Open and Closed Concepts and the Continuum Fallacy


Sandra LaFave
West Valley College


Terms are words or phrases that designate classes.

General terms designate classes with more than one member, e.g., common nouns such as “book” or “tree.”

Singular terms designate individuals, e.g., proper nouns (“The Taj Mahal”), or proper names (“Princess Diana”).

Non-denoting terms refer to the empty class (also known as the “null set”), e.g., “mermaid.”

The denotation of a term is, for general terms, the class of things in the world to which the term correctly applies. For example, the denotation of the term “book” is all the books. Philosophical synonyms for “denotation” are “reference” and “extension.”

The connotation of a term is the list of membership conditions for the denotation. Philosophical synonyms for “connotation” are “sense,” “intension,” and “real definition.” Note this sense of “connotation” differs from the literary sense.

The connotation of the general term “square” is “rectangular and equilateral.”

The denotation of the general term “square” is all squares.

A closed concept is one for which it is possible to precisely specify the connotation. The concept “square” is closed. The set of squares has two membership conditions: equilaterality and rectangularity. For something to be in the denotation of “square,” it must satisfy both conditions; in other words, it must be both equilateral and rectangular. Each condition is necessary and together the conditions are sufficient to make something a member of the denotation of “square.” 

Few concepts are closed. Most concepts are open. This is not a big deal.

An open concept is one for which the connotation cannot be precisely specified; rather, we recognize members of the class by their resemblance to paradigms of the concept.  

Example: “disabled person”

We know the paradigms of disabled persons (people like Stephen Hawking and Christopher Reeve), and we judge whether or not another individual should be called “disabled” on the basis of the resemblance between this person and the paradigms. (I.e., we argue by analogy.)

NOTE: When we say that the concept “disabled person” is open, we mean simply that it is hard to decide in some cases whether that concept should be applied to an individual person. We do NOT mean to say that the concept is useless. Just because it’s hard to say in some cases doesn’t mean there’s no such thing as a disabled person! Arguing that way would be committing the continuum fallacy (see below).

All the interesting social, legal, and philosophical notions are open, and require argument by analogy to decide borderline cases: “person,” “good,” “religion,” “rich,” “sexual harassment,” “obscenity,” “minority,” etc. Such argument is important because it matters what we call things.


The Continuum Fallacy consists in  arguing that a concept is useless or a debate is pointless because some of the key concepts in the discussion cannot be defined precisely. The phrase often heard in continuum fallacies is “Where do you draw the line?”

For example: “I don’t see how you can argue for accommodating people with disabilities. Where do you draw the line between disabled and able-bodied people? You can’t draw the line precisely. So the whole debate about accommodating people with disabilities is pointless.”

A variant of the continuum fallacy is arguing that if you can’t draw the line precisely between X and non-X, there’s no distinction at all between X and non-X. This is exemplified in the famous argument of the beard. We think there’s a difference between having a beard and not having a beard. We also think it’s possible for a man who doesn’t have a beard to have a beard at a later time. But the argument of the beard says a man who doesn’t have a beard can never have one, since it’s not possible to say exactly where to draw the line between a beard and a non-beard:

“If a man has one whisker on his face, he doesn’t have a beard. If he grows a second whisker, he still doesn’t have a beard. A third whisker won’t give him a beard either. So it doesn’t matter how many whiskers a man has: he still won’t have a beard.”

Another example of the continuum fallacy: “Smoking one cigarette won’t harm my health. And two cigarettes is only one more than one. So the second cigarette won’t hurt me.  And three cigarettes is only one more than two.  So that third cigarette won’t hurt me either. Where do you draw the line? You can’t. So I can smoke as many cigarettes as I want and it won’t ever hurt me.”


Defining Our Terms

People who commit the Continuum Fallacy often think they have won the argument by showing that a concept is open. For example, people sometimes argue that all questions of obscenity or sexual harassment or date rape or disability are nonsense, because people can’t say in every case what counts as obscenity or sexual harassment or date rape or disability. These are open concepts with fuzzy borders, and as society changes, those borders can change. So we can’t draw the line precisely. Suppose you point this out. The person committing the continuum fallacy now thinks she has triumphed. She says, “Well, since you can’t draw the line precisely, you don’t really know what you’re talking about at all. You must define your terms, you know. We can’t begin to discuss something we can’t define.”

But this is nonsense, no? We can and do engage in meaningful debate about the application of open concepts. That is the center of law and philosophy. You should answer, “I can’t draw the line precisely, and I don’t have to. An open concept is still meaningful!”

It’s important to realize that open concepts are normal and useful, even if we can’t define them as precisely as we can define closed concepts. It IS important to define terms when we argue. But sometimes we have to settle disputes about paradigms before we can begin to argue specific cases.

For example, suppose you are debating someone over whether people with perfume allergies should count as disabled. You say, “Well, people with perfume allergies are certainly much less disabled than people like Christopher Reeve or Stephen Hawking. People with perfume allergies can still move their arms and legs at will. If they’re disabled, they’re certainly not severely disabled.” Now suppose your opponent says, “Christopher Reeve isn’t disabled at all! He can do anything he wants – in his mind! In fact, no one who can think is disabled, because thoughts are free!”

Clearly, you and your opponent are using very different definitions of “disability” – different paradigms.  You’re not going to get anywhere if you don’t first come to some agreement about the paradigm cases. That’s where it’s important to “define your terms.”  What does your opponent think “disabled” means? The burden of proof is on your opponent in this case to argue for his notion of disability. He must show first that it makes sense (is any conscious person disabled, on his view? if no one counts as disabled, does the term “disabled” mean anything anymore?).

People are of course free to use words any way they like. But in contexts where argument matters – in the university, in science, and in public life – people must defend their eccentric word usage with arguments. In other words, people must give good reasons.



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