Self-referential reasoning in the light of extended truth qualification principle Mohammad Reza Rajati, Hamid Khaloozadeh, Alireza Fatehi Dept. of Electrical Engineering, K. N. Toosi University of Technology, Tehran, Iran [email protected], {h khaloozadeh, fatehi}@kntu.ac.ir

Summary. The purpose of this paper is to formulate truth-value assignment to self-referential sentences via Zadeh’s truth qualification principle and to present new methods to assign truth-values to them. Therefore, based on the truth qualification process, a new interpretation of possibilities and truth-values is suggested by means of type-2 fuzzy sets and then, the qualification process is modified such that it results in type-2 fuzzy sets. Finally, an idea of a comprehensive theory of type-2 fuzzy possibility is proposed. This approach may be unified with Zadeh’s Generalized Theory of Uncertainty (GTU) in the future.

1 Introduction In the era of information and communications, truth and reliability play a significant role and intelligent machinery are demanding new tools of reliable data interpretation and processing. Now, human beings tend to develop new means to handle the great bulk of available information. Unquestionably, mimicry of human’s ability to process the incoming data intelligently and deduce conclusions about its reliability as well as “meaning” is of equal importance with obtaining information, if not more important. After the development of fuzzy computation theory, the great contribution of L. A. Zadeh [1] revealed the fact that there might be rigorous tools via which the “meaning” of the analyzed information could be manipulated besides its statistical nature. He investigated the “meaning-oriented” approach to information processing via the well-founded theory of fuzzy computation and established a new theory of possibility as a counterpart of the probability theory. One of the novel applications of fuzzy possibility is the resolution of the liar paradox (and henceforth, self-referential sentences bearing a paradox). Zadeh proposes a method to assign a truth-value to the liar sentence [2], [3]. In this paper, we investigate the application of Zadeh’s method to resolve the paradox borne by self-referential sentences and propose an extension to

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Zadeh’s method, such that it can be useful in a broader sense. This extension is based on the concept of type-2 fuzzy sets. We first introduce type-2 fuzzy sets, possibility theory, and the concept of self-reference, and then based on Zadeh’s method we try to assign truth-values to self-referential sentences. In the next step, we extend Zadeh’s truth qualification principle to handle the existing uncertainties in fuzzy possibilities. Finally, a comprehensive theory of type-2 fuzzy possibility is touched.

2 Preliminaries Here, we introduce some basic concepts which are necessary for our paper to be understood, including type-2 fuzzy sets, the concept of fixed points, and possibility theory. 2.1 Type-2 Fuzzy Sets The basic notion of type-2 fuzzy sets was suggested by Zadeh [4] to illustrate uncertainty in the membership values of a fuzzy set. Mendel and John distinguish four types of uncertainty in type-1 (conventional) fuzzy systems [5]: (1) The meanings of the words which are used in the premises and conclusions of the rules can be uncertain (words mean different things to different individuals). (2) Conclusions may have a histogram of values assigned to them, especially if the rules are obtained from a group of people who do not agree (and it is almost always the case). (3) Measurements that activate a type-1 fuzzy system may be noisy and hence uncertain. (4) The data that are used to tune the parameters of a type-1 fuzzy system may also be noisy. Mendel and John then conclude that all of these kinds of uncertainty could be interpreted as uncertainties about the membership functions and the best way to model them is making use of type-2 fuzzy sets. Although these types of uncertainty are directly related to rule-based fuzzy systems, other applications of fuzzy computation may also encounter either such uncertainties or other types of uncertainty. We will see this fact in the sequel. Below some definitions and concepts related to type-2 fuzzy sets are presented. The expressions and concepts are somewhat similar to those used by Mendel and John [5]. It is worth noting that there are several papers on basic set theoretic operations of type-2 fuzzy sets [6], centroid of a type-2 fuzzy set [7], and composition of type-2 relations [8], [9]. Also, [10], [11], [12], and [13] developed the use of interval-valued sets in fuzzy logic. The approach of [14] is somewhat related to our work in this paper, because it discusses “interval-valued degrees of belief”. Definition 1): A type-2 fuzzy set A˜ is associated with a type-2 membership function µA˜ (x, y) ∈ [0, 1] where x ∈ U (universe of discourse) and y ∈ Vx ⊆

