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What are Chris Mortensen number systems?

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See the paper Chris Mortensen, Inconsistent Number Systems, Notre Dame Journal of Formal Logic, Volume 29, Number 1, Winter 1988.

This paper continues the study of so-called "inconsistent" number systems, that is, structures considered in a non-classical logic that allows at least some statements to be both true and false (bizarre, I know). He uses a three-valued logic containing true, false and a third value fruitfully interpreted as true-and-false, but please see his paper for the details. He is particularly interested in finite structures in which all the classical theorems of PA remain true (and some of them also false). He writes, "it is one purpose of this paper to...[display] inconsistent theories which contain various well-known classical consistent complete subtheories."

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    $\begingroup$ From the introduction: "A model-theoretic framework is employed, but I suggest that this is a consequence of the fact that intuitive inconsistent thinking is undeveloped (though not entirely absent) among mathematicians and logicians." $\endgroup$
    – Adam Epstein
    Commented Jan 2, 2013 at 23:39

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