# [FOM] Possible worlds

Ignacio Nattochdag inattochdag at gmail.com
Wed Dec 20 23:42:57 EST 2006

```In reply to my posting:

"The trivalent calculus is a calculus of propositions with three
different values: true, false, and possible...

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."

Niel Tennant wrote:

"No, it does not. The proposition

Grass is purple

is false, but it is possible that grass might have been purple (rather
than green)."

I don't want to be a pedant, but the false statement "Grass is purple"
works nice in the Lukasiewicz calculus both intuitively and
technically: the statement "Grass is purple" is false in the trivalent
calculus and the statement "It is possible that grass is purple " is
also false in the trivalent calculus, I don't see any problem here. (I
asked my 5 year old son what color was the grass and he said: "green";
I then asked him "could the grass be purple" and he said: "no": we can
take this answer as representative of common sense).

Neil Tennant wrote:

"Modal logic has always striven to characterize a notion of possibility on
which contingent falsehoods, though false, are nevertheless possible."

That's right: but that development differs from Lukasiewicz's original
investigation: in his system if a proposition is false it is
impossible, that's it.

Neil Tennant wrote:

"Moreover, if (as you claim) the third truth-value of Lukasiewicz's
calculus is to bear the interpretation "possible", and if (as you claim)
that interpretation accords with our modal intuitions, then one is at a
loss to understand why Tarski should have gone to the alleged length
(which you endorsed earlier) of offering CNpp as a "definition" of Mp in
this trivalent calculus."

I do not "claim" anything: Lukasiewicz does; I was simply registering
his findings. The fact that Tarski offered the CNpp definition of
possibility when he was a student at Warsaw is mentioned by
Lukasiewicz in his paper, I just pointed out that it was an
"interesting" fact: why did Tarski offered that definition and what
influence(if any) it had in his later thought is a question that we
could discuss for years(and we should). I never claimed that
Lukasiewicz's notion of possibility "accords with our modal
intuitions": I just said that it worked nice with our intuitions; to
my mind, the creature called "the first man in the street" has
intuitions rather different from the modal ones.

Neil Tennant wrote:

"Another problem is this: you say that the third truth-value is "possible";
and you also talk about the truth and/or falsity of claims of the form "A
is possible". How is the *truth-value* called "possible" connected to the
*metalinguistic predicate* "... is possible"? Do you simply use a Slupecki
operator, which you have omitted to mention?"

You judge the lukasiewicz system from the perspective of modern modal
logic(in which you seem to be an expert): but the early Lukasiewicz
system is just a polyvalued complete system: it is closer to proof
theory than to modal logic. For what I know the paper of Lukasiewicz I
was refferring to is not available in english: If I ever make a pdf
translation I will send it to you and you will see immediately what I
meant(If you read spanish I can scan you the spanish version that I
have). Glance through the papers of Giovanni Panti available at Arxiv:
that is closer to the spirit of the original Lukasiewicz.

Regards,

I. Nattochdag.
```