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[0, 1] : A˜ = {((x, y), µA˜ (x, y))|∀x ∈ U, ∀y ∈ Vx ⊆ [0, 1]}

(1)

A˜ can also be expressed as: Z Z x∈U

y∈Vx

µA˜ (x, y)/(x, y), Vx ⊆ [0, 1]

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RR Here, denotes union over all x, y in U, V . See [5] to understand this notion of union in which, every element of a type-2 (or correspondingly, type1) fuzzy set is viewed as a subset,Pdistinctly. In case of a discrete universe R of discourse, the is replaced by . (Thus, each of U, V or both could be discrete.) Definition 2): A secondary membership function is the basic tool to show the nature of uncertainty in the membership functions of type-1 and is defined as: Z µA˜ (x = b, y) = µA˜ (b) = fb (y)/y Vb ⊆ [0, 1] (3) y∈Vb

It should be noted that a secondary membership function is obtained by retaining x = b and evaluating the membership value of every member of Vb . ( Vb defines all membership values which could be assigned to b) . Now, it is obvious why we call µA˜ (b) a secondary membership function. Definition 3): The uncertainty in the primary memberships of a type-2 fuzzy set A˜ builds a region which is called the footprint of uncertainty (FOU). It is defined as: ˜ = {(x, y)|µ ˜ (x, y) > 0} F OU (A) (4) A Speaking more precisely, FOU could be defined as the closure of the abovementioned set, i.e. the smallest closed set which contains it. FOU is essentially the result of aggregation of all of the intervals associated with the members of the universe of discourse. FOU can contain some points on its boundary or not, depending on V ’s. 2.2 Fixed Points of Real-Valued Mappings Consider a mapping f : W → W where W ⊆
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Given a fuzzy set A in U , the sentence “x is A” provides a possibility distribution assigned to x, denoted by πx : ∀x ∈ U πx (u) = µA (u)

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For example, “John is young” is equivalent to “Age (John) is A”, in which Age is an attribute of “John” and A is a fuzzy set denoting the concept of young. So, we can define: πAge (u) = µA (u). The universe of discourse could be taken as U = [0, 130]. Zadeh establishes rather a simple, but extremely useful mathematical basis to extend the theory of possibility. The development of this theory is wellknown in the fuzzy computation community, so we avoid repeating the material and suggest the interested readers to study Zadeh’s original work deeply, as a contribution which is still highly readable. Also, there are other approaches to the theory of possibility, which are axiomatic and based on Dempster-Shafer theory of evidence [16]. Despite the fact that they are very helpful, the approach of Zadeh is very capable in the “meaning-oriented” possibility, and as we denote in this paper, it is very wellbehaved for interpretation of self-referential sentences. So we adhere to his approach and try to extend it. 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 these principles, we reconsider them briefly in the sequel. Truth Qualification Principle This principle tries to judge the truth of sentences which speak about the truth of other sentences. If x is A → πx (u) = µA (u) then x is A is τ → πx (u) = µτ (µA (u)) in which τ is a linguistic truth-value such as: true, very true, more or less true, false, etc. Probability Qualification Principle If we need to assign a possibility to a sentence which asserts some facts about the probability of a sentence we have: If x is A → πx (u) = µA (u) then R R x is A is λ → πx ( U p(u)µA (u)du) = µλ ( U p(u)µA (u)du) where λ is a linguistic probability value, e.g. likely, very likely, etc. p(u)µA (u)du is R the probability that the value of x falls in the interval [u, u + du] and so p(u)µA (u)du is actually the probability of fuzzy event A˜ [17]. The principle U assigns a possibility to a probability density.

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Possibility Qualification Principle It is obviously realized that this principle qualifies sentences speaking about the possibilities of other sentences and events: If x is A → πx (u) = µA (u) then R R x is A is α − possible → π ˜x = u∈U v∈Vu µπ˜ (u, v)/(u, v) Vu = [min(α, µA (u)), min(1, 1 − µA (u))] Here α − possible is a linguistic possibility value, and π ˜ is a type-2 fuzzy set. This type-2 fuzzy set is constructed to show the uncertainty caused by weakening the proposition by a linguistic possibility value. Zadeh, proposes that µπ˜ = a, a ∈ [0, 1], i.e. an interval-valued fuzzy set. Replacing α = 1 for “completely possible” makes Vu reduce to [µA (u), 1]. Zadeh suggests some other versions of the possibility qualification principle, but for the sake of brevity we don’t mention them. The basic concept of assigning a type-2 fuzzy set as the possibility of a proposition suffices us.

