A set is a collection of things that have a common characteristic. The things are called members of a set.
So all humans have the common characteristic of belonging to the human species. Humans are called members of the set of human species.
In order to know whether or not something is a member of a given set, we use a decision procedure which is a series of steps to verify the membership to something.
Decision procedure to verify membership to the set of even numbers.
Step 1. Take object and check whether the object when divided by 2 gives an integer.
Step 2. If the object gives an integer, then it is a member of the set of even numbers. If the object does not give an integer, then it is not a member of the set of even numbers.
All even numbers have the common characteristic that when divided by 2, they give an integer.
What is the decision procedure for membership to a set of electronic objects?
Step 1. Check whether the object uses electricity to manipulate information.
Step 2. If the object gives uses electricity to manipulate information, then it is a member of the set of electronic objects. If the object does not use electricity to manipulate information, then it is not a member of the set of electronic objects.
Let us try again. What is the decision procedure for membership to the set letter “A”?
The decision procedure would check whether the object “consists of two more or less vertical lines, joined at the top, and crossed in their middle by a horizontal bar”.
All the characters above belong to the members of the uppercase “a” but for some of the characters it takes a while to check whether or not they belong the set of members of the uppercase “a”.
What is the decision procedure for membership to the set red color?
“It is the color of the wavelength of light from approximately 620–740 nm on the electromagnetic spectrum.”
The decision procedure would be to check whether the object has a wavelength of light approximately inside the range 620-740.
What is the decision procedure for membership to the set heap of sand?
This is trickier. Do 2 grains of sand count as a heap of sand? Do 200 grains of sand count as a heap of sand?
What is the decision procedure for membership to the set the jazz genre?
What is the decision procedure for membership to the set art?
The above sets have one thing in common: they are notoriously difficult to define and as a consequence it is notoriously difficult to devise decision procedure for membership to them. The concept of decision procedure and the concept of definition are similar.
The word “definition” comes from a variant of Latin definire, from de- (expressing completion) + finire ‘finish’ (from finis ‘end’).
Defining something means marking the boundaries of something.
And that’s what a decision procedure does. It literally helps us to compare the boundaries of the object and the set. If they have the same boundaries (i.e. the same trait) we say that the object is a member of the set.
If there is a trait that is shared by set A and object B, we say that object B is member of set A. In the picture we see that A and B share trait C, so if A is a set and B is an object, we could say that B is a member of A. When the boundaries of both the object and the set are clear, a decision procedure for membership to the set can be built and viceversa.
However, there are many cases in the real world where the boundaries of a set are ambiguous. We will call them “fuzzy sets” (btw, it has no direct relation to the mathematical objects with the same name). Music genres, art, colours, heap of sand are fuzzy sets. How do we tackle them? We stay away from the borders.
- Heap of sand
If the collection of grains has a triangular shape, then it is a heap. If it does not, then it is not a heap. Of course, this decision procedure is not perfect but it is still useful.
Humans can only perceive colours within particular wavelength ranges (about 390 – 700 nanometres). Different colours have different wavelength ranges. So while the exact value at which red becomes orange cannot be pinpointed, you know that there are certain values inside which you know with certainty whether or not the wavelength belongs to the set of wavelengths that are called red.
- Music genres/Art
Unlike the previous two, there is no decision procedure to determine whether a given object belongs to a particular music genre or whether it is art. The reason for this is that the definition of art is subjective. Art is whatever one believes is art. If one were to put all jazz musicians inside different rooms and were to ask them to list all the properties of jazz, you would likely end up without a single common trait shared by all the musicians. The same applies to art. The reason for this might be found in the way we conceptualise art. Art is a creative endeavour where one is supposed to push the boundaries. When one pushes the boundaries, one changes the boundaries. So while the early jazz musicians might agree that jazz can only be produced with a set of instruments. The new generations might disagree and consider that you can produce jazz with a computer. This is one of the reasons why the claim that computers are not creative is literally meaningless. Since creativity is what ones considers to be creative, if a computer considers that a pattern of sound is creative, no one can challenge the computer. Of course, there might be a lot of discussions regarding whether it is the computer or the computer programmer the one who considers what is art. But implementing random parameters should solve the problem. Then arbitrary parameters would generate the decision procedure to decide whether a pattern of sound created by a computer is art. People might say that the criteria is arbitrary, but one might say the same of human artists. They don’t seem to use (or if they do, they can’t prove it) a non-arbitrary criteria to decide whether or not a given pattern of sound is art. And this is more obvious in experimental art. An experimental musician called John Cage produced a composition of 4 minutes and 33 seconds without any actual sounds played by instruments. The author gives some sort of rationale for the absence of music. Any computer could also generate one. And there should not be any difference between both rationales. The point is that art is subjective and arbitrary so the decision procedure is also subjective and arbitrary. Let us call fuzzy sets where the decision procedure is subjective and arbitrary SA fuzzy sets.
There are many SA fuzzy sets like art where the decision procedure is not objective or logical. SA fuzzy sets are related to human constructs like religion, nationality and any other thing that is subjective.
I think the idea of SA fuzzy sets is even more interesting in religion and nationality due to its social nature. Let us explore them a bit.
Here “nationality” does not refer to the notion of having particular documents that assert that a particular individual has a particular nationality. It refers to the notion of “ethnicity” and “culture” that is often associated by the main ethnic group in a particular society. Let us say, French. What is the decision procedure for membership to the group of people that call themselves “French”? Is it birth in the geographical location known as France? Is it upbringing in the geographical location known as France? Is it sharing “French cultural values” (If that is the case, then we would need a corresponding decision procedure to establish what classifies as “French cultural value”)? Is it having descendants from France (If that is the case then we would need a decision procedure to establish the number of descendents needed to classify as “descendants“)? Eventually, as we trace back decision procedures of decision procedures of decision procedures,etc we will reach a point where will be unable to realise that the decision procedure all the derivative decision procedures are dependent on parameters that are arbitrary and subjective.
Everything I said about nationality applies to religion (talking here about religions with large number of members). When trying to decide whether a particular person belongs to a particular religion, we will eventually find that the criteria used to carry out that decision is arbitrary and subjective. This can be quite interesting when a member of a religion R performs an action that damages the reputation of religion R. Members of religion R will quickly want to disassociate themselves and their religion R from this particular individual by stating that the individual is not an actual member of religion R. But we know that the statement can’t be true, because there is no objective decision procedure to decide who belongs to religion R. Thus, all we have is a collective of individuals voicing their opinion on something that is subjective.