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Fuzzy logic is typically used in control theory and engineering applications, but is it connected fundamentally to classification systems?

Once I have a trained neural network (multiple inputs, one output), I have a nonlinear function that will turn a set of inputs into a number that will estimate how close my set of given inputs are to the trained set.

Since my output number characterizes "closeness" to the training set as a continuous number, isn't this kind of inherently some sort of fuzzy classifier?

Is there a deep connection here in the logic, or am I missing something?

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They are unrelated.

There is a possibility of interpreting fuzzy values as probabilities, but strictly speaking they are different: fuzzy values are vague, while probabilities reflect likelihood (see Wikipedia entry for Fuzzy Logic)

While rolling a particular number on a six-side die has a probability of $1 \over 6$, a roll can actually only ever have one outcome.

A fuzzy value "quite old" can simultaneously be member of a number of fuzzy sets with different degrees of membership, eg "young" with 0.001, "adolescent" with 0.1, "old" with 0.4, "ancient" with 0.7. Unless it is "defuzzified", it is simultaneously contained in all the sets.

Defuzzyfication is a way of interpreting the result of a series of fuzzy operations and finding the set that best matches, but it is not a clearly defined process such as picking a random number according to a set of probabilities (or rolling the die).

I am not sure that the sum of all fuzzy set membership values of any given fuzzy value has to add up to 1.0; whereas this condition has to hold for probabilities.

[EDIT: to clarify - probabilities are not a set; I refer here to all possible outcomes of a random event which have a certain probability of being realised. The sum of all possible event probabilities has to be 1.0]

One alternative interpretation for your application could be the confidence that the input set is identical to the training set. Which could be a fuzzy value if you wanted to do something else with it, eg by combining it with other fuzzy variables.

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  • $\begingroup$ Are the outputs of neural nets explicit probabilities? Could you maybe link me to something that makes clear? Also, intuitively if you trained a neural network on pictures of bald men, somehow it's hard for me to believe that the "baldness" output is actually a probability of being bald. (Relative frequency makes some sense though) $\endgroup$
    – Steven Sagona
    Commented Feb 19, 2019 at 23:43
  • $\begingroup$ It's not a probability, it's a weight score. Basically, you put in some feature values, theses are combined, weighted, and filtered through the activation function, and then come out at the output nodes. So they are neither probabilities nor fuzzy values. $\endgroup$
    – Oliver Mason
    Commented Feb 20, 2019 at 9:29
  • $\begingroup$ Okay but now I'm a bit lost as to what your argument is. You say that fuzzy logic isnt connected to neural nets. But your reasoning is that it's because fuzzy logic isn't the same as probability. But you give no reason as to why or how probabilities are connected to neural networks. So what your point in talking about probabilities in the first place? $\endgroup$
    – Steven Sagona
    Commented Feb 20, 2019 at 21:17
  • $\begingroup$ Right, I see; it's not very clearly formulated. Neither fuzzy values nor probability are related to the output values of neural networks. For some reason I must've misinterpreted/misread your original question. $\endgroup$
    – Oliver Mason
    Commented Feb 20, 2019 at 23:04
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They're pretty much the same thing - in that the underlying logic of neural networks is fuzzy. A neural network will take a variety of valued inputs, give them different weight in relation to eachother, and arrive at a decision which normally also has a value. Nowhere in that process is there anything like the sequences of either-or decisions which characterize non-fuzzy mathematics, almost all of computer programming, and digital electronics. Back in the 1980s there was a debate about what AI would eventually look like - some researchers tried to program 'common sense' with huge bivalent decision trees, while others used neural networks which pretty soon found their way into a multitude of electronic devices. Obviously the underlying logic of the latter approach is radically different to the former, even if neural nets are built on top of bivalent electronics. However, the use of the term 'fuzzy logic' seems to have been downplayed since the 80s, perhaps because colloquially it sometimes implies uncertainty. This is shame because it offers a more accurate way to model complex situations.

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