On 13 Mar 2014, at 17:56, meekerdb wrote:
On 3/13/2014 8:19 AM, Bruno Marchal wrote:
On 12 Mar 2014, at 21:14, meekerdb wrote:
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.
t is equivalent with (p -> p), it is the constant boolean valued
function "true". So "t" is an admissible atomic formula and <>
applies to all formula.
In the arithmetical interpretation (of the modal logic G), <>t is
consistent('~(0=1)'), that is ~beweisbar('~(0=1)').
NOT PROVABLE FALSE = CONSISTENT TRUE.
~[]f = <>t
This is standard use, in both modal logic and meta-arithmetic.
"A is possible" means A refers to the state of some world.
No. It refers to a state, or to a world, or to a number, or to a
cow. At this abstraction level, "some world" looks like a 1004
distracting pseudo-information. We are not doing metaphysics, just
math, which then is applied to formulate the comp measure problem,
and get quantum logic from there.
I don't see that "t" or "1=1" refers to some world, they are just
tautologies, artifacts of language.
t is indeed a tautology, that is a proposition true (by definition)
in all possible "worlds" (a world here is simply a function from
the set of atomic sentences letter in {0, 1}, or {false, true}.
But "1=1" cannot be deduced from logic alone, and you need
primitive terms, like s and 0, to name the non trivial object s(0),
and you need some axioms on equality, "=". Usually x = x, is an
axiom.
In particular "1 = 1" does refer to a reality, which is the usual
(standard) model of arithmetic, denoted by the
mathematical structure (N, +, x).
"1=1" is supposed to refer to that (mathematical) reality.
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?
Because it preserves the hope that there is a reality to which you
are connected.
If you prove "1=1" in classical logic, you can prove anything, you
get inconsistent. There might still be a reality, but you are not
connected to it.
Above you deflect the criticism of a category error by saying, "We
are not doing metaphysics, just math, which then is applied to
formulate the comp measure problem, and get quantum logic from
there." But then it turns out you really are doing metaphysics.
You are taking a tautology in mathematics and using it to infer
things about reality and your relation to it.
Exactly. (Although I prefer the term "theology" to "metaphysics", but
that's a vocabulary topic).
I was just distinguishing the two usages, and we have to do both, when
we apply logic, or any technic, to a field like theology, or machine's
theology (what the machine can believe).
What makes it possible is the natural link between computationalism
and computer science.
Bruno
Brent
You are in "a cul-de-sac world", when seen in Kripke semantics of
G. But don't take this in any literal way, except in terms of the
behavior, including discourse of the machine.
The theory is correct for any arithmetically effective machines
having sound extension beliefs of those beliefs:
0 ≠ (x + 1)
((x + 1) = (y + 1)) -> x = y
x + 0 = x
x + (y + 1) = (x + y) + 1
x * 0 = 0
x * (y + 1) = (x * y) + x
+ the induction axioms.
Bruno
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