[FOM] Possible worlds

[Ignacio Nattochdag]

In reply to my previous posting Neil Tennant wrote:

"What is the "trivalent" calculus?

In the classical calculus, this definition would render every possibilium
true."

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
calculus.

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.

Regards,

I. Nattochdag