between mathematics, philosophy, logic, and linguistics. Some famous examples are Kripke’s theory of truth and semantic theory of truth (Givant and McKenzie 1986; Kripke 1975). The semantic theory of truth has the following general principle for a given language: ‘ p’ is true if and only if p where ‘ p’ is a reference to the sentence (the sentence’s title), and p is just the sentence itself.
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Logician and philosopher Alfred Tarski developed the theory for formal languages such as formal logic. Here he restricted it in this way: that no language could contain its own truth predicate, that is, the expression “is true” could only apply to sentences in some other language. He called the latter an object language, the language being talked about. (It may, in turn, have a truth predicate that can be applied to sentences in still another language.) The reason for his restriction was that languages that contain their own truth predicate will contain paradoxical sentences like the liar: “This sentence is not true”. This statement, also known as the paradox of the liar, is deduced to be false, if considered to be true and vice versa, and this circular argument continues ad infinitum. As a result, Tarski held that the semantic theory could not be applied to natural languages, such as English, because they contain their own truth predicates (Givant and McKenzie 1986; Tarski 1944). On the other hand, Kripke’s theory of truth allows the existence of truth predicates in languages. Kripke builds a language in the following manner: Firstly, he begins with a subset of sentences in a natural language which do not contain expressions “is true” or “is false”. In the next step, he defines a truth predicate for such sentences. Then the definition of truth is generalized so that “is true” or “is false” could be applied to the sentences which are obtained in the preceding step. This process could be continued ad infinitum without any occurrence of liar-like inconsistencies. Note that Kripke’s theory implies the existence of a third truth value (say, “neither true nor false”) for sentences which are not built by his method. As a result, the liar is “neither true nor false”. Philosophical investigations about truth might be considered as the essence of philosophical challenges, and could be sought for in the early works of many schools of traditional and modern philosophy. One may refer to the works and scopes of Socrates, Plato, Aristotle, Avicenna, Aquinas, Kant, Kierkegaard, Nietzsche, James, Dewey, Belnap, and Habermas, among others. Treatment of such philosophical investigations is beyond the scope of this paper. Their relationship to the philosophical and logical aspects of the theory of fuzzy sets is an interesting topic which may be investigated in the future. The mathematical view of paradoxes of self-reference is highly relevant to that of mathematical and philosophical logic and linguistics. Perhaps Russell’s paradox is the most famous paradox of self-reference in mathematics, which is closely related to the liar paradox. Russell’s paradox is the result of accepting an axiom (due to Frege) in set theory, called the axiom of (unrestricted) comprehension: if Ψ is a predicate in the language of set theory, then there is a set that contains exactly those elements x such that Ψ (x). In other words, {x|Ψ (x)} is a set itself. Therefore assuming Ψ (x) to be x ∈ / x, one obtains Russell’s paradox: Let us consider = {x|x ∈ / x}
(1)
It is easily observed that every naïve set is a member of . However, investigating whether ∈ reveals the paradox. If it is true, then from the definition of , it is inferred immediately that ∈ / . If it is false, similarly it is inferred that ∈ . This circular and paradoxical debate quite resembles the argument on the truth or falseness of the liar. There have been several attempts to solve this paradox which implies that the very foundations of mathematics contain inconsistency. An example is the theory of types (Russell 1902) insisting that properties fall into different types, and that the type of a property is never the same as the entities to which it applies. Thus, the question never even arises as to whether a property applies to itself. Hence, the theory elaborates on illustrating the hierarchies of properties. A logical language that divides entities into such a hierarchy is said to employ the theory of types. The theory of types could be viewed as a mathematical counterpart of Tarski’s results. Another attempt was by Zermelo (1908) who formulated set theory to avoid the paradoxes of naïve
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set theory by replacing arbitrary set comprehension with weaker existence axioms, such as his axiom of separation (Aussonderung). Modifications to this axiomatic theory proposed in the 1920s by Abraham Fraenkel, Thoralf Skolem, and by Zermelo himself resulted in the axiomatic set theory called ZFC. This theory became widely accepted once Zermelo’s axiom of choice ceased to be controversial, and ZFC has remained the canonical axiomatic set theory until now (Kunen 1980). Some efforts have been performed to build a set theory in which a universal set exists which contains itself (Church 1974; Forster 1995; Oberschelp 1973; Quine 1937). Since our paper studies self-reference in the framework of fuzzy logic, it is appropriate to inquire that if fuzzy set theory is consistent with the existence of a universal set. It has been shown by White that infinite-valued predicate logic of Lukasiewicz is consistent with the axiom of comprehension and hence the existence of Russell’s set of all sets (White 1979).
3 Liar’s paradox in fuzzy logic Since the seminal work of Zadeh (1965), fuzzy logic has introduced itself as a dynamic area of research in many disciplines. From the early years of its formation, it has been subject to continuing arguments and philosophical, mathematical and technological investigations. For detailed discussions, see for example (Bˇehounek 2008; Cornelis et al. 2006; Entemann 2002; Hajek 1998; Novak 2006; Pelletier 2004; Trillas and Alsina 2001). Interestingly, there were a number of controversial yet sometimes not fully grounded and legitimized opinions (Elkan 1993, 1994a,b, 2001). The paradox of liar was also a subject investigated by researchers in the area of fuzzy logic. Zadeh has shown that the liar’s paradox could be captured by fuzzy logic (Zadeh 1979; Klir and Yuan 1996). As it will be seen in the sequel, following Zadeh’s notion, some researchers have developed methods for self-referential reasoning via fuzzy logic. On the other hand, Hajek, Paris, and Shepherdson (Hajek et al. 2000) have attempted at extension of Tarski’s results to the case of an infinite valued truth predicate. The crux of their approach is the result that a logical system with crisp Peano arithmetic and an infinite valued truth predicate can be consistent. Although their approach is merely logically inclined, it seems that it has been inspired by the approach of Zadeh. As they stress, if the arithmetic is kept to be two-valued, and the unary truth predicate Tr(.) is allowed to be multi-valued, the liar need not be contradictory as long as it is considered to possess the truth value of 1/2 in the context of predicate Lukasiewicz logic. Yet another logically grounded treatment of the liar is the approach of Zhang and Zhang (2004), which introduces a novel fuzzy logic. In their proposed fuzzy logic, they aim to model the internal states of an agent, or interactive relationships among agents as “bipolar” facts. The truth of the “negative” pole of each fact could be any value in [−1, 0] (or −V with the notations of this paper) and the truth of the positive pole could be any value in [0, 1] (or V). They start from the heuristic rule “(self-negation, self-assertion) ⇒ (self-consciousness, self-assurance)” in which the antecedent and the conclusion are bipolar facts about an agent. It is a self-negative statement, therefore we have the following fact for the liar: (self-negation, self-assertion) = (−1,0). Then, by the so called bipolar modus ponens ([(ϕ − , ϕ + ) ∧ ((ϕ − , ϕ + ) ⇒ (Ψ − , Ψ + ))] ⇒ (Ψ − , Ψ + )), it is concluded that the liar should be considered as an honest entity which quite reasonably talks about its mental state: “Sorry! I am wrong”, with the bipolar truth value (−1, 1). However, such a treatment is obviously difficult to be extended to a set of multiple self-referential statements, since there are no ways to extract such heuristic rules for an arbitrary set of metacognitive agents having beliefs
