Part Of: Language sequence
Followup To: An Introduction to Generative Syntax
Content Summary: 800 words, 8 min read

Explaining Substitution

Consider the sentence “I bought this big book of poems with the red cover”.

In everyday language, we often replace words and phrases with indexing words like “one”. Call this indexing replacement.The meaning of these words can be obtained from the context.

At first glance, indexing replacement seems to target a branch in the syntax tree. For example:

• I bought that big one of poems with the red cover (“one” replaces the noun)
• I bought one (“one” replaces the entire noun phrase)

But there are several other substitutions don’t follow from branch replacement:

• I bought that big one.
• I bought that small one
• I bought that big one of poems with the blue cover

Perhaps our notion of noun phrases is too flat. Perhaps we need additional nodes to describe structure within the noun phrase. We will call these intermediate nodes N’, (where N → N’ → N’’ = NP):

This new tree successfully predicts all substitution phenomena, by modeling “one” as replacing various “N-bar” nodes:

We can similarly introduce depth to our verb phrases (VPs), by using intermediate V’ (“V-bar”) nodes:

The X-Bar syntax tree provides a simple explanation of the “do so” substitution effects:

• I will do so in the office before the party.
• I will do so before the party.
• I will do so.

A General Theory of Phrases

We can revise our original NP and VP rules to reflect our intermediate N’ and V’ nodes:

What if noun and verb phrases are instantiations of a more general phrase structure? Just as group theory identifies overlap in the axioms of addition and subtraction, X-bar theory explores the similarity between NP and VP rules.  

There are only four kinds of phrase constituents:

1. The head carries the central meaning of the phrase. Consider the sentence “The tall student who is wearing the red shirt asked questions of her professor, after the lecture.” The central meaning is retained if we remove all non-head words: “student asked questions”.
2. The specifier points to the head. For nouns, specifiers include determiners (“the”) and possessives (“her”). For verbs, adverbs occasionally fill this role (“quickly”).
3. The complement tends to feel intimately related to the head of a phrase (e.g., “of poems” in “a book of poems”).
4. Adjuncts, on the other hand, tend to feel more optional (e.g., “big” in “big book”).

Adjuncts vs Complement

Given that adjuncts and complements both often inhabit prepositional phrases, it is perhaps surprising that they should behave differently. The distinction between adjuncts and complements explains why this should be the case. Let us look at four behavioral differences:

Difference #1. Adjuncts can be reordered freely. 

Consider our example verb phrase:

This rule means that our two adjuncts can be shuffled, but the complement NP must retain its original position

• I will read the letter in the office before the party (Original order: valid)
• I will read the letter before the party in the office (Adj reorder: valid)
• *I will read in the office before the party the letter (Compl reorder: invalid)

Difference #2. Indexing replacement cannot strand the complement.

For example,

• I will do so in the office before the party (Adj is stranded: valid)
• *I will do so the letter before the party (Compl is stranded: invalid)

Consider another part of speech we have not yet considered: conjunction words like “and” and “or”.

Difference #3. Conjunction words bind adjuncts together, and complements together. But adjunct-complement bindings are non-grammatical.

Consider our example noun phrase:

Three examples to illustrate how conjunction works:

• I bought the book of poems and of short stories. (Compl-compl conjunction: valid)
• The book with the red cover and the black spine. (Adj-adj conjunction: valid)
• *The book of poems and with the red cover. (Compl-adj conjunction: invalid)

What X-Bar Theory Tells Us About Memory

Earlier, I introduced the distinction between episodic and semantic memory:

• Semantic: ability to remember facts and concepts (e.g., hands have five fingers)
• Episodic: ability to remember events or episodes (e.g., dinner last Tuesday night)

Concepts are learned by extracting commonalities from episodic memories. If you see enough metallic blocks moving around on four cylinders, you’ll eventually consolidate these objects into the CAR concept:

In philosophy, I suspect the concepts of necessity and contingency relate to semantic and episodic memory, respectively.

In linguistics, I suspect complements help locate concepts in semantic memory, whereas adjuncts assist episodic localization. In the sentence “I bought the book of poems with the red cover”, the complement helps us activate the concept POEM-BOOK, whereas the adjunct creates sense-predictions that locate it within our episodic memory.

