On 08 Feb 2014, at 22:05, LizR wrote:
On 8 February 2014 08:43, Bruno Marchal <marc...@ulb.ac.be> wrote:
On 07 Feb 2014, at 02:29, LizR wrote:
On 7 February 2014 09:14, Bruno Marchal <marc...@ulb.ac.be> wrote:
On 06 Feb 2014, at 07:39, LizR wrote:
<snip>
OK, having had a look at what you say below, let's have another go.
Start from p -> q being equivalent to (~p V q)
That gives us ~p -> q equiv (p V q) and from the above ~p is (p ->
f) so p V q is (p -> f) -> q which I seem to remember is what you
got. OK so far.
p & q --- well, p -> q is ~(p & ~q), so ~(p -> q) = (p & ~q) and
~(p -> ~q) = (p & q)
so ~(p -> (q -> f)) which I guess is ((p -> ( q -> f)) -> f) = (p &
q)
Does it?!?! Looking below, I see that it does. Wow.
I knew you can do that.
With hints.
So with "->" and "f" we can define all connectors.
Is there a connector (like "&", "V", "->", ...) such that all
connectors can be defined from it?
This is a facultative exercise. Only for possible raining sundays. We
will not use this in the sequel.
But that doesn't make sense, because & requires two arguments, so
it would have to be something like ... well, p -> q is (~p V q)
and it's also ~(p & ~q), which contain V and & ... I'm not sure I
know what you mean.
Like for "~", to define "&" and "V" to a machine which knows only "-
>" and "f". You can use the "~", as you have alredy see that you
can define it with "->" and "f".
I reason aloud. Please tell me if you understand.
First we know that "p -> q" is just "~p V q", OK?
So the "V" looks already close to "->". Except that instead of "~p
V q" (which is p -> q) we want "p V q".
May be we can substitute just p by ~p: and p V q might be then ~p -
> q,
Well, you can do the truth table of ~p -> q, and see that it is the
same as p V q.
To finish it of course, we can eliminate the "~", and we have that
p V q is entirely defined by (p -> f) -> q.
OK?
And the "&":
Well, we already know a relationship between the "&" and the "V",
OK? The De Morgan relations.
So, applying the de Morgan relation, p & q is the same as ~(~p V
~q), (the same "logically", not pragmatically, of course).
That solves the problem.
But we can verify, perhaps simplify. We can eliminate the "V" by
the definition above (A V B = ~A -> B),
~(~p V ~q) becomes ~(~~p -> ~q), that is ~(p -> ~q). Or, to really
settle the things, and define & from -> and f:
p & q = ((p -> (q -> f)) -> f).
OK?
Apparently, yes.
OK. (Not sure what you mean by "apparently", though).
Well, even though I did it, the result still looks rather strange to
me!
Cantor said "I see but don't believe it". it is normal. von Neuman
said "nobody understand math", mathematicians get only used to it.
Each world, once "illuminated" (that is once each proposition
letter has a value f or t) inherits of the semantics of classical
proposition logic.
This means that if p and q are true in some world alpha, then (p &
q) is true in that world alpha, etc.
in particular all tautologies, or propositional laws, is true in
all illuminated multiverse, and this for all illuminations (that
for all possible assignment of truth value to the world).
OK?
Question: If the multiverse is the set {a, b}, how many
illuminated multiverses can we get?
I suppose 4, since we have a world with 2 propositions, and each
can be t or f?
Answer: there is three letters p, q, r, leading to eight
valuations possible in a, and the same in b, making a total of 64
valuations, if I am not too much distracted. I go quick. This is
just to test if you get the precise meanings.
Oh, OK. So a and b are worlds, not ... sorry. I see.
Good.
So that is 2^3 x 2^3 because a has p,q,r = 3 values, all t or f,
as does b. OK now I see what you meant.
OK.
Of course with the infinite alphabet {p, q, r, p1, q1, r1,
p2, ... } we already have a continuum of multiverses.
I can't quite see why it's a continuum. Each world has a countable
infinity of letters, and the number of worlds is therefore 2 ^
countable infinity! Is that a continuum?
Yes. We proved it, Liz.
Yes I had a sneaky suspicion we did. It seems familiar ... a bit.
Understanding is good.