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about truth or falsehood of themselves and/or other agents. We do not compare this line of research with other methods of self-referential reasoning in this study, due to the fact that it could not be extended to multiple self-referential statements. In Kehagias and Vezerides (2006); Vezerides and Kehagias (2003), an algorithmic way is developed for converting a set of self-referential statements to a system of nonlinear equations and finding consistent truth value assignments to them. Although Vezerides and Kehagias have been successful in representation of a quite general class of self-referential statements, their approach does not determine how to ‘select’ a particular set of consistent truth value assignments, when there are more than one solution to the corresponding system of nonlinear equations. 4 Preliminaries In this section, we concisely present some preliminaries with intent to make the study selfcontained. In particular, we recall the notions such as of type-2 fuzzy sets, linguistic variables, linguistic truth variables, fuzzy logic, and possibility theory. 4.1 Fuzzy logic According to Zadeh,1 in fuzzy logic (Klir and Yuan 1995) for the expression “x is A”, the numeric degree of truth Tr(“x is A”) is defined as the degree of compatibility of x with the soft constraint A. Assume for example that: p := John is young Given that John is 25, the truth of p, Tr( p) is μ young (25). 4.2 Type-2 fuzzy sets Type-2 fuzzy sets are a special category of L-fuzzy sets introduced firstly by Goguen (1967). They were formulated in the primary form by Zadeh (1975a) to model the uncertainty about the membership values of a fuzzy set. Definition 1 Assume that U is the universe of discourse. A type-2 fuzzy subset of U, A˜ is defined by the membership values: μ A˜ : U → FV μ A˜ (u) = ϕu
(2)
wherein FV is the family of all fuzzy subsets of the unit interval V = [0, 1] and ϕu denotes a member of the family whose membership function is defined by: μϕu : Gu → V
(3)
In (3), Gu is a subset of V and x ∈ Gu is equivalent to μϕu (x) = 0, i.e. Gu is the support of ϕu . In other words, any function which is defined from Gu to V−{0} can serve as the membership function, depending on the fuzzy set of interest. Actually, a fuzzy set ϕu is assigned as the membership of each u instead of a unique membership value, to stress that the membership value itself is uncertain, and can be called 1 It should be noted that fuzzy logic is treated formally as a legitimate logical system obeying a set of axioms
(Cintula and Hajek 2010). For the purposes of this paper, the above broad interpretation of fuzzy logic, which is inspired by possibility theory, is sufficient.
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the fuzzy valued membership grade of A at u. μϕu can be called the secondary membership function of A at u. When for all u ∈ U, μϕu is a constant and Gu is an interval, especially when μϕu = 1, the type-2 fuzzy set is called an interval-valued fuzzy set. 4.3 Linguistic variables and linguistic truth variables For a better treatment of the concept of truth in fuzzy logic, it is instructive to recall the definition and concept of a linguistic variable (Zadeh 1975a,b,c). Definition 2 A linguistic variable is formally denoted by a quintuple (X , T (X ), U, G, M). X is the name or the label of the variable. T (X ) is a set containing the names of linguistic values related to X. G is a syntactic rule for generating the names X ∈ T (X ). M is a semantic rule (function) assigning a meaning M(X ) to each X . This meaning is a fuzzy subset of the universe of discourse U. Definition 3 A linguistic variable X is called structured if T (X ) and M could be characterized algorithmically. Thereby, the elements of T (X ) and their corresponding meaning assigned by M could be derived in an algorithmic fashion. This issue makes the manipulation of structured linguistic variables easier when the number of elements of the set T (X ) is infinite. Truth variables play a key role in everyday deductions, since it is of interest to characterize the degree of truth of statements by expressions like very true, true, fairly true, more or less true, absolutely false, and false. Correspondingly, truth, as a linguistic variable and the linguistic truth values such as ‘true’, ‘very true’, ‘false’, and ‘absolutely false’ have found an important role in approximate reasoning. Essentially, a linguistic truth variable is a structured linguistic variable, and its term set could be derived via algorithms. The term set of the linguistic variable “truth” contains linguistic truth values which could be derived in an algorithmic fashion. Furthermore, a semantic rule M could be found to assign meanings (and hence fuzzy sets) to each truth value. This view of fuzzy truth values is essentially possibilistic, and represents an uncertainty in the value of a numeric truth value. Hence, a linguistic truth value could be seen as a possibility distribution, and is not actually a genuine truth value leading to a genuine logical system. On the other hand, treating truth values as genuine truth values leads to a fuzzy linguistic logic (Dubois et al. 2007). As a result, we could define logical operations on genuine fuzzy truth values. Definition 4 Let τ1 and τ2 be linguistic truth values. The following operations are defined (Zadeh 1975b): The negation of τ1 , ¬τ1 is defined as: ¬τ1 = 1 τ1 μ¬τ1 (u) = μτ1 (1 − u)
(4)
in which denotes the extended subtraction operation. The conjunction τ1 ∧ τ2 and disjunction τ1 ∨ τ2 of τ1 and τ2 , are respectively defined as: τ1 ∧ τ2 = min (τ1 , τ2) μτ1 ∧τ2 (z) = sup min μτ1 (x), μτ2 (y) z=min(x,y)
τ1 ∨ τ2 = max (τ1 , τ2) min μτ1 (x), μτ2 (y) μτ1 ∨τ2 (z) = sup z=max(x,y)
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in which τ1 ∧ τ2 and τ1 ∨ τ2 are determined by fuzzy extensions of min and max, respectively. As noticed by Zadeh, such definitions are semantically different from the conventional complement “ ”, union “∪”, and intersection “∩” encountered in fuzzy set theory. More precisely, when τ1 and τ2 are linguistic truth values of two statements p1 and p2 , τ1 ∨ τ2 and τ1 ∧ τ2 are the truth values of “ p1 or p2 ” and “ p1 and p2 ” respectively. Moreover, ¬τ1 is the truth value of “not p1 ”. On the other hand, A1 ∩ A2 and A1 ∪ A2 are the ‘meanings’ of “A1 and A2 ” and “A1 or A2 ”, for two arbitrary fuzzy sets A1 and A2 . Similarly, A1 is the meaning of “not A1 ”. Essentially, a truth value as a genuine truth value is pertinent to type-2 fuzzy logic. In type-2 fuzzy sets, we assign a type-1 fuzzy membership value to each element, instead of a numeric value. Therefore, the truth functionality (Eqs. 4, 5) applies: it is equivalent to Zadeh’s fuzzy logical notion of truth of a statement. On the other hand, when dealing with truth as a linguistic variable, we must handle it like possibility distributions, and it represents uncertainty about numeric truth values. As possibility theory is the meaning-oriented notion of information processing, this view is equivalent to Zadeh’s notion of meaning of a statement. We must apply conventional complement, union, and intersection of possibility distributions, and truth functionality is not permitted. For an elaborate discussion on this issue, see Dubois et al. (2007); Haack (1979). 4.4 Possibility theory Possibility theory is initiated with the concept of a fuzzy restriction (Zadeh 1975d). If x is a variable which takes values from the universe of discourse U , and A is a fuzzy set in U , the sentence “x is A” is considered as introducing a fuzzy restriction on x where such an ‘elastic’ restriction is characterized by the membership function of A. This leads us to consider μ A (u) as the degree of possibility of x=u. Given a fuzzy set A in U , the sentence “x is A” provides a possibility distribution function assigned to x, denoted by πx : ∀u ∈ U πx (u) = μ A (u)
