Gibbs: The Poetics of Mind, ch. 4 (1994)

Chapter 4 of Raymond Gibbs' book contains some very, very long discussions of the use of metaphor in various domains as well as some evidence.

The chapter is by far the longest in the book (one and a half time longer than chapter 5, the second longest) and makes up around 20% of the whole text.

However, only 6 out of a total of 87 pages in the chapter are used to present psychological evidence for his cognitive claims (pp. 161-67). Much of this evidence is the same as that which he presents in chapter 6.

In the last section before the conclusion of the chapter, Gibbs addresses the objections against cognitive metaphor theory that Naomi Quinn raised in 1991.

Structure of the chapter
Chapter 4 of Poetics of Mind has the following sections and subsections:

1. Metaphor in language (pp. 122-145)
1. The ubiquity of metaphor (pp. 122-124)
2. The communicative functions of metaphor (pp. 124-134)
3. Social functions of figurative language (pp. 134-140)
4. Metaphor and politics (pp. 140-145)
2. The Metaphorical structure of everyday thought (pp. 146-206)
1. Systematicity of literal expressions (pp. 146-154)
2. Novel extensions of conventional metaphor (pp. 154-157)
3. Polysemy (pp. 157-161)
4. Psychological evidence (pp. 161-167)
5. An alternative to metaphor in thought and language (pp. 167-169)
6. Metaphor in science (pp. 169-179)
7. Metaphor in law (pp. 179-183)
8. Metaphor in art (pp. 183-187)
9. Metaphor and myth (pp. 187-192)
10. Metaphor in culture (pp. 192-206)
3. Conclusion (p. 207)

Below, I will summarize the contents of each of these sections. Following that, I will also zoom in on two particular subsections and comment on the arguments given in those.

Contents of the chapter
Here's a brief telegraphic summary of what each section contains:

1. Unnamed introduction: People use metaphor a lot.
2. Metaphor in language
1. The ubiquity of metaphor: Some quantitative estimates of how many metaphors a person produces per minute of conversation -- a somewhat meaningless number from my perspective.
   
2. The communicative functions of metaphor: People use metaphors in order to be more expressive, compact, and vivid. Further, "metaphors facilitate prose comprehension" (p. 131). They are also easier to remember (p. 132).
3. Social functions of figurative language: Language signals group membership.
4. Metaphor and politics: People talk about politics in terms of war, sport, etc.
3. The Metaphorical structure of everyday thought: "The ubiquity of metaphor in everyday discourse is not due to sophisticated rhetorical abilities of ordinary speakers; rather, it is motivated by the [sic] persuasiveness of metaphor in everyday thought" (p. 146).
1. Systematicity of literal expressions: Expressions form thematic clusters. Conceptual mappings can be ordered in a hierarchy of generality (p. 152).
2. Novel extensions of conventional metaphor: Elaborations of metaphors in poetry count as "evidence for the metaphoric nature of everyday thought" (p. 154)
3. Polysemy: Claudia Brugman and George Lakoff's analysis of over also counts as "evidence for the metaphorical nature of thought" (p. 157).
4. Psychological evidence: See below.
5. An alternative to metaphor in thought and language: Gibbs mentions Jackendoff's theory as an alternative to cognitive metaphor theory, but doesn't really discuss the difference in any depth.
6. Metaphor in science: Some Kuhnian observations. Also, a reference to an interesting 1985 historical study that Dedre Gentner and Jonathan Grudin did of the change in metaphors in psychology.
7. Metaphor in law: Essentially a long reference to Winter (1989).
8. Metaphor in art: Film and painting use cognitive mappings, too (e.g., Magritte).
9. Metaphor and myth: Myth use path metaphors etc.
10. Metaphor in culture: References to Claudia Brugman's 1983 paper on Mixtec and Quinn's 1991 paper on marriage. See also below.
4. Conclusion: "[M]etaphor is a fundamental mental capacity by which people understand themselves and the world through the conceptual mapping of knowledge from one domain onto another" (p. 207).

