On 27 Jan 2014, at 23:57, LizR wrote:
On 27 January 2014 06:11, Bruno Marchal <[email protected]> wrote:
On 26 Jan 2014, at 01:56, LizR wrote:
On 25 January 2014 23:56, Bruno Marchal <[email protected]> wrote:
if p is true (in this world, say) then it's true in all worlds
that p is true in at least one world.
You need just use a conditional (if). The word asked was "if".
OK?
OK. I think I see. p becomes "if p is true" rather than "p is true"
Yes.
Rereading a previews post I ask myself if this is well understood.
I have tended to work on the basis that 'p' means 'p is true'
That is correct.
- to make it easier to get my head around what an expression like
"[]p -> p" means.
?
p -> q means: if p is true then q is true. (or means, equivalently
'p is false or q is true')
In fact "p -> q" is a sort of negation of p. It means "p if false
(unless q is true)".
OK, I think I misunderstood something you said which made me think
I'd previously misunderstood ... but actually I hadn't. I got it
right the first time.
Yes. I talk too much :)
Consider all typo and incorrect statements of me as exercises!
(OK, it is a bit easy for me to say this, but expect errors. It is
frequent and normal).
I realise it could also mean "if p is false in all worlds, that
implies it is false in this one"
Here you talk like if "p -> q" implies "~p -> ~q".
But "p -> q" is equivalent with "~q -> ~p", not with "~p -> ~q"
"Socrates is human -> Socrates is mortal" does not imply "Socrates
is not human -> Socrates is not mortal". Socrates could be my dog,
for example.
But "Socrates is human -> Socrates is mortal" does imply "Socrates
is not mortal -> Socrates is not human"
Keep in mind that p -> q is ~p V q. Then (if you see that ~~p = p,
and that p V q = q V p).
~p -> ~q = ~~p v ~q = p V ~q = ~q V p = q -> p. (not "p -> q"). OK?
Yes.
OK.
You said that we cannot infer anything from Alicia song as we don't
know if his theory/song is true.
But the whole point of logic is in the art of deriving and
reasoning without ever knowing if a premise is true or not. Indeed,
we even want to reason independetly of any interpretation (of the
atoical propositions).
Yes, I do appreciate that is the point. I was a bit thrown by the
word usage with Alicia, "if A is singing...everybody loves my
baby...can we deduce..." I mean, I often sing all sorts of things
that I don't intend to be self-referential (e.g. "I am the Walrus")
so I felt the need to add a little caveat.
OK.
Let me try to be clear.
From the truth of "Everybody loves my baby & my baby loves nobody
but me" you have deduced correctly the proposition "everybody loves
me". (with me = Alicia, and, strangely enough, = the baby).
From the truth of "Alicia song "Everybody loves my baby & my baby
loves nobody" ", we can only deduce that everybody loves Alicia or
Alicia is not correct. In that last case either someone does not
love the baby, or the baby does not love only her, maybe the baby
loves someone else, secretly.
OK.
OK.
That error is done by those who believe that I defend the truth of
comp, which I never do.
In fact we never know if a theory is true (cf Popper). That is why
we do theories. We can prove A -> B, without having any clues if A
is false (in which case A -> B is trivial), or A is true.
I will come back on this. It is crucially important.
I agree. I think psychologically it's hard to derive the results
from a theory mechanically, without at least having some idea that
it could be true. But obviously one can, as with Alicia.
You are right. Most of the time, mathematicians are aware of what
they want to prove. They work topdown, using their intuition and
familiarity with the subject. To be sure, very often too, they will
prove a different theorem than the one they were thinking about. In
some case they can even prove the contrary, more or less like Gödel
for his 1931 result. He thought he could prove the consistency of
the Hilbert program, but the math reality kicked back.
Ooh, really?! Well that really IS maths "kicking back big time". I
must remember that as an example of how maths really can kick back
unexpectedly.
Another famous example is the proof of the irrationality of sqrt(2).
Although the fact that Pythagorus killed 100 cows, and one disciple
(!) belongs plausibly to a legend, it is still a kicking back result,
showing that not all length on the plane are commensurable.
of course there are many others.
Nevertheless, the level of rigor in math today is such that in the
paper, you will have to present the proof in a way showing that
anyone could extract a formal proof of it, whose validity can be
checked mechanically in either directly in predicate first order
calculus, or in a theory which admits a known description in first
order predicate calculus, like ZF, category theory.
All physical theories admits such description (like classical
physics, quantum mechanics, cosmology, etc.).
Yes you need what I would call a formal theory, or whatever I should
call it.
Actually those theories does not even climb very high on the ordinal
vertical ladder (of set theory).
???
We can come back on this later. Have you read my post to Craig, of the
23 january. I copy and paste it below. It explain the beginning of the
ordinals, which is the beginning of that vertical ladder.
You don't need this for the modal logic.
So, the concrete rational talk between scientists consists in
"proofs" amenable to the formal notion of proofs, which is indeed
only a sequence of formula obtained by the iteration of the modus
ponens rule.
technically, some proofs in analysis can be obtained or analysed in
term of iterating that rule in the constructive transfinite, but
this will be for another day.
But for now, we are not really concerned with deduction, as we look
only at the semantics of CPL and propositional modal formulas.
A good example is Riemann Hypothesis (RH). We don't know if it is
true, but thousand of papers study its consequence.
If later we prove the RH, we will get a bunch of beautiful new
theorem.
If we discover that RH leads to a contradiction, then we refute RH,
and lost all those theorems, but not necessarily the insight
present in some of the proofs.
Yes, I understand. (But I bet some of those people really, really
wish that the RH will turn out to be true!)
You can bet on that.