(6)
Zadeh establishes a rather simple, but extremely useful mathematical basis to extend the theory of possibility. The development of this theory is well-known in the fuzzy computing community, so we avoid repeating the material and refer the interested readers to the Zadeh’s original work (Zadeh 1978), as a contribution which is highly illuminating. Also, there are other approaches to the theory of possibility, which are axiomatic and based on Dempster–Shafer theory of evidence (Klir and Yuan 1995). Despite the fact that they are helpful, the approach of Zadeh is very capable in the ‘meaning-oriented’ possibility, and as we denote in this paper, it is very well-behaved for interpretation of self-referential sentences. So while trying to extend it, we will adhere to the essential features of the existing approach. In his paper, Zadeh proposes three principles for analyzing the information content of sentences, namely truth qualification principle, probability qualification principle and possibility qualification principle. As our paper concerns truth and possibility qualification principles, we reconsider them briefly. 4.5 Truth qualification and possibility qualification principles Truth qualification principle tries to judge the truth of sentences which speak about the truth of other sentences. If:
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x is A → πx (u) = μ A (u) then x is A isv τ → πx (u) = μτ (μ A (u)) in which τ is a linguistic truth value (a veristic constraint) such as: true very true, more or less true, and false. “→” shows that the statement induces a possibility distribution function. The truth qualification principle has been applied to the medical diagnosis problem by the aid of relational equations (Sanchez 1996). Recently, there have been some studies on the truth values (Gera and Dombi 2007, 2008; Vychodil 2006; Walker and Walker 2005) and the truth qualification principle (Own 2009; Zadeh 2008). It is evident that the possibility qualification principle qualifies sentences speaking about the possibilities of other sentences and events. If: x is A → πx (u) = μ A (u) then x is A is α-possible → π˜ x (u) = ϕu Gu = [min(α, μ A (u)), min(1, 1 − μ A (u) + α)] ⊆ V Here α-possible is a linguistic possibility value (α ∈ [0, 1]), and π˜ x represents a type-2 possibility distribution. This type-2 fuzzy set is constructed to show the uncertainty caused by weakening the proposition by a linguistic possibility value. Zadeh, proposed that μϕu (x) = a ∈ [0, 1] ∀x ∈ Gu , i.e. an interval-valued fuzzy set. Replacing α = 1 for ‘completely possible’ Gu reduces to: Gu = [μ A (u), 1]. Other versions of the possibility qualification principle could be envisioned as well in which we assign type-2 fuzzy sets as the possibility distribution of statements.
5 Fuzzy truth values, generalized theory of uncertainty and self-reference There are some studies which consider self-referential reasoning in the scope of fuzzy logic. As we noted, Zadeh’s work on the resolution of the paradox of the liar via truth qualification principle is the first work in this field (Klir and Yuan 1996; Zadeh 1979). It follows his notion of information processing as manipulation of fuzzy constraints (Zadeh 1975d). This view on information processing was fertilized with the establishment of possibility theory, especially via the qualification principles we reviewed briefly (Zadeh 1978) and is now more fruitful because of Zadeh’s recent works on development of a Generalized Theory of Uncertainty (GTU) (Zadeh 2005, 2006, 2008). Generalized Theory of Uncertainty tries to model information, whatever its form, as a generalized constraint. A generalized constraint is represented as: X isr R where X is a constraint variable, R is a constraining relation and r is an indexing variable denoting the semantics of the constraint. Different kinds of constraints are distinguishable on information; possibilistic (r = blank); probabilistic (r = p); veristic (r = v); usuality (r = u); random set (r = rs); fuzzy graph (r = fg); bimodal (r = bm) and group (r = g). Such constraints could be qualified, combined, and propagated, leading to the Generalized Constraint Language.
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The centerpiece of this paper is the concept of veristic fuzzy sets and veristic constraints, so we follow Zadeh’s recent notation. When we want to accommodate a veristic constraint τ on a statement p j we have the following statement pi : pi := p j isv τ in which τ is a truth value. Taking a glance at the liar sentence: p := p isv not true It could be concluded that this sentence puts a constraint on its truth. Thus its truth could be evaluated via the truth qualification principle. The proposition is reformulated as: p := The truth value (constraint on verity) of p isv τ in which τ is equal to ‘not true’ or false. Actually, this statement assigns a linguistic truth value τ to the verity distribution2 of the statement p. Considering the membership function of the constraint on the truth of p (i.e. the verity distribution associated with p) as v p (u), it is easily observed that by the truth qualification principle, we have: 1 (7) 2 Remark It should be noted that the truth qualification principle is not about assigning linguistic truth values to numeric truth or possibility values. Therefore, we did not use the truth qualification principle to derive a numeric truth value v = 1/2 for the liar. However, some authors (Kosko 1996) prefer to view the truth qualification procedure for the liar as yielding a numeric truth value for it. v p (u) = 1 − v p (u) ⇒ v p (u) =
It is worth emphasizing again that v p (u) represents a verity distribution, and therefore, should be treated by the calculi of possibility theory. Actually, Zadeh accepts the fixed points of the membership function μτ (u) as the value of the membership function v p (u) associated with the verity distribution of the self-referential statement, for all u ∈ [0, 1]. 5.1 Semantics imposed by Zadeh’s method of self-referential reasoning It is by Zadeh that the truth value represented by the membership function μ(u) = 1 could be defined as the truth value unknown (Zadeh 1975a). He then tries to relax the ‘ignorance’ imposed by the term ‘unknown’, via defining the set of truth values ‘α − unknown’. μ(u) = α is representative of these truth values. One may conclude that the reasoning procedure in (7) yields the truth value 1/2-unknown for the liar sentence. However, it should be noted that this type of conclusion is not very useful. Actually, when we are seeking the truth space, we demonstrate that, when a truth value is unknown, it can be each member of the truth space. It is just the same as this trivial semantic interpretation: “when a truth value is unknown, it is equally possible to be any member of V = [0, 1]”. On the other hand, consider the following reasoning process: p := John is young q := The truth value of p isv unknown q models complete ignorance about the truth of p. Truth qualification yields: π Age(J ohn) (u) = 1. Therefore, John’s age may be any arbitrary value in the universe of discourse. By the notion of fuzzy logic presented in Sect. 4.1., the truth value of q, T r (q), is the compatibility 2 A verity distribution is essentially a possibility distribution on the space of numeric truth values V = [0, 1].
Therefore, as a linguistic value is analogous to a possibility distribution, a linguistic truth value is analogous to a verity distribution.