Psychological evidence
The subsection on psychological evidence (pp. 161-167) puts forward the following types of data:

1. The classroom study by Dedre Gentner and Donald R. Gentner (1983) showed that whether students were taught about electricity in terms of water metaphors or crowd metaphors would affect their performance on subsequent tests (p. 162).
2. People form mental images of idioms that are consistent with the words in the idiom; this allegedly contrasts with literal sentences, although no evidence is given for that claim (p. 163).
3. Priming subjects with an instance of a conceptual mapping allowed for more rapid reading of another instance of the same metaphor in one of Gibb's studies (p. 163).
4. When people are interviewed about the cause, volition, and manner of literal explosions, they make the same judgments as they do for metaphorical explosions (pp. 163-164).
5. Gesture studies show, according to Gibbs, that metaphor precedes speech (pp. 164-167).

Some Comments on "Cognitive Topology"
The interview studies mentioned in point 4 are a strange and dubious thing to cite for a cognitive psychologist. In Gibbs' mind, these "reflected mappings of source to target domains that preserved the cognitive topology of the source domain" (p. 163).

The reason is that if you ask someone whether a bomb explodes out of its own accord, they say "no," and if you ask someone whether people explode with anger out of their own accord, they also say "no". Hence, metaphor preserves "cognitive topology," according to his logic.

Lucky that Gibbs didn't ask about the speed, consequences, or reach of explosions instead, one might add."When a person explodes with anger, does he push the air away from him in a sudden shockwave?" "When a person explodes with anger, do people risk shrapnel wounds?" His findings proves more about his methods than about cognition.

Some Comments on the Gesture Studies
The gesture data that Gibbs cites as evidence for his cognitive metaphor theory is really bizarre. Much of it is hardly interpretable as metaphorical, and when it is, it doesn't correlate with speech in any simple way.

Let's look at the examples he quotes from David McNeil (1992), as there are only a few. This will give a sense of how thin the evidence is. Here they are, with slight changes in typography:

• It was a Sylvester and Tweety cartoon
  hands rising to offer an object (p. 165)
• I want to ask you a question
  hand forms a cup (p. 165)
  
• That book is packed with meaning
  one hand pushes against the palm of the other (p. 166)
  
• I've got to tell you something.
  palm-up hand moves toward hearer (p. 166)
• He's trying to masquerade
  both hands spread out and forward with a rotation (p. 167)

I can see that the example with packed with meaning might be taken as support for cognitive metaphor theory if the subject did indeed cram an imaginary object tightly into a container. But how on earth does Gibbs come to see the other cases as "evidence"?

Since when has cognitive metaphor theory related cups to questions? No one has reported that people use phrases like hand you a cup to mean "asking," not has anyone speculated that they might. It seems that these new gesture idioms can only be explained if we postulate a whole new set of mappings to deal with the new modality. That makes gesture data irrelevant as evidence for mappings underneath speech.

This point to the general problem that cognitive metaphor should first commit itself to a single model of what hangs together in the cognitive universe, and only then predict that this structure will be found in language, too. For instance, how are the concepts of speed, certainty, difficulty, hardness, and tallness related in cognition? After you've committed yourself to an answer to that question, cognitive metaphor theory will produce a prediction regarding speech.

Gibbs vs. Quinn
Naomi Quinn has objected to cognitive metaphor theory on the grounds that although people use a whole range of different incoherent metaphors to talk about marriage, they actually live in a pretty stable and coherent way. Often, they will even try out different verbal metaphors to find one that betters fits what they want to express. This constitutes evidence that culture precedes metaphor.

