Toward a Quantum Theory of Humor

This paper proposes that cognitive humor can be modeled using the mathematical framework of quantum theory. We begin with brief overviews of both research on humor, and the generalized quantum framework. We show how the bisociation of incongruous frames or word meanings in jokes can be modeled as a linear superposition of a set of basis states, or possible interpretations, in a complex Hilbert space. The choice of possible interpretations depends on the context provided by the set-up vs. the punchline of a joke. We apply the approach to a verbal pun, and consider how it might be extended to frame blending. An initial study of that made use of the Law of Total Probability, involving 85 participant responses to 35 jokes (as well as variants), suggests that the Quantum Theory of Humor (QTH) proposed here provides a viable new approach to modeling humor.


Introduction
Humour has been called the "killer app" of language [1]; it showcases the speed, playfulness, and flexibility of human cognition, and can instantaneously put people in a positive mood. For over a hundred years scholars have attempted to make sense of the seemingly nonsensical cognitive processes that underlie humour. Despite considerable progress with respect to categorizing different forms of humour (e.g., such as irony, jokes, cartoons, and slapstick) and understanding what people find funny, there has been little investigation of the question: What kind of formal theory do we need to model the cognitive representation of a joke at the instant it is understood?
These quantum inspired approaches make no assumption that phenomena at the quantum level affect the brain; they draw solely on abstract formal structures that, as it happens, found their first application in quantum mechanics. The common approach is to utilize the structurally different nature of quantum probability. While in classical probability theory events are drawn from a common sample space, quantum models define states and variables with reference to a context, represented using a basis in a Hilbert space. This results in behaviour such as interference, superposition and entanglement, and ambiguity with respect to the outcome is resolved with a quantum measurement and a collapse to a definite state.
This makes the quantum inspired approach an interesting new candidate for a theory of humour. Humour often involves ambiguity due to the presence of incongruous QUANTUM THEORY OF HUMOUR 4 schemas: internally coherent but mutually incompatible ways of interpreting or understanding a statement or situation. As a simple example, consider the following pun: "Time flies like an arrow. Fruit flies like a banana." This joke hangs on the ambiguity of the phrase FRUIT FLIES, where the word FLIES can be either a verb or a noun. As a verb, FLIES means "to travel through the air".
However, as a noun, FRUIT FLIES are "insects that eat fruit". Quantum formalisms are highly useful for describing cognitive states that entail this form of ambiguity. This paper will propose that the quantum approach enables us to naturally represent the process of "getting a joke".
We start by providing a brief overview of the relevant research on humour.

Brief Background in Humour Research
Even within psychology, humour is approached from multiple directions. Social psychologists investigate the role of humour in establishing, maintaining, and disrupting social cohesion and social status, developmental psychologists investigate how the ability to understand and generate humour changes over a lifetime, and health psychologists investigate possible therapeutic aspects of humour. This paper deals solely with the cognitive aspect of humour. Much cognitive theorizing about humour assumes that it is driven by the simultaneous perception [29,30] or 'bisociation' [31] of incongruent schemas. Schemas can be either static frames, as in a cartoon, or dynamically unfolding scripts, as in a pun. For example, in the "time flies" joke above, interpreting the phrase FRUIT FLIES as referring to the insect is incompatible with interpreting it as food travelling through the air. Incongruity is generally accompanied by the violation of expectations and feelings of surprise. While earlier approaches posited that humour comprehension involves the resolution of incongruous frames or scripts [32,33], the notion of resolution often plays a minor role in contemporary theories, which tend to view the punchline as activating multiple schemas simultaneously and thereby underscoring ambiguity (e.g., Martin [34]; McGraw and Warren [35]). QUANTUM THEORY OF HUMOUR 5 There are computational models of humour detection and understanding (e.g., Reyes et al. [36]), in which the interpretation of an ambiguous word or phrase generally changes as new surrounding contextual information is parsed. For example, in the "time flies" joke, this kind of model would shift from interpreting FLIES as a verb to interpreting it as a noun. There are also computational models of humour that generate jokes through lexical replacement; for example, by replacing a 'taboo' word with a similar-sounding innocent word (e.g., [37];Valitutti et al. [38]). These computational approaches to humour are interesting, and occasionally generate jokes that are laugh-worthy. However, while they tell us something about humour, we claim that they do not provide an accurate model of the cognitive state of a human mind at the instant of perceiving a joke. As mentioned above, humour psychologists believe that humour often involves not just shifting from one interpretation of an ambiguous stimulus to another, but simultaneously holding in mind the interpretation that was perceived to be relevant during the set-up and the interpretation that is perceived to be relevant during the punchline. For this reason, we turned to the generalized quantum formalism as a possible approach for modelling the cognitive state of holding two schemas in mind simultaneously.