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of the truth of p with unknown which is always 1, whatever the truth value of p is. For self-referential statements, both of p and q are the same. Because the sentence itself contains information about its truth, the result of the reasoning process needs more effort to be interpreted. For example, for the liar sentence, referring to the definition of fuzzy logic in Sect. 4.1., supposing the truth value of p be any value x in V = [0, 1], it is deduced that the truth value of p is the degree of compatibility of x with v p , i.e. v p (x) = 1/2. Then there is no way but to assert that p is half true, because any assumption will yield that: Tr( p) = 1/2. It is possible to interpret Zadeh’s solution of the liar paradox more easily. Concerning the concept of fuzzy logic presented in Sect. 4.1., it is easy to show via the truth qualification principle that when the truth value of a statement p is known to be Tr( p), the truth value of the statement q : p is τ is μτ (T r ( p)). Therefore, assuming the numeric truth value of liar to be v, one can easily infer that: v =1−v ⇒v =
1 2
(8)
This could be a justification for dealing with the numeric truth value of the liar, as Kosko suggests (Kosko 1996). Another helpful idea is the viewpoint of two different kinds of truth values we presented in Sect. 4.3.: truth values as members of the term set of the linguistic variable TRUE, and truth values as genuine logical entities which are pertinent to type-2 fuzzy logic, i.e. they can be employed as fuzzy membership values (secondary membership grades) in type-2 fuzzy sets. Obviously, here we deal with the first notion of fuzzy truth values which is possibilistic. Therefore, the result v p (u) = 1/2 says that, the verity (analogous to the possibility) that the numeric truth value of the liar is u, assuming that u is any member of the unit interval, is 1/2. However, one may prefer the first interpretation of Zadeh’s solution to the liar, since despite its intricacy, it perfectly illustrates the special condition of the liar in which the subject and object are the same, and both are the statement itself: It starts with an unknown verity distribution function for the liar, and shows that after the truth qualification procedure, assuming any numeric truth value for the liar leads to the conclusion that its truth value is 1/2. Henceforward, in this paper, we assume verity distribution functions for self-referential statements. Another important fact reveals itself here: essentially the truth qualification principle manipulates the ‘meaning’ of the statement which it tries to qualify, which comes from its possibilistic nature. As we noted in Sect. 4.3., ‘not true’ could be treated in two ways: when one refers to the meaning of ‘not true’, μnot tr ue (u) = 1 − μtr ue (u). In contrast, when ‘true’ is the linguistic truth value of a statement p, the truth value of ‘not p’ is characterized by μtr ue (1 − u). The nature of truth qualification principle is meaning-oriented and it is quite compatible with the semantics of natural languages. Consequently, when Zadeh applies the truth qualification principle to the liar which states a fact about its truth value, he preserves the ‘meaning-oriented’ nature of the truth qualification principle. Therefore, he does not interpret the liar as: v p (u) = v p (1 − u)
(9)
which implies an infinite number of linguistic truth values for the liar. This interpretation is not semantically valid. Accordingly, when connectives are present in the statements, one must use conventional connectives of fuzzy sets rather than logical operations on fuzzy truth values. This fact will be used in Sect. 9.
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6 Representation of self-reference via dynamic semantic maps Inspired by sequential reasoning about the liar, Grim views the liar sentence as a dynamical system (Grim 1993). He asserts that the oscillating nature of reasoning about the liar, whose truth results in its falsehood and vice versa, might be modeled by a simple dynamical system: tn+1 = 1 − tn
(10)
He concludes that this bivalent map, which oscillates between 0 and 1 (true and false) is also valid when the initial value of tn is assumed to be a member of [0, 1]. The behavior of the map is still oscillatory although it can take values in the interval of [0, 1]. This discussion leads to interpretation of self-referential reasoning as a dynamical system. For example, in Grim’s paper, the following proposition is called the fuzzy logistic: p := It isv very false that p isv true iff it isv false Utilizing the fuzzy equivalents of ‘very false’, ‘true’ and ‘false’ and generalizing the previous discussion, the following map is obtained: tn+1 = (1 − (1 − abs(tn − (1 − tn ))))2
(11)
Grim’s approach is extendable to multiple sentences, each of which is speaking about the truth of itself or the others. They also yield dynamical mappings whose state variables are the truth values of the sentences. The fuzzy inconsistent dualist is a sound example: p1 := p1 isv true iff p2 isv true p2 := p2 isv true iff p1 isv very false The corresponding dynamical system is: tn+1 = 1 − abs(tn − rn ) rn+1 = 1 − abs(rn − (1 − tn )2 )
(12)
In the rest of his article, Grim inspects many sets of self-referential sentences and derives their resultant dynamical systems. He shows that the so-called semantic systems may be chaotic or have strange attractors. The main goal of his studies is to illustrate that fuzzy logic does not remedy the problem of self-reference and the dynamical systems which model the reasoning process may not converge. Grim does not address the work of Zadeh in his article; instead, he reaches the ‘dynamical system’ approach via generalization of the cyclic reasoning about the liar. Furthermore, he considers numeric truth values as the states of the dynamic systems pertinent to self-referential statements, in contrast to truth qualification approach which essentially deals with verity distributions (although, as we mentioned in Sect. 5.1., truth qualification procedure can somehow support considering numeric truth values for the liar and other self-referential statements). However, about the liar (and as it could be seen, all other self-referential sentences), the work of Grim, verifies Zadeh’s insight indirectly. The abovementioned dynamical systems may be chaotic or oscillatory unless they are initiated from one of their fixed points. Hence the best solution is to accept to start the reasoning process from the fixed points of the related dynamical system. They are proved to exist under some rather weak assumptions as we see in Sect. 9. This is the same as accepting the fixed point itself as the solution, since the system states do not change in this case. Consider the dynamical map generated by the liar sentence: tn+1 = 1 − tn t0 = 1/2 ⇒ tn = 1/2
∀n>0
(13)
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It is obvious that starting from the fixed point of the system, it would not exhibit oscillatory behavior. This is analogous to the approach of Zadeh in accepting the fixed point of false as the membership function of the truth value of the liar.
7 Implicative interpretation of self-reference In another attribution to self-referential reasoning in the framework of fuzzy logic, Chen (1999) revisits the liar. The work is apparently influenced by Zadeh’s method for resolution of the liar. Chen’s paper only considers the liar and does not refer to other forms of selfreference, except in a section in which indirect self-reference in the inconsistent dualist is considered: p1 := p2 isv true p2 := p1 isv false However, Chen views these two sentences as a special form of the liar. The article employs the cyclic reasoning caused by the liar to construct an alternative interpretation of it: q1 := If p isv true, then p isv false q2 := If p isv false, then p isv true This is obviously another aspect of the paradoxical nature of the liar. It generates q1 , q2 which are inconsistent, at least in classical logic. Again, assuming that v p (u) is the membership function of the truth value of p, statements q1 , q2 are translated to the following: vq1 (u) = I (v p (u), 1 − v p (u)) vq2 (u) = I (1 − v p (u), v p (u))
(14)
in which I is a fuzzy implication function:3 I : [0, 1] × [0, 1] → [0, 1] I (0, 0) = I (0, 1) = I (1, 1) = 1 I (1, 0) = 0
(15)
To reflect the paradoxical inherence of p, q1 and q2 should be equally true, thereby it is intuitively realized that: I (v p (u), 1 − v p (u)) = I (1 − v p (u), v p (u))
(16)
It is clear that independent of selecting a particular implication function, v p (u)=1/2 is a solution to (16), due to its symmetric nature. From this viewpoint, the paradox of the liar reveals itself when someone tends to decide whether the liar’s sentence is true or false. Generalizing this argument, one might be inclined to demonstrate whether the self-referential sentence: p := ( p is τ ) is α or β, where α, β and τ denote truth values. We have the following reasoning procedure: q1 := If p isv α then p isv τ isv α q2 := If p isv β then p isv τ isv β vq1 (u) = I (μα (v p (u)), μα (μτ (v p (u)))) vq2 (u) = I (μβ (v p (u)), μβ (μτ (v p (u))))
(17)
3 Implications are also shown by the symbol ‘⇒’ in fuzzy logic. We have used this notation in this paper as
well. The main reason for using I (., .) in this section is its suitability for presenting Chen’s approach.
