On 24 Mar 2012, at 21:21, meekerdb wrote:

On 3/24/2012 12:58 PM, Bruno Marchal wrote:

Google on "theaetetus".
Socrates asked to Theaetetus to define "knowledge". Theatetus gives many definitions that Socrates critizes/refutes, each of them. One of them consists in defining knowledge by belief, in "modern time" the mental state, or the computational state of the belief and the knowledge is the same, and a belief becomes a knowledge only when it is (whatever the reason or absence of reason) true. Another one is the justified true belief, which is the one which you can translate in arithmetic with Gödel's predicate. You can read "Bp & p" by I can justify p from my previous beliefs AND it is the case that p. To give you an example, if the snow was blue, a machine asserting "snow is blue" can be said to know that snow is blue. Indeed, the machine asserts "the snow is blue", and it is the case that snow is blue (given the assumption).

The "problem" (for some) with that theory is that it entails that, when awake, we cannot know if we are dreaming or not, although in dream we can know that we are dreaming, the same for "being not correct". It is not a problem for comp which makes that ignorance unavoidable.

For a machine that "we" know to be arithmetically correct, we know that Bp and Bp & p are equivalent. Yet, the machine cannot know that about herself, and the logic of Bp and of "Bp & p" are different. They obeys to the modal logics G and S4Grz, but I guess you need to read some book or some web pages to see what I mean here.

I find your explications of knowledge to be confusing. You refer to Theaetetus who said knowledge = true belief. But in your modal logic formulation B stands for either provable or proven (Beweisbar). "Provable" and "believed" are too very different things.

Hmm... I am just using Dennett's intentional stance toward machine. A machine believes p, really means only that the machine assert p. I limit myself to machine having their beliefs closed from the modus ponens rule, and obeying classical logic, and applying classical logic on their description etc. I limit myself on classically self- referentially correct machine with respect to some other possible machines or oracles.

I think that knowledge consists of a belief that is both true and causally connected to the thing believed (c.f. Gettier's paradox).

That is not incompatible with the Theaetetus' idea. Somehow the whole problem is there.

Of course belief that is held because the proposition is proven from some axioms does have a causal connection to the axioms.

Yes. And the axioms can be any successful "memes" in the way to unravel a difficulty. Axioms which solves many problems and many class of problems can win local games.

I identify partially a person with the set of its beliefs. More concrete person can revised beliefs (normally), which is not used here, given that I restrict myself of the math of the correct machine.

But that is more than just "believed".

Because you talk from the point of view of a much more complex consistent (let us hope) extension, viewed at some level. For the interview I limit myself to an apparently simple machine with beliefs, that we all believe in (I hope).

The problem then arises when you say things like, "We know there are true but unprovable facts about arithmetic."

No. About the machine. Sometimes that machine is Peano Arithmetic, because it is the better known Löbian theory. It is an axiomatizable theory, which means, by a theorem of Craig, a recursively enumerable set of beliefs.

We only 'know' those things in different meta-system where they do have a causal connection to other axioms we hypothesize as true.

Of course. The points is that the Löbian machine does that too, and can even do it for themselves. This gives them ability to climb the constructive transfinite, and to develop talks and mind tools from beyond the constructive.

But ultimately we cannot 'know' that axioms are true - as you say we just bet on them.

I think so. Even unconsciously, when our brain conceptualizes that there is a reality behind the back of the computer screen. We extrapolate all the times, and we have partial controls and partial responsibilities and those kinds of things.

The nice happening here is that by the incompleteness, Bp can't be confused with Bp & p. By incompleteness Bf does not imply f, so Bf is not the same as Bf & f. (Beliefs can be wrong. Knowledge can't, by definition).

This shows that the same set of believed arithmetical sentences can be seen from many points of views. They provide an arithmetical interpretation of Plotinus, through what happens when a universal machine looks inward.

It applies to us, tangentially, as far as we are self-referentially correct ourselves, tangentially.



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