Hanks: "Metaphoricity is gradable" (2006)

This is Patrick Hanks' contribution to Corpus-Based Approaches to Metaphor and Metonymy (2006). As the title says, its claim is that expressions can be more or less metaphorical.

Hanks grades a number of expressions involving the word sea according to how metaphorical he finds them, but it is unclear how to generalize his methods. In the course of the text, he gives at least two different answers to the question of how one might measure the metaphoricity of a phrase.

The Frequency Measure of Metaphoricity

First, he suggest the absolute frequency of the phrase as a measure (p. 21). An area of research is thus perceived as relatively literal because the string as a whole is quite frequent. By contrast, an oasis of calm is quite metaphorical because infrequent.

Hanks may have a point here, but it's difficult to tell, because he never submits any evidence that the statistic of interest is the absolute frequencies rather than the relative ones. It is true that area of research only accounts for about 0.1% of the 34,824 occurrences of area in the BNC, while the 6 occurrences of oasis of calm add up to 2.4% of the 248 occurrences of oasis.

But on the other hand, these numbers are seriously sensitive to paraphrase: if you include study and search into the analysis,  the tally jumps from 43 to 168, i.e., a near quadrupling. For the oasis of calm, expanding the search to include silence, peace, quiet, tranquillity, repose, and serenity only yields 9 more hits, i.e. a growth of about 50%. It is thus not at all clear which of those numbers we should trust more, and which will be responsible for how we think.

	study	work	research	research	concern	search	analysis	enquiry
area of …	87	61	43	43	40	36	5	4

	calm	peace	tranquility	serenity	silence	quiet	repose
oasis of …	6	3	2	2	1	1	1

The Resonance Measure of Metaphoricity

Second, Hanks suggests that the degree of metaphoricity is determined by certain quality called "resonance," a concept he has picked up from Max Black (pp. 20, 22). Hanks writes:

In the most metaphorical cases, the the secondary subject [source domain] shares the fewest properties with the primary subject [= target domain]. Therefore, the reader or hearer has to work correspondingly harder to create a relevant interpretation. At the other extreme, the more shared properties there are, the weaker the metaphoricity. (p. 22)

The term "resonance" subsequently seems to be used as the inverse of "semantic distance" (p. 22), and a similarity-based prototype theory is taken for granted. Accordingly, he finds that an Antarctic oasis—an ice-free area on Antarctica—is slightly metaphorical because it deviates from the stereotypical cartoon oasis with palm trees etc. (p. 30–31).

Meaning and Personal Experience

I thought I should just cite the summary of cognitive metaphor theory that Hanks gives, because it succinctly sums up some of the concerns that a number of people have had about it:

Lakoff and Johnson's basic thesis about metaphor is that its function is to enable us to interpret concepts (especially abstract concepts) in terms of familiar, everyday cognitive experiences. This is broadly satisfactory, though we might be tempted to substitute 'perceptual experiences' for 'cognitive experiences', and common sense forces us to acknowledge that the 'everyday experience' in question is that of the language community at large, not each individual. (p. 19–20)

In fact, I don't think a metaphor needs to be grounded in anybody's first-hand experience in the general case; and metaphors that involve dying are good illustrations of this phenomenon.

Corts: "Factors characterizing bursts of figurative language and gesture in college lectures" (2006)

Apparently, Daniel Corts and Howard Pollio published a study of the use of metaphor in college lectures in Metaphor and Symbol in 1999. Their conclusions were, unsurprisingly, that college lecturers tend to use figurative language when they have to explain something difficult. In 2006, Daniel Corts then published a new study which seem to cover pretty much the same ground.

What is a Metaphor "Burst"?

The data analyzed by Corts is a pair of videotaped lectures on geology and a pair on Greek mythology. Both pairs were held by a single lecturer.

The kind of "bursts" of metaphors that Corts has in mind are passages like the following, here taken from the geology lectures:

[269] That’s a glass with ice tea in it … this is the tea, and that is the ice … [272] so much sugar in there that it is very sweet … you drink that tea when it first arrives … it tastes very sweet … you come back later, and what’s happened to the iced tea there? … [276] The temperature melts the ice … changes the composition of the tea, and as you pointed out, it’s diluted … [279] That tea has a different chemical composition.

[316] That gets us to the end of … igneous rocks. Now I want to step across this line … move on to Objectives 3 and 4 … [319] This is kind of a gear shift … I want to help you through this gear shift.