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Upon considering that v p1 is equal to v p2 we have: I (μα (v p (u)), μα (μτ (v p (u)))) = I (μβ (v p (u)), μβ (μτ (v p (u))))
(18)
This leads to definition of α–β–τ legitimacy. When we are to reason about the relationship between assigning a truth value of α or β to a self-referential proposition and assigning a truth value of τ to it, it makes sense to try to find a v p (u) such that (18) holds. When a solution v p (u) exists, the self-referential expression is called to be α–β–τ legitimate, i.e. its problematic meaning could be captured in the landscape of fuzzy logic. The solution(s) v p (u) is (are) therefore called the I -truth value(s) of the liar paradox. In this sense, the liar paradox is true-false-false legitimate. Chen tries to find conditions for μτ , μα , μβ , which make (18) have a solution. As stated previously, Chen’s paper devotes a section to indirect self-reference. It considers the inconsistent dualist as a special form of the liar. So it utilizes a similar method, based on implication, to interpret it: p1 := p2 isv true p2 := p1 isv false For p1 we have the inference: q1 := If p1 isv true, then p1 isv false q2 := If p1 isv false, then p1 isv true We could also construct similar sentences for p2 . The following equation is then concluded: I (v p1 (u), 1 − v p1 (u)) = I (1 − v p1 (u), v p1 (u))
(19)
In a general setting, the following pair of sentences could be considered: p1 := p2 isv γ p2 := p1 isv δ in which γ , δ are truth values. Inspecting if p1 and p2 are α or β we can assert that: q1 := If p1 isv α, then p2 isv γ is α, then p1 isv δ isv γ isv α Thereby: vq1 (u) = I (μα (v p1 (u)), μα (μγ (μδ (v p1 (u)))))
(20)
In a similar way: q2 := If p1 isv β, then p2 isv γ is β, then p1 isv δ isv γ isv β whose truth is: vq2 (u) = I (μβ (v p1 (u)), μβ (μγ (μδ (v p1 (u)))))
(21)
As the past, the truth values of both of the fuzzy implications are argued to be identical: I (μα (v p1 (u)), μα (μγ (μδ (v p1 (u))))) = I (μβ (v p1 (u)), μβ (μγ (μδ (v p1 (u)))))
(22)
We have also the following for v p2 : I (μα (v p2 (u)), μα (μδ (μγ (v p2 (u))))) = I (μβ (v p2 (u)), μβ (μγ (μδ (v p2 (u)))))
(23)
Again, the self-referential set of sentences is α–β–γ –δ legitimate if there is at least one solution pair (v p1 , v p2 ) to (22) and (23). It is somewhat difficult to conclude about the existence
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of solutions to this problem generally. However, using Brouwer’s fixed point theorem (see Sect. 9), Chen tries to show that there are solutions for v p1 (u) and v p2 (u), in some special cases such as when α and δ denote true and both β and γ denote τ , and μτ is a continuous isomorphism on V = [0, 1]. Unfortunately, Chen’s approach is not easy to extend for other self-referential sentences. To illustrate this issue, let us consider the following set of statements: p1 := p2 isv γ p2 := p3 isv δ p3 := p1 isv η We are going to decide if p2 is α or β, using Chen’s approach: If p2 is α, then p3 is δ is α, then p1 is η is δ is α, then p2 is γ is η is δ is α. On the other hand, if p2 is β, then p3 is δ is β, then p1 is η is δ is β, then p2 is γ is η is δ is β. At last we have: I (μα (v p2 (u)), μα (μδ (μη (μγ (v p2 (u)))))) = I (μβ (v p2 (u)), μβ (μδ (μη (μγ (v p2 (u))))))
(24)
This procedure becomes more complex and less algorithmic when the number of sentences becomes large, and the logical functions of each sentence go more intricate. For example, this procedure is tedious for the following set of propositions (The Greek letters represent truth values): p1 p2 p3 p4
:= := := :=
p2 isv γ if and only if ( p3 isv γ or p4 isv ε) p1 isv ζ or p3 isv κ p1 isv η p2 isv ω or p4 isv λ
Moreover, finding conditions on the truth values to force the equations to have solutions (making them legitimate in some sense) becomes harder. Thus, taking Chen’s approach as a general reasoning tool is impractical. Furthermore, Chen does not propose a method for ‘optimal’ selection of implication functions (and also other logical functions). He does not offer any criterion for selection of a solution, when more than one solution is found for the truth values. Another problem to his approach is the possibility of the definition of higher orders of legitimacy for any set of self-referential statements. By deriving the same equations as (22), we could judge whether a member of a set of self-referential statements is e.g. α1 -α2 -α3 -α4 -τ legitimate, but this requires generating four sentences q1 , q2 , q3 , q4 for each statement, and try to find a v pi (u) such that it makes vq1 (u) = vq2 (u) = vq3 (u) = vq4 (u). It is debatable that which order of legitimacy is ‘adequate’ for resolution of a set of self-referential statements. Obviously, reasoning about higher order legitimacy of the statements suffers from more severe versions of the abovementioned problems, and this may justify neglecting it. It is also worth noting that assigning numeric truth values to self-referential statements via this approach is imaginable, following our previous discussion on assigning a numeric truth value to the liar, suggested by Kosko (cf. Sect. 5.1.) Chen’s approach could be a tool for generation of dynamical semantic maps, such as those considered by Grim. However, again the abovementioned problems make it an impractical one.