The negation of (p -> q) = ~(p -> q) = ~(~p V q) = ~~p & ~q = p &
~q. That's all. It describes the only line where (p -> q) is false.
p must be false and q true.
Ah, so ~(~p V q) is ~~p & ~q. I would have naively assumed it was
~~p V ~q (though obviously using a truth table would show the error)
I will have to come back on this later!
Many logical laws have names. Here are the laws of de Morgan:
~(A & B) = (~A V ~B)
Aha.
Eh Eh ...
~(A V B) = (~A & ~B)
Aha again! This is important to know when using a rule like (p -> q)
= (~p V q), so the negation is ~(~p V q) which is (~~p & ~q) or (p
& ~q)
Yes, it is rather "important". It is an important classical duality.
It is similar with ~ExP(x) = Ax ~P(x), and ~AxP(x) = Ex~P(x), or
with ~[]A = <> ~A, and ~<>A = [] ~A.
Drawing exercise (which I will not solve, thus) in "modern math":
Compare with, A and B being arbitrary subset of some big set.
The complement of (A intersection B) = the union of the complement
of A with the complement of B.
The complement of (A union B) = the intersection of the complement
of A with the complement of B.
Can you verify this by drawing potatoes?
Yes, I just did it (on paper).
Good!
Soon the sequel ...
Bruno
=======
On 23 Jan 2014, at 15:05, Craig Weinberg wrote: (I added their number
name (28 Jan 2014)).
On Wednesday, January 22, 2014 5:46:26 PM UTC-5, Liz R wrote:
On 23 January 2014 03:13, Craig Weinberg <[email protected]> wrote:
Consciousness uses computation to offload that which is too
monotonous to find meaningful any longer. That is the function of
computation, automation, and mechanism in all cases: To remove or
displace the necessity for consciousness. What is the opposite of
automatic? Manual. What is manual? By hand - intentional, personal,
aware.
See what I mean?
Yes, and it's an interesting viewpoint (and more "far out" than I
expected!)
I agree with Liz, and Craig, here. It is an interesting idea. Not new,
though. But I don't find the reference. I just remember that some
people at IRIDIA works on this (in the frame of research in AI).
The typical example was based on "driving". The young driver is
hyperconscious on all his decisions all the time, and the more older
driver, drive unconsciously, may be solving puzzle in his head, until
the motor breaks, and suddenly his consciousness was back.
As such example implies, consciousness did never disappear, and most
people, including me, consider that the idea defended by Craig here go
more around a theory of attention and focus, than of consciousness per
se, but this is of course debatable.
When Craig says that consciousness offload that which is too
mechanical to be meaningful reminds well the hardness of the
consciousness problem; if everything is mechanical in the brain what
is the need of consciousness. It reminds also to me the reflexion/
comprehension principles in set theory, and often alluded for a
definition of the numbers or even all Cantor ordinals.
You are supposed to comprehend what you see (comprehension), and to
add to the universe what you have just understood (reflexion). I love
to do that with the kids.
At the start there is nothing. You comprehend (= you put a potatoes
around it, or just "{" and "}", as we cannot draw here:
So comprehend nothing, and reflecting it in the universe (which was
empty at the start) gives
{ } = 0
OK?
But now you see that, and that is not nothing. So you comprehend it---
it gives {{ }}, and you add it to the universe, getting
{ } {{ }} = 0, 1
And now you see that, comprehend it, ---that gives {{ } {{ }}), and
you add it in the universe, getting
{} {{ }} {{ } {{ }}) = 0, 1, 2
At some points the kids says that they are bored, and got the point,
which means that they know how to continue, and have a good idea of
the universe. It looks like the sequel is *mechanical*.
But then "creativity" (in in sense close to Post, of jumping out of
the picture) is brought by the next "comprehension". Indeed you can,
and have to say, once the kids get the "mechanic" of the progression,
and believe that the universe is given by
{} {{ }} {{ } {{ }}) ... = 0, 1, 2, 3 ....
You have to say that you see now {} {{ }} {{ } {{ }}) ..., and
comprehension means that you have to encircle it, that is the universe
is { {} {{ }} {{ } {{ }}) ... }, but then reflexion strikes
again, getting
{} {{ }} {{ } {{ }}) ... { {} {{ }} {{ } {{ }}) ... } = 0,
1, 2, 3, ...., omega
And then the next steps:
{} {{ }} {{ } {{ }}) ... { {} {{ }} {{ } {{ }}) ... } {{}
{{ }} {{ } {{ }}) ... { {} {{ }} {{ } {{ }}) ... } }
= 0, 1, 2, 3, 4, ... omega, omega+1
Etc.
Etc?
That "etc" cannot be mechanical, as it it is, by comprehension and
reflexion it miss the next ordinal. This is a nice way to "generate"
the ordinal, and with Church thesis, using Kleene recursion theorem,
there is a sense to say that we, and the machine, can give unambiguous
computable names to those ordinal up to the first non computably
definable ordinal omega_1^CK (CK for Church and Kleene).
Note that omega_1^CK is still enumerable (but not Recursively
enumerable), and actually much smaller that aleph_1, the first non
enumerable ordinal.
Where Craig might be wrong or not enough precise (to invalidate comp),
is in believing that a machine can only name a computable ordinal.
But machine, like us can climb such sequences, get bored, and do the
limit, which here recurs and recurs, in less and less mechanical way,
so to speak. And, we, like machines, can only provide non ambiguous
name to the computable ordinal, which explains in fact the difficulty
we can met with notion like all ordinals, or all cardinals.
But this highlights some non computable aspect in the phenomenon of
attention and focus, which was to be expected with a "non computable-
by itself" first person associated to the machine (by Theaetetus).
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