Axe Of Faith

Bill Vallicella, the Maverick Philosopher, has written a series of posts lately about just what atheism is. In particular his aim has been to rebut the notion that atheists merely lack a positive belief in God, and that the burden of proof naturally falls upon the theist.

I am not going to take up the burden-of-proof argument here, other than to say that my sympathies do not coincide with Bill’s in this case: I see a natural world all around me, and no evidence of any gods, so it seems fair enough to me that the onus probandi lies with those who claim that such intangible entities are real.

But after reading this post and the ensuing discussion, I am left wondering: just what is “belief”, anyway?

In the post itself Bill, always thorough, sets aside some frivolous cases:

Some define atheism in terms of the absence of the belief that God exists. This won’t do, obviously, since then we would have to count cabbages and sparkplugs as atheists given the absence in these humble entities of the belief that God exists. But the following could be proffered with some show of plausibility: An atheist is a person whose psychological makeup is such as to permit his standing in the propositional atttude of belief toward the proposition that God exists, but who as a matter of fact does not stand in this relation, nor is disposed to stand in this relation were he to be queried about the existence of God. Note that it does not suffice to say that an atheist is a person in whom the belief that God exists is lacking for then the neonatal and the senile would count as atheists, which is surely a bit of a stretch.

In the post’s comment thread, philosopher Peter Lupu (who is also an occasional commenter here) argues for a distinction between having the positive belief that God does or does not exist and having no belief about God’s existence at all. He also refers to a “degree” of belief that may vary continuously between abject doubt regarding what is believed, and absolute certainty:

Those who are not dogmatists and attach some degree of probability short of certainty to their respective beliefs do see a point in offering arguments on behalf of their respective beliefs (and against their opponents) because if such arguments are deemed cogent, then they will increase the degree of belief in the respective propositions they already hold. Similarly, viable arguments against the proposition decrease the degree of belief in the respective propositions they already hold. But, (and this is the critical point to keep in mind) in all of these cases and regardless of your place in this epistemic spectrum, you either believe that a theist God exists (and, thus, you are a theist) or you believe that a theist God does not exist (and, hence, you are an atheist). The spectrum represents the *degree of intensity* with which you hold the respective beliefs. But in order to hold a given belief with an extremely high intensely (say up to certainty) or with much less intensity *you must have the belief* as part of your belief-corpus. “Misgivings”, on either side of the theist-atheist aisle, occur and perhaps with a reasonably high frequency; but unless they turn into all out skepticism, they simply change the intensity with which a belief is held (temporarily or permanently) and not the fact that the belief is held.

Now this, I am afraid, does not go down so easily — at least when the subject is not someone for whom the question of God’s existence has never even arisen, but is, rather, someone who is well aware of the issue, and realizes that the question “does God exist?” actually has an answer, regardless of whether we can ever know it with certainty or not. In the comment-thread I made the following remarks:

How would you characterize a person at position 50 in this epistemic spectrum, who really isn’t sure just what he “believes”? How is an “agnostic” at position 50 different, in any meaningful way, from a “theist” or “atheist” at position 50? In one place you speak of “degree of belief”, and in another you seem to treat belief as a binary attribute that is either wholly present or wholly absent. On this view, could you have an “atheist” with a position of 99 on the scale and a “theist” with a position of 1 (i.e. “believing” with only one-percent certainty)?

This all seems a bit too complicated to me; the only thing that matters, I think, is the degree of confidence one has in the truth of the proposition “God exists”. If you are far enough down the low end of the scale, it seems appropriate, in a common-sense way, to start referring to oneself as an “atheist”, while “agnostic” would simply mean that you are more or less in the middle. The difficulty of marking off exact regions (other than 0 and 100) is, I think, at the root of the terminological difficulties being discussed here.

In my own case, my confidence that God exists is so very, very low that “agnostic” seems misleadingly weak, but that confidence still hovers somewhere above zero. In other words, I wouldn’t say that I “believe” that God doesn’t exist — but I do think that’s where the smart money is, and I’d be very, very surprised indeed if it turned out that he did.

Peter is sticking to his guns:

1) Since we are talking here about believing, disbelieving, and having an agnostic posture, we are in fact talking in large part about the doxastic (epistemic) states of people. So let A be an arbitrary agent and P be any proposition whatsoever. We can now start by distinguishing three doxastic states:

(a)A believes that P: A’s belief-corpus (doxastic state) includes P;
(b)A believes that ~P: A’s belief-corpus includes ~P;
(c)A does not believe P and A does not believe ~P: A’s belief corpus includes neither P nor ~P.