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8 Computational verb systems and liar’s paradox In another treatment of the paradox of the liar, computational verb systems are employed by Yang (2001). Computational verb systems are proposed to capture the dynamical semantics imposed by the use of verbs in natural languages. A computational verb α is defined by the following: (1)
A set Eα which is called the evolving system in the context of computational verbs:
Eα = {T, X s , X p , Aα , Bα }
(2)
where T ⊂ , X s ⊂ 3 and X p ⊂ k symbolize time, space and cognition and associate these features with a verb. Aα is a function called the inner system and models self-reflection in the brain: Aα = T × X s × X p → T × X s × X p
(3)
(25)
(26)
The outer system Bα is a model of the existing world and is defined as: Bα = T × X s × X p → T × X s × X p
(27)
Here, T ⊂ , X s ⊂ 3 and X p ⊂ k are the ‘observed’ time, space and cognition. To involve the context in which a verb sentence is asserted, and to evaluate its truth or falsehood, a transformation is made which is called a BE-transformation. It transforms the verb sentence into the following so-called BE-proposition: S: [Subject V er b Object] BE word
(28)
in which word could be an adjective. Now, the numeric truth value of the above statement could be evaluated. To capture this concept mathematically, the canonical BE-transformation is defined as a mapping which transforms a point of the time-space-cognition P to a numeric truth value between 0 and 1 inclusive. Essentially, if the inner system of a verb is known, the BE-transformation I word (V er b, P) could be illustrated as: I word (V er b, P) : × T × X s × X p → [0, 1]
(29)
wherein V er b ∈ is a verb in the set of all of the verbs of the language of discourse and P is a point in the time-space-cognition space. If the outer system of the Verb is known, we have: O (V er b, P) : × T × X s × X p → [0, 1] (30) word wherein Verb∈ is a verb in the set of all of the verbs of the language of discourse and P is a point in the time-space-cognition space. To obtain an interpretation of the paradox of the liar which could be captured by computational verb sentences, Yang starts from Chen’s interpretation which is mentioned in Sect. 7. Then he concludes that Chen’s interpretation can be unified into a single verb statement: p is what p is not Consequently, a general form of the liar is extracted as a verb sentence:
S : [Subject V er b what Subject NOT V er b]
(31)
Yang also offers some examples like the ones below: I say what I do not say
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I do not feel what I feel Then, a BE-transformation is applied to such sentences: S : [Subject V er b] BE word [Subject NOT V er b] BE word
(32)
Now, regarding the context in which the BE-transformation of Eq. (32) is performed, a verb sentence might be paradoxical or non-paradoxical. Yang uses a function : [0, 1]×[0, 1] → [0, 1] to show the paradoxical value of the verb sentence. Therefore, the paradoxical value of a verb sentence could vary between 1 (completely paradoxical) and 0 (non-paradoxical). Let us consider the paradox of the liar, which is transformed to “This sentence is what this sentence is not”. In order to find a reasonable paradoxical value for the liar, let v1 be the numeric truth value of the first part (i.e. the assertion that the sentence is true) and v2 be the numeric truth value of the second part (i.e. the assertion that the sentence is not true). Then Yang concludes that: (1, 0) = (1, 0) = 1 and (0, 0) = (1, 1) = 0. It is obvious that an infinite number of candidates for are available. Yang chooses the following: 1 (v1 − 0.5)(v2 − 0.5) < 0 (v1 , v2 ) = (33) 0 (v1 − 0.5)(v2 − 0.5) > 0 The function (33) returns a Boolean paradoxical value. To obtain a function which yields a continuum of truth values between 0 and 1, Yang proposes the following paradoxical value function: (v1 , v2 ) =
1 − exp(abs((v1 − 0.5) − (v2 − 0.5))) 1 − exp(1)
(34)
Yang tries to classify the paradoxes into three categories. First, he defines the so-called paradoxical strength of a verb sentence: word∈ P∈χ word (V er b, P), word (not V er b, P) Ψ (S) = (35) word∈ P∈χ 1 in which S is the verb sentence of interest, is collection of all possible words which can be used to derive BE-transformations of S.χ is a discrete subset of the time-space-cognition space. Actually it includes all contexts that words can impose. For a continuous χ, obviously the summation is converted to a Riemannian integral: word∈ χ word (V er b, P), word (not V er b, P) d P (36) Ψ (S) = word∈ χ d P A Linguistic Classification is then done according to the paradoxical strength. If Ψ (S) is near 1, it indicates a strong verb paradox. If it is near 0, it indicates no verb paradox. If it is between 0 and 1, it indicates a weak verb paradox. As the previous sections, here we try to analytically compare Yang’s method with the other methods. The main difference is that the approach does not tend to solve the paradox with fuzzy logic. Rather it utilizes fuzzy logic to sort the degree with which a sentence is paradoxical. The method claims to include context in the reasoning process. The main problem with such an approach is that it is not justifiable to see the liar paradox similar to the sentence (31). In classical logic, sentences like “This sentence is what this sentence is not” are simply false and this will not result in a circular reasoning continuing ad infinitum.
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Since this approach relies on interpretation of self-referential statements by Chen’s method, it suffers from the same limitations as his approach, and is not amenable to be generalized to multiple statements. Multiple statements should be translated to a single statement of the form (31) and then using BE-transformations, their paradoxical value : [0, 1]m → [0, 1], in which m is the number of multiple statements, is determined. This approach is not quite algorithmic and comprehensible.
9 Self-reference needs extended truth qualification In this section, we present a unified framework for manipulation of self-referential expressions via fuzzy logic as follows: For simplicity, assume that we have sentences in which every sentence only speaks about the truth of itself and other sentences in the set. First, we start with a rather general form of such sentences. Assume that: Si j := H j1 p j isv τ ij1 •H j2 p j isv τ ij2 • · · · •H jm( j) p j isv τ ijm( j) (37) Here, Si j is an expression which contains m( j) statements about the truth of statement p j . Definitely, each τ ijk , k = 1, . . ., m( j), is a truth value. The sign • represents any of the logical connectives ∧, ∨, ⇒, ⇔ . H represents a set of hedges. Each sentence in the set of self-referential statements is of the form: pi := Hi (Si1 •Si2 • · · · •Siq(i) )
(38)
Hi stands for a set of hedges. It should be noted that the aforementioned form of self-referential statements is not the most general case, albeit it can cover a vast range of self-referential statements. It is presented merely for obtaining a better understanding of the problem. For example, the following sentence could not be interpreted in the form of Eq. (38): p1 := p2 isv true and it isv fairly false that ( p3 isv very false and p3 isv fairly false) However, it is quite straightforward to derive its logical interpretation:
v p1 (u) = v p2 (u)
1 − (1 − v p3 (u))2 1 − v p3 (u)
(39)
wherein we have used the common definitions for fairly and very and algebraic product for the logical connective ‘and’. It is worth noting that Kehagias and Vezerides have previously attempted at building a language which can transform the members of a quite general class containing sets of selfreferential statements to nonlinear equations (Kehagias and Vezerides 2006; Vezerides and Kehagias 2003). One may be interested in the exact algorithmic way of converting a set of self-referential statements to a system of nonlinear equations. As a general rule, one must build a language whose atomic terms are statements of the form p j isv τ ij1 and derive a suitable algorithm. However, it seems that such a conversion is easily performed by an expert, although building formal languages is quite possible by extending the approach of Vezerides and Kehagias or by means of the results concerning the concept of fuzzy formulas (Shen et al. 1988). As a comprehensive example, consider the following set of self-referential statements:
