F && F    =    T,     T && T     =    F

To those who have studied formal Logic, the above statements should provoke hysteria—and not just because I’m using whitespace instead of parentheses.

Normally in Propositional Calculus, one considers statements like “I’m shopping and I don’t have time to talk about this right now.”  If either (A) the speaker isn’t shopping, or if (B) the speaker does have time to talk about this right now, then the statement is considered false.  The statement is abbreviated A and B, or A ^ B, and considered formalistically.  True and False can also be thought of as 1 and 0, with AND ^ working like multiplication •.  That is, a legal functor maps ({1,0}, •) to ({T,F}, AND).

T^T=T;   T^F=F, F^T=F, F^F=F

Nothing could be more obvious than this interpretation of the word “and”.  However, obvious doesn’t cut it in quantum mechanics.

Not only does False AND False = True in QM, but negative probabilities abound, too!  Yipes!  Well it’s not that surprising coming from the field that brought us quantum tunneling.

Pitowsky represents probabilities as convex hulls which live inside the [0,1] boundary.  Logic inhabits only the boundary; probability lives in the interior.  Oh, just go get yourself a copy!  I don’t want to write for too long about the technicalities.

Quantum Probability, Quantum Logic by Itamar Pitowsky.

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