Takeaways

• With flat syntax trees, it is difficult to explain indexing substitution (e.g., “bought a book” → “bought one”)
• If we make syntax trees binary, by introducing intermediate  X’ (“X-Bar”) nodes, substitution becomes more straightforward.
• Noun and verb phrases thus parameterize a more general phrase structure.
• Phrases have four kinds of constituents: head, specifier, complement, and adjuncts.
• The differences between complements and adjuncts are instructive:
  • Only adjuncts can be reordered.
  • Indexing replacement cannot strand the complement.
  • Conjunction cannot bind across categories
• In human cognition, complements and adjuncts may correspond to semantic and episodic memory, respectively.

Followup To: Logic Structure: Connectives in IPL
Part Of: Logic sequence
Content Summary: 300 words, 3 min read

Motivations

Last time, we looked at Intuitionistic Propositional Logic (IPL). In IPL, there are five connectives, and hence five introduction-elimination pairs:

What if you had to design a new logic from scratch? Suppose we were to invent five new connective symbols. Would you start by defining their introduction rule, and use these to infer elimination? Or would you instead define elimination first?

This choice reflects different ways to interpret the semantics of logic:

• The verificationist starts with introduction first. For them, the meaning of a connective is in their constructor (introduction rules).
• The pragmatist starts with elimination first. For them, the meaning of a proposition is how you use it.

But if introduction and elimination rules agree, then a logical system has harmony.

How do we evaluate harmony in practice? Harmony is defined as two propositions:

• Local soundness: if I introduce and then eliminate a connective, do I gain information? If so, the elimination rule is too weak.
• Local completeness: if I eliminate then re-introduce connective, do I lose information? If so, the elimination rule is too strong.

Demonstrating Harmony in IPL

We can show that conjunction rules exhibit harmony.

Note that we have only shown soundness for left-elimination. But demonstrating soundness for right-elimination is highly analogous.

Implication rules also exhibit harmony.

So does disjunction.

It is trivial to demonstrate the harmony of truth and falsity. Thus, we can say that IPL, as a formal system, has harmony.

Takeaways

In this article, we have discussed harmony, which helps us evaluate how useful a given formal system is. This notion may seem straightforward in IPL; however, it will prove useful in designing new logics, such as linear logic.

Another more subtle point to consider is that the soundness demonstration also seems to reflect a logic of simplification. This point will return when we discuss the Curry-Howard-Lambek correspondence, and the deep symmetries between logic and computation.

Until next time.

Part Of: Neural Architecture sequence
Content Summary: 700 words, 7 min read

Dual-Process Theory

Dual process theory identifies two modes of human cognition: a fast, parallel System 2 and a slow, serial System 1.

This distinction can be expressed phylogenetically:

But this is incorrect. We know that engine of consciousness is the extended reticular-thalamic activating system (ERTAS), which implements feature integration by phase binding.  Mammalian brains contain this device. Also, behavioral evidence indicate that non-human animals possess working memory and fluid intelligence[C13].

Conscious, non-linguistic animals exist. We need a phylogeny that accepts this fact.

The Tripartite Mind

Let’s rename System 1, and divide System 2 into two components.

• The Autonomous mind is a subsymbolic neural network.
• The Algorithmic Mind constructs perceptual object via the Global Workspace.
• The Linguistic Mind applies linguistic processing to conscious contents.

This allows us to conceive of conscious, non-verbal mammals:

This lets us refresh our view of property dissociations:

Most dual-process theorizing (for example, our theory of moral cognition) maps neatly to the autonomous and linguistic mind, respectively.

But the Tripartite Mind theory cannot bear the weight of all behavioral phenomena. For that, we need the more robust language of two loops. In fact, we can marry these two theories as follows:

This diagram reflects the following facts:

• The Autonomous Mind constitutes most of the brain.
• The Algorithmic Mind is perceptual, and processes Autonomic representations.
• The Linguistic Mind receives Algorithmic (but not Autonomic) information.

Boundaries on the Linguistic Mind

The Linguistic Mind creates cultural knowledge. It is the technology underlying the invention of agriculture, calculus, and computational neuroscience. It is hard to see how such a device could be not only biased, but in some respects completely blind.

But the Linguistic Mind does not have access to raw sensorimotor signals. It only has access to the intricately curated working memory. You cannot communicate mental experiences outside of working memory. You can try, but that would be confabulation (unintentional dishonesty). As [NW77] describe in their seminal paper Telling more than we can know, in practice, human beings are strangers to themselves.