Brief Background in Generalized Quantum Modeling
Classical probability describes events by considering subsets of a common sample space [39]. That is, considering a set of elementary events, we find that some event e occurred with probability p e . Classical probability arises due to a lack of knowledge on the part of the modeller. The act of measurement merely reveals an existing state of affairs; it does not interfere with the results.
In contrast, quantum models use variables and spaces that are defined with respect to a particular context (although this is often done implicitly). Thus, in specifying that an electron has spin 'up' or 'down', we are referring to experimental scenarios (e.g., Stern-Gerlach arrangements and polarizers) that denote the context in which a measurement occurred. This is an important subtlety, as many experiments QUANTUM THEORY OF HUMOUR 6 have shown that it is impossible to attribute a pre-existing reality to the state that is measured; measurement necessarily involves an interaction between a state and the context in which it is measured, and this is traditionally modelled in quantum theory using the notion of projection. The state |Ψ representing some aspect of interest in our system is written as a linear superposition of a set of basis states {|φ i } in a Hilbert space H which allows us to define notions such as distance and inner product. In creating this superposition we weight each basis state with an amplitude term, denoted This state is separable if a ij = a A i a B j . It is inseparable, and therefore an entangled state, In some applications the procedure for describing entanglement is more complicated than what is described here. For example, it has been argued that the quantum field theory procedure, which uses Fock space to describe multiple entities, gives a kind of internal structure that is superior to the tensor product for modelling QUANTUM THEORY OF HUMOUR 7 concept combination [5]. Fock space is the direct sum of tensor products of Hilbert spaces, so it is also a Hilbert space. For simplicity, this initial application of the quantum formalsm to modelling humour will omit such refinements, but such a move may become necessary in further developments of the model.
Quantum models can be useful for describing situations involving potentiality, in which change of state is nondeterministic and contextual. The concept of potentiality has broad implications across the sciences; for example, every biological trait not only has direct implications for existing phenotypic properties such as fitness, but both enables and constrains potential future evolutionary changes for a given species. The quantum approach been used to model the biological phenomenon of exaptationwherein a trait that originally evolved for one purpose is co-opted for another (possibly after some modification) [40]. The term exaptation was coined by Gould and Vrba [41] to denote what Darwin referred to as preadaptation. 1 Exaptation occurs when selective pressure causes this potentiality to be exploited. Like other kinds of evolutionary change, exaptation is observed across all levels of biological organization, i.e., at the level of genes, tissue, organs, limbs, and behavior. Quantum models have also been used to model the cultural analog of exaptation, wherein an idea that was originally developed to solve one problem is applied to a different problem [40]. For example, consider the invention of the tire swing. It came into existence when someone re-conceived of a tire as an object that could form the part of a swing that one sits on.
This re-purposing of an object designed for one use for use in another context is referred to as cultural exaptation. Much as the current structural and material properties of an organ or appendage constrain possible re-uses of it, the current structural and material properties of a cultural artefact (or language, or art form, etc.) constrain possible re-uses of it. Here, we suggest that incongruity humour constitutes another form of exaptation; an ambiguous word, phrase, or situation, that was initially interpreted one way is revealed to have a second, incongruous interpretation. 1 The terms exaptation, preadaptation and co-option are often used interchangeably.

