Formalizing Irony with Doxastic Logic

Formalizing Irony with Doxastic Logic WANG ZHONGQUAN National University of Singapore April 22, 2015 1 Introduction Verbal irony is a fundamental rhetoric device in human communication. It is often characterized by a speaker s act of saying something other than what he or she means, with the intention to create an effect of humor or emphasis. The persistence presence of verbal irony in user-generated content has given rise to computational irony, a research topic within sentiment analysis. The motive of computational irony is to address the difficulties that verbal irony brought to the current schema of sentiment analysis - the surface meaning of an utterance, which many methods in sentiment analysis are aiming to detect, is not equivalent to the utterance s latent meaning if it is intended ironically. In fact, it has been reported that the presence of verbal irony is responsible for many of errors made by state-of-the-art methods for sentiment analysis [1]. A formalization of verbal irony, if possible, will improve methods in sentiment analysis greatly. More broadly, the formalization of verbal irony is necessary for any computational process that aims at automating the analysis and simulation of human communication. This paper has two parts. The first part introduces the theories of verbal irony in rhetoric and literary to the reader. It also discusses the prime example of computational irony - the model of pragmatic context. In the second part, the current paper builds on the introduced theories and presents a different formalization of verbal irony with formal logic. The author argues that, utilizing the properties of doxastic logic, it is feasible to construct a formalization of verbal irony which will allow the detection, construction and interpretation of ironic proposition on the semantic level. 1

2 Theories of verbal irony In this section, I will discuss two characterizations of verbal irony that have been proposed in the realm of rhetoric and literary: the maxim-violation theory and the pretense theory of irony. The intention here is not to derive a perfect characterization, but to shed light on characteristics that a formalization of verbal irony should capture. British philosopher Paul Grice regards verbal irony as the discerned intentional violation of the maxim of Quality, one of the four rules of conversation that he proposed to interlocutors in his work Logic and Conversation. The maxim of Quality prescribed that do not say what you believe to be false [3]. Given that, a discernible intentional violation of the maxim of Quality is the expression of a proposition that the audience know the speaker does not believe. It can be illustrated by the example below: Example 2.1. In a restaurant, 1. Alice is a vegetarian 2. Alice told Bob before that she is a vegetarian 3. Alice says to Bob: How I love this piece of meat they gave me! According to maxim-violation theory, Bob recognizes the ironic utterance because Bob believes that Alice is a vegetarian; moreover, Bob believes that Alice believes that Bob believes that Alice is a vegetarian. This discerned intentional violation of the maxim of quality motivates Bob to reject the surface meaning of Alice s statement and seek alternative interpretation. Grice suggests that the negation of the ironic proposition often becomes the most intuitive interpretation of the ironist s intended meaning after the surface meaning is rejected [3]. Another characterization of verbal irony is the pretense theory proposed by American psychologies Herbert Clark and Richard Gerrig in 1984. The pretense theory revolves the argument that verbal irony necessarily postulates two groups of audience: those equipped to detect the ironical voice and decode the intended latent meaning of an utterance, and those who will accept it at its surface meaning [2]. In this theory, verbal irony works as follows: the ironist pretends to hold a belief; he or she expresses this belief or its consequences as a proposition, addressing the group of understanding audience who will consider this proposition as ironic; through 2

asserting the absurd, the ironist mocks those who would sincerely hold the ironic propositions and thereby create humorous effect. Shared by the two theories above is the argument that the audience recognize verbal irony if and only if they recognize the disparity between a made proposition and their beliefs about the speaker. In Grice s theory, such a disparity serves as the cue of the intentional violation of the maxim of Quality; in the pretense theory of irony, to recognize such a disparity is to recognize the absurdity of the ironist s proposition, and therefore is a necessary means for one to become a member of the understanding audience. The two theories both suggest that the formalization of verbal irony is necessarily beyond the syntactic analysis of the given proposition. The formalization of verbal irony must be supplemented by the representation of facts that are not contained in the given utterance but held as beliefs by the speaker and audience. The importance of external facts in the communication of verbal irony can be further illustrated by the difference between a lie and an ironic proposition. Syntactically, a lie and an ironic proposition can be identical. Consider the example below: Example 2.2. In a restaurant, 1. Alice is a vegetarian 2. Bob does not know that Alice is a vegetarian 3. Alice says to Bob: How I love this piece of meat they gave me! In Example 2.1. and 2.2., the exactly same proposition is expressed by Alice. However, this proposition will only be perceived as ironic by Bob in Example 2.1., the situation in which Bob acquired the necessary belief to discern the disparity between Alice s proposition and his own beliefs. As shown in this example, the lie-irony distinction, like the difference between ironic and non-ironic utterance, can be contextual instead of syntactic. 1 3 Existing (Semi-)formalization of verbal irony In computational irony, various formalizations of verbal irony have been proposed, such as Akira Utsumi s model of failed expectation and Byron 1 An interesting question pertaining to the lie-irony distinction would be, is there a lie that does not sound ironic to anyone? 3