3 Self-Referential Sentences and Paradox Self-referential sentences have been a source of debate in logic and mathematics, mainly because some of self-referential sentences make logical paradoxes. Since the time of ancient Greek, the paradoxical nature of self-referential sentences was known [18], [19], [20],[21]. Maybe the best known self-referential paradox is the liar, which could be formally expressed as the following statement, known as the modern form of the liar: p := Sentence p is false where := stands for “is defined to be” “This sentence is false” is also a form of the liar. The paradoxical nature of this sentence is obvious: its truth leads to its wrongness and its wrongness leads to its truth and this circular debate will continue ad infinitum. So it could not be determined whether this sentence is true or false. It is beleived that the works of Eubulides of Miletus, are the most ancient resources on the liar. He included it among a list of seven puzzles: “A man says that he is lying. Is what he says true or false?” Epimenides, in his works, reportedly uttered the following sentence: “All Cretans are liars” although he was a Cretan. It can be shown that there are other self-referential sentences which bring about paradox. A collection of sentences containing reference to each other or themselves can generate paradox. The paradox is due to an indirect selfreference, so we call them self-referential sentences as well. An example is the inconsistent dualist: p1 : =Sentence p2 is true p2 : =Sentence p1 is false

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It is worth noting that self-referential sentences may be consistent, i.e. without any paradoxical nature. Good examples are the truth-teller: p1 : =This sentence is true and the consistent dualist: p1 : =Sentence p2 is true p2 : =Sentence p1 is true These sentences generate no paradox, but the problem faces when we realize that we can consider them either false or true! Other kinds of self-referential sentences are those which assign numerical truth-values to each other, in the framework of fuzzy logic. See the following example: p1 : =The truth-value of p2 is 0.9 and the truth-value of p3 is 0.2 p2 : =The truth-value of p1 is 0.8 and the truth-value of p3 is 0.3 p3 : =The truth-value of p1 is 0.1 Another example is the following sentences containing biconditionals: p1 : =The truth-value of p2 is 0.9 if and only if the truth-value of p1 is 0.6 p2 : = The truth-value of p2 is 0.7 if and only if the truth-value of p1 is 0.2 It will be confusing to imagine how we can assign truth-values to the abovementioned set of sentences. Considering the vagueness in the natural language expressions, we can also investigate self-referential sentences containing vague hedges or uncertain expressions. For example see the following sentences: Modest liar 1 [22]: p : = p is at least a little false Modest liar 2 [23]: p : = p is more or less false Modest and emphatic truth-teller [23]: p1 : = p1 is less and more true p2 : = p2 is very true Fuzzy logistic [23]: p : =It is very false that p is true if it is false Two fuzzy equivalents of the inconsistent dualist are [23]: p1 : = p1 is true if and only if p2 is true p2 : = p2 is true if and only if p1 is very false and: p1 : =It is very false that p1 ↔ p2 p2 : =It is less and more false that p1 ↔ ¬p2 The so-called triplist: p1 : =It is very false that p1 ↔ ¬(p1 ↔ p2 ) p2 : =It is very false that p1 ↔ ¬p3 p3 : =It is very false that p3 ↔ ¬(p1 ↔ p3 )

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3.1 Zadeh’s Method for Resolution of the Liar Zadeh in [2] proposed a method based on truth qualification principle to resolve the paradox of the liar. Almost all of the work done in the fuzzy community to cope with paradoxes is inspired by the work of Zadeh [22], [23], [24]. To use the landscape of possibility theory, Zadeh interprets the modern form of the liar as: p : = p is τ The truth qualification principle asserts that for the liar: πp (u) = πτ (πp (u)) u ∈ U

