Bruno, Stathis, list,

I tend to agree with Bruno about the meanings of words like "atheist" and 
"agnostic." But I do agree with Stathis that we should consult with those who 
hold with various beliefs / disbeliefs, etc. I might add, however, that it 
appears that, even when we do regard it etymologically, the form _atheos_, 
"Godless" appears to precede longer forms, such that "atheism" means not 
"lacking atheism" but "Godless-ism." I think that Stathos' distinctions lead to 
taking into account degrees of commitment, interest in the question of God's 
existence, etc., which Bruno's formalism isn't meant to do.
Under the modal-logical assumptions for Bruno's "Bg" formalism, which seem 
quite reasonable for its purposes, there's no difference anyway. 
Coarse-grained, yes. Convenient, also yes. I would also agree that Bruno's 
formalism is not all-purpose, it really is too coarse-grained to capture all 
the shades of meaning involved in real-life theism, atheism, agnosticism, etc.

As for belief and knowledge, I stand by my remark that we have too few words 
for the number of ideas involved and that one may distinguish belief and 
knowledge in different ways, more than one of which appear pertinent to such 
questions as that of how it is that mathematicians "correctly believe" or know 
that Peano Arithmetic is consistent (i.e., my discussion of ampliative 
induction in connection with it, and, though, below, I won't be returning to 
that issue, it's among the things which I'm keeping in mind below.) So, at the 
start of Bruno's development of G & G*, one set of views about belief and 
knowledge may be worked out and used, good for that theoretical development so 
far as it goes. But the question of what are belief and knowledge needs to be 
revisited toward understanding what G* really is in terms of belief and 
knowledge in the broader world in terms of which we do things like apply names 
to fields or disciplines.

In one sense, "knowledge" is simply the achievement-word version of "belief." 
But, in another sense, one's knowledge is one's having verified, proven, 
confirmed, corroborated, etc. One could distinguish degrees and kinds of 
knowledge in terms of degrees and kinds of proof, confirmation, corroboration, 
etc. We do not have to treat those all as achievement words in order to 
distinguish their meanings from those of words like "belief" and 

It is enough that the provers, confirmers, etc., themselves distinguish their 
proofs, confirmations, etc. as achievements from things which they regard and 
treat as merely their beliefs, interpretive understandings, etc. The 
confirmations, etc., may be formal or informal. They may be hardly conscious. 
(I do think that we, as onlookers on that scenario, can't easily avoid taking 
up some view as the "real" view of what's happening in it , sort of like the 
view of "the" observer in a relativity example in which numerous variously 
moving observers are supposed. I also tend to suppose that if we try to avoid 
taking up some such view, nevertheless if the scenario is represented with 
sufficient detail, our nevertheless implicitly adopted view will tend to become 

Anyway, we don't have to treat "confirmation" etc. or even "knowledge" as 
achievement words in order to distinguish them from belief. For instance, let's 
say I catch sight of an eclipse and notice that the sun's rim is quite visible 
all the way around the moon. I now have a representation of this eclipse, a 
belief, and, if I happen not to question it, then I don't seek to verify it. I 
may have been sufficiently attentive in my experience at the time in order to 
regard it as a firm observation. But then I draw an interpretive inference, 
that the moon must have been maximally far in its orbit from the Earth at the 
time. Now I have more motivation to verify. Did I really observe it correctly 
in the first place? Now I regard that observation merely as a representation, a 
belief, sufficient for my actions here and now toward things as they appear, 
but not sufficient such that I suppose that I would factor its truth into my 
basis for action in all _conceivable_ "suddenly" arising !
 circumstances. And other questions about my interpretation of this event also 
arise. If I sufficiently verify my original seeming-observation and my 
interpretation, then I recognize and acknowledge them as true, valid, sound, 
etc. I still might just be wrong about the whole thing. But in a pretty 
distinct sense, I recognize and know the moon as having been maximally far in 
its orbit during the eclipse. I regard this as a second sense of words like 
"recognize" and "know," though I acknowledge that the meaning of "know" as an 
achievement word is the primary sense in everyday language. This can happen 
even in deductive reasoning if the deducer does not adequately check and 
observe the structure of his/her reasoning. (Some may note that this all sounds 
Peircean, but I augment "object - representation - interpretation" with a 
fourth stage, one of recognition, confirmation, etc., correlated vaguely to the 
verificatory recipient in info theory. Certain four-fold processes and stru!
 ctures like that are why I'm interested in Tegmark's picture o!
 f a four
-level Multiverse.) As a practical matter, not only can intepretive results be 
interpreted in their turn, but substantiations can in turn be substantiated -- 
or overturned --, which happens almost naturally to at least some extent in 
science when science is healthy, building on firm (or firm-seeming) results. 
What's built on a supposedly firm basis should start to have problems if the 
building is valid but the basis is in fact infirm.

Best, Ben Udell

----- Original Message ----- 
From: "Bruno Marchal" <[EMAIL PROTECTED]>
To: "Stathis Papaioannou" <[EMAIL PROTECTED]>
Sent: Tuesday, January 10, 2006 8:24 AM
Subject: Re: Paper+Exercises+Naming Issue

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 
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. 

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 

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 


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