A Quantum Inspired Model of Humour
A quantum theory of humour (QTH) could potentially inherit several core of from previous cognitive theories of humour while providing a unified underlying model.
Considering the past work discussed in section , it seems reasonable to focus on the notion that cognitive humour involves an ambiguity brought on by the bisociation of internally consistent but mutually incongruous schemas. Thus, cognitive humour appears to arise from the double think that is brought about by being forced to reconsider some currently held interpretation of a joke within the light of new information: a frame shift. Such an insight opens up humour to quantum-like models, as a frame shift of an ambiguous concept is well modelled by the notion of a quantum superposition described using two sets of incompatible basis states within some underlying Hilbert space structure.
In what follows we shall introduce some features that would be required in a formal QTH as we start to sketch out a preliminary quantum inspired model of humour.
This initial model provided enough insight to formulate an experimental procedure aimed at discovering whether humour was likely to behave in a quantum-like manner. A preliminary study is introduced in section , which has provided insights and opened up a rich set of avenues for future investigation. Thus, this paper is a first step towards the development of a formal theory for a field that to date has not been well modelled.

The Mathematical Structure of QTH
We start our journey toward a QTH by building upon an existing model of conceptual combination first proposed by Gabora and Aerts [8]: the State-COntext-Property (SCOP) model. As per the standard approach used in most quantum-like models of cognition, |Ψ represents the state of an ambiguous element, be it a word, phrase, object, or something else, and its different possible interpretations are represented by basis states. Core to the SCOP model is a treatment of the context in which every measurement of a state occurred, and the resultant property that was measured. These three variables are stored as a triple in a lattice.
QUANTUM THEORY OF HUMOUR 9 The State Space. Following Aerts and Gabora [6], the set of all possible interpretation states for the ambiguous element of a joke is given by a state space Σ.
Specific interpretations of a joke are denoted by |p , |q , |r , · · · ∈ Σ which form a basis in a complex Hilbert space H. Before the ambiguous element of the joke is resolved, it is in a state of potentiality, represented by a superposition state of all possible interpretations. Each of these represents a possible understanding arising due to activation of a schema associated with a particular interpretation of an ambiguous word or situation. The interpretations that are most likely are most heavily weighted. The amplitude term associated with each basis state represented by a complex number coefficient a i gives a measure of how likely an interpretation is given the current contextual information available to the listener. We assume that all basis states have unit length, are mutually orthogonal, and are complete, thus i |a i | 2 = 1.
The Context. In the context of a traditional verbal joke the context consists primarily of the setup, and the setup is the only contextual element considered in the study in Section . However, it should be kept in mind that several other contextual factors not considered in our analysis can affect perceived funniness. Prominent amongst these is the delivery; the way in which a joke is delivered can be everything when it comes to whether or not it is deemed funny. Other factors include the surroundings, the person delivering the joke, the power relationships among different members of the audience, and so forth.
As a first step, we might represent the set of possible contexts for a given joke as Each possible interpretation of a joke comes with a set f i ∈ F of features (or properties), which may be weighted according to their relevance with respect to this contextual information. The weight (or renormalized applicability) of a certain property given a specific interpretation |p in a specific context c i ∈ C is given by ν. For example, ν(p, f 1 ) is the weight of feature f i for state |p , which is determined by a function from the set Σ × F to the interval [0, 1]. We write: (q, e, p) → µ(q, e, p).
Thus, a first step towards a full quantum model of humour consists of the 3-tuple (Σ, C, F), and the functions ν and µ. However, we have yet to address the core questions that should be asked of any cognitive theory of humour: what is the underlying cognitive model of the funniness of a joke?