The evidence suggests that working memory does not contain any information about your judgments and decision making. All attempts to describe this aspect of our inner life fail. Introspection on these matters cannot secure direct access to the truth of the matter. Rather, we guess at our own motives, using the exact same machinery we use to interpret the behavior of other people. For more on the Interpretive Sensory Access theory of introspection [C10], I recommend this lecture.

Sociality and the Linguistic Mind

Per the Social Brain Hypothesis [D09], humans are not more intelligent than other primates; we are rather more social. In other words, the Linguistic Mind is a social invention, which facilitates the construction of cultural institutions which allow propriety frames to be synchronized more explicitly.

On the argumentative theory of reasoning, social reasoning is not independent of language. It is the purpose of language.

While the Linguistic Mind evolved to satisfy social selection pressures, not all primate sociality is linked to this device. Social mechanisms have arrived in stages:

• Primary emotions as social behavior network can be traced back to ray-finned fish.
• In New World primates, body language evolved as an extension to our autonomic nervous system, as described in the polyvagal theory. [P03]
• Certain human-specific social mechanisms evolved within the neuroimmune axis as a defense mechanism to parasites [TF14].

All of these mechanisms can be attributed to the Autonomous Mind. But since the Linguistic Mind is driven by our motivation apparatus (just like everything else in the brain), its behavior is sensitive to the wishes of these “lower” modules. This doesn’t contradict our earlier assumption that its content is divorced from Autonomous data.

References

• [C13] Carruthers 2013. Evolution of working memory.
• [C10] Carruthers 2010. Introspection: Divided and Partly Eliminated
• [TF14] Thornhill, Fincher (2014). The Parasite-Stress Theory of Sociality, the Behavioral Immune System, and Human Social and Cognitive Uniqueness.
• [P03] Porges (2003). The Polyvagal Theory: phylogenetic contributions to social behavior
• [D09] Dunbar (2009). The social brain hypothesis and its implications for social evolution

Core Sequence

• Continued Fractions
• Modular Arithmetic
• Octonions

This sequence is simply a bucket that collects some of my older, “pre-theoretic” views on psychology. YMMV. 🙂

Random musings

• Robot’s Rebellion [5 posts]
• Exploring Biofeedback
• Helen Keller
• Cognitive Semiotics
• Phonological Loop

Books I enjoyed

• [Review] Daniel Kahneman – Thinking, Fast and Slow
• [Review] Mihaly Csikszentmihalyi – Flow: The Psychology of Optimal Experience
• [Review] Daniel Siegel – The Neurobiology of “We”
• [Review] Ramachandran – The Tell-Tale Brain
• [Quotes] Freud: On The Interpretation Of Dreams
• [Review] Metzinger – The Ego Tunnel
• [Review] Jaynes – The Origin Of Consciousness And The Bicameral Mind

Part of: Demystifying Consciousness sequence

This sequence explores attention, and how its neural organs relate to the rest of the brain.

Suggested Prereading:

• Three Stream Hypothesis
• Two Cybernetic Loops

Core Sequence:

• An Introduction To The Attentional Spotlight
• Attention As Gatekeeper
• WTA: How Meat Decides
• Salience Maps: The Auction For Awareness

A talk I gave this week, which localizes and extends the two-stream hypothesis.

Three Stream Hypothesis

Until next time.

Part Of: Sociality sequence
Followup To: Agent Detection: Life Recognizing Itself
Content Summary: 1500 words, 15 min read

David Hume once observed:

There is an universal tendency among mankind to conceive all beings like themselves, and to transfer to every object those qualities of which they are intimately conscious. We find faces in the moon, armies in the clouds; and, by a natural propensity, if not corrected by experience and reflection, ascribe malice or goodwill to every thing that hurts or pleases us … trees, mountains and streams are personified, and the inanimate parts of nature acquire sentiment and passion.

How did we acquire the ability to discover minds in the world? Let’s find out.

Equifinality: Awakening To Goals

Obviously enough, living things obey the laws of physics. But living things (which we called agents) also have properties unique to them: characteristic appearances (e.g., faces) and behaviors (e.g., self-propelled movement). In Life Recognizing Itself we saw how this allows the brain to build better prediction machines around these agents. We saw that differentiating agents does not require representing other minds. This fact allows us to appreciate the gradual maturation of the Agency Detection system as we travel across the Tree of Life.