In the section headed "Metaphor in culture", Gibbs replies to this by saying that Quinn's analysis "confuses metaphor with idealized cognitive models" (p. 204). His idea of how an "Idealized Cognitive Model" can be both metaphorical and non-metaphorical at the same time is quite confusing, so I have to quote it at some length:

[...] there may possibly be one general cognitive or cultural model for marriage, just in the way Quinn suggests, that is motivated by a cluster of contiguous conceptual metaphors. The variety of expressions people use to speak of marriage reflect their different conceptual metaphors for different aspects of their experience of marriage. Yet the variety of expressions do not mean that there is not some sort of cultural model of marriage based on a complex configuration of different types of conceptual metaphors. [...] As is the case for anger, people use different metaphors, even within the same narrative, because each metaphor reflects a different aspect of their metaphorical understanding of some experience. One's cognitive model of marriage may consist of various metaphors that capture different aspects of our understanding of marriage, such as compatibility, mutual benefit, and lastingness. These metaphors may be contiguously linked, perhaps as a kind of radial structure, yet need not be internally consistent. For example, we may at times see marriage as being a container but at other times as being like a manufactured product. (pp. 204-205)

The best I can do in terms of interpretation is to say that Gibbs imagines that we have a chimera of metaphors in our head, so that marriage might have pigeon wings but a lion head. This doesn't explain how people can use two distinct metaphors for the same aspect (the head is both pigeon and lion?). It also doesn't explain how the conflicts between inconsistent metaphors are resolved.

Quinn's theory, on the other hand, is quite simple: People have a cultural model which is not a metaphor, and they then choose metaphors to express aspects of that model. It is revealing how this simple model contrasts with the "complex configuration" that Gibbs proposes.

Bruno Latour: What Is the Style of Matters of Concern? (2008)

This little booklet has been lying around the office here at Oude Turfmarkt since I moved in here, and since I love Bruno Latour, I thought I should have a look at it some time. Today was as good a day as any.

The booklet consists of two lectures that Latour delivered in Amsterdam as the Spinoza lectures of 2005. I have so far only read the first one. The upshot of the lecture is that he wants science and science studies to create knowledge without splitting up the world in the a cold realm of nature-in-itself and a comfy but scientifically irrelevant world of nature-for-us.

Two Heroes Going With the Flow
Latours argument is driven by a central metaphor: Instead of coping with the gap between language and world by building a bridge from one bank to the other, he wants us to "go with the flow" down the river that separates the two realms (pp. 14-15). In this way, he hopes that we will be able to overcome the "bifurcation of nature" (a phrase he borrows from Whitehead) into primary and secondary qualities.

In order to meet this challenge, he invokes another one of his heroes, the French 19th-century sociologist Gabriel Tarde. Tarde was very explicit about employing a methodology "almost the exact opposite of [...] Monsieur Durkheim's" (quoted on page 19). That is, instead of privileging the bird's-eye, whole-sale view of a society or aggregate, he privileges the view from the inside, the meaningful detail.

Fleshy Signals in Biology and Computer Science
I wonder whether all of this might have an application to the philosophy of information. Latour, with Tarde, hints at some consequences that this new thinking might have on evolutionary thought (pp. 16-17): The "information" reproduced in a particular shift of generations is under a constant pressure to produce difference, that is, decide on a content/form distinction.

This might be elaborated into a more general theory in which the metaphysical conception of pure, abstract information yields to some kind of contextual or naturalized conception of "content" or "meaning," perhaps akin to that of biosemiotics.

The Nirenberg-Badiou debate over Being and Event

In the latest issues of Critical Inquiry, one can watch a philosophical critique of Alain Badiou's book Being and Event degenerate into an academic mudslinging contest with a distinctly testosterone-laden tone.

Ricardo and David Nirenberg proposed a criticism of the way Badiou employs mathematics in the summer 2011 issue. In the most recent issue, the philosophers Adam Bartlett and Justin Clemens as well as Badiou himself responded with quite violent counterattacks.