Wallace s model of pragmatic context. As attempts to improve the accuracy of sentiment analysis, these models mainly focus on the automatic detection of verbal irony. In this section, I will use the model of pragmatic context as an example to illustrate how computational irony provide reasonable yet limited formalization of verbal irony. 3.1 Motivation The model of pragmatic context is proposed as an improvement of previous machine learning approaches in computational irony. Previously, machine learning approaches have been designed to infer P{I(U)}, the probability that an utterance U is intended ironically. In these methods, a statistical inference is made by extracting the syntactic features of U to form a vector Sx(U), and estimating the conditional probability P{I(U) Sx(U)} with reference to existing data. P{I(U)} is solely a function of this conditional probability. In other words, these approaches assume the following relationship: P{I(U)} P{I(U) Sx(U)} [5] From the theories discussed Section 2, it is clear that the above relationship is insufficient for the detection of verbal irony. It is often the presence or lack of certain external beliefs, rather than syntactic features, that determines whether the audience consider a proposition ironic. This insufficiency presented by existing machine learning approaches motivates Wallace to propose a model that utilize both contextual information and syntactic analysis to detect verbal irony. 3.2 Construction Central to present method is the identification of pragmatic context, a set of elements that encode(s) expectations by capturing what we believe the speaker would likely (not) say [5]. Elements of a pragmatic context can be assignments of truth values to propositions - two such elements are the assignment of "True" to the propositions Alice is a vegetarian and A vegetarian does not like meat in Example 2.1.; elements can also be estimates along a spectrum, such as one s political leaning or an user s historic rating for Stanley Kubrick s films. The relevant aspect of the speaker s pragmatic context, denoted by a, is a vector formed by combing relevant elements in 4

the pragmatic context. It serves as an important basis of judging whether a given utterance is intended ironic. For further elaboration on Wallace s construction, we shall introduce the set of notations below: - U S : an utterance from the speaker S - I(U S ): the event that the utterance U from the speaker S is intended ironic - a: an aspect, the vector comprises of elements in the pragmatic context that is pertaining to the utterance U S - M(a, S): estimation of the vector a, with regard to the speaker S - Var{M(a, S)}: an indication of the model s confidence in its estimation of M(a, S). Less confidence is indicated by a larger variance. - Sx(U): the vector formed by the extracted syntactic features of the utterance U The main goal of the pragmatic context model is to infer P{I(U S )}, the probability that an utterance from the speaker S is an intended verbal irony. This probability, according to Wallace, can be estimated through the relationship below: P{I(U S )} P{I(U S) M(a, S)} Var{M(a, S)} P{I(U S ) Sx(U S )} [5] The above relationship provides a feasible schema for the automatic detection of verbal irony using machine learning. The likelihood of an utterance being ironic is function consisting of two terms. The numerator of the first term is the conditional probability that the utterance is ironic given that the relevant aspect of speaker S s pragmatic context was estimated to be M(a, S). This probability is weighted by Var{M(a,S)}, an indicator of how much confidence the model has in its estimation of M(a, S). The second term, P{I(U S ) Sx(U S )}, is the conditional probability that the utterance is ironic given that syntactical analysis of the utterance U S evaluates to Sx(U S ). Its value can be obtained by various methods in sentiment analysis, such as the bag-of-words model [5]. With methods in data mining, the relationship above can be easily estimated and cross-validated with the input of training data. 5