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p1 := p2 isv true and ( p3 isv very false or p3 isv fairly true) p2 := It isv very true that ( p2 isv very true iff ( p3 isv fairly false and p4 isv fairly true)) p3 := ( p1 isv very true iff p3 isv very false) and p4 isv fairly true p4 :=It isv very very true that ( p1 isv false iff ( p1 isv true or p2 isv fairly false or p3 isv false)) Its resultant map is thereby: f : R4 → R4 f (v p1 (u), v( p2 (u), v p3 (u), v p4 (u)) = ( f1 , f2 , f3 , f4 ) 2 , v (u)) (u)) f 1 = v p2 (u) × max((1 − v p3 p3
f 2 = (1 − abs(v p2 (u)2 − ( 1 − v p3 (u) × v p4 (u))))2 f 3 = (1 − abs(v 2p1 (u) − (1 − v p3 (u))2 )) × v p4 (u)
f 4 = (1 − abs(1 − v p1 (u) − max(v p1 (u), 1 − v p2 (u), 1 − v p3 (u))))4
(40)
In the logical interpretation of the statements, the conventional functions for hedges ‘very’ and ‘fairly’, maximum for ‘or’ and algebraic product for ‘and’ are used arbitrarily. One has many other choices for them. We discuss this fact in this section more thoroughly. Following Zadeh’s approach in accepting the fixed point of the mapping generated by the liar as the solution, the fixed point(s) of the mapping generated by the set of equations (we derived such a mapping in (38)) could be considered as the solution(s) of self-referential sentences. By Brouwer’s fixed point theorem, it is obvious that for the existence of a solution it is sufficient that the t-norms, t-conorms, c-norms and hedge functions are continuous on V = [0, 1]. It is rather a mild condition and we could always use many well-known and continuous functions in the reasoning procedure: Theorem 1 (Brouwer’s Fixed Point Theorem). Let g(x) : D → D be a continuous surjective function, and D ⊂ Rm be closed and connected. The equation g(x) = x has at least one solution x0 ∈ D which is called a fixed point of g(x) (Rudin 1976). In such an approach, the maps may have many (and maybe an infinite number of) fixed points. Also, there might be different mappings for the sentences due to various choices of t-norms, t-conorms and c-norms, and hence different sets of fixed points. An important question arises here: Which solution to select? One might be inclined to define a measure to select the solution, or make use of the opinion of an ‘expert’. But it seems that it is somewhat meaningless to select a solution and refuse others. Also, it is difficult to find an objective function to evaluate the solutions. These barriers lead us to model the uncertainty in the selection of proper solutions via fuzzy logic tools. Type-2 fuzzy sets are efficient tools in modeling uncertainty in membership functions of fuzzy sets. Verity distributions are represented by veristic fuzzy sets (fuzzy truth values). Therefore, we could use a ‘set of solutions’ as the membership values of the veristic fuzzy set. This approach is quite consistent with the conventional methods of fuzzy logic. Essentially, this is an extension to the truth qualification procedure. Furthermore, it is yet another step to enrich the ability of fuzzy logic such that it can consistently assign truth values to sets of self-referential expressions. For non self-referential sentences, it is easy to apply the truth qualification principle and other laws of fuzzy logic to obtain the possibility distributions. For self-referential sentences, the output of the qualification process may be a type-2 fuzzy set. Assume that F = {V 1 , . . ., V 2 , . . ., V k , . . .} is the set of all fixed points of the mapping f generated by a set of self-referential statements. Thereby each V i could be shown as:
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T j j j V j = v p1 (u), v p2 (u), . . . , v pn (u) F may be a countable or uncountable, finite or infinite set. The truth value of each sentence becomes: v pi (u) = Gui = v 1pi (u),v 2pi (u),…,v kpi (u), . . . (41) This is a multi-valued or infinite valued membership function for each veristic fuzzy set. This approach is plausible and consistent with the methods of reasoning in fuzzy logic, but it is not flawless. Its main problem is that it does not always yield one kind of type-2 fuzzy set. In some cases, v pi (u) is a uni-valued (thus type-1) or multi-valued membership function and in the other occasions, it becomes an interval valued fuzzy set. It may be sometimes more convenient to interpret the verity distributions as interval valued fuzzy sets, especially in case they happen to be used in the design of a knowledge based system, or are exploited in another reasoning procedure. In this case, we suggest the following verity distribution functions: v pi (u) = inf Gui , sup Gui (42) This approach is similar to the method of considering uncertainty bounds in membership functions of rule-based fuzzy decision making and control (Wu and Mendel 2002). The next task is assigning a membership function to Gui to make the set valued membership functions be fuzzy set valued ones: v pi (u) = iu
(43)
iu is a fuzzy subset of Gui . To accomplish this task, Mendel’s proposals for modeling uncertainties via type-2 fuzzy sets are helpful (Mendel 2003). He believes that asking persons to assign anything other than a uniform weighting to the set valued memberships would be difficult. Thereby, we are guided to choose: μiu (x) = c x ∈ Gui , u ∈ U
(44)
c is a constant, and especially it may be taken 1. Mendel also suggests to model uncertainties existing among a group of people by means of aggregation of each person’s equally weighted Gui ’s. If some of the experts are more trustworthy, their opinion might be weighted more than the others. Thus, instead of assigning a predetermined μ to the members of Gui , we could refer to a group of experts in the area of fuzzy reasoning and logic and build a type-2 fuzzy set according to Mendel’s prescriptions. It should be noted that discussion about proper choices for μ is less important than the previous issues on taking the uncertainties about selection of the solutions into account. To present an illustrative yet simple example, let us consider the so-called truth-teller: p : p isv true. The truth-teller is known to be logically problematic, and some authors (Goldstein 1992) consider it as a counterpart of the liar. This statement seems to produce no paradox. However, it could be assumed to be both true and false! To capture its truth value via the proposed method, we have the following equation by truth qualification principle: v p (u) = v p (u)
(45)
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The set of admissible solutions to the above truth value assignment problem is [0, 1], i.e. any number in the unit interval could be assigned as a truth value to the truth-teller. To be more precise, as the solution v p (u) is a fuzzy set over the unit interval V, not only is any point c in [0, 1] a solution (v p (u) = c), but also the membership function of any fuzzy set v p over V satisfies Eq. (45). But it seems that the reasoning procedure is incomplete: Finally, what is the truth value of the truth-teller statement? The provisional method discussed above proposes that all of the admissible truth values are aggregated into an interval-valued fuzzy set, and assigned to the truth-teller as its truth value: ∀u ∈ V v p (u) = Gu = [0, 1]
(46)
In this problem, the fuzziness in the truth of the self-referential statement is induced by existence of more than one fixed point for the mapping which represents the self-referential statement. Accepting a set of truth values rather than one truth value is quite in accord with the conventions of fuzzy logic. This problem illustrates the strength of Zadeh’s method of treating self-reference. Although the truth-teller is not paradoxical like the liar, any numeric truth value could be ascribed to it in the realms of multiple-valued and two-valued logic. Applying Zadeh’s method reveals the problematic nature of this statement, and depicts that any verity distribution could be ascribed to the truth-teller.
