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A few of my friends and I were playing around with math (more specifically, why (-1)(-1)=1) and we figured out that multiplication (with regards to signs) was an "nxor" operation (I.E. If we treat "1" as "true" and "-1" as "false," than the values of multiplication, again with regards to signs, are the same as the "nxor" operation.) Now, we've begun thinking about redefining multiplication as other logical operations (For example: under "and" (-1)(-1)=-1) My questions are these: is this line of thought similar to any current area of mathematical research? If so, where can I go to find more information on it. I am especially interested in any proven theorems or open conjectures on this topic.

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  • $\begingroup$ You may want to take a look at truth-tables. Note, though, that any resemblance that your logic functions have to arithmetic functions depends on how you choose to represent "true" and "false". In your case "1" and "-1". Under your interpretation "and" is the same as "min" and "or" is the same as "max". $\endgroup$
    – James
    May 10, 2016 at 15:59
  • $\begingroup$ more conventional is 1 for true and 0 for false, and multiplication for "and". $\endgroup$
    – Doug M
    May 10, 2016 at 16:03

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" My questions are these: is this line of thought similar to any current area of mathematical research?"

Yes, absolutely. The area you rediscovered is called algebraic logic. I think it is not a very active area of research any more, but in 50's and 60's it was rather active. Especially Tarski school of logic did many things in this area. You may want to check this wkikipedia page https://en.wikipedia.org/wiki/Algebraic_logic

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