Zillions of Beliefs?

Here’s a fun one:

  1. Anyone who knows basic maths knows that 2+2=4.
  2. If someone knows that 2+2=4, then that person believes that 2+2=4.
  3. The known proposition in premises 1 and 2 (i.e. that 2+2=4) can be replaced by a very large number of other propositions (e.g. that 2+3=5 or that 5-1=4) while maintaining the truth of the premises.
  4. Therefore, anyone who knows basic maths has a very large  number of beliefs (countably-many?).
  5. Regular people do not have a large number of occurent beliefs.
  6. Therefore, many of the beliefs of regular people are non-occurent.

I’ve heard a few people complain that this idea of a non-occurent or implicit belief is non-sensical or elusive.  If you’re one of those people, which premise do you reject?


35 Responses to Zillions of Beliefs?

  1. Steve Nayak-Young says:

    OK, I’ll bite. I’d reject 3. All people run out of known “basic” arithmetical propositions at some point, and many run out quickly. They could discover more such knowledge, e.g., 36+52=88, but I don’t think that they already know or believe it in virtue of knowing basic maths.

    I don’t have a problem with non-occurrent knowlege/beliefs, e.g., my belief that radishes grow in bunches was non-occurrent until a second ago. But I’m not sure about the implicit. For starters, I bet that some of the basic logic I already know implies a bunch of unspeakably complex stuff that I hope I never know.

  2. Bryan says:

    Do you know the truth of the proposition 1234+5678=6921?

    I didn’t. So premise 3 seems to be false. In fact, I don’t know the majority of propositions involving addition.

    Knowing basic math just means I can come to know such propositions (usually only briefly), because I believe the Peano axioms, plus the recursively defined rules for addition and multiplication, and the division algorithm. But this requires me to believe less than 50 propositions about arithmetic. (I also know a few tables of arithmetic, but all in all less than a couple hundred propositions.)

    Of course, I’m afraid the notion of ‘non-occurrent’ belief is not something that occurs in my belief structure. Perhaps you can help me?

  3. Colin Caret says:

    Gotta echo the sentiment that the opponent of implicit belief will probably reject (3) here on the grounds that replacement instances of (1) are often false, e.g. for the examples given by Steve and Bryan.

    But here is another strategy. The opponent of implicit belief might accept your first three premises, but reject premise (5). Perhaps the spin on this could be that the falsity of (5) is a philosophical discovery which was only revealed by an enlightened rejection of the notion of implicit belief.

  4. Jon S. says:

    I’m pretty sure I’ve never had even 50 occurrent beliefs at once. Maybe we just need a good characterization of what it is for a belief to be non-occurrent.

  5. jmsytsma says:

    I would think that the opponent of non-occurrent belief would object to premises (1) and (2). Let’s assume that it is silly to deny that occurrent knowledge is (or implies) occurrent belief. One could nonetheless note that there are two different versions of the second premise:

    (2a) If someone has the occurrent knowledge that 2+2=4, then that person has the occurrent belief that 2+2=4.

    (2b) If someone has the non-occurrent knowledge that 2+2=4, then that person has the non-occurrent belief that 2+2=4.

    It seems that it is (2b) that is relevant to the argument. Accepting that, premise (1) now needs to be changed:

    (1b) Anyone who is able to do basic math has the non-occurrent knowledge that 2+2=4 when not thinking about that fact.

    Now the opponent of non-occurrent belief can deny either premise (1b) or (2b). Denying premise (1b) is to deny that there is non-occurrent knowledge; denying (2b) is to deny that non-occurrent knowledge is a type of (or implies) belief (and if we are assuming that it is silly to deny that occurrent knowledge is (or implies) occurrent belief, then it is to essentially take a sort of disjunctivist stance on knowledge). One might justify taking a different stance on occurrent versus non-occurrent knowledge, here, by arguing that non-occurrent knowledge is non-propositional, or know-how, but that it is only propositional knowledge that is (or implies) belief.

  6. dtlocke says:

    I reject (2). Belief required for knowledge? Pshaw!

  7. Jonathan Livengood says:

    I think that if you allow “<” as part of “basic maths” and treat “2 < 2 + 6” as distinct from “2 < 3 + 5” and allow sums of arbitrary length, then anyone who knows basic maths actually has uncountably many beliefs. To see this, take N = {1, 2, 3, …}. Allow “0” as part of basic maths and note that 0 < 1, 0 < 1 + 2, 0 < 1 + 3, …, 0 < 2 + 3, 0 < 2 + 4, …, 0 < 1 + 2 + 3, 0 < 1 + 2 + 4, … and so on. Now, think of “+” as encoding a set. So, 1 + 2 + 3 encodes {1,2,3}, 2 + 7 encodes {2,7}, etc. We can encode every subset of N with sums like this, though a lot of these will have to be infinite (and the encoding only goes one way). Since |N| < |P(N)|, the collection of right-hand sides of the inequalities above is uncountable.