The Humour of a Joke
As the listener hears a joke, more context is provided, and in our model their understanding (i.e., the cognitive state of the listener) evolves according to the transition probabilities associated with the cognitive state and the emerging context.
When the listener interprets the joke the listener is perceiving a bisociation of meaning.
That is, the first interpretation that the listener ascribes to the joke changes because two meanings are possible for the core concept in the joke. A projective measurement onto a funniness frame is the mechanism that we use to model the likelihood that a given joke is considered funny.
Thus, in our model, funniness plays the role of a measurement operator, and it is affected by the shift that occurs in the understanding of a joke with respect to two possible framings: one created by the setup, and one by the punchline. The probability of a joke being regarded as funny or not is proportional to the projection of the individual's understanding of the joke (|Ψ ) onto a basis representing funniness. This means that the probability of a joke being considered as funny, p F is given by a projection onto the |1 axis in H 2 F , a 2 dimensional Hilbert sub-space where |0 QUANTUM THEORY OF HUMOUR 11 represents 'not funny' and |1 represents 'funny'.
Similarly, the probability of a joke being regarded as not funny is represented by Note that |Ψ evolves as the initial conceptualisation of the joke is reinterpreted with respect to the frame of the punchline. This is a difficult process to model, and we consider the work in this paper to be an early first step towards an eventually more comprehensive theory of humour that includes predictive models.
To start in this journey towards a QTH, we will now present two examples in which two specific instances of humour are considered within the perspective of this basic quantum inspired model. First the approach is applied to a pun. Second, the approach is applied to a cartoon that is a frame blend. Both scenarios will help to deepen our understanding of the significant complexity of humour, and the difficulties associated with creating a mathematical model of this important human phenomenon.

Example 1: A Pun
Consider the pun: "Why was 6 afraid of 7? Because 789." The humour of this pun hinges on the fact that the pronunciation of the number EIGHT, a noun, is identical to that of the verb ATE. We refer to this ambiguous word, with its two possible meanings, We note that these two spaces should not be considered as mutually orthogonal, but that they will overlap. If they were orthogonal then the funniness of a joke would be independent of the interpretation that a person attributes to it.
With this added mathematical structure, we can represent the interpretation of the joke as a superposition state in where a n and a v are amplitudes which, when squared, represent the probability of a listener interpreting the joke in a noun or a verb form (|n and |v ) respectively. This state is depicted in Fig. 1(a), which shows a superposition state in the semantic space.
When given no context in the form of the actual presentation of the joke, these amplitudes represent the prior likelihood of a listener interpreting the uncontextualized word (i.e. EYT) in either of the noun or verb senses (e.g. a free association probability, see [12] for a review). However, we would expect to see these probabilities evolving throughout the course of the pun as more and more context is provided (in the form of additional sentence structure). Throughout the course of the joke, the state vector |Ψ therefore evolves to a new position in H 4 .
Since the set-up of the joke,"Why was 6 afraid of 7?", contains two numbers, it is likely that the numbers interpretation of EYT is activated (a situation represented in Figure 1(a)). The listener is biased towards an interpretation of EYT in this sense, and so we would expect that a n >> a v . However, a careful listener will feel confused upon considering this set-up because we do not think of numbers as beings that experience fear. This keeps the interpretation of EYT shifted away from an equivalence with the eigenvector |n . As the joke unfolds, the predator interpretation that was hinted at in If we consider the set of properties associated with EYT then we would expect to see two very different prototypical characteristics associated with each interpretation.
For example, the EIGHT interpretation is difficult to map into properties such as 'food' denoted f 1 , and 'not living' denoted f 2 (since when something is eaten it is usually no longer alive). Because 'food' and 'not living' are not properties of EIGHT, We can now start to construct a model of humour that could be correlated with data. If jokes satisfy the law of total probability (LTP) then their funniness should satisfy the distributive axiom, which states that the total probability of some observable We can manipulate the interpretation that a participant is likely to attribute to a joke by changing the semantics of the joke itself. Thus, changing the joke should change the semantics and so affect the humour that is attributed to the joke. We shall return to this idea in Section .

QUANTUM THEORY OF HUMOUR 14
This section has demonstrated that a formal approach to concept interactions which has been previously shown to be consistent with human data [5] can be adapted to the simultaneous perception of incongruous meanings of an ambiguous word or phrase in the understanding of a pun. What other aspects of cognitive humour might this new mathematical apparatus be applied to?