Bulldozer Ethics?
It is quite evident from his work that Badiou holds himself in high esteem. Depending on taste, one may see this as a case of megalomania or simply of "refreshing immodesty" (as Bartlett and Clemens see it, p. 372). In either case, grooming one's legacy is legitimate enough, even if it gets a little thick at times.

However, other aspect of his style are a little harder to forgive. His attempt to secure himself a place in history is not just expressed through a "refreshing immodesty," but at times also through cruder bullying tactics. Sometimes this just involves him holding up the authority of mathematics as a shield in front of himself, but at other times, he gets more aggressive with the intimidation and the name-calling.

For instance, there is nothing discreet about his attack on the Nirenbergs (his "adversaries," p. 362). They are "stupid," "ignorant," and "careless" (p. 364, 362, and 363), and they make "beginner's errors" (p. 363) while a "constant confusion dominates their thought" (p. 364).

Their critics, Bartlett and Clemens, are on the other hand "clear, informed, and independent" (p. 362).

"Insinuating Rhetoric"
This last remark is a little surprising in the light of the actual style of the article by Bartlett and Clemens. While they scorn the Nirenbergs for "promulgation of an insinuating rhetoric, whose ideological affiliations are at once evident and obscured" (p. 367), they themselves do not seem to mind a bit of ridiculing name-calling: "we would like to renominate Nirenberg & Nirenberg, Nirenberg & Son, as Nini" (p. 366).

One should read these allegations of rhetoric and insinuations against the backdrop of their own style, including quotes like this:

Among much, we will ignore Nini’s effort to demonstrate a familiarity with French philosophy. Badiou’s engagement with the philosophical ‘tradition’ is so extensive, fundamental, and so beyond anything cited here by Nini as to make the latter’s contribution puzzling. (p. 371-72)

Or, in the style of classical rhetoric: "Now, other people might have drawn attention to my opponent's lack of intelligence, to his fat body, or to his embarrassing military record, but I, on the other hand, value the proper tone of the open debate ... (etc.)"

This is all a little disturbing, given that Badiou claims to have something to say about love and politics. It's one thing that he overstretch the significance of his mathematical metaphors a little, but is this really the kind of person we want to take our ethical advice from?

Gigerenzer and Brighton: "Homo Heuristicus" (2009)

Somewhat couterintuitively, Gerd Gigerenzer and Henry Brighton argue in this article that quick-and-dirty decision algorithms can actually have higher predictive accuracy than more informed methods. They explain this as an effect of the so-called "bias-variance dilemma."

The Risk of Overfitting
Put slightly differently, the issue is that a fitting process with many degrees of freedom will be more likely to respond to random noise as well as genuine patterns than a fitting process with less freedom. A large model space will thus create a better fit to the data, but possibly at the cost of responding to accidents.

They illustrate this with an example by fitting polynomials of increasing degrees to a set of data points recording the temperature in London in the year 2000. As expected, the achieve higher fit with higher degree; however, they also acquire more idiosyncrasies and less predictive power with the increasing degrees (Gigerenzer and Brighton 2009: 118):

As they note, the predictive power of the best-fit model essentially flows from the environment, not from an inherent quality of the model space itself. In a different environment, a polynomial of degree 4 could thus easily have been wildly inadequate.

Ecological Rationality
The cognitive consequence that the authors draw from these considerations is that the kind of heuristics that people use to make judgment may in fact be highly adapted to the kind of environment they live in, even though they may seem statistically crude.

They conceptualize the different learning strategies that an organism can have using two concepts from machine learning, bias and variance (p. 117). As far as I can tell from their informal description, the difference between these two is the summation order:

Bias is the distance between the average best-fit function and the underlying data-generating function (averaged over all data sets). Variance is the average distance between the individual best-fit functions and the underlying data-generating function (again, average over all data sets).

I have here assumed that function distance is a matter of summing a number of squared differences.

