Notes for 4/27/2026

  

4/27/2026

 

Would you rather understand a few things perfectly or many things imperfectly?

 

A “sorites paradox” is a paradox of vagueness.

 

Suppose you pour a container of salt on the table to form a heap of salt.

 

Rule1: If you remove 1 grain of salt, you will still have a heap.

 

We know that if we remove enough grains of salt,  Rule1 will fail (certainly when there are zero grains left).

 

Now start putting grains of salt back on the table.

 

Rule2: If you add 1 grain of salt, you won’t have a heap.

 

We know that if we add enough grains of salt, Rule2 will fail (certainly when the whole container has been added back).

 

The question is what “enough” means in both Rule1 and Rule2.

 

There are different basic theories of vagueness.

 

Metaphysical Vagueness: The world has vague objects and vague boundaries.

 

 

This model of a hydrogen atom is an idealization. In reality, the electron doesn’t have such a neat orbit. The reality is more like this:

The electron’s orbit is not precise, but vague.

Not only that, but until measured, the electron has no definite position at any time. Rather, it exists in a “superposition” of multiple possible probabilistic locations and only acquires one position definitively when measured.

 

The metaphysical view takes this as evidence that vagueness is “out there” as an objective feature of reality.

 

 

Conceptual Vagueness: Vagueness isn’t a feature of how things are, but rather of the concepts we use to represent how things are.

 

There is a definite fact as to how many grains of salt are on the table, but the concept of “heap” has no single correct “what, exactly” answer. Exactitude is impossible because the concept isn’t exact.

 

 

 

Epistemic Vagueness: Vagueness is really a form of ignorance. There is a point where Rule1 or Rule2 fails, but we are not generally capable of knowing where that point is.

 

The epistemic theory has little initial appeal. But here is a supporting argument:

 

Devil’s Tower, WY

 

 

Suppose you are standing on top of Devil’s Tower (=ON). If you move 1cm to the right, you will still be on DT. If you repeat this rule, there will a point where you are clearly no longer on top of DT (=OFF). For every point in between ON and OFF, there are only 3 possibilities:

     You are still on DT.

     You are off DT.

     It is indeterminate whether you are on or off DT. (Neither ON nor OFF =IND)

 

Now consider all the points between some IND and OFF. For every point in between IND and OFF, there are only 3 possibilities:

You are still on an IND point.

     You are off DT.

     It is indeterminate whether you are on an IND point or off DT. (And since IND = neither ON nor OFF, this = IND.)

Now, since there is a point where you are definitely off DT, there must be some movement where you transition from IND to OFF.

But this means the concept of OFF must have at least a clear case where it applies and where it doesn’t. That is, the concept isn’t vague; vagueness arises from our not knowing whether or not it applies in some cases (which is the epistemic view).

One approach to “solving” vagueness is called “supervaluation.”

 

We’ve seen a number of cases where concepts can have different boundaries under different conditions or in the minds of different competent language users.

 

To give a defensible “what, exactly” answer that defines a boundary for a concept can be called a “precisification” (way of making precise).

 

A statement is "supertrue" if it comes out true under all defensible precisifications, and "superfalse" if false under all of them.

 

So, for example, “I am now on top of Devil’s Tower” is superfalse and “I am now in Akron” is supertrue.

 

Supervaluationism is one way of defending the conceptual theory of vagueness against the argument above.

 

 

Defenders of the metaphysical theory could take the view that the reason WHY there are multiple conceptual variations possible in vagueness cases is exactly because those cases are vague in themselves. If two people draw their conceptual boundaries in different ways, this is because there are no precise objective boundaries that determine whether or not any given set of conceptual boundaries is correct.

 

Which of the 3 views most appeals to you? Metaphysical, conceptual, or epistemic?