On 26 Jan 2014, at 01:56, LizR wrote:
On 25 January 2014 23:56, Bruno Marchal <marc...@ulb.ac.be> 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)".
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?
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.
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.
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.).
Actually those theories does not even climb very high on the ordinal
vertical ladder (of set theory).
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)
~(A V B) = (~A & ~B)
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?
This invites an algebraical semantics for propositional logic, where
you interpret proposition by sets, the "&" by intersection, the "V" by
the union, the "~" by the complement. Cf George Boole (The laws of
thought).
Bruno
--
You received this message because you are subscribed to the Google
Groups "Everything List" group.
To unsubscribe from this group and stop receiving emails from it,
send an email to everything-list+unsubscr...@googlegroups.com.
To post to this group, send email to everything-list@googlegroups.com.
Visit this group at http://groups.google.com/group/everything-list.
For more options, visit https://groups.google.com/groups/opt_out.
http://iridia.ulb.ac.be/~marchal/
--
You received this message because you are subscribed to the Google Groups
"Everything List" group.
To unsubscribe from this group and stop receiving emails from it, send an email
to everything-list+unsubscr...@googlegroups.com.
To post to this group, send email to everything-list@googlegroups.com.
Visit this group at http://groups.google.com/group/everything-list.
For more options, visit https://groups.google.com/groups/opt_out.
