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I have been reading Wolfram's A New Kind of Science, and as I was reading the section on systems of logic and axioms, I came across this axiom, for which all of the normal axioms of Boolean logic can be proved: ((a.b).c).(a.((a.c).a))=c, where . is the Nand (not and) operator. I believe it is true, but I would like to see a proof of some of the axioms of boolean logic from it.

I have tried to prove some of them myself, but I couldn't figure out how to prove lemmas from that axiom that are (even remotely) helpful.

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  • $\begingroup$ I might be missing something, how can we deduce a=a from this single axiom? $\endgroup$
    – Wojowu
    Dec 3, 2014 at 12:13
  • $\begingroup$ @Wojowu: You have equational logic at your disposal. $\endgroup$
    – Emil Jeřábek
    Dec 3, 2014 at 12:52
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    $\begingroup$ Good luck with that. Wolfram says he used Waldmeister to find proofs from the axioms he discussed, and those proofs have the ridiculous proportions shown on pp. 810-811: wolframscience.com/nksonline/page-810 $\endgroup$
    – user44143
    Dec 4, 2014 at 19:39

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