[142] We have planes of weakness parallel to slope. Yeah, the best way to do that is a deck of cards. Take this deck of cards [places hands parallel to each other horizontally] and you tilt it [tilts hands]. Now, they’ll start sliding off one right after the other [after hands tilt, the top hand ‘slides’ off the other]. That’s in essence what we’re doing here – we’re taking a deck of cards … water gets in here...and then they’ll just get and go off these surfaces [hand slides off again].

The quotes are taken from his Table 4 on page 222. The numbers in square brackets refer to line numbers in the transcript.

As Corts notes, it is characteristic for all of these analogies that they are quite lengthy, quite novel, quite coherent, and that they are accompanied by suitable gestures. One might also add that they are quite explicit in nature, in the sense that no one listening to this lecture would doubt that we are talking in terms of analogies here.

Does Gesture Cohere With Speech?

According to Corts, one  of the conclusions of the original 1999 study was that gesture and speech appeared coherent in the college lectures:

When figurative language and gestures did overlap, they presented the same metaphorical concept rather than two different or independent representations of the concept. (p. 212)

This is true, he claims (p. 227), even for stock phrase metaphors such as

• "Here's where we've been so far … and here's where we're going…"

or

• "on one hand … but on the other…"

That's not quite consistent with the widespread speech/gesture incoherence that Daniel Casanto has reported in a number of studies. In fact, he claimed that the before/after dimension was almost always "conceptualized" as left/right by English speakers, not by behind/front, as Corts claims.

Wax Off, Wax On

It's difficult to assess whether this disagreement in the literature has come about because Corts has been overlooking contrary evidence, since we don't have his data. But at least one of his examples (p. 225) seems to indicate problems with a stricter version of the coherence claim:

Literature, especially oral literature [right hand loops backwards], reflects the way [right hand moves back and forth, flipping from palm up to palm down] the world is. It doesn’t generate the world [hands flip backward, palm down], it reflects it [hand flips forward, palm up].

OK, so if I'm a mirror, then I can "reflect" the world by holding up my hand, palm out, like a policeman stopping a car. But how does flipping my hand backwards, like I'm closing a cigar box, correspond to "generating" the world? This reminds me more of the weird examples Ray Gibbs cited in "support" of cognitive metaphor theory (1994, p. 165–67).

The real reason for this is probably rather that the gesture indicates contrast than that it has any inherent meaning on its own. Any sort of opposing hand movements could probably have done the same job (e.g., wave to the left, wave to the right).

And that's OK — it's only a problem if we claim that the gesture somehow reveals how this professor thinks about the subject, window-to-the-mind style. In that case, we would be forced into a highly awkward reading of the situation when we saw the "shovel water backwards" move next to the word generated.

Metaphor Comparison and Negation

Incidentally, this point is also confirmed by another of Corts' observations: In the discussion, he notes (p. 228) that we often find mixed metaphors in places where two analogies are explicitly compared, or one is negated:

• "Drug abuse is not a disease, it is a game"
• "Literature does not create the world as much as it reflects it"

The fact that we get the point of such comparisons, and especially of such negated metaphors, seems to indicate that we are less captives of our metaphors than the orthodox "conceptual structure" interpretation will have it. If anything, it is rather the metaphors that serve our purposes than the other way around.

Shen and Balaban: "Metaphorical (In)Coherence in Discourse" (1999)

One consequence of the strong "conceptual structure" view of metaphors is that metaphorical expressions are always understood through activations of an underlying metaphor or source domain.

As a consequence, one would expect that writers and speakers were primed to use several instantiations of the same metaphor once they has used a single instantiation of it: We have come a long way since our humble beginnings, we were stuck, but now we're back on track, etc., etc.

But this doesn't seem to be the case, Yeshayahu Shen and Noga Balaban show in this paper. Their methodology pivots around a comparison of a corpus of "planned" or highly explicit uses of a metaphor, and corpus of spontaneous uses. Using a number of different measures, they find that the "planned" newspaper clippings in their corpus exhibit much, much higher metaphorical coherence than the unplanned.

