Ben, here is a comment to Stathis's post which can serve as a preliminary for the synthesis I will try to do this week and which should answer some of your comments.

Le 10-janv.-06, à 01:46, Stathis Papaioannou a écrit :

If you remain true to the Greek roots of the words, atheists lack a belief in the existence of God, as agnostics lack knowledge of whether God exists: "a-" = without, "theism" = belief in God (by later convention, a personal God), "gnosis" = knowledge. It is not quite the same as saying that God definitely does not exist, or that knowledge about the existence of God is impossible, respectively, although a few atheists and rather more agnostics would go on to make these stronger claims.

OK but this is perhaps too much etymological and it could be confusing to use "knowledge" already in tis setting, unless we give already some minimal theory of belief and knowledge right at the start. Even in that case we are at risk to mix a too much precise notion of knowledge with a too much imprecise notion of God. And if you look at the page referred too by Jesse, it seems that the more common acceptation for the meaning of "agnostic" is: "I don't know/believe if G exists or not", and for "atheist" it is: I think/belief that God does not exists. Nobody will say "I know that God does not exist", because it seems ridiculous or too much arrogant. Also, the idea that "knowledge about the existence of God is impossible" can indeed be defended by some atheist and agnostic, but is also the favourite affirmation of many mystical theists, including some Neoplatonist, but also some Buddhist.

In any case, it will be useful to agree on some axioms for belief and knowledge: I hope everyone will agree that both knowledge and belief makes the following formula true. I assume a classical (platonist) background, and by "p -> q" I mean that either p is false or q is true. It is the IF ... THEN ... of the classical mathematicians.

B(p -> q) -> (Bp -> Bq)     (named K, for Kripke)

In "French": if I believe that (p -> q) then if I belief p then I will belief q if I know that (p -> q) then if I know p then I will know q.

Note that this axioms is already criticised in some AI approaches to knowledge, because it gives rise to a form of omniscience. This is not a problem given that I will extract the measure on the comp histories from the limitation of that omniscience, and Bp is really more "p is believable" or "p is knowable" instead of "believes" or "knows". Having said that, I hope people will agree with the following axioms, again both for *believability* and *knowability*:

Bp -> BBp (named 4, by modal logicians, in honour of C.I. Lewis)

In "French":   if I believe p then I will believe that I believe p
                 if I know p then I will know that I know p.

In the classical background all classical tautologies are accepted, and I will suppose, at least in the beginning that the theories are closed for the rule of modus ponens and the rule of necessitation, and some substitution rules. The first one means that if I have already prove p and (p -> q), I am entitled to derive q. The second one says that if I have already prove p then I am entitled to derive Bp.

What does, then, distinguish knowledge and belief (or better knowability and believability)? By definition we just cannot know something false. Nobody will say "I knew that (a+b)^2 = a^2 + b^2, but I was wrong". People say: "I believed that (a+b)^2 = a^2 + b^2, but I was wrong", and this is a symptom that the most basic distinction between knowledge and belief is that we just cannot know a falsity. Of course we can believe falsity (in dreams, in the state of error, or in trusting misfortunately some liar for example). So to get an (axiomatic) theory of knowledge it is enough to add the axioms Bp -> p (either you don't know p or p is true).

To sum up the (very standard) notions of believability and knowability are given by the theories K4 and KT4 respectively:

K4  (believability, or introspective belief):

K  B(p -> q) -> (Bp -> Bq)
4  Bp -> BBp
+ rules of modus ponens, necessitation and substitution

KT4 (better known as S4, it is the fourth modal system of C.I. Lewis, knowability or introspective knowledge):

K  B(p -> q) -> (Bp -> Bq)
T   Bp -> p
4   Bp -> BBp
+ rules of modus ponens, necessitation and substitution

Then I propose, if only for the sake of the discussion to define agnostic by the ~B~g & ~Bg clause, letting open the question if B if for belief or knowledge.

Exercises for those who want (I am so sorry, but it will be useful for a latter deepening): 1) What is your feeling about formal provability by a consistent or even correct (sound) machine: I mean does it follows K4, KT4. Is formal provability closer to knowledge or to belief? (This is tricky!) 2) Remember and explain to your children or husband/wife or parents what is a Kripke multiverse and what it means for a proposition to be true at a world in a *illuminated* multiverse and what it means to be valid in a multiverse.
3) Show that K is valid in all Kripke multiverse
4) Show that 4 is valid in all transitive multiverse
5) Show that T is valid in all reflexive mutltiverse
6) Show that modus ponens and necessitation preserves validity
7) if that seems hard, don't panic we will come back on this but train yourself to ask questions (after having (re)read the definitions perhaps).


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