A Theory of Vagueness

Beginning with Reality

Start with sand.

A pile of sand on a table. How many grains? Let us say ten thousand. Each grain has a position, a size, and a shape. At the level of physics, this is a precise configuration. We could, in principle, specify the exact coordinates of every grain. Call this complete specification a microstate.

The microstate is exact. There is no blur in reality. Grain number 4,217 is either here or there, not vaguely somewhere in between. The universe, at bottom, deals in precision.

Now remove a grain. A new microstate. Remove another. Another microstate. Each configuration is as precise as the last. The universe does not become less precise as we remove grains. It simply moves from one exact state to another.

Here is the ancient puzzle. Ten thousand grains is a heap. One grain is not. Yet there is no particular grain whose removal transforms heap into non-heap. We cannot find the boundary. The boundary seems not to exist. And yet surely there is a difference between a heap and a single grain.

This is the problem of vagueness. I want to suggest that we have been thinking about it wrongly. The puzzle is not about finding a hidden boundary or tolerating a fuzzy one. The puzzle dissolves once we understand what vagueness actually is.

A Thought Experiment: Infinite Minds

Imagine a being with infinite cognitive capacity.

Such a being could represent every possible configuration of sand distinctly. Not “heap” and “pile” and “few grains.” Those are groupings, compressions, and lossy summaries. Instead, Configuration-7,531,842 and Configuration-7,531,843, each with its own unique label. Every possible arrangement of matter in the universe would be named individually.

For such a being, the question “is this a heap?” does not parse. There are no heaps in its ontology. There are only configurations with names. It might understand the question the way we understand “is this number gruesome?” as a question that presupposes a category that, for us, does not exist.

What would vagueness look like to such a being? It would not look like anything. There would be no borderline cases, because there would be no borders. No fade zones, because no fading. Every question of the form “is this an X?” would have a determinate answer. Either the configuration is Configuration-7,531,842 or it is not. Yes or no. Always.

The infinite being cannot experience vagueness because vagueness requires grouping, and the infinite being does not group. It has a unique representation for every distinct state of affairs.

This thought experiment establishes something important. Vagueness is not in the world. The pile of sand is precise. The configurations are exact. Vagueness enters, if it enters at all, somewhere else.

What We Are

We are not infinite beings. We are finite processors navigating an effectively infinite world.

The space of possible configurations is combinatorially explosive. Consider just the sand. The number of ways to arrange ten thousand grains in three-dimensional space is astronomical. And sand is simple. Consider all the configurations of matter you might encounter in a lifetime. Consider all the configurations of social situations, emotional states, and abstract ideas.

A finite system cannot assign unique representations to each possible state of affairs. There is not enough room. Not enough memory. Not enough processing capacity. The infinite being’s trick, one label per configuration, is unavailable to us.

So we group. Many configurations map to a single representation. We call this “abstraction” or “categorization” or “concept formation,” but at bottom it is compression. Lossy compression. Information is discarded. Distinctions are collapsed. Many become one.

This is not a defect. It is a condition of thought itself. Without grouping, no generalization. “This configuration preceded food” is useless. “This kind of situation precedes food” is actionable. Without generalization, no prediction. Without prediction, no intelligence. Grouping is not a limitation we might overcome. It is constitutive of what it means to be a finite mind thinking about an infinite world.

The question is this. How does grouping work? What determines which configurations fall under the same concept? Why is this configuration a “heap” and that one a “pile” and another one “a few grains”?

How a Child Learns “Bottle”

Consider how concepts actually form.

A child is in a kitchen. Her father hands her an object and says “bottle.” She looks at it. Plastic. Cylindrical. Has a cap. Contains liquid. She does not yet know which of these features matter.

Later, a different object. “Bottle.” Glass this time. Different shape. Still has a cap. Still contains liquid.

Later still, another object. “Cup.” Hmm. Also contains liquid. But no cap. Different shape. Open at the top.

From these examples, and dozens more, hundreds more, over months and years, a concept forms. The child does not learn a definition. No one recites necessary and sufficient conditions for bottlehood. Could you? Try to define “bottle” precisely. You will find it surprisingly difficult. The child learns from instances.

Some instances matter more than others. The clearest cases, encountered earliest or most frequently, anchor the concept most firmly. Call these paradigms. The child’s paradigm bottles, the prototypical, central, obvious bottles, shape what “bottle” means for her.