They conclude:

None of the analyses [in the paper] supports the claim that, as a default, unplanned discourse exhibits a coherent distribution of metaphorical expressions. […] The use of metaphors in unplanned discourse appears more like free, uncontrolled "navigation" between a large number of root metaphors than a consistent elaboration of any unifying root metaphors. Indeed, special planning seems to be required to make discourse metaphorically coherent. (p. 151)

As a consequence, it would seem that kliché metaphors have a different cognitive status than deliberate, explicit, and overt metaphors do. Conventional metaphors do not necessarily activate any interesting cognitive architecture:

Rather, the fact that conventional metaphorical instantiations of a given metaphor appear in a given discourse does not necessarily seem to reflect a corresponding activation (in the producer's mind) of the root metaphor. Perhaps this means that the use of conventional expressions in real discourse is autonomous in that, under such conditions of natural discourse production, they do not rely on the functional activation of the entire root metaphor and, in that sense, their meanings are divorced from the underlying conceptual metaphors posited by the CM theory. (p. 152)

Thus, no role for "conceptual structures" with respect to conventional metaphors.

Shen and Balaban also note in passing that "metaphorical shifts in the unplanned passages occurred not only at the intersentential level but also at the intrasentential level" (p. 148). They found, for instance, inconsistent metaphors packed closely together in phrases like "cook up a trap" (p. 150) or the following examples (p. 148):

The peace process is moving forward and ripening (FOREIGN POLICY IS A JOURNEY/FRUIT).

The first step [of the Oslo agreement], the real embryonic one, actually works well (FOREIGN POLICY IS A JOURNEY/A PERSON/A MACHINE).

Violence is eating away the foundations of democracy (THE POLITICAL SYSTEM IS A FOOD/A BUILDING).

Given the reading speed of an ordinary adult, it seems reasonable to assume that these metaphor pairs cannot possibly be comprehended through a process involving mental imagery.

Kimmel: "Why we mix metaphors" (2009)

This is a quantitative and qualitative study of mixed metaphors by Austrian linguist Michael Kimmel (not the American gender researcher of the same name).

Kimmel's topic is cases of metaphor spaghetti like the following, taken from the Guardian:

Tony Blair’s criticism of EU regulations […] would be laughable if it were not so two-faced. While preaching the pro-business gospel, he has done nothing to stop the tide of EU rules and red tape from choking Britain. (quoted on p. 109; boldface in the original)

Such mixed metaphors are common and rarely cause much confusion. In the quantitative part of the paper, Kimmel finds that about three quarters of all pairs of adjacent and topic-specific metaphors are of a mixed type (p. 102 and his table 6).

Metaphors Are Still "Deep"

This poses the question of "why mixed metaphors are cognitively successful" (p. 110).

Kimmel's answer seems to be that we don't experience a rhetorical tension between metaphors because they are mostly processed locally, so we "forget" the image in one sub-clause before we merge it with its neighbor:

[…] I hypothesize that the tightness of connection of ‘‘carrier’’ clauses crucially affects our tendency to process the metaphors integratively or not. Only the close syntactic integration of two metaphors within a clause can enforce or foster a close integration of their semantic content qua imagery. Where mixed metaphors occur across clauses no ontological clashes will be felt to begin with and secondary mechanisms to keep the clash at bay are dispensable. If this is correct, making sense of mixed metaphors is a natural by-product of default clause processing. (p. 110)

So "metaphors with regard to their imagistic meaning must be primarily understood within their proper clause units and not between the units" (p. 112, emphasis in original).

This seems to imply that metaphor mixtures are coherent if and only if they appear in separate clauses, which is not quite the case. Kimmel thus adds the possibility of "shallow processing" (in the sense of Gibbs 1999; see Kimmel's page 112). He can then say:

How jarring such a clash of images becomes in someone’s perception probably depends on the depth of metaphor processing (Gibbs, 1999), as will be discussed later. (p. 109)

Flashes of Metaphor

But "shallow processing" seems to be a last resort for Kimmel, and it seems that he wants to defend a strongly "cognitive" view of metaphor for as long as possible. In many cases that contain inconsistent metaphors, he thus resolves the tension by postulating a super-process that decides what gets into the final parse of the sentence, and what doesn't:

In my view, we may think of a metaphor cluster in such a way that a given conceptual metaphor temporarily "flashes up" in the cognitive unconscious when the first metaphorical expression is processed. In some cases this activation – without being expectational in the strict sense – influences the selection of the subsequent metaphor(s), whereas in many other cases this activation fades or is overridden by other discourse devices. In that sense temporarily active conceptual metaphors are part of a field of multiple discourse attractors that vie for influence. (p. 113)

So the ambiguous conclusion is that

non-metaphoric linguistic devices invariably blend into metaphor processing (p. 109)

and that

conceptual metaphors as cognitive, but perhaps not fully discourse-governing conceptual structures. (p. 109)

Specifically, he emphasizes the general flow of an argument as a force that can impose order on an array of diverging metaphors (cf. p. 109).

