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Is there a unifying abstraction that links Boolean algebra and probability theory?

Both Boolean algebra and probability provide us the means to answer questions about set participation. On the one hand, Boolean algebra is an absolute, binary view of set participation -- either you're in or you're out (0 or 1). With probability we can think of expected "degrees" of participation within a set according to a real-valued probability that spans between 0 and 1.

Is this about as far as I can draw a comparison? I'm interested to know if there has been work done to develop a deeper connection between a probability variable to a Boolean variable. Are there any sources that describe Boolean algebra as a limiting case of probability theory?

Dan Kowalczyk
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  • A boolean variable can be seen as a random variable with only two possible values. – Peter Oct 31 '14 at 23:04
  • I don't know if it is what you are thinking about, but http://en.wikipedia.org/wiki/Fuzzy_logic – yago Oct 31 '14 at 23:05
  • @YannHamdaoui Yes, seeing that made me feel warm and "fuzzy", har har. I'll have to read up. Thank you for sharing. – Dan Kowalczyk Oct 31 '14 at 23:11
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    @Dan you're welcome. Actually I rather wanted to link to http://en.wikipedia.org/wiki/Fuzzy_set (which is the same idea but applied to set theory) – yago Oct 31 '14 at 23:18

2 Answers2

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Take a look at “Probability Theory: The Logic of Science” by Jaynes. He casts probability as an extension of logic, developing it as Boolean algebra with uncertainty.

Alex
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I'm not sure how real mathematicians would do it but I have a technique I use.

I represent the probabilities as binary literals and let the boolean algebra work as it normally would.

For example,

Accept some probability that a literal is true: P(ε) = 0.65 and then just plug that into the boolean algebra the way you normally would: !ε.s + ε.!s = F ε s | F 0 0 | 0 0 1 | 1 1 0 | 1 1 1 | 0

John Forbes
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