Example 2: A Frame Blend
Although our first example used a pun for simplicity, we believe that quantum inspired models can be fruitfully applied to more elaborate forms of humour, such as jokes involving incompatible frames or scripts. Our second example is a cartoon that was initially analysed in terms of the concept of a frame blend, which involves the merging of incongruous frames [42]. The cartoon is shown in Figure 2(a) and the frame blend analysis is shown in Figure 2 In a QTH, the two interpretations of the incongruous situation represented by the scene in Figure 2, as a dating scene and as an octopus scene, would be designated by the unit vectors {|d , |o }. The cognitive state of perceiving the blended frames is represented as a superposition of the two frames, however the underlying dynamics behind this joke are likely to be different. That is, rather than being led "down the garden path" by the setup and subsequent re-interpretation in light of the punchline, in this scenario the humour appears to result from the immediate simultaneous presentation of seemingly incompatible frames, which creates a similar tension or bisociation as eventually arose in the previous example. As with phenomena such as conceptual combination, in this scenario there are likely to be constraints on how the frames can be successfully blended, and it will be necessary to consider this when constructing a model of humour for this scenario. We reserve further exploration of this interesting class of humour for future work.

Probing the State Space of Humour
Returning to the question raised by equation (7), a QTH should be justified by considering whether humour does indeed violate the Law of Total Probability (LTP) [3].
However, the complexity of language makes it difficult to test how humour might violate the LTP using a method similar to those followed for decision making [11]. The model discussed above brings us to a position where past work on humour is unlikely to yield the data that is required to perform tests such as this. For example, we currently have no experimental understanding of how the semantics of a joke interplays with its perceived funniness. It seems reasonable to suppose that the two are related, but how?
We are not aware of any data sets that provide a way in which to evaluate this relationship. This is problematic, as there are a number of interdependencies in the framing of a joke that make it difficult to construct a model (even before considering factors such as the context in which the joke is made, and the socio-cultural background of the teller and the listener). In this section we present results from an exploratory study designed to start unpacking whether humour should indeed be considered within the framework of quantum cognition. As an illustrative example, consider the following joke: V O : "Time flies like an arrow. Fruit flies like a banana." As with the joke discussed in section , the humour arises from the ambiguity of the words FRUIT and FLIES. The first frame (F 1, the set-up), leads one to interpret FLIES as a verb and LIKE as a preposition, but the second frame (F 2, the punchline), leads one to interpret FRUIT FLIES as a noun and LIKE as a verb. A QTH must be able to explain how the funniness of the joke depends upon a shift in the semantic understanding of the two frames, F 1 and F 2.
We now outline a preliminary study that has helped us to explore the state space of humour.

Stimuli
We

Participants
The participants in this study were 85 first year undergraduate students enrolled in an introductory psychology course at the University of British Columbia (Okanagan campus). They received partial course credit for their participation.

Procedure
Participants signed up for the study using the SONA recruitment system, and subsequently responded at their convenience to an online questionnaire hosted by FluidSurveys. They were informed that the study was completely voluntary, and that they were free to withdraw from the study at any point in time. They were also informed that the researcher would not have any knowledge of who participated in the study, and that their participation would not affect their standing in the psychology class or relationship with the university. Participants were told that the purpose of the study was to investigate humour, and to help contribute to a better understanding the cognitive process of 'getting' a joke. Participants were asked to fill out consent forms. If they agreed to participate, they were provided a questionnaire consisting of a series of jokes and joke variants (as described above) and asked to rate the funniness of each using a Likert scale, from 1 (not funny) to 5 (hilarious). The questionnaire took approximately 25 minutes to complete. They received partial course credit for their participation.

Results
The mean funniness ratings across all participants for the entire collection of jokes and their variants (as well as the jokes and variants themselves) is provided in the Appendix.