This is how most concepts form. Not by definition. By paradigm.

“Love” is never defined. What would the definition even be? It is learned from instances, from experiencing it, from observing it in others, from stories and songs and a thousand indirect encounters. “Chair,” “game,” “tall,” “heap,” “justice,” and “friend” are all learned this way. We encounter examples. The concept forms around them.

Even concepts that seem definitional often are not. Ask someone what a “bird” is. They might say “an animal with feathers that flies.” But penguins do not fly. Ostriches do not fly. Bats fly but are not birds. The “definition” is a post-hoc rationalization, an attempt to articulate boundaries that were never learned as boundaries. What was learned was robins, sparrows, eagles, and pigeons, which are paradigm birds. The concept formed around them.

Where Paradigms Come From

Paradigms need not come from direct experience.

“Unicorn.” You have never seen one. Yet you have a concept of unicorn. Where did the paradigms come from? Pictures. Movies. Descriptions. Stories. The concept formed from these indirect encounters, just as “bottle” formed from direct ones.

“Democracy.” You did not encounter it directly. You learned about it through language, through examples, and through descriptions of how it works and where it exists.

We can even form concepts on the fly by combining existing ones. “A purple elephant wearing a top hat.” You have never encountered one, but you can think about one. The concept is assembled from parts, each of which traces back eventually to paradigms.

The source of paradigms varies. The structure is the same. Concepts are anchored by paradigms, whether those paradigms come from experience, from language, from images, or from combination.

The Key Point: Centers, Not Edges

Here is what I want to say.

Paradigms determine centers, not edges.

When you learn “heap” from examples, you learn what a clear heap looks like. The paradigm heaps, the ones you encountered and the ones that shaped the concept, fix what counts as central, obvious, and unambiguous. Ten thousand grains piled high is clearly a heap. Your paradigms tell you this.

But what do your paradigms tell you about fifty grains? About twenty? About five?

Nothing.

The paradigms are silent. They did not include fifty grains. They did not include twenty. They fix the center. They do not fix the periphery. They determine what is clearly in. They do not determine where membership ends.

A boundary would require something to constitute it. What could that be?

A rule? “Heaps have more than n grains.” But no such rule was learned. No one told you the number. You could not state it if asked.

A precise similarity threshold? “Anything more than 73.2 percent similar to the paradigms counts.” But similarity does not come with a percentage. And even if it did, who set the threshold?

The complete pattern of usage in your linguistic community? But that pattern is itself generated by individuals using paradigm-based concepts. It cannot be more precise than the concepts that generate it.

Something metaphysical, independent of minds and language? But what would that even mean? What mind-independent fact could make fifty grains a heap or not a heap?

I do not think any of these options work. And if nothing constitutes a boundary, there is no boundary.

This is not epistemicism, the view that there is a boundary but we cannot know where it is. It is the denial that there is a boundary at all. Not hidden. Absent. Nothing to know.

What There Is Instead: The Gradient

If there is no boundary, what is there?

A gradient.

The concept “heap” applies strongly to configurations very similar to the paradigms. It applies less strongly to configurations that are less similar. It applies barely, or barely fails to apply, to configurations that are much less similar.

Picture it as a field of force. Strong near the paradigms. Weakening with distance. Eventually indistinguishable from zero, but with no line marking where.

This is not a container with fuzzy edges. It is not a sharp boundary that has been smeared. It is a gradient that never had edges. Strong in the middle. Fading outward. The fading is the structure. There is nothing else.

Near the paradigms, there is clear membership. A thousand grains piled high is clearly a heap. The gradient is strong. The concept applies robustly.

Far from all paradigms, there is clear non-membership. A single grain is clearly not a heap. The gradient has faded to nothing. The concept does not apply.

In between lies the fade zone. Fifty grains. The gradient is weak. The concept sort of applies. It sort of does not.

What Happens in the Fade Zone

What is it like in the fade zone?

Not ignorance. It is not that there is a fact about whether fifty grains is a heap and we simply cannot access it. What would that fact consist of? The paradigms do not determine it. The similarity structure does not determine it. Nothing determines it. There is nothing to be ignorant of.

Not ambiguity. Ambiguity is when a word has multiple precise meanings and context does not select between them. “Bank” can mean a financial institution or a riverbank. But “heap” in the fade zone is not switching between multiple precise meanings. It is one meaning, the paradigm-based gradient, that simply does not determine membership in the fade zone.