The fourth state I have distinguished, the case of the ignoramus, is a case where A could not epistemically have either P or ~P as part of his belief-corpus because he cannot even understand these propositions. Let us keep this case in the background.

2) It is important to see that there is a sharp difference between the case of (c), on the one hand, and cases (a) and (b) with respect to the pair of propositions P and ~P. While in the later cases A’s belief-corpus includes either P or ~P (we assume not both), in the case of (c) A’s belief-corpus includes neither. So if we imagine an assignment of degrees-of-belief (or intensity of belief) to the propositions in a belief-corpus, then in the case of (a) we will have a certain positive assignment to P (depending on how intensely A believes P) and ~P will receive 0. So suppose that both A and B belong to case (a). We can now compare the degree-of-belief they have in P. In the case of (b) exactly the opposite will happen. But in the case of (c), both P and ~P will receive 0 degree-of-belief (or degree of intensity), since neither P nor ~P are present in A’s belief-corpus. Therefore, (c) is not somewhere in between cases (a) and (b) as you seemed to suggest. (c) is a totally separate case where both P and ~P receive the value 0.

One point that I think it is important to keep in mind is that this is a “zero-sum” situation. If we are certain that either P or ~P, then to the extent that we doubt P, we must affirm ~P. The two do not, despite what Peter suggests, move along separate axes. In other words, and despite Peter’s suggestion that the issue is simply a matter of logical “scope operators” (for which see here), nobody who understands the problem can give both P and ~P a simultaneous degree-of-belief of 0.

I replied as follows (this comment, however, did not make it onto the first page, and I think was seen by nobody, because the TypePad blogging software that Bill uses makes it easy not to notice that there are further pages of comments):

Regarding the proposition G (“God exists”): we will all agree, I think, that any informed and rational person should, regardless of his belief about G itself, believe with 100% certainty in the truth of the metaproposition I’ll call X: “Either G or ~G”. I think it is safe to say that *nobody* is in a position of witholding belief in X; i.e. nobody is “agnostic” regarding X.

I am interested in distinguishing these three cases:

1) The person who is “agnostic” regarding G. He is equally, and symmetrically, disposed toward accepting both G and ~G, but has no doubt that one or the other must in fact be the case. He simply does not feel that he has sufficiently compelling reasons to move to one side or the other.

2) The “atheist” who “believes” ~G, but with 50% confidence (“degree-of-belief”). He too knows with certainty that X, but the lack of truly convincing arguments against G leaves him in the middle.

3) The similar case of the “theist” who believes G, but also with only 50% confidence. Like the others, his confidence in X is certain: he believes, absolutely, that God must either exist or not exist, but the lack of any evidence, or even compelling arguments, for G leaves him right at 50 also.

Each of these three knows (because, remember, they all believe confidently that X) that to the extent they lack confidence in G they must have confidence in ~G, and vice versa. They know for sure that either G or ~G must be true.

Given that belief is a psychological disposition, how do these people differ? In particular, how is the agnostic at the middle of the scale meaningfully different from the theist or atheist in the middle? Both are disposed in the same way towards both propositions G and X.

Simply put, I think the word “belief” is losing, on the philosopher’s logic-chopping block, some of its connection with what really goes on in people’s heads. And we should bear in mind that there is nothing more to belief than what goes on in people’s heads. In other words, is there really anything more to our “belief” in G than our relative levels of confidence in G versus ~G?

Any thoughts?

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  1. David Brightly says


    Forget God for a minute and think about my mum. If I tell you P, that her birthday is on 12 April, with what confidence will you believe that’s true? You really have no evidence either way so your degree of belief in both P and ~P ought to be 0, Peter’s agnostic position. You seem to be claiming that this state is no different from one in which you believe P and ~P with equal confidence of 0.5. But do you really think I’m telling you the truth with 50% probability? I might have rolled a die to decide whether to tell the truth or not, or chosen a date at random.

    Perhaps an ideally rational believer would assign degree of belief as

    weight of evidence for
    weight of evidence for + weight of evidence against

    What happens when there is no evidence for or against?

    Posted March 19, 2009 at 10:30 pm | Permalink
  2. Malcolm says

    Hi David,

    The testimony of experts and eyewitnesses is evidence enough to convict people in a court of law, and I shall assume that you are an expert witness on the subject of when your mother’s birthday is.