Towards a test of the QTH
Recall that the Law of Total Probability (LTP) as represented by equation (7) suggests that the mean funniness of a joke should be equal to the sum of its funniness as judged under all possible semantic interpretations. This is not an equality that we can directly test given our current understanding of language and how it might interplay with humour. However, the dataset reported here gives us some initial ways in which to consider this question. With a methodology for converting the Likert scale ratings into projective measurements of a a joke being funny or not, we can start to consider the relative frequency that an original joke is judged as funny and comparing this result with the individual components.
We start by translating the Likert scale responses into a simplified measurement of funniness, by mapping the funniness ratings into a designation of funny or not. In order to run a quick comparison between the relative frequencies that participants decided the full joke (V O ) was funny when compared to the simple components of the joke (V S and V P ), we took the mean value of the components for each subject. Given that puns are not generally considered particularly funny (a result backed up by our participant ratings) we used a fairly low threshold value of 2.5 (i.e. if the mean was less than 2.5 then the components were judged as unfunny, and vice versa). Exploring the results of this mapping gives us the data reported in Figure 3 for the V O , V S and V P variants of the jokes, listing the frequency at which participants judged the joke and subcomponents funny. A mean value for the joke fragments is also presented. All data uses confidence intervals at the 95% level.
QUANTUM THEORY OF HUMOUR 19 We see a significant discrepancy between the funniness of the original and the combined funniness of its components. This is not a terribly surprising result, jokes are not funny when the set-up is not followed by the punchline, and participants usually rated V S and V P variants as unfunny (i.e. scoring them at 1). Table 1 in the Appendix shows that in the participant pool of 85, the set-up and punchline variants of the joke rarely had a mean funniness rating above 1.5. However, to extract a violation of the LTP for this scenario, we would need to construct expressions such as the following How precisely could such a relationship be tested? Two forms of data are required to test whether the simple puns used in our experiment actually violate the LTP: 1. We have demonstrated a method for extracting the funniness ratings above. How might we obtain data for the semantic probabilities? First we must consider the precise interpretation of what these probabilities might actually be. Firstly, we note that it seems likely participants will interpret just a set-up or a punchline in the sense that the fragment represents. The bisociation that humour relies upon is not present for a fragment, and so a person hearing a fragment will be primed by its surrounding context towards interpreting an ambiguous word in precisely the sense intended for that fragment. Indeed, the incongruity that results from having to readjust the interpretation of the joke, and the resulting bisociation, lies at the very base of the humour that arises.
Free association probabilities will not give these values. To test the LTP, it would be necessary to extract information about how a participant is interpreting core terms in QUANTUM THEORY OF HUMOUR 20 the joke as it progresses; some form of nondestructive measurement is required, and a new experimental protocol will have to be defined. We reserve this for future work.
However, the significant difference between the rated funniness of the fragments and that of the original joke allows us to formulate an alternative mechanism for testing equations of the form (7) and (8). We can do this by asking whether there is any way in which the semantic probabilities could have values that would satisfice the LTP? An examination of Figure 3 for the setup and punchline variants of the jokes suggests that there is no way in which to chose semantic probabilities that will satisfy the LTP. Thus, we have preliminary evidence that humour should perhaps be treated using a quantum inspired model.