Tallying and Take-the-best Algorithms
As an illustration of the power of heuristics, Gigerenzer and Brighton consider an algorithm that I think is quite interesting in its own right. The idea behind the learning algorithm is to look for the single most telling difference between to objects and use that as a predictor of some quality of interest.

In the concrete example they discuss (pp. 113-116) the task was to teach a program to pick the larger of two cities based on a number of cues.

The cues were binary variables "such as whether the city had a soccer team in the major league" (p. 113). A city was thus described by a vector of binary numbers, and the training set consisted in a number of comparisons between such binary strings of the type "(0, 1, ..., 0) > (1, 1, ..., 0)".

Based on the data set, one can order the cues in decreasing order of "validity," where validity is defined is the relative frequency with which a certain difference in cues (one city has a university, the other does not) is correlated with one of the two possible judgments (the city with a university is the larger).

Confronted with a new example, the program should then runs down through the sorted list of cues looking for the first one that applies to the pair and then based its judgment solely on that single cue. (Note that cues are frequently highly dependent, so Bayesian methods cannot be employed directly.)

The results are quite impressive: As the training set approaches 50 objects, the accuracy of the program approaches 75%. I don't know, however, whether "50 objects" means 50 comparisons or 50 x 49 = 2450 comparisons. If the latter is the case, the the growth rate of the accuracy levels look a little less impressive.

Upcoming talks

In the course of the next two and a half months, I will be giving lectures or presentations on the following occasions:

Date	Venue	Place	Topic
February 24	Adventures in Multimodality	Amsterdam	Cognitive metaphor theory
March 6	Philosophy and Ethics seminar	Eindhoven	Philosophy of linguistics
April 4	Structures for Semantics guest lecture	Amsterdam	Game theory and semantics
April 26-27	Feminist Materialisms	Copenhagen	Philosophy of technology

Gibbs: The Poetics of Mind (1994)

This is the book that everyone in cognitive metaphor theory refers to when they get nervous. Written by a psychologist and full of compact summaries of experimental evidence, it provides just the kind of authority that a linguist needs when people start asking skeptical questions.

Some of the assertions about metaphor that have been claimed to have their empirical support in this book do indeed. Others do not. And in between, there are disputable points and conclusions that others have stretched beyond what they can carry.

General Outlook and Ideas
The book is pretty orthodox in its embrace of cognitive metaphor theory. Here are some illustrative quotes from the introduction, with parentheses added by me:

• "Language (is not independent of the mind but) reflects our perceptual and conceptual understanding of experience." (p. 16)
• "Figuration (is not merely a matter of language but) provides much of the foundation for thought, reason, and imagination." (p. 16)
• "Metaphorical meaning is grounded in nonmetaphorical aspects of recurring bodily experiences or experiential gestalts." (p. 16)
• "[...] metaphorical understanding is grounded in nonmetaphorical preconceptual structures that arise from everyday bodily experience." (p. 17)

Just to add a few from later in the book:

• "[...] it seems certain now that the study of clichéd idiomatic expressions can provide significant evidence on how people think metaphorically in everyday life." (p. 318)
• "[…] the mind itself is primarily structured out of various tropes." (p. 434)
• "Figuration (is not merely a matter of language but) provides much of the foundation for thought, reason, and imagination." (p. 435)
• "Similar cognitive mechanisms drive our understanding of both literal and figurative speech." (p. 435)
• "[…] we metaphorically conceptualize our experiences through very basic sensory experiences that are abstracted to form figurative thought." (p. 444)

So, in summary, we still have the three-stage model of cognitive metaphor theory:

Experience

→

Thought

→

Speech

Mental Images as Empirical Evidence
Gibbs enthusiastically embraces the use of introspective reports as a way of investigating psychological aspects of metaphor: "One way to uncover speakers’ tacit knowledge of the metaphorical basis for idioms is through a detailed examination of speakers’ mental images of idioms," he explains (p. 292).

