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http://plato.stanford.edu/entries/peirce-logic/#TV
Cite:
"In three unnumbered pages from his unpublished notes written before
1910, Peirce developed what amounts to a semantics for three-valued
logic. This is at least ten years before Emil Post's dissertation,
which is usually cited as the origin of three-valued logic. A good
source of information about these three pages is Fisch and Turquette
(1966), which also includes reproductions of the three pages from
Peirce's notes."
How far this exploration in 1910 has been going, I don't know.
Cite
"This much would probably be accepted by most interpreters.
What the restriction is, however, and just what motivates
it are matters of wide disagreement. It has been proposed,
for instance, that Aristotle adopted, or at least flirted
with, a three-valued logic for future propositions, or that
he countenanced truth-value gaps, or that his solution
includes still more abstruse reasoning. The literature is
much too complex to summarize: see Anscombe, Hintikka, D.
Frede, Whitaker, Waterlow."
So three valued logic was already in the air much more earlier.
But I doubt that three value logic is useful for reasoning.
Because we can get away without, and still have undefined.
For example the empty theory neither entails p nor ~p,
so this might be interpreted as undefined.
The application I had for three value logic was in logics
for computation. So A+ means the program halts with yes for
the query A. And A- means the program halts with no for the
query A. And ~A+&~A- means the program neither halts.
So on the object level we have binary programs and not
"truth". Respectively on the object level we have the
"truth" of programs halting or not halting, and if halting
then answering yes or no. It is very natural to have a
3 valued logic here.
But it can be broken down into a two valued logic. I.e.
A+ and A- which behave purely two valued.
Bye
Confutus wrote:
Jan Burse wrote:
Hi
Confutus wrote:
> It's [](_ -> _} and [](_< -> _} applied to
> the Lukasiewicz conditional that seem to have
> gone unappreciated.
What are the curly braces above ("}"), and the
plonked less than sign ("<") ?
Pardon my typos. That should have been a parenthesis, not a curly
brace, in both cases, and the <-> is meant to be the biconditional,
that is (A <-> B) = (A -> B) & (B -> A)
> Granted, but without a well-behaved conditional
> this is only a fragment of a full-featured logic.
BTW:
(A->B)+ = A- v (A+ & B+) v (~A- & ~A+ & ~B-)
(A->B)- = A+ & B-
Which can easily be read off from your 3-valued
table, thus if you had the constant u in your
system L3 you could represent A->B by the other
connectives. Namely by combining (A->B)+ and (A->B)-
as I suggested in my other post.
There are a couple of ways to do it. I think
(A -> B)+ = A- v B+ v (~A- & ~A+ & ~B- & ~B+)
also works. But these both look to be rather cumbersome places to
start. Why would anyone expect these formulas to be successful, if they
didn't have some clue to start with?
But you dont need the constant u in your system.
You can use an arbitrary propositional variable p,
then we have:
u == ~p+ & ~p-
Or when we translate it back to your "modal" operators,
we then have:
u == ~[]p & <>p
further notice that ~[]~A == <>A. Thus your whole
system could be reduced to &, v, ~ and []
Since & and v are interdefinable, as in classical logic, only one of
them is necessary. It's also possible to take -> and ~ as primitive,
and define everything else.
The question isn't whether these definitions can be made and the
consequences explored. As far as I can tell, they _could_ have been
done any time since 1920 and from more than one starting place. It's
whether anyone has actually made the definitions, explored the
consequences, and recognized their potential significance. If so, who,
when, and where is it published, and why isn't it better known already?
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