What is crisp logic (in the area of classification)?

Question

I have a brief question. I am having trouble finding a good reference that explains what crisp logic is.

What I think:

I have two classification models, a decision tree and a ruleset, which I think are crisp models. Since they say that an instance is either class A or not.

I have another classification model, a logistic regression, that is not crisp, since it gives the probability that an instance belongs to this class.

I want to mention this in a presentation that I have later today. I tried looking up this subject, but it would be great if somebody could just confirm how I interpreted it. I could also use a good reference to use in my paper.

Thanks,

Answer 1

Crisp vs Fuzzy Logic

As far as I remember, crisp logic is the same as boolean logic. Either a statement is true or it is not, meanwhile fuzzy logic captures the degree to which something is true.

Consider the statement: "The agreed to met at 12 o'clock but Ben was not punctual."

• Crisp logic: If Ben showed up precisley at 12, he is punctual, otherwise he is too early or too late.
• Fuzzy logic: The degree, to which Ben was punctual, depends on how much earlier or later he showed up (e.g. 0, if he showed up 11:45 or 12:15, 1 at 12:00 and a linear increase / decrease in between).

I do not exactly know who first used the term "crisp", but I have seen it multiple times in the closely related Fuzzy Set Theory, where it has been used to distinguish Cantor's set theory from Zadeh's. So if you are looking for a reference, the original work of Zadeh or one the textbooks in the area might be a way to go.

... in Machine Learning

In Machine Learning most of the classifiers produce so called scores, which are in general more or less rough estimates of the probability that the scored instance belongs to a particular class.

To the best of my knowledge, these scores have no explicit link to Fuzzy Logic. Both Fuzzy Logic and Probability Theory are close to each other, but technically they are not the same (Fuzzy_logic#Comparison_to_probability (english wikipedia)).

So it is not correct to label the output of Logistic Regression as fuzzy. Aside, the mentioned decision tree also calculates scores (the subjective probability that an instance in the leaf belongs to the particular class), which often results in a majority-decision for the leaf.

Summary

But if you are willing to drop the difference between Fuzzy Logic and Probability for the sake of simplicity, you may say that the scores produced by a suitable classifier are fuzzy, meanwhile the decision for a class based on the score is crisp. For example in a direct mail campaign, you can calculate a score how likely it is that a customer will respond, but in the end you have to perform a crisp decision which customers you will send an actual letter.

This paper might be interesting (Eyke Hüllermeier- Fuzzy Sets in Machine Learning and Data Mining). From the abstract:

Over the past years, methods for the automated induction of models and the extraction of interesting patterns from empirical data have attracted considerable attention in the fuzzy set community. This paper briefly reviews some typical applications and highlights potential contributions that fuzzy set theory can make to machine learning, data mining, and related fields. The paper concludes with a critical consideration of recent developments and some suggestions for future research directions.

Answer 2

• crisp / fuzzy is used in fuzzy logic
• hard / soft is sometimes used for continuous classifier scores in [0, 1] as well, e.g. in the remote sensing community.

Interpretation of continuous [0, 1] scores varies:

• mixtures of pure, hard/crisp classes that are not resolved by the measurement
• cases that are truly in between classes
• probability, that a case belons (completely) to the class.

Note that fuzzy logics usually is not about probabilities but about genuinely belonging to more than one class (partially), i.e. the first two bullet points.

Reichenbach developed a probability logic (in the 1930s).

You may want to check out the manyvalued and fuzzy logic articles at the Stanford Encyclopedia of Philosophy.

I have a paper upcoming (about using samples with soft/fuzzy reference for classifier validation), but it is not yet through the review process so I cannot yet make it publicly available here. If you're intrested in the manuscript, send me an email (Claudia dot Beleites at ipht minus jena dot de) The paper will be available at arxiv once it is accepted.

Answer 3

The statement which is either true or false but not both is called a proportion is denoted by an upper case letter of alphabets , a simple proportion is also known as an atom, in order to represent complex information one has to build a sequence of proportion link using connectives or operators There are five major operators are as follows .... 1. AND (^) 2. OR (/) 3. NOT ~ 4. IMPLICATION => 5. EQUALITY =

Please ignore my spell mistake as i am a student so please dont mind if you find any spell mistakes in my answer