Once an organism is detected, the Agent Classifier strains additional information out of its perceptual signature. But there are many other improvements we can make to our prediction machines. Consider, for example, the perspective of an infant who every night is picked up and placed in her crib.

On some nights she may be crawling around the kitchen, in others the family room, in others sitting in her parent’s lap on the couch. Despite these very diverse beginnings, the outcome of the putting-to-bed process is always the same. From the perspective of the neurons in her eyes, very different beginning states always result in the same end state. This property is known as equifinality, and it is not unique to nursery rooms: it is ubiquitous among living things (another example: the behavior of water buffalo around watering holes).

How might prediction machines anticipate equifinality? An efficient explanation of equifinality comes from ascribing goals to an agent.

Dominance Hierarchies: Awakening To Belief

Minds capable of computing situation-agnostic equifinality already have an advantage: perhaps the lion just ate and is not hungry enough to give chase, but the gazelle is just as well modeling the lion as universally hungry. But there is room to inject a little contingency. Return back to our nursery example: the infant knows that putting-to-bed desire is only activated at night. How best to anticipate contingent desire? An efficient explanation for equifinality contingency is ascribing beliefs to an agent.

In fact, there are myriad benefits towards possessing belief inference systems besides equifinal contingency.  Another important reward gradient stems from most important social structure of the animal kingdom: the dominance hierarchy. Wolf packs, for example, feature an alpha male, who is given special feeding & reproductive privileges. In species like the baboon, this hierarchy becomes much more detailed: every individual male knows their who is above & who is below their place.

Dominance hierarchies can arguably exist without its members having theories of mind. But considering that agility in climbing the hierarchy is under strong selective pressure, and that effective navigation requires tremendous psychological prowess. For example, in his book A Primate’s Memoir, Robert Sapolsky recounts tales of his baboons forming alliances in their attempts to dethrone the sitting alpha. The ability to model the beliefs and desires of one’s alliance partner is surely a boon; hence discovering mind can be viewed as a selective consequence of the dominance hierarchy.

Intentional Stance

This tendency to ascribe beliefs and desires onto other agents together known as the intentional stance. Let us name the module responsible for this ability the Agent Mentalizer. The intentional stance appears in children between 6 and 12 months (Gergely et al, 1994).

As we might expect from any algorithm in the Agency Detection system, we might expect the intentional stance to be quite vulnerable to false positives. And that is exactly what we find. For an extreme demonstration of this, consider the following video (taken from Heider & Simmel. (1944)):

While the Agent Mentalizer was designed to understand the minds of other animals, it had no trouble ascribing beliefs and goals to two dimensional shapes. This is roughly analogous to your email provider accepting a tennis ball as a login password.

Impression Management Via Secondary Models

In your brain, you have a set of beliefs about the world. These beliefs may be stored in many different memory systems, but all of them together improve the power of prediction that you wield over your environment. Your knowledge, your web of belief, I call a World Model. For every significant individual in your life, you have a collection of beliefs about them, provided by your relationship modeling system. Some of these beliefs are non-mental, obtained from systems like the Agent Classifier. Call these beliefs collectively your primary model. For every significant individual in your life, you have a primary model for them; your World Model contains many primary models.

However, the Agent Mentalizer evolved to infer mental states (beliefs, goals) of other individuals as well. But simulating mental states is not like simulating objects – mental states are about objects. That is, we require a new type of model – a secondary model – to simulate the primary model of another person.

Confused? An example should help.

Imagine a three year old child, Bonnie, and her best friend Clyde. Since she was very young, Bonnie has been accumulating shared experiences with Clyde, and is now able to recognize him by appearance. These memories and knowledge of Clyde are stored in a primary model.

At the twelve month mark, Bonnie acquired the ability to simulate beliefs/goals of other people. This awakening has greatly improved her understanding of Clyde. She began noticing when Clyde was not in the mood to play, and his opinions of various toys. She even became able to crudely simulate Clyde’s response to situation X, even if Clyde hadn’t encountered X in real life.

During this time, of course, Clyde went through the very same maturation process. We can model their co-understanding as follows:

The above graphic underscores the “egotistical” subset of secondary models: simulation of the impression they are making on one another. The intentional stance is the birthplace of impression management.

