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.
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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