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 <marchal.domain.name.hidden> 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/
Received on Wed Jan 04 2006 - 09:04:25 PST