10 Remarks on extended truth qualification process It is notable that we can take two approaches in modeling the uncertainty about choosing the logical functions and representations of hedges: The first approach is reasoning with different sets of logical operators and including the fixed points obtained by each set of interest in the resulting type-2 fuzzy set. The second approach is considering the semantics of logical operators. It is reasonable to choose a unique set of operators for interpreting logical functions, because of different semantics of the operators. Thus one may reason that using different functions for negation, hedges, t-norms and t-conorms, illustrate different strengths and meanings and yields different versions of a sentence. However, this discussion is a complement to our argument on aggregation of the type-1 truth values of the sentences into type-2 fuzzy sets. It does not affect the generality of our argument at all. It is worth noting that our extended truth qualification method removes some ‘information’ from the reasoning process. We illustrate this with a simple example: Assume that V 1 = {v 1p1 , v 1p2 , v 1p3 }, V 2 = {v 2p1 , v 2p2 , v 2p3 }, V 3 = {v 3p1 , v 3p2 , v 3p3 } are three sets of solutions found for the equations corresponding to three self-referential statements, p1 , p2 , and p3 . When these solutions are combined (via constructing type-2 verity distributions), we assign a ‘set of truth values’ to each sentence, and therefore we have a set-valued (and therefore type-2) fuzzy set as the verity distribution of each statement: v p1 (u) = W 1 = v 1p1 , v 2p1 , v 3p1 (47) v p2 (u) = W 2 = v 1p2 , v 2p2 , v 3p2 v p3 (u) = W 3 = v 1p3 , v 2p3 , v 3p3 If one is interested in finding the original set of solutions from (47), one should begin a try and error search to extract V 1 , V 2 , V 3 from other invalid selections like {v 1p1 , v 2p2 , v 2p3 } which do
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not satisfy the original equations, i.e. test all three permutations of members of W 1 , W 2 , and W 3 in the equations to find V 1 , V 2 , and V 3 . The ‘vanished’ information is arbitrarily paid to make a valid reasoning process resulting in type-2 fuzzy sets which is in agreement with the conventions of fuzzy logic about modeling the uncertainties about membership functions with type-2 fuzzy sets. In case of using Eq. (42), the original set of truth values could not be recovered at all. Our example, truth-teller, is an appropriate illustration of the fuzziness induced by the existence of more than one solution to the truth qualification. The proposed reasoning method is simply a trade-off: some ambiguity is presented to the truth values of the sentences to make the reasoning process have a unique solution of the from of a type-2 fuzzy set. This ambiguity and the abovementioned vanished information about the truth values compensate the problem with finding an ‘optimal’ set of truth values for the statements. Henceforth, it is quite reasonable if one assumes the proposed reasoning process as a rational ‘suggestion’. This is quite extendable to some other reasoning methods encountered in fuzzy logic, for example possibility qualification principle (See Sect. 4.4). Zadeh (1978) noted that possibility qualification should be considered as speculative, especially until when the nature of possibility qualification is clarified very well. Interestingly, possibility qualification principle states that a possibility qualification procedure induces a type-2 fuzzy set, due to the weakening effect of a linguistic possibility assigned to a soft constraint characterized by a type-1 fuzzy set. To present how the proposed provisional method assigns the verity distributions to a set of multiple self-referential statements, we consider some examples: The following two sets of self-referential statements are considered to be analogous to the truth-teller (Goldstein 1992): p1 : p2 isv true. p2 : p1 isv true. The corresponding mapping is: f : R2 → R2 f (v p1 (u), v p2 (u)) = ( f 1 , f 2 ) f 1 = v p2 (u) f 2 = v p1 (u)
(48)
g : R2 → R2 g(vq1 (u), vq2 (u)) = (g1 , g2 ) g1 = 1 − vq2 (u) g2 = 1 − vq1 (u)
(49)
and q1 : q2 isv false. q2 : q1 isv false. Its associated mapping is:
The solution for the fixed point of the mappings f and g leads to the following equations: v p1 (u) = v p2 (u) vq1 (u) + vq2 (u) = 1
(50)
It is apparent that there is an infinite number of solutions for both equations. We aggregate all the solutions in a type-2 verity distribution to capture the uncertainty due to multiple solutions for the equations:
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v p1 (u) = v p2 (u) = [0, 1] vq1 (u) = vq2 (u) = [0, 1]
(51)
As a more complicated example, consider the following set of self-referential statements: p1 : It isv very false that p1 ⇔ ( p1 ⇔ p2 ). p2 : It isv very false that p1 ⇔ ¬ p3 . p3 : It isv very false that p3 ⇔ ( p1 ⇔ p3 ). Its corresponding mapping is: f : R3 → R3 f (v p1 (u), v p2 (u), v p3 (u)) = ( f 1 , f 2 , f 3 ) f 1 = abs(v p1 (u) − (1 − abs(v p1 (u) − v p2 (u))))2 f 2 = abs(v p1 (u) − (1 − v p3 (u)))2 f 3 = abs(v p3 (u) − (1 − abs(v p1 (u) − v p3 (u))))2
(52)
which admits at least the following fixed points: [0.29969, 0.19997, 0.35284]T , [1, 0, 0]T , [0.29295, 0.12715, 0.35047]T . It is somewhat difficult to asses whether the mapping has other fixed points or not. However, since our approach does not concern the method of finding the fixed points, we assume that these are the only fixed points of the mapping. Our method suggests that we consider the following (set-valued) verity distributions for the statements: v p1 (u) = {0.29295, 0.29969, 1} v p2 (u) = {0, 0.12715, 0.19997} v p3 (u) = {0, 0.35047, 0.35284}
(53)
We prefer to assign crisp sets as the truth value of each statement, since there is no reason to prefer a particular solution to the others, and assign a fuzzy set to each statement as its truth value. It should be noted that by such a truth value assignment, we do not mean to assign inconsistent truth values such as the following to the aforementioned set of self-referential statements. v p1 (u) = 1 v p2 (u) = 0.12715 v p3 (u) = 0
(54)
Instead, we try to show that the problem imposes an ambiguity in selecting a set of consistent truth values for a particular set of statements. In this way, some information on the possible sets of consistent truth values is missed, which makes our suggestion a provisional one. However, no other reasonable method could be envisioned to aggregate all possible consistent truth value assignments.
11 Conclusions and future works We have considered different methods of self-referential reasoning by fuzzy logic tools in this paper. We showed that the previous studies are not adequate both in representing multiple self-referential statements and in deciding the set of truth values to be accepted, unless the truth qualification process is modified for sets of self-referential statements resulting in mappings with multiple fixed points. It appeared to be reasonable not to simply ‘choose’ a set of truth values for each statement, but to aggregate all correct truth values for a statement in a
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type-2 fuzzy set. Although this process reduces some amount of information about multiple self-referential statements, it completes the reasoning process for them based on the previous approach of Zadeh in solving the paradox of the liar via fuzzy logic. Self-referential reasoning by fuzzy logic has many potential applications. It could be helpful in truth maintenance in knowledge-based systems and conflict management in multiagent systems. It could prevent the occurrence of logical inconsistency in decision-making and uncertainty management. In this paper, we considered only statements which speak about the truth and falsehood of each other. However, in more realistic cases, the sentences might be the internal beliefs of many agents or game players, which bring about an inconsistent system of beliefs. Hence, each agent may have other statements in the set of its beielfs together with judgments on the truth or falsehood of its beliefs or others’ beliefs. A simple example is: Agent1: Agent2 is young. What Agent2 believes isv false. Agent2: Agent1 is old. What Agent1 believes isv very true. The next step in developing methods for handling metacognitive reasoning is finding solutions for such problems, which seem to be quite reachable following the approach of this paper. A somewhat similar problem of inconsistency detection is considered recently in the framework of fuzzy logic and classical AI, however, without any direct attribution to the paradoxes arising in cases in which the agents are reasoning about the truth of the beliefs of each other (Patching et al. 2006). It would be interesting to study the implications of existence of a universal set in fuzzy set theory and fuzzy logic. In this way, one may refer to the previous studies on axiomization of fuzzy set theory (Hajek and Hanikova 2003), and more generally, the studies on mathematical fuzzy set theory and mathematical fuzzy logic (Gottwald 2005, 2006).
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