Hi John,

I think you may have problems because you are not used neither trained in axiomatic thinking. The idea consists in NOT defining the objects we want to talk about, and keeping just some needed properties from which we prove other theorem.

Let me give an example with the idea of knowledge. Many philosophers agree that knowledge should verify the following law, and I take it as the best definition of knowledge we can have:

1) If I know some proposition then that proposition is true
2) If I know some proposition then I know that I know that proposition
3) If I know that some proposition a entails some proposition b, then if I know a, I will know b.

Then you don't need to look in a dictionary for a definition of knowledge, because mathematician will *defined* it by anything which obeys the laws above. Also the laws above are made more easily readable and memorisable when written in a shorter way like

1) Bp -> p
2) Bp -> BBp
3) B(p -> q) -> (Bp -> Bq)

Biologist, Chemist, and many Physicists just does not use the axiomatic method, and sometimes they feel they cannot understand math or logic because they feel they do not know the definition, when in fact there is just no definition at all: just axioms and rules describing some behavior of undefined terms or formulas.

Actually I have already try to explain the systems G and G* in a non axiomatic way; notably when discussing with Hal Finney in the course of some trip into Smullyan Knight Knaves Island.

And this settles also Stathis' natural question: given that there are many modal logics and many corresponding notion of multiverses, how to choose the "right" modal logic.

The concrete answer I can give is that we will just interview a naive chatting Platonist machine which believe in enough arithmetical truth. With such a machine we can take the following expression as synonymous:

- The machine will print that 1+1=2
- The machine will prove that 1+1=2
- The machine will believe that 1+1=2
-The machine will say that 1+1=2,   or more simply (more shortly):
- B"1+1=2"

This is very concrete. You must imagine yourself in front of some concrete machine/device which print propositions from time to time.

Now, if the machine is enough rich in its language ability then the proposition "The machine will print 1+1=2" is itself a proposition of the machine language. It could be that one day the machine will print the proposition "the machine will print 1+1=2". In that way, the machine is able to talk about itself. Do you understand the difference between Bp (the machine print p) and BBp (the machine prints Bp) ?

And the questions are then:
A) what laws do the machine printability (provability ...) obey to?
B) what laws do the machine prints

And the answer for A will be given by a modal logic know as G*, and the answer for B will be given by a modal logic known as G.

I stop here in case you want already make a remark. Believe me, what I try to say is probably much more simple than you may imagine. There is probably less to understand than what you think. The hardest part of logic for beginners consists in understanding how simple it really is.

Best,

Bruno


Le 29-déc.-05, à 17:56, John M a écrit :

your brief added last par is of great help. I would
NEVER mix provability and probability, I am not
Spanish (b=v?) and think in semantical rather than
formal meanings. I wish I knew what is a "modal logic"
(G and G*) and am a bit perplexed of your (??) logic
defining G* as beeing 'something or not'. (Like: "F"
is =,<, or >, of "B")- Then again "true" may not
exist, indeed. (1st pers?)
Similarly it does not help me, if I get a lot of other
'names' for something I don't know what it is to begin
with. I like WORDS.
(I also like word-puzzles, but only solvable ones in
my domains).
*
I glanced over the Stanford blurb and found exciting
titles. When clicked, they overpoured me with
equational lettering and I had no idea about their
meaning. Even if I had a vocabulary of those letters,
it is practically (humanly) impossible to "read" a
text
and follow those equations by looking up every letter
for the meaning and content (with, of course clicking
after all the  connotations galore). Besides it is
full of signs I cannot even read out and have nothing
similar on my keyboard (maybe they are in some hidden
modes as are the French accents).
***
As a comparison: here is a description of a statement
from my old profession about something I did:
"when mixing the DVB and St in a DBP catalysed 1:3
stoichiometry
it exotherms and has to be temp-controled. At
reaction-startup I added the DEB and then dispersed
the mix in an aqueous medium with PVA stabilizer. The
beads were then WV-boiled off and filtered.
They showed a controllable macroporous structure with
large sp. surface internally for adsorptive sites.
Then came the transform by polymeranalogous reactions
to introduce polar or ionic sites."

And so on. It made perfect sense in my profession.
(Never mind)
No modal or out of modal logic, no 'ABC... with signs'
equations.
***
How does the "provability" (no b) jibe with Poppers
scientific 'unprovability'? Is falsifiability =
provability?

Bruno, I like what you SAY, I like YOUR logic, not
somebody else's. I don't want to 'give up' on you
because of a system so strange to me. I am 'fishing'
for word-hooks in your writings. In 1940 I took
philosophy (to major chemistry) and sociology. I
should have taken logic instead of the Br. of
Brandenstein.
Of course it would have been of little use now, 65
years later.
With friendship

John

--- Bruno Marchal <[EMAIL PROTECTED]> wrote:

Hi John,

To search informations on the net on G and G*, it is
easier to search
on "logic of provability".

G is also called KW, KW4, L, GL, PRL in other papers
or book.
G* is also called G', PRL^omega, GLS

The Stanford entry is rather good:
http://plato.stanford.edu/entries/logic-provability/

In brief words G is a modal logic which describes
what a classical
theory or machine can prove about its own
provability abilities. And G*
is a modal logic which describes what is true
(provable or not by the
machine) about its own provability abilities.

Don't confuse "provability" with "probability".
Careful when typing
because the "b" and the "v" are close on the
keyboard!

Bruno


Le 29-déc.-05, à 00:48, John M a écrit :

Bruno, could you include some BRIEF words for the
profanum vulgus about that ominous  "G - G*" magic
as
well? I searched Google, Yahoo, Wikipedia, but
could
not find any reasonable hint.
 You and other savants on the list apply this
magic
many times always. Am I the only one who missed
that
in grammar school?

John



http://iridia.ulb.ac.be/~marchal/





http://iridia.ulb.ac.be/~marchal/


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