Against Ternary Models

If secondary models simulate primary models, can ternary models simulate secondary models? Well, let’s not close ourselves to the possibility. Here’s what such a thing looks like:

So… ternary models are hard to understand! Let’s simplify:

• A’s model of B = How B Appears To A
• B’s model of A’s model of B = How B thinks B appears to A

Make sure the construction logic of the above image is clear: you should have no trouble constructing a quaternary graphic, quinary graphic, etc. Doesn’t the raw ability to reason about such higher-order logics mean that your brain is capable of infinitely-nested meta-models?

A useful analogy here is the natural numbers (0, 1, 2, …). How can your brain hope to count arbitrarily large numbers of things? After all, folkmathematics is fueled with biochemistry; the number of representations it can store is finite. So… how many numbers can it count?

The correct answer is four.  That is, your subitizing module renders your ability to count up to four is almost instantaneous; for larger numbers, response times rise dramatically, with an extra 250-350 ms added for each additional item beyond about four items. We see a similar time difference in recursive reasoning: most people are easily able to simulate of other people (primary modeling) and conduct impression management (secondary modeling). But imagining the impression-management of other people takes conscious effort; and thus cannot be part of your default Theory Of Mind machinery.

With these insights in mind, we see that ternary models out not be admitted into our mental architecture simply because they are conceivable.  Such a solution would not be parsimonous since these complex behaviors can be built from these atomic components. In fact, in the simplification above, you can even see hints of your two-level brain “translating” ternary concepts into a more direct language that it better understands. 

Takeaways

Here are the ideas I want you to walk away from this post:

• Agents are able to steer different situations towards one outcome. Seeing a world of desire, a world of goals, is how children explain equifinality.
• Social animals typically compete for resources in the shadow of a dominance hierarchy. Modeling the beliefs of their peers about them improves social acumen, and with it, fitness.
• Ascribing beliefs and desires to other agents is known as the intentional stance. It appears very early in humans, between six and twelve months.
• We can formalize the above notions in representation theory. Thinking about other people reside in primary models, thinking about other people’s thoughts go in secondary models.
• Ternary models are unlikely to exist for the same reason that computers don’t feel inclined to count to infinity. Our brains can get there by recursively invoking more simple components.

Relevant Resources

• Gergely et al (1994) Taking the intentional stance at 12 months of age
• Heider & Simmel. (1944) An experimental study of apparent behavior
• Trick, L.M., & Pylyshyn, Z.W. (1994). Why are small and large numbers enumerated differently?

The Nature Of Metaphor

Just for fun, let me open today’s discussion with a few aphorisms:

1. It feels more natural to say “her smile is warm” than “her body warmth is a smile”. Metaphor is asymmetrical. 
2. Abstraction is wedded to metaphor.
3. Inference flows from the concrete to the abstract. Metaphor relocates inference.
4. The flow of inference is constrained. When we say “that lawyer is a shark” our brains decide which of our shark inferences are relevant.
5. Idiom is a form of metaphor. Like idiom, metaphor can go stale.
6. Metaphor relocates affect, even after the stream of inference dries up.
7. Metaphor imbues communication with affective flair or style.
8. Metaphors are hierarchical, with complex themes (e.g., “a Purposeful Life is a Journey”) made of smaller metaphors.
9. In my language, I say that metaphor is narrative. That is, weaving metaphorical hierarchies is narrative paint.
10. Metaphor is not yet differentiated sufficiently to compose well with the rest of cognitive science.

Primary Sensorimotor Metaphor

Okay, time to delve a little deeper. Consider the following metaphors.