3.3 Evaluation The model of pragmatic context improves on previous machine learning methods in computational irony. Its emphasis on the context is relevant to both the maxim-violation theory and the pretense theory of irony. Moreover, the statistical inference can be viewed as an abstraction of the mental process in which a group of audience with certain beliefs receive, interpret, and discern verbal irony within a proposition. The model of pragmatic context can be viewed as a feasible formalization of the detection of verbal irony. However, the model of pragmatic context is not a formalization of verbal irony. To be specific, a formalized method for detecting verbal irony is only part of the formalization of verbal irony itself. A full formalization will allow computational process to perform harder tasks, such as constructing ironic proposition, recovering the latent meaning of ironic propositions, and even facilitate proofs of the existence of verbal irony given a specific context. Currently, none of these tasks can be achieved using the model of pragmatic context, which is built on statistical rather than semantic identification of verbal irony. 4 Formalizing verbal irony with doxastic logic In this section, the author proposes an alternative formalization of verbal irony that, instead of statistical inference, utilizes doxastic logic. The goal is to obtain a formalization of verbal irony which possesses the capacity to: 1. symbolically represent the maxim-violation theory and the pretense theory of irony 2. provide an intuitive representation of the necessary and sufficient conditions for the communication of verbal irony 3. identify the set of ironic propositions with regard to a specific group of audience 4. facilitate the semantic interpretation of an ironic proposition 6

4.1 Construction The current construction recognizes that there are two stages in the communication of verbal irony - the speaker s act of expressing an ironic proposition, followed by an interpretation on the side of the audience. An ironic proposition is successfully communicated if and only if the proposition is intended ironically by the speaker, and perceived as ironic by the audience. Having identified these two stages, the objective of the current construction becomes formalizing the answers to two questions: 1. Under what circumstances a proposition is intended ironically by the speaker? 2. Under what circumstances a proposition will be perceived as ironic by the audience? To answer the these two questions formally, it is necessary to adopt a framework that can represent the beliefs of the speaker and the audience simultaneously. This can be achieved by doxastic logic, the semantic of propositional modal logic that formalizes the discussion of beliefs of individual agent. A model M in doxastic logic can be defined as below: Definition 1. (Doxastic models) Let a set of propositional variables P and a finite set of agents N be given. A doxastic model is a triple M = (W, V, R) where W is a set of world, V : W P(P) assigns a valuation to each world w W, and R : N P(W 2 ) assigns an accessibility relation to each agent i N, satisfying transitivity, seriality and euclideanness [4]. Doxastic logic employs the possible world semantic to interpret R i, which is illustrated in Definition 2.: Definition 2. R i (w, w ) if and only if for agent i, the world w is a possible world of w. Once the binary accessibility relation R i is interpreted, we can give a semantic to the modal operator in the Kripke structure. In the current construction, the representation of the beliefs of individual member of the audience necessitates the usage of modal operator B i in doxastic logic, which can be defined as below: Definition 3. In doxastic logic, an agent A in w i believes ψ, denoted by w i = B A ψ, if and only if for all w W with R A (w i, w), we have w = ψ. 7

In section 2, an observation was made by the maxim-violation theory that the audience recognize verbal irony if and only if the made proposition contradicts what the audience believe the speaker believes. Moreover, the audience should believe that this contradiction is intentionally made discernible. These necessary and sufficient conditions for the audience to perceive the proposition as ironic can be translated into an axiom in the language of doxastic logic: Axiom 1. (Perception of Verbal Irony) In a model M, a proposition ψ expressed by speaker S is perceived as ironic by an agent A in w if and only if w = B A B S ψ and w = B A B S B A B S ψ. On the other hand, we need to identify the circumstances that a speaker intends a proposition to be ironic. Recall that in Section 2, the pretense theory of irony articulates that the act of expressing an ironic proposition is always addressing two groups of audience: those who will regard the proposition as ironic, and those who will take away its surface meaning. This condition, again, can be axiomized using doxastic logic, with reference to Axiom 1.: Axiom 2. (Intention of Verbal Irony) In a model M, a proposition ψ is intended as verbal irony to J by speaker S in w if and only if w = (B S B J B S ψ) (B S B J B S B J B S ψ) and K such that w = (B S B K B S ψ). 4.2 Lemmas and Proofs One advantage of the current formalization using doxastic logic is that we can derive lemmas and results from the two axioms and existing properties of doxastic logic. Some of these lemmas and their proofs are listed below: Lemma 1. (Extended Perception of Verbal Irony) In model M, if φ = ψ and the proposition ψ is perceived as ironic by agent A in w, then φ is perceived as ironic by agent A in w. Proof. 1. = φ ψ (Premise) 2. = ψ φ 8

