[FOM] Possible worlds

Ignacio Nattochdag inattochdag at gmail.com
Mon Dec 18 22:42:14 EST 2006

```Timothy Y. Chow wrote:

"is there any interesting way in which the relevant sense of
possibility can be axiomatized?"

There is a paper of Lukasiewicz(member of the Warsaw circle) on
polivalent logics that offers a quite satisfying and rigorous
axiomatization of the notion of possibility: it is an interesting fact
that the notion of posibility in lukasiewicz's paper is due to his
student Tarski. I quote from Lukasiewicz's paper:

"The following definition was discovered by Tarski in 1921, when he
was attending my seminars as student of the university of Warsaw. The
definition of Tarski is as follows:

Mp = CNpp     (This should be read : "p is possible equals if not p then p") "

The fundamental result in the paper is the following theorem:

All the traditional theorems for modal propositions are established
free of contradiction in the trivalent propositional calculus, using
as base the definition "Mp = CNpp".

The Lukasiewicz polyvalent logic was shown to be "complete" later on:
there is a geometrical demonstration for infinite valued logics
published in the JSL by Giovanni Panti.

The paper of Lukasiewicz is titled: "Philosophische Bemerkungen zu
mahrwertigen systemen des assagenkalkuls" (Philosophic observations
about polyvalent systems of the propositional calculus". I never found
neither the original nor any english translation: what I found was a
spanish edition by "Revista de occidente" of papers and essays of
Lukasiewicz that includes the above paper.

The paper, although written in the 1920s, is closer to modern
"complexity theory" than to philosophy: I think it already shows that
the notion of possibility should and can be linked to polynomial
algorithmic questions: this can be done in a way that is satisfying
both intuitively and formally.

Regards,

I. Nattochdag.
```