    I will leaven my confidence, perhaps, with the thought that you might be delirious, or oddly forgetful, or motivated for some unfathomable reason to lie to me, but I’ll still come away with a fairly high degree of confidence.

    (Well, given the context here, I might want to follow up with a question like “is that really her birthday, or are you just making that up?” But you get the idea.)

    If there were absolutely no evidence at all (and you seem to be suggesting that I should treat your assertion about her birthday as providing none), then ordinary probability comes into play, and I would have a degree of confidence of somewhere around 1/365 that her birthday is April 12th (the day before mine, by the way).

    What happens when there is no evidence for or against, and the odds are even? You just sit in the middle, or slightly to one side if you are prejudiced in some way. If we must have a formula, I think it would be better if we moved the neutral point to 0, as you suggested at Bill’s, and did something like:

    degreeOfBelief = 0 + (weightFor – weightAgainst)

    Posted March 19, 2009 at 10:37 pm | Permalink
  3. JK says

    Hey David?

    Neither do I know you, nor really Malcolm. I certainly don’t know your Mum.

    Mine tells me to send Birthday cards (don’t tell her I wrote this) on a certain day in November. Her Presbyterian friends insist she “doesn’t look a day over fifty.” Her Baptist friends tell her she looks to be about forty-eight. Her Methodist friends seem to agree with her Baptist friends.

    I look at myself in the mirror and my eyes tell me, “Not a day over thirty.”

    (Nevermind that some of my Professors ask me, “What did you do in Viet Nam?”)

    (Lacking an answer, since I never was in Viet Nam, I reply “Nothing.”)

    But since the Professor asked me in open class, my three decades younger or so classmates all believe I am old enough to have served in Viet Nam. (Notice the date of this post.) One asked, “Why did you do that at My Lai?”

    Mum’s under fifty and yet some honestly believe I served in Viet Nam. While I cannot distinctly recall chasing tail during the Viet Nam era, I believe I probably was.

    Actually I’m pretty sure I was because I have a grandson who purportedly is sixteen.

    David? I’d appreciate some fancy equation.

    Po po tweet?

    Posted March 20, 2009 at 4:19 am | Permalink
  4. David Brightly says

    Hi Malcolm,

    Where has the theorising left you? You seem to be coming round to saying that you are more than 99% sure that P is false. Would you bet on that? What is your state of belief in P? Does it make sense to say you have one? Yet my mum does have a birthday.

    Posted March 20, 2009 at 5:38 am | Permalink
  5. Malcolm says

    Rambling a bit there, JK.

    Posted March 20, 2009 at 10:55 am | Permalink
  6. Malcolm says

    Well, David, let me reassure you that I am 100% confident that your mum has a birthday. If you reassure me that “no kidding, it really is April 12th”, then I will have a high degree of confidence that it is. Would I bet my house on it? No way.

    This is how it is with everything. What else can we do? Not all evidence is created equal. If I see a mouse in the kitchen, yes, I will be very confident that there is in fact a mouse in the kitchen. If you tell me there is a mouse in the kitchen, I will be somewhat less certain. If you tell me there is a moose in the kitchen, less so still — and if you tell me Moses is in the kitchen, I’ll think you’ve gone right off your rocker.

    So: do I “believe” that your mum’s birthday is April 12th? Kind of. My point is that the meaning of this word “believe” is so vague as to fall apart under close inspection. The psychological fact, it seems to me — and the psychological facts are all the facts there are about belief, save for neurological ones — is that all that we ever really have is varying degrees of confidence in various propositions.

    Posted March 20, 2009 at 10:58 am | Permalink
  7. JK says

    Sorry ’bout that. It was late and I was dehydrated.

    Posted March 20, 2009 at 2:17 pm | Permalink
  8. Addofio says

    Just to toss in a few other notions regarding your question about the nature of belief:

    Religious belief is not, I think, quite the same as everyday belief in mere propositions about everyday facts. Consider the difference between saying you believe something, vs. saying you believe in something. You, Malcolm, believe that Peter’s mom has a birthday–but do you believe in that proposition? Seems kinda silly, or at least odd, to say such a thing, doesn’t it? Yet I think it’s no stretch to say that you both believe and believe in the possibility and importance of rational thought.

