Yes I am particularizing things... But "the end justifies the means". I
am being positivist, trying to express these rules as a function of an
observer. In any case, once the specific example is worked out, we can
fall back on the general case.
Your feedback about "exist" not really being adequate to express truth
is well noted. Let me change the proposed rules to express truth as a
function of an axiomatic system A existing as data .... either in the
memory of M .... or as a axiomatic substrate for a simulated world
W..... Let's try the following:
In a world W simulated according to the axiomatic data system A, there
is a machine M, data p and data q such that
1) If M has access to p (possibly in its memory), then p exists in W.
(exist=being simulated in W according to A )
2) If M has access to p, then M has access to the access point to p.
3) If M has access to the information relating or linking p to q then if
M has access to p, it also has access to q.
Now we can make the statements reflexive ( I don't know if this is the
right word) by setting data p = Machine description M.
In a simulated world W following the axiomatic data system A there is a
machine M=p and data q such that
1) If M has access to M then M exists in W. (reflexivity?)
2) If M has access to M, then M has access to the access point to M.
(Infinite reflexivity? - description of consciousness?)
3) If M has information describing q as a consequence of M in accordance
with A, then if M has access to M, it also has access to q. (This is a
form of Anthropic principle)
I am not sure if this is leading anywhere, but it's fun playing with it.
Maybe a computer program could be written to express these staqtements.
Bruno Marchal wrote:
> George, you can do that indeed, but then you are particularizing
> things. This can be helpful from a pedagogical point of view, but the
> advantage of the axiomatic approach (to a knowledge theory) is that
> once you agree on the axioms and rules, then you agree on the
> consequences independently of the particular instantiation you think
> about. Word like "machine", "access", "memory", "world", data, are,
> fundamentally harder than the simple idea of knowledge the modal S4
> axioms convey. Using machines, for example, could seem as a
> computationalist restriction, when the axioms S4 remains completely
> neutral, etc. Also, acceding a memory is more "opinion" than knowledge
> because we can have false memory for example. (And then what are the
> inference rules of your system?).
> S4 is a normal modal logic with natural Kripke referentials
> (transitive, reflexive accessibility relations).
> A bit more problematic is your identification of "true" with "exist".
> This hangs on possible but highly debatable and complex relations
> between truth and reality. This is interesting per se, but imo a bit
> out of topics, or premature (in current thread). Perhaps we will have
> opportunity to debate on this, but I want make sure that what I am
> explaining now does not depend on those possible relations (between
> truth and reality).
> Le 24-nov.-07, à 21:23, George Levy a écrit :
> Bruno thank you for this elaborate reply. I would like these three
> statements to make use of cybernetic language, that is to be more
> explicit in terms of the machine or entity to which they refer.
> Would it be correct to rephrase the statements in the active
> tense, using the machine as the subject, replacing proposition p
> by the term data and replacing "true" by "exist"? The statements
> would then be:
> In a world W there is a machine M, data p and data q such that
> 1) If M has access to p (possibly in its memory), then p exists in W.
> 2) If M has access to p, then M has access to the access point to p.
> 3) If M has access to the information relating or linking p to q
> then if M has access to p, it also has access to q.
> I assumed that the term "has access" means "in its memory"... but
> it does not have to.
> I also assumed in statements 3 that the multiple uses of M refers
> to the same machine. I guess there may be cases where multiple
> machines can have access to the dame data.
> Same with statement 4
> Bruno Marchal wrote:
> 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
> 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
> 3) if it is knowable that p entails q, then if p is
> knowable then q is
> (+ 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.
> 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.
> "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
> Oops, I must already go. Have a good week-end, George, and all
> of you,
> PS Marc, Thorgny: I will comment your post Monday or Tuesday.
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