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Intuition and Existence of a Feasible Set




This post is about the use of intuition in a conceptual analysis. I will argue that without some method of quantification of the strength of the conditions we are trying to establish, we have no way to know whether the conditions we wish to seek exists in the first place.

In trying to come up with a set of conditions for a term T, our goal is usually to find one such that the conditions are individually necessary and jointly sufficient for T.
In this way, our methodology is non-axiomatic in nature. One natural question to ask is thus whether it is possible to even arrive at a solution in such a case.

Here's a simple example to demonstrate what I am trying to say:
Suppose we are to establish the conditions for the term "Tall" (in an average full-grown human point of view).
I might first go ahead and say that if a person's height is any less than 140cm, I wouldn't call him tall. In this case, I am effectively claiming that whatever the condition of tallness may be, at least it should be greater than or equal to 140cm (and of course we can debate about this number, but that's not the point of this example), I am establishing a lower bound. We arrive at the following diagram where a = 140:

Let us call the shaded area the feasible set.

In a similar way, and also for the sake of argument, I might say that if a person's height is any more than 260cm, I wouldn't call him tall, but perhaps genetically extraordinary. Here, I am establishing an upper bound where b = 240:


We may have some other conditions here and there that refine these bounds, but in the end, for the condition x for a term T, we get 2 inequalities that must be satisfied, namely: x > a and x < b.
For any x satisfying these 2 inequalities, we can then say that T holds. This is the shaded area:


We can work in this way until we've arrived at a satisfying set of conditions, but it is non-obvious that a < b!
This is the question that I've been referring to at the beginning of this post; if x must be greater than a, and it also must be less than b, we can have no such x if our lower bound is greater than our upper bound:

This is equivalent to saying that the height should be greater than 220cm to be considered tall, while also saying that it should be less than 210cm because any higher than that would be considered unnatural, say (the numbers are just for demonstration purposes). In such a case, the bounds we've established doesn't allow for anyone to be 'tall' at all! No one can be higher than 220cm and lower than 210cm.

But this example looks plainly stupid. The problem, however, is that we rarely ever have a quantification of the 'stength' of the conditions we've come up with. Most of the times the only clue we have to go on with is whether so and so conditions are too weak (it allows us to say that a person with height 140cm is tall, say), or that they are too strong (it doesn't allow us to say that a person with height 200cm is tall, say). Under these circumstances, we make adjustments to the bounds a and b accordingly.

That is to say in determining whether the conditions we have are too weak or too strong, we are effectively playing around with an analogous picture with a lower bound a and an upper bound b. But without having a quantification of their strength, we have no way to know whether a < b (that there is a feasible set) at the end of it.
We may start out hoping so because we do make use of the term T (like 'tall') after all, but at least intuitively so. This post tried to show that it is not obvious that our intuitive use of any term T will even allow for a consistent set of conditions for T.



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