    Nor is belief simply a matter of the weight of the evidence, not even for the most determinedly rational of us. We all hold beliefs in thousands of unexamined propositions. And with regard to some propositions, we might respond “I don’t care.” Philosopher X claims an infinite number of angels can dance on the head of a pin; Philosopher Y claims only one can; I say “I don’t care”. What is the state of my belief, in your scheme, regarding angels dancing on the head of a pin? (You may substitute capital letters as variables in that if it seems more intellectually respectable to you.)

    As for belief in God? I am an agnostic in a sense that you don’t address. Not only do I declare that I do not know whether or not God exists, I am quite certain that no one else knows either, though many are sure one way or another. So I claim there is yet a third possibility with regard to P or ~P: “undecidable”. How would your linear scheme accommodate that way of thinking?

    Posted March 22, 2009 at 11:17 am | Permalink
  9. Malcolm says

    Hi Addofio,

    How would you characterize the psychological distinction between “Mary believes that P” and “Mary believes in P”? (And by the way, the use of symbols is not for the sake of respectability, but rather brevity and clarity).

    It seems to me that “believes in” adds a normative element, and unpacks to:

    “Mary would like P to be true, and would like others to want P to be true as well. So she will voluntarily dispose herself to behave as if her confidence in P is very high (in many cases, far higher than it may actually be) because it makes her happier, gives her hope, etc.”

    So yes, not only do I believe with a high degree of confidence that rationality is useful and effective, but I also want it to be as useful and effective as it appears to be, want others to behave rationally also, etc. — in other words, I think we ought to be rational — so I believe in it as well.

    As for how many angels can dance on the head of a pin, we are talking about a different sort of case from the question “Does God exist?”, as I am not even the least bit confident in the truth of the metaproposition “There is some definite number of angels that can dance on the head of a pin.” It is not a “zero-sum game” the way belief in God is; first we must be confident that angels exist, that they dance, etc.

    I doubt very much that angels exist at all, and my degree of confidence in the composite proposition (for example) that “Angels do exist AND there is a particular number of them that can dance on the head of a pin AND that number is eleven” is very, very low indeed. I imagine yours is too. Indeed, it is so low that the whole thing just seems ridiculous, which is why I (and you) give it a “don’t care”. But what that “don’t care” means, I think, is that the composite proposition stacks so many improbabilities one upon the next, and is so far removed from anything that seems normatively important, that it isn’t even worth bothering about. If, however, you were a medieval theologian who had a high degree of confidence in the antecedent propositions, and for whom there were a number of normative issues at stake, you’d see it very differently.

    I think I do address your agnosticism in my model; indeed I think my position is probably similar to your own. Like you, I agree that no solid evidence of God’s existence or nonexistence is in the offing — so yes, ultimately the proposition may be undecidable. But that doesn’t mean we don’t have good reasons to consider the truth of “God exists” more or less likely than its negation. And that is what determines your position on the confidence scale.

    Finally, belief is a matter of psychology, and it would be disingenuous for me to pretend that my own normative prejudices don’t affect the behavior I will manifest regarding various propositions. For example, I think religion is a persistent delusion that causes a great deal of trouble in the world, and that it is time for our species to outgrow it; this will of course affect my disposition toward religious propositions generally. Not only do I have a very low degree of religious belief; I also have a very low degree of belief in religious belief.

    Posted March 22, 2009 at 12:35 pm | Permalink
  10. Addofio says

    I guess what I was trying to do was reject the zero-sum nature of propositions when it comes to belief. I don’t even accept the proposition “P or ~P” in all cases when considering truth, and we aren’t talking about truth, but belief. You seem to be saying that one must come down on one side or the other, and that only one’s confidence level may vary. I’m saying I don’t have to commit in the first place. Not when it comes to belief. Both “I don’t know” and “I don’t care” are possible, and perfectly reasonable, positions when it comes to belief.

    You also imply that the existence or non-existence of God is a simple proposition, and that therefore P or ~P applies to it. Well, maybe. But as I am sure others have pointed out to you before, the God you do not believe in may or may not be the God someone else does believe in. That is, you have some concept of God that you are rejecting, and it may or may not come close to another’s concept of God. The simplicity of the proposition is an illusion.

    Posted March 23, 2009 at 10:50 am | Permalink
  11. Malcolm says

    Hi Addofio,

    Yes, you are quite right that before we can form the proposition “God exists”, we need to establish what “God” refers to. Presumably we are able to do that for ourselves; whether it is the same as somebody else’s concept is rather beside the point as far as our own state of belief is concerned. Likewise, you can also weigh in on someone else’s definition; for example, I can describe my degree-of-belief in the proposition “Quetzalcoatl exists” (it is very low).

