[FOM] Possible worlds
inattochdag at gmail.com
Tue Dec 19 19:35:10 EST 2006
In reply to my previous posting Neil Tennant wrote:
"What is the "trivalent" calculus?
In the classical calculus, this definition would render every possibilium
The trivalent calculus is a calculus of propositions with three
different values: true, false, and possible. This trivalent calculus
is equivalent in all non-trivial matters to the infinite valued
calculus of propositions, which was shown to be complete years ago:
see the paper: "A geometric proof of the completeness of the
Lukasiewicz calculus" published in the JSL.
In the trivalent calculus if the proposition A is true, then the
proposition "A is possible" is also true; if the proposition B is
false, then the proposition "B is possible" is also false; this works
quite nice both with our intuitions and our calculating methods. Note
the fact that every true proposition A is merely "possible" but not
viceversa: this fits with common sense.
Fundamentally, Lukasiewicz's aim was to prove the two following thesis:
1) All propositions that work free of contradiction in the classical
bivalent calculus also work free of contradiction in the trivalent
2) A proposition from the bivalent calculus fails in the trivalent
calculus only if it already fails in the bivalent calculus.
To my mind, this is a highly admirable development consistent both
with modern complexity theory and Tarski's investigations on truth.
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