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Regarding the definition of possibility, we should assume that U = [0, 1] or at least U ⊆ [0, 1] . From the definition of a fixed point, we can realize that πp (u) is a fixed point of πτ : [0, 1] → [0, 1] . If τ is replaced by “true”, Zadeh’s method is trying to evaluate the “truthteller”. So πτ (v) = v and then the resultant mapping becomes πp (u) = πp (u), which says that p is true for any u ∈ U . It could be supposed that πp (u) is a possibility distribution which can take any value in [0, 1] . Evaluating the liar in the same way we have: πp (u) = 1 − πp (u)

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This induces a possibility distribution on U defined by: πp (u) = 1/2 ∀u ∈ U

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This result determines that the possibility for the truth-value of p to be u is 1/2 for all u ∈ U . This interpretation is somewhat different from saying that “the liar is half-true”, which assigns a truth-value to the liar. The same discussion applies to other self-referential sentences as well. However, the result is heuristically interpreted as a truth-value assignment to such sentences.

4 Truth-Value Assignment to Self-Referential Sentences Zadeh’s method is a proper way for truth-value assignment to self-referential sentences. First, we try to use his interpretation to assign truth-values (or possibility distributions) to self-referential sentences. Actually, Zadeh accepts the fixed point of the mapping πτ : [0, 1] → [0, 1] as the solution of the liar paradox. It is a natural extension to use the fixed points of the mappings generated by self-referential sentences as their solutions. However, as we show, the mappings could be assumed in different ways. First, observe the following theorem about fixed points:

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Theorem 1(Brouwer fixed point theorem): Any continuous mapping f : W → W has at least one fixed point provided that W is a nonempty, compact and convex subset of
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πp1 (u) = πp2 (u) = a ∈ [0, 1]

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So the solution is:

For self-referential sentences which assign truth-values to each other in the framework of multiple-valued logic, we can still utilize Zadeh’s method. Consider the sentence p : =The truth-value of q is 0.9 Naturally, we can interpret it as: πp (u) = π0.9 (πq (u))

(12)

π0.9 is a fuzzy number which reflects the uncertainty about the meaning of “is 0.9”. It could be taken for example by the approach of Vezerides and Kehagias [26]. They assume that for p : =The truth-value of q is a we can assert that: πp (u) = 1 − abs(πq (u) − a) Thereby equation (12) results in πp (u) = 0.95. Accordingly, the fixed point for the liar is obtained: πp (u) = 1 − abs(πp (u) − 0) ⇒ πp (u) = 1/2 Applying this rule for the truth-teller: πp (u) = 1 − abs(πp (u) − 1) ⇒ πp (u) = a ∈ [0, 1]

(13)

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In the same way, Zadeh’s method could be directly applied to selfreferential sentences with fuzzy hedges and vague expressions. Consider the set of sentences which we called triplist in section 3. We use √ g(v) = v 2 for “very”. However, there are f (v) = 1 − v for “more or less” and √ 3 many other choices (e.g. (f (v) = 1 − v ) and g(v) = v 3 . The corresponding mapping is: πp1 (u) = abs(πp1 (u) − (1 − abs(πp1 (u) − πp2 (u)))2 πp2 (u) = abs(πp1 (u) − (1 − πp3 (u)))2 πp3 (u) = abs(πp3 (u) − (1 − abs(πp1 (u) − πp3 (u)))2 We found some fixed points for the mapping numerically: (0, 0, 1), (0.29969, 0.19997, 0.35284), (1, 0, 0), (0.29295, 0.12715, .35047) Correspondingly, we could assign a mapping to any set of self-referential statements containing linguistic and numerical truth-values.

