Monday, December 22, 2008

Accessibility and the Issue of Vagueness

Ambiguity, as opposed to vagueness. Also, to stretch the applicability of this comic, an example of language used to express an idea instead of merely categorize it for personal use

In the last post, I toasted the shortest day of the year by noting a famous poem by Catullus. However, I neglected to put a translation. Hence, as a commentator noted, it was odd to have the Latin but no explanation.

This of course reminds me of the idea that language is primarily exercised as a way to communicate reality to others, not merely order it for oneself. This is an inductive statement about human society, not a prescription. As the Woman on the Island scenario suggests, a lone person does not need language, but she can design a system of glyphs to archive her findings. Nor does a group of people need language, though language aids their interaction (and from many theories, is inevitable). And that language must have consistent rules and common vocabulary in order to reliably communicate ideas from one person to another.

The entirety of the poems by Catullus are gibberish to someone without any knowledge of Latin, or even its derivative tongues. Likewise, my humble scrawlings cannot communicate with those who cannot read English.

However, language necessarily creates division and distinction, causing the question to arise: are these distinctions reliable? Words such as "big" or "small" are relative. In a tribe of people who always grow above six feet, "tall" takes on a different meaning than it would for a tribe who never exceed five foot five.

To say that language sets up expectations is to cast doubt on the usefulness of vague words such as "big." As a teacher, I occasionally have students whose handwriting is far too small to be easily legible, and others who write too large. However, if I simply say "Write larger," how large is large? Usually, I end the discussion by providing them with an example of proper size, or some framework. To simply say that their handwriting should be larger raises the question: "How large do you need it to be?"

If a fiction writer describes a character as "tall, handsome, and quick in step," this risks creating a plethora of incarnations of that character who are all beholden to individual readers and their conception of "tall," "handsome" and "quick."

It is interesting to note that there is not much vagueness (not much vagueness? How vague) in the fifth poem of Catullus. After all, he does not say "many kisses;" he specifies the numbers. He uses the word "brief," true, but he contrasts the idea (brevis lux, brief light) with "endless" (nox...perpetua, endless night), creating a clear distinction. The poem even alludes that they seek to create vagueness in the minds of their antagonists, with the certainty of the expression of their love belonging to only the two lovers.

Vagueness is not ambiguity. These two words are often used interchangeably, but I think creating a distinction between them is inherently useful. A word that is vague does not clearly state to what extent it must be applied. A parent who says "Don't go outside until it warms up" will likely be asked for specifics. How warm is warm? When the snow melts? When the thermometer reads that the temperature has gone up ten degrees? A word that is ambiguous has more than one meaning, and the meaning being used is not readily apparent. If I tell another teacher that a student had "wise remarks" to offer during a class discussion, they may ask me to clarify if I mean she had something profound to say, or was simply being sarcastic (wisecracking).

In pre-college logic classes, students often learn the process of crafting proper definitions when describing terms or states with words or phrases. One thing they are instructed to do is avoid vagueness. But is vagueness avoidable in all cases?

Wikipedia summarizes the famous Sorites paradox of vagueness fairly well, though the article has gotten rather messy. In a nutshell, the paradox speaks of a "heap." A heap does not have a solid definition. It is not literally 10,000 grains of sand, nor it is 200,000 grains of sand. But words like "pile," "heap," "pool," "crowd," etc. are used everyday.

If a group of people agreed that a pile of sand was a "heap," and one of them removed a single grain with tweezers, would it still be a heap? Generally, the answer is yes. But if a second is removed, and then so forth, eventually there will only be one grain. Is a single grain a heap?

Many logicians say that the problem with defining what a "heap" is comes from the fact that the word is inherently vague. To call a thing a human may still be broad (is it a man? A woman? Young? Old?) but it is not vague. There is a clear set of characteristics that all must be had to be a human. Not so with words like "vague," or descriptions such as "tall" or "small," "hot" or "cold." In that view, perhaps the best solution to the Sorites paradox is to say that it is no longer a heap when an intelligent mind no longer needs to think of it as a heap.

However, the question of vagueness in logic is more than language. It can also be of perception.

Let's say I roll around in the heap, and parts of it adhere to my skin. Are they a part of me? They have a different source, it seems - but what am I but matter, as the heap is matter? Once, I didn't exist. I was conceived, and matter that came from my mother fueled my development. My constituent elements share many things in common with the ground, though I would not say I am the ground.