It turns out that the kind of support he has in mind concerns the coherence of imagery across subjects:

If people’s tacit knowledge of idioms is not structured by different conceptual metaphors, there should be little consistency in participants’ responses to questions about the causes and consequences of actions in their mental images of idioms with similar nonliteral interpretations. (p. 293)

That is, supposing that the coherence cannot be brought about be any other means than a conceptual metaphor. This heavy but implicit assumption is never questioned or examined and puts his whole edifice in danger of become a load of introspective nonsense.

He does, of course, find that people are pretty consistent in the way they picture scenes like spilling the beans. From this he concludes that "the figurative meanings of idioms are motivated by various conceptual metaphors that exist independently as part of our conceptual system" (p. 295).

However, he is understandably uncomfortable with the idea of the mental image being the meaning of an idiom. To avoid this natural next step, he is at pains not to have the mental image be a byproduct of the conceptual mapping, not the flesh and blood of it:

The empirical evidence in support of this conclusion does not in any way suggest that people actually form mental images of idioms as a normal part of their online understanding of idioms. The data simply, and significantly, demonstrate how people’s common metaphorical knowledge provides part of the motivation for why idioms have the figurative meaning they do. Traditional theories of idiomaticity have no way of accounting for these imagery findings, because they assume that the meanings of idioms arise from metaphors that are now dead and no longer a prominent part of our everyday conceptual system. (p. 295)

 So oddly, he recognizes that the mental images that his subjects described may have been constructed on his cue -- but he insists that this drawing up of a picture from a sentence can only occur in a consist way if people already think in terms of conceptual mappings.

From those premises, sure, this data proves the existence of cognitive mappings.

Dead Metaphors and Shallow Processing
On of the basic tenets of cognitive metaphor is that there are no dead metaphors. If cognitive metaphor theorists concedes that some metaphors were dead, it would undermine the "deep" analyses of all the others. If kick the bucket is dead, why should we believe that see the point isn't?

It is therefore somewhat surprising that Gibbs does in fact allow for dead metaphors:

[...] the dead-metaphor and conceptual views of idiomaticity should not be seen as competing theories. Many idiomatic phrases could very well be dead or have meanings that are arbitrarily determined by as matters of convention. (p. 308)

Even more disturbingly, he states that people can in fact get by pretty well without conceptual mappings:

[…] listeners may not always instantiate specific conceptual metaphors that motivate an idiom’s meaning when understanding some phrase in conversation. Similarly, people may not always analyze the literal word meanings of idioms during comprehension. There will be occasions when people do tap into an idiom’s conceptual foundation. Readers may also process the individual word meanings when they attempt to comprehend certain kinds of idioms. But it is a mistake to assume that some types of analysis will occur each and every time someone encounters an idiomatic expression.

So one may ask what exactly it is that conceptual mappings do if they aren't really central to understanding an idiom, but aren't images either?

Both of these points certainly cast a somewhat troubling light the reference Lakoff and others make to Gibbs' book as if it supports the "aliveness" of any speculative story they might cook up.

Analyzable and Unanalyzable Idioms
Gibbs makes a distinction between idioms that are understood in a compositional fashion and idioms that aren't, and he states that "there is reasonable consistency in people’s intuitions of the analyzability of idioms" (p. 279).

This claim is, again, based on introspective reports from his subjects. In a 1989 article he wrote with Nandini Nayak, a list of idioms are thus categorized according to a number of parameters, for instance the possibility of passivization (e.g., *the bucket was kicked by him).


This gives them a list of "decomposable" idioms (including break the ice, let off steam, play with fire, and clear the air) and a list of "nondecomposable" ones (including kick the bucket, chew the fat, raise the roof, and play the field).