Not truth-value gaps in the sense of missing semantic machinery. The word “heap” has a meaning. The sentence “this is a heap” is grammatical and interpretable. The machinery is all there. What is missing is a fact for the machinery to latch onto.

In the fade zone, there is no fact about membership. Not hidden. Not ambiguous. Not semantically defective. Just absent.

If you are asked, “is this a heap, yes or no?”, you must respond. The question demands an answer. But you are not discovering a hidden truth. You are creating a categorization where none existed. The answer was not waiting to be found. It comes into being with your response.

This is what vagueness is. Vagueness is the fade zone. It is where paradigm-based concepts, being gradients without edges, fade to a level that determines nothing.

The Fade Zone Has No Sharp Boundary Either

A natural question arises. Where does the clear zone end and the fade zone begin?

There is no sharp boundary there either. The theory predicts this.

The gradient fades gradually. There is no line marking where “clearly a heap” becomes “sort of a heap.” The fade zone fades into existence just as the concept fades out of applicability.

This is sometimes called “higher-order vagueness,” meaning vagueness about where the vagueness begins. Some theories struggle with it. This one does not. The gradient structure handles it automatically. There is no boundary between zones because there are no zones. There is only a continuous gradient, strong in the middle, fading in all directions, with no lines anywhere.

A Second Thought Experiment: Gort

Consider a community that speaks a language much like ours, with one addition.

They have a word, “gort,” for piles of roughly 500 to 1,500 grains. Their paradigm gorts are configurations that, for us, fall somewhere in the fade zone between “pile” and “heap.” Where we see borderline cases, they see clear cases of gort.

Now consider a configuration of 1,000 grains.

For the gort-speakers, this is a clear case. It sits squarely in the center of the gort gradient. There is strong membership and no vagueness. Someone asks “is this a gort?” and the answer is obviously yes.

For us, this same configuration is borderline. It falls in the fade zone between our concepts. It is not clearly a heap. It is not clearly not a heap. It is vague.

Same configuration. Same pile of sand. Same microstate. But vague for us and not vague for them.

What does this show?

It shows that vagueness is not in the world. The configuration is precise. Every grain is somewhere specific. The vagueness is not a property of the sand.

It also shows that vagueness is not in the concept alone. “Heap” is vague, but the vagueness of a particular case depends on whether that case falls in the fade zone. A thousand grains is vague relative to our concepts. It is not vague relative to theirs.

Vagueness is relational. It is a three-place relation. A configuration is vague relative to a set of concepts for a processor. Change the concepts and the verdict changes. The same configuration can be clear or borderline depending on what concepts are available.

Could we eliminate vagueness by adding “gort” to our vocabulary?

No. We would relocate it. There would now be fade zones between pile and gort, and between gort and heap. The borders move. They do not disappear. Add another word and fill in another gap, and new fade zones appear at the new boundaries.

The only way to eliminate vagueness entirely would be an infinite vocabulary, with a unique word for every possible configuration. But then it would not be a vocabulary. It would be an inventory. Language requires grouping. Grouping by paradigms produces gradients. Gradients have fade zones.

Vagueness is inevitable for finite minds navigating a precise world with finite vocabularies. It is not a problem to be solved. It is a structural feature of the situation.

The Exception: Rule-Based Concepts

Not all concepts are vague.

“Even number” has no borderline cases. A number is even or it is not. There is no fade zone, no gradient, and no vagueness.

Why?

Because “even number” is not paradigm-based. You did not learn it by encountering examples and grasping a similarity structure. You learned a rule. A number is even if and only if it is divisible by two. When you categorize, you do not compare to paradigms. You check the rule. It is satisfied or it is not.

Rule-based concepts have stipulated boundaries. The stipulation does what paradigms cannot. It determines an edge. Where exactly does “even” end and “odd” begin? At divisibility by two. Precisely there. The rule says so.

This is why legal systems often convert vague concepts into rule-based ones. “Adult” is vague, learned from paradigms with a gradient and a fade zone. When does childhood end and adulthood begin? For practical purposes, the law stipulates eighteen years old. The cutoff is arbitrary. Why eighteen and not seventeen? But it is precise. Vagueness is eliminated by fiat.