Imagine a heap made of my skin cells shed over the past twenty years. I feel no sense of loss when they die and flake off, so I have not been seeking them. But let's suppose they are gathered using advanced technology, and presented as a heap of dust before me. If I were to press my arm against them, and they became stuck to my skin, are they now part of me? What if I ingest them? What if I ingest anything? Does the water I take that is dispersed to fuel my body become part of me? Does the nutrients I take in become part of me?

If I open up my vacuum cleaner and make a heap of what's inside, certainly some of that heap would be my own dead skin cells.

The distinction of what is or is not me perhaps cannot be made based on whether I have conscious control over it. After all, the growth of my skin as well as most internal functions does not depend on a consciousness directing them. Putting food in my mouth or taking medicine does require the conscious action of a mind; my heart beating is involuntary.

But what of my beard? It is part of me, but if I clip it, is it still part of me?

The Woman on the Island who does not have language still has a mind, and can still exercise logical thought. When she sees a scorpion running up her leg, she desires to kill or remove it. If it were on the wall, she could swat it with a stick. But if it is on her leg, she will approach the matter differently. After all, a distinction can be made between harming a wall, and harming herself. Both risk destroying something useful to her, but the latter will create much more immediate pain.

This does have a bearing on logic. It may seem more like philosophical or even biological speculation, but logic asserts as axiomatic that A = A. If it exists, it exists. And the extension is the Law of Non-contradiction, which asserts that something cannot both be and not be. In symbols, this could be rendered ~(A & ~A).

These laws may have no conflict with the idea that I am both myself and part of a larger system. It does raise the issue of language's usage in ordering reality.

Considering the related paradox of the Ship of Theseus. If we had a ship, and over the course of years replaced each part with a new one in order to preserve it for tourists, can we really claim it's the same ship? But when did it stop being the ship? When only one original part remained? Only one plank? But when only two original planks remained, is that really a major difference than when only one remains?

Take the variant involving a river. One can never step into the same river twice, it is said. A river is constantly flowing. It is not a single object that can be picked up and studied.

However, having one label to describe it, such as the Tiber River or the Tigris River, is inherently useful. Just as it may be entirely impossible to exactly define what is part of me, and what is not - but having a conception of me is useful to myself, and understanding what is considered me is useful to conscious minds interacting with me.

These ideas may seem odd for introductory logic, but I would argue they are not. My experience is limited to the students I have had in the junior high logic course I teach (or should it be courses? One can never teach the same class twice, after all), but they seemed to handle it.

It allowed the introduction of an important idea: language is artificial. The problem of vagueness is not a problem with reality, but simply with the tools we have to symbolize and describe it. The word "problem" may even be misleading. Is it really a problem? If we didn't have a general conception of "bald," we would perhaps require a different definition for each head of hair. If we didn't have a general idea of "heap," we may have to produce a different word for each grouping of sand we see. Such an undertaking may be within divine ability, but not ours. So having a number of words that are vague may not be problematic. Instead, it may be necessary for finite minds.

However, one may point to the fact everything that exists is part of a larger system. To attempt to create distinction, or to exercise Laws of Thought, may seem futile. Of course, this does not invalidate the Laws of Thought. After all:

If the system cannot be accurately separated via description, it cannot be accurately separated via description.


A system cannot be described by language AND not be able to be in any way, shape, or form described by language.

Even a universe where 2+2=5 does not invalidate the Laws of Thought. But that's a discussion for another time.

Perhaps the companion or derivation of Gödel's incompleteness theorem is the idea that a language that is not merely consistent but accessible can never be exhaustive, but must have some segment of its vocabulary (possibly a large segment of it) be built on vagueness.

A language built for one may seek to be exhaustive, but one that is meant to be understood by others may require sensitivity to the issue of accessibility.

Of course, how can one resolve a question of vagueness? If I hold a man to be tall and a friend does not regard them as tall, how can we resolve the issue? Presumably, we will eventually settle that "he is tall to me, but not tall to you." Categorizing that man differently does not have an inherent risk (that I can see, yet). However, if I hold a bottle to contain poison, and my friend says it does not contain poison, we cannot say "it poison to me, but not poison to you." If he drinks that bottle, and it is poisonous, it will have impact on him.

Perhaps it is best to say that language meant for usage by two or more people requires both concrete words that separate objects and concepts from each other, and vague words that attempt to describe perceptions. The former is making a statement that is inherently universal; the latter is one that relies on personal judgments.

In other words, we can have a rational discourse about a heap of sand even if I consider it a heap and you do not, as long as we both agree that there is sand before us. We can discuss whether a man before us is handsome or not as long as we both first grant that there is a man before us.

But I could be wrong. I would like to soon discuss Aksapada Gautama's views on inference as opposed to perception.

I reserve the right to be full of nonsense,

x Peregrinus

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