On Vagueness

How do we measure and classify things quantitatively? What is the true difference between a pile, heap, handful, and speck of sand? Are these true distinctions or conventions of language? At what point does one become the other if one grain of sand is being removed? This is a problem one can find in many features of our life and the way we use language. So many of our concepts are loosely defined instances of a qualitative pragmatism designed to enable us to have a coherent conversation without defining our terms every step of the way. To really get to this is a problem I would like to discuss something that gets at this issue of vagueness: sorites paradox.

Sorites paradox is a useful tool to getting at this concept of vagueness, it is attributed to Eubulides of Miletus, a contemporary of Aristotle (so Wikipedia tells me). It goes something like this: consider a heap of sand, then remove exactly one grain, it surely must still be considered a heap of sand. If you continually remove exactly one piece of sand over and over again until one grain is left, where did it stop becoming a heap of sand? Surely you cannot say that one grain of sand can still be considered a heap. You could point at the heap when there is a mere few thousand grains left and comfortably say it is not a heap, likewise when a few thousand are removed you could comfortably call it a heap. The hard case is exactly when it stops becoming a heap. If I were to proclaim that “more than 6,097,898 grains of sand constitute a heap while less than that is not” you would rightly condemn this absurdity. There could be two answers to this; that the number of grains of sand is not what we mean by this concept (there is another measure) or that it is vague. 

What this paradox aims to point out is the vagueness present in some of our concepts. This problem is present when a concept, such as ‘heap’, is insufficiently defined or measured, meaning a contentious plurality of things could be defined in this way. An example of this is democracy; it is a loaded term lacking precise bounds. I am sure most students alongside me have been asked to define democracy at some point only to have a brief panic; what actually is it? The way we usually get around this problem in ordinary language and conversation is to point at (obvious) examples of what we mean by the concept, “look at New Zealand, THATS what a democracy is” or we give properties and characteristics of the concept, “a democracy has free and fair elections, and meaningful competition!” In fact this process marks the basis of a whole slew early Platonic dialogues. For example, Socrates attempts to draw out what courage (Laches) or virtue (Meno) really is. The structure of these dialogues is often the interlocutor merely offering examples of X or simple definitions that fall to the least scrutiny; only to end up inconclusive. It seems once again elusive to try to reach the true Form of such a concept; it seems purely intuitive, or merely a word existing to define its own phenomena. 

While at least there is some intuitive sense of what virtue or courage is in this way, this can cause problems when we wish to make seemingly important and concrete distinctions among things placed on a continuum. Categorising any instance of a quantitative measure P in a dichotomy of X vs.Y (vs. Z etc.) is not always easy in practice. Take a simple dichotomous example of freedom in states to illustrate this point: Imagine Freedom House (a popular and prestigious measure of freedom in states) were to draw the "free/not-free" line at 5/10 average score based on multiple measures. Now consider a situation where nation X scores 5.01 and nation Y scores 4.99. It seems arbitrary does it not. What makes X free and Y not-free? If nation X and Y were classified in degree, “X is slightly more free than Y” we are comfortable. When translating these scores to a dichotomy though, calling X free and Y not-free seems unfair and arbitrary. Or, in short:

Imagine that instance P can be scored on a univariate continuum, measuring variable v, from 1-10 where if: 

• P<5, P is an X 

• P>5, P is a Y 

If P1 scores 5.01 and P2 scores 4.99 It seems merely arbitrary. What makes P1 an X and P2 a Y? They are both measured by variable v (this can even be a multitude of things of things) but are placed into two entirely different categories.

There really is no clear and distinct way to go from degree to dichotomy in continuous measures of anything. There must first be intuitive boundary cases of states that are clearly free or not-free; such as New Zealand and North Korea, then we go from there making comparisons and weighing different characteristics about what we think we mean by freedom (which invites its own disagreement).  This may suggest that it would be easier to look at specific instances of P, or, to call a duck, a duck. Sticking to qualitative classifications of any instance P has its own problems though; we see the same duck from different angles.

"Euthyphro: I can't explain to you what I mean, Socrates. Whatever definition we put forward seems to keep going round in circles somehow. It won't stay in the place where we put it."