You have a heap of sand and take a grain of sand from it. Is it still a heap? If you answered “yes,” then, by virtue of induction, you will be forced to admit that one grain of sand is also a heap, as well as no grains of sand, since I can recursively ask you the same question for a million times and you will be forced to give me the same answer. This is the famous Sorites paradox and it is one of the many reasons there are for abandoning classical logic.
Classical logic can be thought of as common-sense logic. For each statement, there are two possibilities: it is either true or false. There is no such thing as a half-true statement. This seemingly obvious requirement actually causes problems, such as the paradox above. It is precisely this which allowed me to mathematically prove immortality is moral a week or so ago: by not allowing the sentence “living for X years” to be more or less true, the reader got stuck with a black or white choice.
Fuzzy logic, invented in the seventies by Lofti Zadeh, aims to change all of that. The idea is extremely simple but very powerful: expand the concept of truth. Let a sentence be 70% true or 30% false, depending on your aesthetic preference.
Unlike what it may seem, this has nothing to do with probabilities. That a sentence is 70% true does not mean that it has a 70% percent probability of being true: that would be falling back into old, classical logic. What it means is precisely what it says: the sentence is partly true and partly false. The perfect example is a glass which is 70% full. Is it full? Well, kind of. Is the sentence true? Well, kind of.
Fuzzy logic solves the Sorites paradox by allowing the truth-value of a sentence to decrease gradually. If I remove a grain from a heap of sand, is it still a heap? Yes, but less so than before. For example, we could say that the truth value of “this is a heap” decreases by 1 divided by the total number of grains of sand, for example.
There seem to be several issues with this. For example: how do you assign the truth-values? Isn’t it arbitrary to say that a sentence is 53.4% false? How do you know it’s not 53.5?
Here there are several points to be made. Firstly, the beauty of fuzzy logic is that the particular numbers do not matter, but only the relationship between them. That is: any transformation that leaves the hierarchy of truths untouched will not affect the outcome of our operations. Secondly, the sentence “this sentence is 53.4% true” is also fuzzy: there is no point in asking whether it is true or not, but only in asking how true it is. In this sense, one could think of a meta-fuzzy logic on fuzzy predicates themselves.
Fuzzy logic is an offshoot of another branch of mathematics called “fuzzy set theory.” In this case, the idea is even simpler: one element can belong to a set to a certain extent. Using the example from before, we could say that a glass which is 70% full belongs to the set of full glasses to a 70% degree. This will be important later, when defining fuzzy integrals.
Fuzzy logic is not speculative mathematics and it is not open to debate, in the same way that Riemannian geometry is not open to debate. It is routinely used in electronics and artificial intelligence and powers almost every washing machine in the planet, as well as most likely the brakes of your car. So it is not just an idea put forward by some detached mathematician: it is being used every day to run stuff in your everyday life.
Fuzzy logic can also be used in philosophy for a number of things. A lot of philosophical problems can be dealt with by realizing we’re using classical logic instead of fuzzy statements. Take, for example, individuality. I think I am an individual; however, my two brain hemispheres are not. If you and I were connected with the same bandwidth as my two hemispheres (sharing thoughts, memories, perceptions and the like) we would most likely feel as if we were one individual. One could use this to justify there is no such thing as an individual, since we cannot draw the line between individual and non-individual.
However, this can be easily overcome using fuzzy logic. One can allow for the sentence “X is an individual” to be fuzzy, thus having truth-values between zero (totally false) and one (totally true.) In fact, it is possible to define the degree of individuality as:
I = 1 – (Information exchanged externally) / (Information exchanged internally)
Which, as you can check for yourself, behaves properly for the extreme cases. Bear in mind, though, that any other definition preserving the truth hierarchy would work just as well.
On to more abstract stuff. If you are allergic to math, feel free to skip the next two paragraphs!
You may be familiar with something called an “integral” in mathematics. An integral is just a sum of some value over some region of space. A perfect example is the height of a mountain at different coordinates. An integral adds up each of these values (the height at point A plus the height at point B and so on), multiplied by the (infinitesimal) area element they are on.
You can imagine an integral as adding up all the values that belong to a certain set, defined by the area or volume where we are performing the addition. But what if we make this set fuzzy? What if we say “the points in the middle count for sure, but for the ones at the border we’re not so sure”? In this case, the number we will get at the end will be a fuzzy number, determined by how strongly each point belongs to the set. This can be particularly useful for determining areas of regions which are not well delimited, for example.
Fuzzy logic is not the only alternative to classical logic. There are other contenders, such as intuitionist logic (which is pretty similar, though) or quantum logic, each of which has its own merits, demerits and areas of application. I am particularly attached to fuzzy logic because I derived it independently using something called “tensors” in my early twenties and was fascinated (and a little bummed) to discover it was already invented. I also think it may be the answer to predicaments such as the one brought about by Gödel’s theorem, an Earth-shaking mathematical result that I will tackle at some other point.