Le 22-nov.-07, à 20:50, George Levy a écrit :
> Hi Bruno,
>
> I am reopening an old thread ( more than a year old) which I found
> very intriguing. It leads to some startling conclusions.
>
> Le 05-août-06, à 02:07, George Levy a écrit :
>
> Bruno Marchal wrote:I think that if you want to
>> make the first person primitive, given that neither you nor me can
>> really define it, you will need at least to axiomatize it in some
>> way.
>> Here is my question. Do you agree that a first person is a knower,
>> and
>> in that case, are you willing to accept the traditional axioms for
>> knowing. That is:
>>
>> 1) If p is knowable then p is true;
>> 2) If p is knowable then it is knowable that p is knowable;
>> 3) if it is knowable that p entails q, then if p is knowable then q
>> is
>> knowable
>>
>> (+ some logical rules).
>>
>
> Bruno, what or who do you mean by "it" in statements 2) and 3).
The same as in "it is raining". I could have written 1. and 2. like
1) knowable(p) -> p
2) knowable(p) -> knowable(knowable(p))
In this way we can avoid using words like "it", or even like "true".
"p" is a variable, and is implicitly universally quantified over.
"knowable(p) -> p" really means that whatever is the proposition p, if
it is knowable then it is true. The false is unknowable (although it
could be conceivable, believable, even provable (in inconsistent
theory), etc. The "p" in 1. 2. and 3. is really like the "x" in the
formula (sin(x))^2 + (cos(x))^2 = 1.
"knowable(p) -> p" really means that we cannot know something false.
This is coherent with the natural language use of know, which I
illustrate often by remarking that we never say "Alfred knew the earth
is flat, but the he realized he was wrong". We say instead "Alfred
believed that earth is flat, but then ...". . The axiom 1. is the
incorrigibility axiom: we can know only the truth. Of course we can
believe we know something until we know better.
The axiom 2. is added when we want to axiomatize a notion of knowledge
from the part of sufficiently introspective subject. It means that if
some proposition is knowable, then the knowability of that proposition
is itself knowable. It means that when the subject knows some
proposition then the subject will know that he knows that proposition.
The subject can know that he knows.
> In addition, what do you mean by "is knowable", "is true" and
> "entails"?
All the point in axiomatizing some notion, consists in giving a way to
reason about that notion without ever defining it. We just try to agree
on some principles, like 1.,2., 3., and then derives things from those
principles. Nuance can be added by adding new axioms if necessary.
Of course axioms like above are not enough, we have to use deduction
rules. In case of the S4 theory, which I will rewrite with modal
notation (hoping you recognize it). I write Bp for B(p) to avoid
heaviness in the notation, likewize, I write BBp for B(B(p)).
1) Bp -> p (incorrigibility)
2) Bp -> BBp (introspective knowledge)
3) B(p->q) -> (Bp -> Bq) (weak omniscience, = knowability of the
consequences of knowable propositions).
Now with such axioms you can derive no theorems (except the axiom
themselves). So you need some principles which give you a way to deduce
theorems from axioms. The usual deduction rule of S4 are the
substitution rule, the modus ponens rule and the necessitation rule.
The substitution rule say that you can substitute p by any proposition
(as far as you avoid clash of variable, etc.). The modus ponens rule
say that if you have already derived some formula A, and some formula A
-> B, then you can derive B. The necessitation rule says that if you
have already derive A, then you can derive BA.
> Are "is knowable", "is true" and "entails" absolute or do they have
> meaning only with respect to a particular observer?
The abstract S4 theory is strictly neutral on this. But abstract theory
can have more concrete models or interpretations. In our lobian
setting, it will happen that "formal provability by a machine" does not
obey the incorrigibility axiom (as Godel notices in his 1933 paper).
This means that formal provability by a machine cannot be used to
modelize the knowability of the machine. It is a bit counterintuitive,
but formal provability by a machine modelizes only a form of "opinion"
by the machine, so that to get a knowability notion from the
provability notion, we have to explicitly define knowability(p) by
"provability(p) and p is true". (Cf Platos's Theaetetus).
Here provability and knowability is always relative to an (ideal)
machine.
I will come back on this in my explanation to David later. But don't
hesitate to ask question before.
> Can these terms be relative to an observer? If they can, how would you
> rephrase these statements?
An observer ? I guess you mean a subject. Observability could obeys
quite different axioms that knowability (as it is the case for machine
with comp).
Just interpret "knowable(p)" by "p is knowable by M".
"M" denotes some machine or entity belonging to some class of
machine/entity in which we are interested.
> One more question: can or should p be the observer?
"p" has to refer to a proposition. Of course in english (at least in
french) we often use similar word with different denotation or meaning.
For example, you can say "I know Paul". And Paul, a priori is not a
poposition. But such a "know" is not the same as in "I know that Paul
is a good guy".
S4 is a good candidate for the second "know". The "know" (in "I know
Paul") has a quite different meaning, somehow out of topic (to be
short). Actually "I know Paul" really means humans variate and
pragmatic things like "I met Paul before", or "I know Paul is not the
right guy to hire for this job", etc.
With the epistemological sense of "knowing", we cannot know a knower,
nor an observer. We can only know propositions. Those proposition
copuld bear on a knower: like in I know that Paul know that 17 is
prime. Sure.
Of course we can observe an observer. This illustarte already that
observations and knowledge obeys different logics; hopefully related,
of course, as it is with the arithmetical hypostases).
Oops, I must already go. Have a good week-end, George, and all of you,
Bruno
PS Marc, Thorgny: I will comment your post Monday or Tuesday.
http://iridia.ulb.ac.be/~marchal/
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Received on Fri Nov 23 2007 - 09:54:34 PST