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. I think that knowledge consists of a belief that is both true and causally connected to the thing believed (c.f. Gettier's paradox). Of course belief that is held because the proposition is proven from some axioms does have a causal connection to the axioms. But that is more than just "believed". The problem then arises when you say things like, "We know there are true but unprovable facts about arithmetic." We only 'know' those things in different meta-system where they do have a causal connection to other axioms we hypothesize as true. But ultimately we cannot 'know' that axioms are true - as you say we just bet on them.


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