Epstein: The Theory of Gambling and Statistical Logic (1977)

In his book on games, Richard Epstein opens one of the chapters with the following theorem:

Theorem 1: If a gambler risks a finite capital over a large number of plays in a game with constant single-trial probability of winning, losing, and tying, then any and all betting systems lead ultimately to the same value of mathematical expectation of gain per unit amount wagered. (p. 53)

This is pretty badly written. I could also be interpreted in several different ways, some of which lead to false statements.

Epstein briefly discusses the details of the game he has in mind, but not enough to dispel the most important ambiguities. In a infuriating footnote, he even adds

Proofs of the theorems have been developed in the first edition of this book. (p. 52)

Wonderful. When I find my time machine, I'll read them.

What Does the Theorem Say?

Style aside, here's what seems to be clear:

We are assuming a situation in which a gambler starts with one unit of capital (X₀ = 1), and we then interested in the limit behavior of E[X_n], the expected capital after n games. The claim is that this depends only on the probabilities and odds of the game, not on the bets.

The problem is that it's not quite clear what Epstein takes to be permissible betting strategies. Depending on what you mean by "risking a finite capital over a large number of plays," his theorem can be read at least two very different ways, and only one of them is true.

The Stockpile Reading

The first reading would be one in which the gambler splits a starting capital of 1 into n piles, and then wagers those piles, one at a time, without reinvestment.

This corresponds essentially to playing n identical games at the same time with n small bets. By the linearity of expectations, this will give the same expected gain as a single game with the entire capital (although a higher variance).

So if that's the intended reading, then the theorem is true, but also quite uninteresting. In any realistic setting, a gambler (or investor) has the possibility of reinvesting past gains, and the sum of the bets over time can thus easily add up to more than the size of the starting capital.

The Percentage Reading

Another reading is that the gambler at any specific time can bet any percentage of the capital accumulated at that time. The claim would then be that however the series (b₁, b₂, b₃, …, b_n) of percentages would look, the expected capital after n games will be the same.

That's a false claim. I can choose the constant betting strategy b₁ = b₂ = b₃ = … = b_n = 0 and get an expected loss of 0 regardless of the game. In fact, even if I must bet a positive fraction of my capital in each game, I can still limit my losses to any arbitrarily small number a by betting the fractions

b₁ = (1 – a)/2,   b₂ = (1 – a)/4,   b₃ = (1 – a)/8,   b₄ = (1 – a)/16, …

So even with some quite generous assumptions, the "theorem" is simply false under this reading.

What We Can Say

Here's one thing which is true, though:

Any particular game of this kind will have a doubling rate W which is computable from its odds and its probabilities. After n games, the expected capital of the gambler will lie somewhere between 1 and 2^Wn, regardless of the betting strategy. (This is true both for positive and negative growth rates, although 2^Wn will be larger than 1 in the former case and smaller in the latter.)

If we like, we can thus get a bound on the expectation of the capital growth. But it's an exponential bound, and it can't be improved: For every number within that bound, there is a betting strategy that gives me an expected gain of exactly that amount.

Postscript: A Comment by Peter Griffin

Now, after having written this, I also see that there is a paper by Peter A. Griffin which addresses the ambiguity in Epstein's theorem. He writes that quantity which Epstein claims to be constant is "E(Win)/E(Bet), the average win divided by the average bet," as opposed to "E(Win/Bet), the average of individual win rates" (p. 1541).

This is consistent with my first interpretation of Epstein's claim: A static game gives, of course, a static expected gain per dollar wagered. As Griffin puts it, this rate will quantify "in what direction and how fast the money is flowing," and this flow rate determined solely by the odds and probabilities of the game.

The alternative that one might be interested in is "how, on average, individuals perceive their win rates." This can indeed vary according to betting system, and for the gambler interested in maximizing returns, this quantity is of course the figure to keep your eyes on.