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I am not sure if the names of these logics are right, but I saw that you can generalize logic with probability theory. Then I found 3 ways of reasoning with probabilistic/smooth logic, Fuzzy, Bayes/probabilistic and quantum. Can someone explain what are the major differences and expressiveness of such logics? Are they useful in normal logic?

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    $\begingroup$ That seems too broad to me. Why don't you read some standard background on each of those logics, try to identify the differences yourself, and see if you have any specific question that you're not able to answer? $\endgroup$ – D.W. Aug 25 '18 at 15:30
  • $\begingroup$ Yes but I would like to hear an answer from someone who already has background in those areas, before I start looking for myself :D $\endgroup$ – pedroth Aug 25 '18 at 17:46
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    $\begingroup$ I have to agree with D.W.; I think you really should start “looking for yourself”, and start asking questions when you cannot make progress in spite of a reasonable amount of effort. $\endgroup$ – PJTraill Aug 25 '18 at 18:20
  • $\begingroup$ Yes, probably will follow your advise, It was a nice try nonetheless $\endgroup$ – pedroth Aug 26 '18 at 0:12
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This is a good way to understand quantum probability theory and how it relates to classical probability theory: https://arxiv.org/abs/quant-ph/0101012

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Russell and Norvig catalogue a few formal languages in the following table[*]:

\begin{array} {lll} \hline \text{Language} & \text{Ontological Commitment} & \text{Epistemological Commitment} \\ & \text{(What exists in the world)} & \text{(What an agent believes about facts)} \\ \hline \text{Propositional logic} & \text{facts} & \text{true/false/unknown} \\ \text{First-order logic} & \text{facts, objects, relations} & \text{true/false/unknown} \\ \text{Temporal logic} & \text{facts, objects, relations, times} & \text{true/false/unknown} \\ \text{Probability theory} & \text{facts} & \text{degree of belief $\in [0, 1]$} \\ \text{Fuzzy logic} & \text{facts with degree of truth $\in [0, 1]$} & \text{known interval value} \\ \hline \end{array}

(Note this is within the context of artificial intelligence.)

So the most important difference between fuzzy logic and probabilistic logic seems to be that in probabilistic logic you do not deal with degrees of truth, only degrees of belief.

I couldn't tell you very much about quantum logic other than that it appears to have originated in this paper by Birkhoff and von Neumann.

[*] Russell, S. and Norvig, P. (2009). Artificial Intelligence: A Modern Approach (third edition). Pearson. Chapter 8, First-Order Logic.

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  • $\begingroup$ It would be great if you can add a reference to the article where Russell and Norvig made that table. $\endgroup$ – Apass.Jack Jun 28 at 13:00
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    $\begingroup$ @Apass.Jack Sorry. Reference added. Although the book doesn't really say anything else about fuzzy logic, I thought the table might be useful anyway to anyone with this question. $\endgroup$ – user2846495 Jun 29 at 18:43
  • $\begingroup$ As it highlights the essential difference between probabilistic logic and fuzzy logic. $\endgroup$ – user2846495 Jun 29 at 18:49

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