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.
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
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
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