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herbzet wrote:
> Confutus wrote:
> >
>
> [...]
>
> > Then it occurred to me that what was good for the biconditional was
> > good for the conditional itself, and that I could define a strict
> > Lukasiewicz conditional as p => q :: [](p -> q).
> >
> > And there was the key that unlocked everything. Shazam!!
>
> OK, I'm interested.
>
> Could you please post the new system all in one post, i.e, the
> definitions for "and", "or", "not", "->", "=", "<->" "[]", "<>".
>
> I gather that to make a modal assertion "necessarily p" or
> "possibly p" or their negations, is to make an assertion that
> cannot take the third truth value.
>
> This is not a criticism, just an observation.
>
> How does this system differ from standard 2VL? Other than its
> being able to cope with "uncertain" propositions? Do "paradoxical"
> formula arise for the strict Lukasiewicz conditional, such as
> p=>(q=>p)?
>
> Are there tautologies that "ought" to occur for the strict conditional,
> but don't?
>
> I've got plenty of questions, but I'll stop here.
>
> > First of all, this immediately explained not only why MP and the
> > transitivity didn't work, but why they shouldn't. The Lukasiewicz
> > conditional may have a value of U and permit doubtful implication, and
> > chains of doubtful inferences can lead to false conclusions. The fix
> > was to forbid doubtful conditionals. MP and transitivity do work for
> > the strict conditional.
> > Second, and related, was that the strict Lukasiewicz conditional can
> > be taken as an ordering relation, that the truth value of P is less
> > than or equal to the truth value of Q. It is sometimes too easily
> > forgotten that logic was invented for the purpose of ensuring that true
> > premises do not lead to false conclusions. In two valued logic, the
> > material conditional suffices for this purpose. In a three valued
> > logic, it's just as important to keep true premises from leading to
> > doubtful conclusions and doubtful premises from leading to false
> > conclusions.
>
> Hear, hear.
> .
> > Third, comparing these conditionals, the Lewis strict implication is
> > too strict. It doesn't account for uncertain or doubtful statements at
> > all. Neither does the ordinary material conditional. The strict
> > Lukasiewicz conditional is closest in behavior to the material
> > conditional in 2 VL. The unmodified original Lukasiwicz conditional is
> > too permissive and allows doubtful inference.
> >
> > I could not believe that no one had come up with this before.
> > Surely, I thought, in 70 years, someone has either done all this, or
> > demonstrated conclusively what's wrong with it. I've been searching
> > the literature in vain for either one. Instead, all I've seen is now
> > close various people have come...and missed.
>
> --
> hz
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