    This leads me to deny the unstated premiss:

    (0) Regular people know basic maths.

    Premiss (0) is clearly false: no “regular person” knows basic maths in the required sense. Indeed, no one does. (Except maybe God.) The falsity of Premiss (0) does not depend on my version of “basic maths”. Even if you restrict to three-place predicates of the form _ = _ + _, where the _s range over N, no one knows basic maths, for the reasons pointed out by Steve and Bryan.

    Anyway, I think this just highlights that “basic maths” is equivocal. If you mean it in the sense necessary for the proof, then no one knows it. If you mean it in the sense of everyday speech, where everyone *does* know it, then you have something like what Bryan describes, and you don’t get the results of the proof.

    I’d also like to echo Bryan’s question, though. What are occurrent/non-occurrent beliefs?

  8. Thanks for the thoughtful replies everyone!

    Some replies are in order:

    Occurent/Non-Occurent Belief:
    Sorry, I should have said something about this in the post. An occurent belief is a belief which is presently before the mind (i.e. that I am conscious of at the moment). For instance, if I am standing before the Sears Tower, I may look up at the tower a think to myself “Wow, that’s tall.” At that time, I’m having the occurent belief that the tower is tall. A non-occurent belief is a belief that is not occurent.

    Re: Rejecting Only Premise 3
    I do not think this approach is satisfactory without also rejecting 1 or 2. All that 3 claims is that there’s a bunch of other propositions that would have worked just fine in 1 and 2. Here’s a bunch: 1+1=2, 1+2=3, 1+3=4, 1+4=5, 1+5=6, 1+6=7, 1+7=8, 1+8=9, 1+9=10, 2+1=3, 2+2=4, 2+3=5, 2+4=6, 2+4=6, 2+5=7, 2+6=8, 2+7=9, 2+8=10. Now if I had written 1 and 2 using any one of those propositions instead, you would have still accepted it, right? If so, then you should accept 3. (Notice that 3 does not commit you to knowing complex things like Bryan’s example of 1234+5678=6921.) So, I think we should stick with 3 if we stick with 1 and 2.

    Re: Rejecting 5
    Premise 5 was not meant to express a “philosophical” discovery (except maybe in the loose naturalistic sense in which philosophy and psychology are continuous). Premise 5 is a result from psychological research, which says that we can have about 6 or 7 occurent beliefs at a time at most. I’m relying on my naturalistically inclined friends for this info (and I’m pretty sure that I recall Pollock and Cruz citing work that maintains this in their book). This fact also seems to fit well with the phenomenon. Really, at any moment, I’m at most thinking about a few things.

    Re: jmsytsma’s suggestion
    I’ll accept both 2a and 2b. 1b on the other hand is a little different. 1b adds this new element of being able to do math that is not present in 1 (there it’s that one must know math). If we replace your 1b with the following it would better line up with your move with 2:

    (1b’) Anyone who non-occurently knows basic maths non-occurently knows that 2+2=4.

    Now, one can deny 1b’ without denying that there is non-occurent knowledge.
    That said, I’m not sure how this responds to the argument at hand. You say that the opponent of non-occurent belief would reject 1 or 2, and I agree with this, but rejecting either 1b’ or 2b does not commit you to rejecting 1 or 2 since neither 1b’ nor 2b is implied or is implied by 1 or 2. Also, rock on with the knowledge-how, but as I said, I think that responds to the wrong puzzle. (As a side note, some hold that knowledge-how is propositional, e.g. Stanley and Williamson.)

    Re: dtlocke
    Yo Dustin, I’m sure you meant this jokingly given your presentation of it, but some may actually go with this option, e.g. David Lewis. In that case, I think the puzzle could be reformulated in terms of non-occurent knowledge to have a similar conclusion.

    Re: Jonathan Livengood
    Really? You think it’s false that regular people know basic maths? I find this intuitively unacceptable. But, I guess that’s an option for denying the puzzle. I guess I’m inclined to think that knowing basic maths does not require that you know even “three-place predicates of the form _ = _ + _, where the _s range over N”. My intuition is that knowing basic math only requires that you know some of these instances at most. You claim “If you mean it in the sense necessary for the proof, then no one knows it”, but I think this view is only due to a misunderstanding of the requirements of premise 3 (see that above).

    (Also as a technical point, I do not buy your proof of the uncountability of the number of truths the knower (in the strong sense) is committed to. If we allow that + can be generalized to countable sets of numbers (which your proof requires), then we should allow that + can generalize to arbitrary sets (i.e. say that +S where S is a set of numbers is the infinite sum of the numbers in S). But, if + generalizes in this way, we could generate one known proposition for each transfinite cardinal, and hence the class of things known would be a proper class. Another way to see this (which is easier) is to notice that for each cardinal c, the knower would know that |c|<|P(c)|. But that would mean that there is a known fact for each number and hence the `size’ of the collection would be the `number’ of numbers, i.e. a proper class. So, either the knower knows at most countably-many things or she knows more things than there are numbers.)