3. = B S ( ψ φ) 4. = B S ψ B S φ (K, 3) 5. w = B A B S ψ B A B S B A B S ψ (Premise) 6. w = B A B S φ B A B S B A B S φ (4, 5) 7. φ is perceived as ironic by agent A in w (Axiom 1.) Lemma 2. (Satisfiability of Intended Irony) In model M, if a proposition ψ is intended as verbal irony by a speaker S, ψ is satisfiable in M. Proof by contradiction. 2 1. ψ is unsatisfiable in M 2. B S ψ is unsatisfiable in M (R s is serial, 1) 3. K, B k B S ψ is unsatisfiable in M ( k R k is serial, 2) 4. K, B S B K B S ψ is unsatisfiable in M (R S is serial, 3) 5. K, such that w = B S B K B S ψ (premise, Axiom 2.) 6. contradiction (4, 5) 7. ψ is satisfiable in M Lemma 3. (Introspection of the Ironist) In a model M, if the speaker A in w believes in ψ, then A perceived his or her own expression of ψ as verbal irony. Proof. 1. w = B A ψ (premise) 2. w = B A B A ψ (B A is transitive) 3. w = B A B A B A ψ (B A is transitive) 4. w = B A B A B A B A ψ (B A is transitive) 5. ψ is perceived as verbal irony by A in w (Axiom 1., 2, 4) 2 A similar proof can be constructed to prove that proposition perceived as verbal irony by an agent in M must be invalid in M. 9

4.3 Discussion The construction proposed in Section 4.1 is an attempt to formalize the communication of verbal irony in complex, multi-agent setting. The two axioms are proposed with the intention to reflect the two characterizations of verbal irony discussed in Section 2. Respectively, these axioms specify the necessary and sufficient conditions for a speaker to intend a proposition as verbal irony and the audience to perceive a proposition as ironic. Recognizing these two axioms enable one to utilize theorems in doxastic logic to semantically detect, construct and reason about verbal irony within an existing formal framework. The current construction needs to be subjected further examination. More working examples and proofs can be provided to showcase the practical and theoretical quality of this preliminary attempt to formalize verbal irony. On the other hand, the author also notes that new questions arise as doxastic logic is chosen as the framework. Particularly, it is questionable that whether doxastic logic is the most appropriate framework for the current purposes, given that there are appealing alternatives such as multimodalities framework involving both B i and K i and dynamic doxastic logic which is able to characterize the changes of the beliefs of the agent upon the communication of certain propositions. These issues are central to the current construction and should be examined by further investigations. 5 Conclusion A review of the theories of verbal irony and the model of pragmatic context suggests that there is currently a lack of semantic analysis in computational irony. In response to this lack, the author proposes a characterization of verbal irony within the framework of doxastic logic. This characterization, reflecting the maxim-violation theory and the pretense theory of verbal irony, will be able to formalize the detection, construction and interpretation of ironic proposition on the semantic level. 10

References 1. Paula Carvalho, Luís Sarmento, Mário J Silva, and Eugénio de Oliveira. Clues for detecting irony in user-generated contents: oh...!! it s so easy;-). In Proceedings of the 1st international CIKM workshop on Topic-sentiment analysis for mass opinion, pages 53 56. ACM, 2009. 2. Herbert H Clark and Richard J Gerrig. On the pretense theory of irony. 1984. 3. Herbert P Grice. Logic and conversation. Harvard Univ., 1970. 4. Hans van Ditmarsch, Jan van Eijck, Floor Sietsma, and Yanjing Wang. On the logic of lying. Games, Actions, and Social Software, page 53, 2010. 5. Byron C Wallace. Computational irony: A survey and new perspectives. Artificial Intelligence Review, pages 1 17, 2013. 11