    Having established the referent under consideration for the term “God”, though, I think you are staking out a lonely and difficult position if you are saying, as you seem to be, that you will not assent to (i.e., believe with high confidence that it is true) the metaproposition

    “Either God (according to my working definition of the term) exists, or does not.”

    How can that be? You have decided what you mean by “God”, and are wondering whether such an entity actually exists. It’s fine if you have no way of knowing whether it does or not, but surely it either does or doesn’t, no?

    You say

    “I don’t even accept the proposition “P or ~P” in all cases when considering truth…”

    This seems quite remarkable, if P is made sufficiently clear. Can you give us an example?

    Of course, there are those who are outside the picture altogether; for example, Sentinelese Islanders likely have no degree-of-confidence at all regarding the truth of Goldbach’s Conjecture. In this sense they are no different, as regards their belief-attitude toward this proposition, from an infant, a dog, or a brick. If this is what you mean by being “uncommited”, then that’s fine. But I am talking about “belief” in the sense of someone who understands the proposition in question. At that point, I quite agree that you can say “I don’t know” (which in my scheme means “I have no reason to be any more confident in P than ~P”), and I also agree, of course, that you can always say “I don’t care” — which just means that you have no normative or intellectual interest in how it turns out.

    Posted March 23, 2009 at 11:05 am | Permalink
  12. Addofio says

    I said “I don’t even accept the proposition “P or ~P” in all cases when considering truth…”

    to which you replied “This seems quite remarkable, if P is made sufficiently clear. Can you give us an example?”

    There’s often the possibility that P is “undecidable”–which in my book makes neither P nor ~P true. For example, those pesky statements in mathematics that Godel demonstrated could be either true or false without creating contradictions in the system. The Axiom of Choice has this property, if memory serves me. So is the axiom true or is it false? There are circumstances in which one must choose to regard it as one or the other in order to proceed with further mathematics–but it is not possible to establish either its truth or falsity. It’s truth value is “undecidable”.

    And then there are cases in which someone has asked me a seemingly simple question that contained so many misconceptions or perhaps preconceptions that differ from my own that to answer either “yes” or “no”, “right” or “wrong”, would be deceptive. If I were clever or had a perfect memory, I’d supply an example, but I can’t–I just know it happens. I know these always simply silence me–and seem to me to contain statements whose truth value is neither True nor False.

    And if my own conception of God is very fuzzy and not clear–I see no reason I would then have to accept the metaproposition “Either God (according to my working definition of the term) exists, or does not.” I could be right, wrong, partially both, or so far off base that the metaproposition doesn’t even make sense, for all I know. Some people may have a nice, clear conception of God that they can then clearly accept or reject–but I’m pretty sure I don’t.

    But take heart–I often do accept the law of the excluded middle. I’m not completely irrational according to your criteria :-)

    Posted March 23, 2009 at 9:57 pm | Permalink
  13. Malcolm says

    Hi Addofio,

    We must remember that Gödel’s seminal 1930 paper was called “On Formally Undecidable Propositions of Principia Mathematica and Related Systems”, and the whole point of it was that there are statements expressible in formal systems that are actually true, but remain unprovable within the confines of the formal system itself. So it isn’t as if the propositions have no truth value; it’s simply about the inherent limitations of formal systems. Indeed, the actual truth of the special proposition he designed was central to Gödel’s argument.

    I entirely agree with your next paragraph. If we cannot form coherent propositions, then it is meaningless to talk about believing them to be true. That’s central to the point I’ve been trying to make here. How can we “believe that P” if we cannot say what P is?

    But, as far as belief in God goes, I think the question “do you believe the God exists?” is more meaningful to you than “do you believe Freen exists?”, and that this is because despite your disclaimer you actually do have some idea of what “God” refers to, and that with a little effort to narrow or broaden that definition we could formulate a proposition that you could then assign a degree of confidence to. “God” might be as broad as “some sort of conscious entity that deliberately created the Universe”, or it might pick out a more precise set of attributes, but sooner or later we’d arrive at something suitable. And at that point you could ask where you stand between affirming and denying the proposition; at the very least you would, I think be confident that it was either true or false.

    My point is that otherwise, the notion of “belief” itself is meaningless.

    And yes, not to worry, I know you aren’t irrational!

    Posted March 23, 2009 at 10:33 pm | Permalink