Interestingly, they also feel the need for a third category for idioms that can be analyzed under certain assumptions about the metaphorical reference of the constituents. As the booklet given to the subjects explain,

[...] there are idioms that are decomposable but whose individual words have a more metaphorical relation to their figurative meanings. Thus, the phrase spill the beans means something like ‘reveal a secret.’ Although there is a fairly close relationship between spill and ‘reveal’, the word beans refers to ‘secrets’ in a less direct, metaphorical way. Idioms such as spill the beans are called ‘abnormally decomposable.’ (p. 109)

This category ends up catching a number of idioms including promise the moon, pass the buck, steal one's thunder, and bury the hatchet.

Whatever we think of the psychological relevance and reality of this taxonomy of metaphors, it is certainly clear that Gibbs does not want all idioms to be on par.

He must have an intuition that the empirical evidence can't hold up to Lakoff and Johnson's strong claims about the "aliveness" of idioms, and he thus tries to carve out a more reasonable position for himself by introducing a protective belt of terminology.

Cacciari and Tabossi: "The Comprehension of Idioms" (1988)

In this paper, Cristina Cacciari and Patrizia Tabossi argue that our idiomatic reading of a phrase like shoot the breeze is triggered by a "key." The process by which we understand such phrases can consequently differ depending on the position of the key.

Although they don't give any examples, they probably have string like take the bull..., a silver spoon..., and make a fool... in mind when they talk about "keys."

Experimental Evidence

They support this theory with three priming experiments. These experiments show that:

• When the key is highly predictive of the idiom, and it occurs early in the phrase, the idiom is quickly and automatically activated, but the literal word meanings are not.
• When the key is more ambiguous or occurs late (or both), the opposite occurs: The idiom is not activated, but the word meanings are. However, this presupposes that the decision task that tests the activation levels is posed immediately after the stimulus.
• If one inserts a 300 ms delay before the decision task, though, both the idiom and the word meanings become active. This seems to suggest that the initial ambiguity associated with the late/weak key is resolved after the delay.

Recognizing Idioms From "Keys"

Cacciari and Tabossi don't formalize their notion of a "key," but it seems that it might be done in statistical terms. To illustrate that, take for instance the following idiomatic sentence:

• It is best to take the bull by the horns

This sentence contains a number of incomplete left-segments:

1. it ...
2. it is ...
3. its best ...
4. it is best to ... (etc.)

As we consider bigger and bigger left-segments, it becomes more and more probable that these incomplete sentences are going to be completed as the full sentence it is best to take the bull by the horns. This can be estimated by looking at the number of occurrences of the segment vs. whole sentence.

By thus comparing the predictive strength of the left-segment, we can get a sense of where the the uncertainty tips into certainty:

The numbers behind this graph are based on Google searches. For instance, the bar above the word bull shows the number of hits for it is best to take the bull divided by the number of hits for the whole sentence. The 'keyness' is just the difference between this statistic for the relevant word and its neighbor to the left.

As the picture clearly shows, the big jump occurs with the word bull. So we can expect an average English speaker to switch into "idiom mode" upon reaching that point in the sentence.

Conditional Probability Is Not Redundancy

Note this is different from the redundancy of each completion.

That would be another relevant statistic to base the concept of "keyness" on, but it would also require an estimate of the number of completions of each of these left-segments and their probabilities.

That's not a completely crazy thing to try to estimate based on some kind of language model, but it does require some corpus data that I don't have at hand right now.

Opaque Idioms

I just want to give a list of some of the Italian idioms that Cacciari and Tabossi provide, because they are such great examples of how much uncertainty we face when interpreting unknown idioms. Here's a list of their most striking items:

• He was born with the shirt (= born with a silver spoon in his mouth)
• That would have done him the skin (= killed him)
• He made a whole in the water (= did not succeed)
• She did a job with the feet (= badly)
• The project has gone to the mountain (= failed)
• The assets have been given bottom (= been depleted)
• She had the moon (= was in a bad mood)
• He was at the green (= broke)
• He made himself in four to succeed (= tried hard)
• She was left of salt (= struck dumb)