1. Affection Is Warmth (“her smile is warm”)
2. Important is Big (“tomorrow is a big day”)
3. Happy Is Up (“I feel uplifted”)
4. Intimacy Is Closeness (“we’re beginning to drift apart”)
5. Bad Is Stinky (“this artist stinks”)
6. Difficulties Are Burdens (“finals are weighing me down”)
7. More Is Up (“prices are high”)
8. Categories Are Containers (“do tomatoes go in the fruit category?)
9. Similarity Is Closeness (“these colors are close”)
10. Linear Scales Are Paths (“your IQ goes well beyond mine”)
11. Organization Is Physical Structure (“how do the pieces of this theory fit together”)
12. Help Is Support (“support your local charity”)
13. Time Is Motion (“time flies”)
14. States Are Locations (“close to having an anxiety attack”)
15. Change Is Motion (“car has gone from bad to worse”)
16. Actions Are Self-Propelled Motions (“my project is moving along”)
17. Purposes Are Destinations (“I’m not where I wanted to be”)
18. Purposes Are Desired Objects (“grab the opportunity”)
19. Causes Are Physical Forces (“pushed the bill through Congress”)
20. Relationships Are Enclosures (“this feels confining”)
21. Control Is Up (“I’m on top of it”)
22. Knowing Is Seeing (“see what you mean”)
23. Understanding Is Grasping (“gotten my mind around imaginary numbers”)
24. Seeing Is Touching (“pick my face out of the crowd”)

What similarities between these metaphors do you see? [Footnote 1] Well, these questions are all unidirectional, and explain abstract concepts by appealing to more down-to-earth domains. What do I mean by down-to-earth? Well, all of the above examples appeal to perceptual or motor phenomena!

In terms of the human brain, perceptual and motor (“sensorimotor”) systems tend to reside in the cortical homunculus. In terms of the human memory hierarchy, these types of concepts tend to arise in procedural memory.

Primary Metaphor In Other Memory Systems

Now, human memory contains more than just procedural memory. We can use our understanding of other memory systems to predict other kinds of primary metaphor.

• “Lawyers are sharks” might be better explained by appealing to a culturally-ubiquitous item of semantic memory
• The Biblical metaphor “Sinners are tax collectors” would plausibly draw from a culturally-ubiquitous item of episodic memory.
• Since autobiographical memories are not culturally ubiquitous, we might predict a more personal taste to this type of metaphor.

Metaphor Composition Is Narrative Paint

Human beings conceptualize abstract objects by bringing many primary metaphors into a complex whole. Let me pull an example from Lakoff & Johnson: the concept of Time [Footnote 2].

The Time Orientation Metaphor looks like this:

• The Location Of The Observer → The Present
• The Space In Front Of The Observer → The Future
• The Space Behind The Observer → The Past

Examples: That’s all behind us now. We’re looking forward to your presentation. He has a great future in front of him.

The Moving Time Metaphor interprets times to be objects and the passage of time to be the motion of objects past the observer.  This metaphor really finds its legs when composed with the Time Orientation metaphor. The Time Orientation + Moving Time complex metaphor, then, looks like this:

• The Location Of The Observer → The Present
• The Space In Front Of The Observer → The Future
• The Space Behind The Observer → The Past
• Objects → Times
• Motion Of Objects Past The Observer → The “Passage” Of Time

Examples: The time will come when there are no more typewriters. The time has long since gone when you could mail a letter for three cents. The time for action has arrived. Thanksgiving is coming up on us. Time is flying by. Let’s meet the future head-on.

But abstractions like time are typically underwritten by more than one can of narrative paint. In this case, the Moving Observer Metaphor alternatively imagines location on the observer’s path as times, and the motion of the observer as the passage of time. Here is the Time Orientation + Moving Observer complex metaphor, in full detail:

• The Location Of The Observer → The Present
• The Space In Front Of The Observer → The Future
• The Space Behind The Observer → The Past
• Locations On Path Observer’s Path   → Times
• Motion Of The Observer → The “Passage” Of Time
• Distance Moved By Observer → The Amount Of Time “Passed”

Examples: There’s going to be trouble down the road. What will be the length of his visit? Let’s spread the conference over two weeks. We passed the deadline. We’re halfway through September. His visit to Russia extended over many years.

Takeaways

Today, I gave you examples of “primary” metaphor, which in this case were grounded in the human perceptual/motor systems. Abstract concepts are made by gluing primary metaphors together like Legos. I also left you with several aphorisms, including:

• Metaphor relocates inference.
• Metaphor imbues communication with affective flair or style.
• Weaving metaphorical hierarchies is narrative paint.

Footnotes

1. This question (“do you see”) nicely illustrates primary sensorimotor metaphor #22.
2. Source: http://www.amazon.com/Philosophy-Flesh-Embodied-Challenge-Western/dp/0465056741. For reasons outside the scope of this post, I cannot endorse this text, but I did find its presentation of complex metaphor useful.