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.
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. 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
On Tue, Mar 11, 2014 at 2:18 PM, Bruno Marchal <marc...@ulb.ac.be>
wrote:
Hi Terran,
On 11 Mar 2014, at 17:10, Terren Suydam wrote:
Hi Bruno,
Sure, "consciousness here-and-now" is undoubtable. But the p refers
to the contents of consciousness, which is not undoubtable in many
cases. "I am in pain" cannot be doubted when one is feeling it, but
other felt sensations can be doubted, e.g. see http://www.ncbi.nlm.nih.gov/pmc/articles/PMC2956899/
Such illusions of experience can even be helpful, as in
Ramachandran's Mirror Box therapy for phantom limb sufferers, see http://www.ncbi.nlm.nih.gov/pmc/articles/PMC3468806/
Illusions of experience are evidence that what we experience is of
our brains' constructions, like a waking dream, guided in healthy
brains by the patterns of information streaming from our sense
organs.
Exactly: like a walking dream. That's the root of the Bp & p idea,
in the Theaetetus. To do the math I concentrate to "rich" (Löbian)
machine for the "B", but the idea of defining knowledge by true
belief is an act of modesty with respect to the question if we are
dreaming or not, or more generally, if we are wrong or not.
Brains that are defective in this manner result in schizophrenia
and presumably other dissociative pathologies.
OK.
For me it all casts doubt on whether Bp & p is an accurate
formalization for experience, but I might be missing something.
As I said above, it is a simplest "meta" definition which capture
the "main thing" (the truth of the experience) without needing to
define it.
Also, for the "physical" first person *experience*, Bp & p, which is
only the knower, is not enough, you will need Bp & <>t & p, which by
incompleteness has its own logic, quantum like when restricted to
the sigma_1 truth. You need a reality (<>t).
Can you make sense of Bp & p for a schizophrenic who hears voices?
If a schizophrenic says that he hears voices, and if he hears voice
(mentally, virtually, arithmetically, brain-biologically, ...), then
he knows he hears voice.
An insane guy who says that he is Napoleon does not know that he is
napoleon, but he believes it only. He still might know that he
believes being Napoleon, and be only ignorant or denying that this
is false.
How about your own salvia experiences?
It is very hard to describe, even more to interpret. And I am biased.
It is indeed: [](... what-the-f.) and ... what the f. Most
plausibly.
It is like remembering forgotten qualia since eons.
It might confirms the idea that brains, machines, words, theories
filter consciousness only.
Consciousness would be a close sister of (arithmetical) truth.
Salvia might open the appetite for platonism, but of course it is
also a question of taste.
Bruno
T
On Mon, Mar 10, 2014 at 3:26 PM, Bruno Marchal <marc...@ulb.ac.be>
wrote:
On 10 Mar 2014, at 16:28, Terren Suydam wrote:
Question for you Bruno:.
You say (with help from Theaetetus) that 1p experience is given by
Bp & p. Yet, our experience is often deluded, as in optical
illusions, or in various kinds of emotional & psychological
denial. Can we ever really say that our knowledge, even 1p
experience, refers to anything True?
In public? No.
In private? Yes.
I would say.
Then in the frame of theories about such 1p things, like
consciousness, we can decide to agree on some "property" of the
notion. Then, "consciousness-here-and-now" might be a candidate for
a possible true reference, if you agree consciousness-here-and-now
is undoubtable or incorrigible.
Then we can approximate many sort of truth, by the very plausible,
the probable, the relatively expectable, etc.
If someone complains, is the pain real or fake? Eventually it is a
question for a judge.
The truth is what no machine can really grasp the whole truth, but
all machines can know very well some aspect of it, I think, but
very few in justifiable modes.
Bruno
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