On 2/23/2016 12:17 AM, Bruno Marchal wrote:
On 22 Feb 2016, at 18:00, Brent Meeker wrote:
On 2/22/2016 4:38 AM, Bruno Marchal wrote:
In the formal treatment, this is recovered by the fact that G*
proves the extensional equivalence of all intensional variants of
provability, and G* proves the non intensional equivalence of thoise
variants, by constructively ascribing them different logics
(intuitionistic/epistemological for the first person singular,
quantum logic for first person plural and matter, etc.).
Interesting point, but why should we consider intensional
non-equivalence to be anything more than a difference of description?
Because G* shows them to obey to different logics.
Also, []p & p is not just a different description than []p. Not only
the first obeys to a logic of knowledge (S4Grz if you remember) but it
does not admit any 3p description in the language of the machine (or
in arithmetic, as our pet machine is PA). []p does, it is
beweisbar(x), but []p & p does not: it should be beweisbar(x) & x; but
x is a number and not a proposition, so it should be something like
[]p & true(p) but true cannot be defined in the language of the
machine (by Tarski-Gödel immediate diagonalization. And it can be
shown that no such predicate can exist for knowledge (something shown
by Kaplan and Montague in their "paradox regained" paper, reference in
my theses: I can show it if someone asks, as it is a quasi direct
consequence of Gödel's diagonal lemma, or Kleene second recursion
theorem when formalized in the language of the machine).
I'd like to see - or if you can point to a good reference.
Brent
[]p & <>t does admit a description in the language, and correspond to
the "measure one" on the computations when p is restricted to the
sigma_1 sentences. So you might say here that it is "just" a
difference of description, but again, it is enough to get a different
logic, and actually this gives a quantum logic, like S4Grz provides an
intuitionist logic for the internal logic of the knower.
I hope this answers your question. It is the interest of the G---G*
splitting that it provides different logic for all the intensional or
modal nuances already disserted by the philosophers, from Plato to
Wittgenstein (which made a similar remark to you in its latest
treatise "On uncertainty").
Bruno
Brent
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