5 An Extension to Zadeh’s Truth Qualification Principle Previously, we studied the usefulness of the truth qualification principle in resolution of self-referential sentences and proposed a new method for interpretation of them. However, this approach leaves some questions unanswered. One may be the actual “meaning” of a truth-value less than one for a sentence, say 0.5. Obviously, without the “meaning-oriented” theory of possibility and fuzzy constraints, it seems to be of very little meaning and usage. However, justifications for this interpretation of the liar need to be probed. We provided a justification after reviewing Zadeh’s resolution of the liar. Another problem is that different maps are obtained using different logics, i.e. different t-norms, s-norms, hedge interpreters and etc. In addition, a mapping πτ : [0, 1] → [0, 1] may have several fixed points. So, it is difficult to select a suitable solution to the set of self-referential sentences. It might be firstly imagined to ask an expert to choose a solution or define a proper criterion for selecting it. But how could one accomplish this task when the interpretation of the possibilities seems to be controversial? It is an open problem for further investigation. Another way is using proper tools for modeling uncertainty in the possibilities which are membership functions themselves. The uncertainty of having multiple fixed points somewhat resembles the second kind of uncertainty in membership functions presented in section 2. Therefore, it is quite natural to model the “output” of the whole process of truth qualification as a type-2 fuzzy set. This idea is supported by Zadeh’s possibility qualification principle very well: When he tends to model the uncertainty (or relaxation) induced by the α − possibility constraint in this sentence: “ A is τ is α − possible” he

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makes use of interval-valued fuzzy sets which are obviously type-2 fuzzy sets with constant µπ˜ (x, y). Therefore, in the first step we suggest that the possibility distribution function πp be interpreted as a type-2 fuzzy set: Z X π ˜p = µπ˜ (x, y)/(x, y) (14) x∈U y∈F

where F is the set of known fixed points of πτ . F could be defined in a wider sense as a set of fixed points of all available candidates for πτ . It is an accepted convention (which is quite reasonable [5]) to select µπ˜ (x, y) = 1 or another constant in the interval [0, 1]. This reduces π ˜p to an interval-valued fuzzy set. In his paper [27], Mendel proposes new methods for modeling uncertainties via type-2 fuzzy sets. He believes that asking persons to assign anything other than a uniform weighting to their entire FOU would be very difficult. So, he is inclined to interpret the uncertainties about a word’s membership function as an interval-valued fuzzy set. However, someone may disagree him and assert that a person may be inclined to weight some membership functions for a linguistic variable more than the others. He also suggests to model uncertainties which may exist between a group of people by means of aggregation of each person’s equally weighted FOU. He also mentions that we may trust some people more, so we can weight their suggestions more than the others. Thereby, instead of assigning a predetermined µπ˜ (x, y) to the set of solutions, we could refer to a group of experts in the area of fuzzy logic and build a type-2 fuzzy set according to Mendel’s suggestions. For an application of uncertainty bounds in designing fuzzy systems see [28]. Inspired by the solution of the truth-teller which has an infinite number of fixed points, we try to propose a type-2 fuzzy set, of which the universes of discourse are continuous. We characterize this set as follows: Z Z π ˜p = µπ˜ (x, y)/(x, y) Fˆ = [inf(F ), sup(F )] (15) x∈U

y∈Fˆ

Here F is the set of available fixed points, as before. It is obvious that if inf(F ) = sup(F ) then the type-2 fuzzy set will reduce to the so-called “type2 singleton”1 which is actually analogous to a conventional fuzzy set. For a similar approach see [29], and [30].

6 Toward A Theory of Type-2 Fuzzy Possibility We beleive that the approach of Zadeh to the theory of possibility should be investigated more such that it is fertilized to handle type-2 fuzzy sets. 1

This expression is due to the authors.

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Many types of uncertainty could be present in the interpretation of important linguistic variables such as truth, probability, and possibility. Furthermore, as we showed, the procedure of truth qualification of self-referential statements introduces some uncertainty to the possibility distributions. Therefore, a thorough study and consistent reformulation of Zadeh’s approach is needed for type-2 fuzzy possibility. We beleive that it is yet another step in completing the recently proposed Generalized Theory of Uncertainty [31]. An example of proper tools for type-2 possibility formulation is the concept of a fuzzy-valued measure [32].

7 Conclusions In this paper, we tried to formulate truth-value assignment to self-referential sentences following Zadeh’s approach in interpreting the truth-values as possibility distributions of the truth of sentences. We also established a method for truth-value assignment to sentences which assign numerical truth-values to each other. Then, we broadened the idea of truth-values to type-2 fuzzy sets such that we can handle the uncertainty induced by existence of multiple fixed points for possibility mappings. Immediately, we modified Zadeh’s truth qualification principle taking uncertainties in membership functions into account. At last, we touched a comprehensive theory of type-2 fuzzy possibility to formulate type-2 fuzzy constraints and possibilities.

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