On 3/12/2014 8:33 AM, Bruno Marchal wrote:
Hello Terren,
On 12 Mar 2014, at 04:34, Terren Suydam wrote:
Hi Bruno,
Thanks, that helps. Can you expand a bit on <>t? Unfortunately I haven't had the time
to follow the modal logic threads, so please forgive me but I don't understand how you
could represent reality with <>t.
Shortly, "<>A" most "general" meaning is that the proposition A is possible.
Modal logician uses the word "world" in a very general sense, it can mean "situation",
"state", and actually it can mean anything.
To argue for example that it is possible that a dog is dangerous, would consist in
showing a situation, or a world, or a reality in which a dog is dangerous.
so you can read "<>A", as "A is possible", or possible(A), with the idea that this means
that there is a reality in which A is true.
Reality is not represented by "<>A", it is more "the existence of a reality verifying a
proposition".
In particular, <>t, which is "t is possible", where t is the constant true, or "1=1" in
arithmetic, simply means that there is a reality.
"t is possible" looks like a category error to me. "A is possible" means A refers to the
state of some world. I don't see that "t" or "1=1" refers to some world, they are just
tautologies, artifacts of language.
This, Aristotle and Leibniz understood, but Kripke enriched the notion of "possibility"
by making the notion of possibility relative to the world you actually are.
Somehow, for the machine talking in first predicate logic, like PA and ZF, more can be
said, once we interpret the modal box by the Gödelian "beweisbar('p')", which can be
translated in arithmetic.
First order theories have a nice metamathematical property, discovered by Gödel (in his
PhD thesis), and know as completeness, which (here) means that provability is equivalent
with truth in all models, where models are mathematical structure which can verify or
not, but in a well defined mathematical sense, a formula of classical first order
logical theories.
For example PA proves some sentences A, if and only if, A is true in all models
of PA.
If []A is provability (beweisbar('A')), the dual <>A is consistency
(~beweisbar('~A').
<>A = ~[]~A.
~A is equivalent with A -> f (as you can verify by doing the truth table)
<>A = ~[]~A = ~([](A -> f))
Saying that you cannot prove a contradiction (f), from A, means that A is
consistent.
So "<>t" means, for PA, with the arithmetical translation ~beweisbar('~t'),
= ~beweisbar('f'), that PA is consistent, and by Gödel *_completeness_* theorem, this
means that there is a mathematical structure (model) verifying "1=1".
So, although ~beweisbar('~t'), is an arithmetical proposition having some meaning in
term of syntactical object (proofs) existence, it is also a way for PA, or Löbian
entities, to refer, implicitly at first, to the existence of a reality.
But why should the failure to prove f imply anything about reality?
Brent
Of course, when asked about <>t, the sound machines stay mute (Gödel's *_first
incompleteness_* theorem), and eventually, the Löbian one, like PA and ZF, explains why
they stay mute, by asserting
<>t -> ~[]<>t (Gödel's *_second_* *_incompleteness_*).
This is capital, as it means that []p, although it implies <>p, that implication cannot
be proved by the machine, so that to a get a probability on the relative consistent
extension, the less you can ask, is <>p, and by incompleteness, although both []p and
[]p & <>p, will prove the same arithmetical propositions, they will obey different logics.
More on this later. When you grasp the link between modal logic and Gödel, you can see
that modal logic can save a lot of work. Modal logic does not add anything to the
arithmetical reality, nor even to self-reference, but it provides a jet to fly above the
arithmetical abysses, even discover them, including their different panorama, when
filtered by local universal machines/numbers. As there are also modal logics capable of
representing quantum logic(s), modal logics can help to compare the way nature selects
the observable-possibilities, and the computable, or sigma_1 arithmetical selection
enforced, I think, by computationalism.
Bruno
--
You received this message because you are subscribed to the Google Groups
"Everything List" group.
To unsubscribe from this group and stop receiving emails from it, send an email
to [email protected].
To post to this group, send email to [email protected].
Visit this group at http://groups.google.com/group/everything-list.
For more options, visit https://groups.google.com/d/optout.