“Intoxicated” becomes “blood alcohol above 0.08.” “Speeding” becomes “exceeding the posted limit.” “Murder” is carefully defined with specific criteria. The law cannot function with fade zones, so it stipulates them away.

But notice that this requires explicit stipulation. Someone must make a decision. The precision does not emerge from the concept itself. It is imposed from outside.

Most concepts, most words in natural language and most categories in everyday thought, have no such stipulation. “Bottle,” “tall,” “love,” “game,” “justice,” “friend,” and “heap” are all learned from examples. They are all paradigm-based. They are all gradients. They are all vague.

Vagueness is the default. Rule-based sharpness is the exception, achieved only through explicit stipulation.

Context and Paradigm Activation

“Is she tall?”

The answer depends on context. Tall for a gymnast? Tall for a basketball player? Tall for a child her age? The same person, the same precise height, can be clearly tall in one context and clearly not tall in another.

This is not a problem for the account I am offering. It is predicted by it.

Context activates different paradigms. When you hear “tall basketball player,” you do not summon your general paradigms for tall people. You summon paradigms of basketball players, which form a much taller set. The gradient is shaped by these paradigms. A different gradient produces a different fade zone and a different verdict.

Similarly, consider “expensive restaurant” versus “expensive car,” “old building” versus “old person,” and “fast for a human” versus “fast for a cheetah.” In each case, context selects paradigms, paradigms shape the gradient, and the gradient determines where clarity ends and vagueness begins.

Context-sensitivity is not a separate phenomenon from vagueness. It is a manifestation of the same underlying structure, namely paradigm-based gradients whose shape depends on which paradigms are activated by context.

How We Communicate

Different people have different paradigms. My paradigm bottles are not exactly your paradigm bottles. I encountered different examples at different times, with different salience.

How, then, do we communicate?

Our paradigms overlap enough. We live in a shared world. We encounter similar examples. The bottles I have seen are similar to the bottles you have seen. Our gradients are not identical, but they are close.

This means that we mostly agree on clear cases. Show us a paradigm bottle and we both say “bottle.” Show us a paradigm cup and we both say “cup.”

It also means that we might disagree on borderline cases. Show us something in the fade zone, and my gradient might pull me one way while yours pulls you another. This is not error. It is not miscommunication. It is what happens when slightly different gradients meet a case in the fade zone.

The theory predicts this. Agreement on clear cases and potential disagreement on borderline ones. This seems to match how language actually works.

Concepts and Words

I have been sliding between “concepts” and “words.” Are they the same thing?

Not quite. Concepts form in thought. Words attach to concepts.

A pre-linguistic creature could have paradigm-based concepts, that is, similarity-based categories with gradients and fade zones. It could recognize things as similar to paradigms without having words for them. Vagueness would exist for such a creature, though it could not discuss it.

Language gives concepts names. The word “bottle” attaches to the bottle concept. Both inherit the gradient structure. The word is vague because the concept is vague. The concept is vague because it is paradigm-based.

So vagueness is primarily about concepts. Language inherits it.

What Vagueness Is

Let me gather the threads.

The world is precise. Configurations are exact. Every grain of sand is somewhere specific. Reality does not blur.

We are finite processors. We cannot represent every configuration distinctly. We must group. Grouping is a condition of thought, not a defect.

We group by paradigms. We encounter examples. Concepts form around them. This is how most concepts work, learned from instances rather than definitions.

Paradigms determine centers, not edges. They fix what is clearly in. They do not fix where membership ends. Nothing does. A boundary would require something to constitute it, and there is nothing.

So concepts are gradients. Strong near paradigms. Fading with distance. No sharp boundary anywhere.

In the fade zone, there is no fact about membership. Not hidden. Absent. Nothing to know. A question presupposes a fact, and in the fade zone there is none.

If forced to answer, we create a categorization rather than discover one. The answer comes into being with the response.

Vagueness is relational. A configuration is vague relative to a set of concepts. Different concepts yield different verdicts. The same configuration can be clear for one vocabulary and borderline for another.

Vagueness is the default, because paradigm-based learning is the default. Sharpness is the exception, requiring explicit stipulation.

Vagueness is inevitable for finite minds navigating a precise world. It is not a puzzle to be solved or a defect to be repaired. It is what happens when bounded processors compress an unbounded reality.

In essence: the world is precise. Our minds are not. In the gap between them arises vagueness.