Understanding and memorizing, even with the help of a well presented
diary, is better, as it saves the future possible works.
I agree. I'm sure I started one, too, but I can't find it now. (So
sometimes I have to treat you as my diary...)
Well, I hope you will not lost me too!
Memorizing is good, but only if you manage to keep the memory
accessible. 'course.
Take a the infinite propositional symbol letters {p, q, r, p1, q1,
r1, p2, ... } . They are well ordered. So a sequence of 1 and 0
(other common name for t and f) can be interpreted as being a
valuation. The valuation are the infinite sequences of 1 and 0. Or
the function from N to {0, 1}.
If such a set of function was in bijection with N, i -> f_i, the
function g defined by g(n) = f_n(n) + 1 would be a function f_i,
let us sat f_k, and f_k, applied on k, would gives both f_k(k) + 1
and f_k(k), and be well defined, making 0 = 1.
OK. I think.
Hmm... OK. I think. For now. (That was quick).
I meant it's clear once you assign values to them and make binary
strings that they can be diagonalised. And I remember the above
proof, at least in outline.
OK.
The world of maths kicks back!
Yeah! That's its charm.
Some would disagree...
I guess that they met the bad math teacher who kicks the student
before math kick them, making it impossible for them to understand the
real kicking back of math, and develop the appreciation.
That's bad for the slow student, which sometimes are slow because they
are more demanding in understanding, and it is good for the quick
student, who can learn to solve problem by no more than pattern
matching, without any understanding. Consumerist societies favour
quick students, which aggravate the situation for slow students, and
long term project.
As a math teacher, I try to help the two kinds of student, but it is
not always easy, and to be honest, I favor the slow one.
For me, a valid reasoning with a false answer is better than a false
reasoning with a correct answer. I know that in real life, the
contrary is true.
[]A is true in alpha, by definition, means that A is true in all
world beta *accessible* from alpha.
And
<>A is true in alpha iff there is a world beta; where A is true,
accessible from alpha.
OK. That makes sense, but I'm not sure I can use that fact to work
things out...
Understanding is good.
But understanding + familiarity is better, and that comes with
*some* practice.
Yes, I know. But you are trying to teach an old dog new tricks (as
we say). I have recently learned how cryptic crosswords work and
(maybe not quite so recently) the craft of novel writing. Plus my
work requires me to learn new things, at least if I want to live in
New Zealand (because NZ is too small to have the exact jobs I'm good
at). And I have 2 kids and a husband to look after. So I am trying
to fit in some logic as well ... luckily I have some spare time at
work quite often...
Well, be careful. I would feel sorry if you lost your job because you
do logic.
You might try to make your husband and kids working on the problem,
which gives you the good exercise of being able to pose the problems,
which sometimes gives the main hints to the solution.
But you can also asks for hints, or for solution, or for supplementary
explanations (if interested of course).
Old dogs can learn new tricks, it just take more time (unless
Alzheimer or other health problem).
Oh dear. I don't seem to be able to get my head around this.
That happens. Tell me if the explanations above help. Ask any
question.
Well it does seem to make sense now. The binary relations help, I
didn't really get that.
I am a bit aware of that. But it might be because I go quick.
But I might go quick because yo don't ask enough question, also.
That will be when I don't have time to really engage. I can do all
this, but only when I concentrate on it - it doesn't come naturally.
That is why I feel a bit guilty when I go too much quickly. If I give
too much simple exercise, which helps for the familiarity, I am afraid
to be boring.
The difficulty in explaining AUDA is that it is necessarily technical.
It is not the place to prove everything to you, but a minimal amount
of logic is needed to understand the enunciation of the results.
Now I have to go and make breakfast :)
I am happy you don't forget what is really important in life. Bon
appetit!
Soon, some summing up definition and few "easy" exercises to solidify
the old dog memory, and makes all of this more familiar and more
natural ... as far as it is possible, because logic is not something
natural, and classical logic is a bit counter-intuitive.
Some others seems interested in the thread too, but might be less
courageous for participating, as you need some courage to do a sort of
persistent "exam" online. I can understand. But I know that if I
explain everything ex-cathedra, everyone will be lost somewhere, and
nobody will know where. I do hope some others will participate to
make things lighter on your shoulders.
Bruno
http://iridia.ulb.ac.be/~marchal/
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