Discussion
It would appear that there is some support for the hypothesis that the humour arising from bisociation can be modelled by a quantum inspired approach. Furthermore, the experimental results presented in section suggest that this model might more appropriate than one grounded in classical probability. However, much work remains to be completed before we can consider these findings anything but preliminary.
Firstly, the model presented in Section is simple, and will need to be extended.
While an extension to more senses for an ambiguous element of a joke is straightforward with a move to higher dimensions, the model is currently not well suited to the set of different variants discussed in section . A model that can show how they interrelate, and how their underlying semantics affects the perceived humour in a joke is desirable.
Furthermore, the funniness of the joke was simplistically represented by a projection onto the 'funny'/'not funny' axis. A more theoretically grounded treatment of the Likert data is desirable. For example, the current threshold value of 2.5 was chosen somewhat arbitrarily (although could be justified by a consideration of the mean values for funniness scores reported in the Appendix -see Table 1). A more systematic way of considering the Likert scale measures to allow for a normalisation of funniness ratings at the level of an individual is also desirable. As a highly subjective phenomenon, QUANTUM THEORY OF HUMOUR 21 funniness is liable to be judged by different individuals inconsistently and so it will be important that we control for this effect in comparing Likert responses among individuals.
Considering experimental results, the sample size of the data set is somewhat small (85 participants), although our funniness ratings appear to be reasonably stable for this cohort. A more concerning problem revolves around the construction of a LTP relationship for our simple model. There are many alternative ways in which a LTP could be constructed for puns, and more sophisticated models need to be investigated before we can feel confident that our results do indeed demonstrate that humour must be modelled using a quantum inspired approach. In particular, we require a more sophisticated method that facilitates the extraction of data about the semantics attributed by a participant to a joke. A two stage protocol may be the answer for obtaining the necessary semantic information and so providing a more rigorously founded test of the violation of LTP. It would be useful to construct a systematic study of the manner in which adjusting the congruence of the set-ups and punchlines influences perception of the joke. The quantum inspired semantic space approaches of van Rijsbergen [13], Widdows [43] may prove fruitful in this case, as they would facilitate the creation of similarity models such as those explored by Aerts et al. [44], Pothos and Trueblood [45].
In summary, humour is complex, and it will take an ongoing program of research to gradually understand the interplay between the semantics of a joke and its perceived funniness. However, at this point we might pause to consider the broader question of why humour might be better modelled by a quantum inspired approach than by one grounded in classical probability? To this end we return to the discussion of Section .
As we saw, the humour of a pun involves the bisociation of incongruent frames, i.e., re-viewing a setup frame in light of new contextual information provided by a punchline frame. Moreover, the broader contextuality of humour means that even the funniest of jokes can become markedly unfunny if delivered in the wrong way (e.g. a monotone voice), or in the wrong situation (e.g. after receiving very bad news). Funniness is not a pre-existing 'element of reality' that can be measured; it emerges from an interaction between the underlying nature of the joke, the cognitive state of the listener, and other social and environmental factors. This makes the quantum formalism an excellent candidate for modelling humour, as this interaction is well described by the concept of a vector state embedded in a space which is represented using basis states that can be reoriented according to the framing of the joke. However, this paper only provides a preliminary indication that a QTH may indeed provide a good theoretical underpinning for this complex process. Much more work remains to be done.

Conclusions
This paper has provided a first step towards a quantum theory humour (QTH).
We constructed a model where frame blends are represented in a Hilbert space spanned by two sets of basis states, one representing the ambiguous framing of a joke, and the other representing funniness. The process of 'getting a joke' then consists of a dual stage scenario, where the cognitive state of a person evolves towards a re-interpretation of the meaning attributed to the joke, followed by a measurement of funniness. We conducted a study in which participants rated the funniness of jokes as well as the funniness of variants of those jokes consisting of setting or punchline by alone. The results demonstrate that the funniness of the jokes is significantly greater than that of their components, which is not particularly surprising, but does show that there is something cognitive taking place above and beyond the information content delivered in the joke. A preliminary test to see whether the humour in a joke violates the law of total probability appears to suggest that there is some reason to suppose that a quantum inspired model is indeed appropriate.
Our QTH is not proposed as an all-encompassing theory of humour; for example, it cannot explain why laughter is contagious, or why children tease each other, or why people might find it funny when someone is hit in the face with a pie (and laugh even if they know it will happen in advance). It aims to model the cognitive aspect of humour only. Moreover, despite the intuitive appeal of the approach, it is still rudimentary, and QUANTUM THEORY OF HUMOUR 23 more research is needed to determine to what extent it is consistent with empirical data. Nevertheless we believe that the approach promises an exciting step toward a formal theory of humour. It is hoped that future research will build upon this modest beginning.

2.76
Continued on next page

1.29
Continued on next page  QUANTUM THEORY OF HUMOUR 47 Figure 3 . A comparison of the frequency with which a specific joke and its fragments are considered funny for participants in the pilot trial (using a threshold value of 2.5, n=85). A mean of the set-up and the punchline variants (V S and V P ) is also given.
Confidence intervals are set at 95%.