    Re: Implicit vs. Non-occurent beliefs
    Steve mentioned it, and a couple others hinted at it, that there may be a difference between non-occurent and implicit beliefs. Aaron Bronfman and Steve Campbell suggested that this difference may be here: Non-occurent beliefs are beliefs that happen to not be before my mind. Implicit beliefs are belief-like things that either (1) I would believe were I to consider them (e.g. that no elephant owns 15 shoes), or (2) that follow from other beliefs I hold. This account clearly would need to be worked out further, but by seperating non-occurent from implicit beliefs, we could imagine someone rejecting implicit beliefs without rejecting non-occurent beliefs or visa versa. For instance, in this case, it’s seems plausible that the belief that 2+2=4 that we attribute to basic math knowers may be both non-occurent and explicit. Whereas the belief that 123+123=246 may be only implicit. It’s not clear to me that this distinction/clarification will resolve any of the issues here or solve this puzzle, but it’s benificial to our thinking about the issues none-the-less.

  9. Steve Nayak-Young says:

    Hey Dan, thanks for the clarification. Reviewing the argument now, I find that I don’t disagree with it unless “countably many” means something crazy.

    (I think I once heard Phil Kremer distinguish between countably and uncountably infinite numbers, but I might have been hallucinating from the math fear.)

    If all you mean is that anyone who knows basic math has a “very large” (as in, tens or hundreds) number of *non-occurrent* beliefs, I’m cool with that.

  10. Hey Steve,

    Yea, frankly, I did have a stronger idea in mind when I formulated the argument (hence the “countably-many?” thing). But the argument should be weakened from that original intuition. I think I stand by this new interpretation of the argument.

    Thanks for the feedback,


  11. Jonathan Livengood says:

    Daniel (hope it’s okay to use your first name),

    Of course, lots of people know how to do simple sums. I wasn’t trying to be snide, just a bit tongue-in-cheek. If an occurrent belief is like saying a sentence to yourself (maybe in your head, whatever that means), then I’m sure I agree with you: everyone has non-occurrent beliefs.

    Anyway, since I didn’t really know what an occurrent belief was, I assumed (as it appears most readers did) that you really needed the full strength of the mathematical example. So, I thought you meant “basic” to indicate the kinds of objects and operators at stake, i.e. no integration, no graph theory, no topology, etc. As you state it now, it seems like you just need to give a nice long list of banalities. The sky is blue. The grass is green. Snow is cold. Diamonds are hard. Pillows are soft. The sun is bright. There are lots of stars. People have two arms.

    A nice long list of sentences no one doubts is enough to show that the claim that there are only occurrent beliefs is just silly. Or am I missing something? What do the people you mentioned who think non-occurrent beliefs are incoherent have to say?

    As to the proof, I’m still pretty confident that it works, but maybe I’m just not understanding your complaint. It’s also not important, I was just having a bit of fun. Again, not something that comes across well in type, I’m sure.

  12. James Gray says:

    If we want to defend non-occurent beliefs, then why not try to come up with a common sense defense of it? Is there anything phenomenologically attractive about it.

    The example given here seems to imply that I have infinite non-occurent beliefs. (If 2+2 = 4, then 2+2+2+2 = 8, etc.) The argument is so counter-intuitive that it seems to defeat its own purpose.

    We should try prove one clear-cut case of occurent belief to defend it.

    I would want beliefs to have various strengths. A strong belief that bread is tasty should give me a desire to eat bread at some point. In other words a tie between belief, desire, and action seems to be important to a non-occurent belief hypothesis. We need some kind of evidence that we have these beliefs through phenomenology and/or our behavior.

    Hypnosis is probably involved with some arguments, but it is hard for many people to believe in hypnosis.

  13. Jon S. says:


    Now that Daniel has defined non-occurrent beliefs, and Jonathan Livengood has given a list of 8 things everybody believes that you can add to 2+2=4 (and whatever else you’re willing to grant that everybody believe), do you still really want to claim that none of those things are non-occurrent beliefs? Here’s a common sense defense of non-occurrent belief: you believe all of those things; there is at least one time when you believe at least one of them when it is not ‘presently before the mind’; so you have non-occurrent beliefs. To deny that you have non-occurrent beliefs, you have to deny that you have beliefs that aren’t presently before the mind. That strikes me as prima facie absurd, just because of what the folk psychological notion of a belief is.

    But, if you insist on more being said, how about this: if someone asks you what 2+2 is, you assert 4 without calculating. I don’t know how that’s possible unless you’re guessing (which you aren’t) or you believe that 2+2=4. Nevertheless, you were not thinking about the fact that 2+2=4 until you were asked for the sum.

    Steve N.Y.,

    (Assuming you exclusively an ethics guy and really don’t know what countably many means,) For a set to be countable is for there to be some function lining up elements of that set with elements of the natural numbers. All finite sets are countable, because you can match them up with finite sets of natural numbers (by simply enumerating the elements–call one the first, another the second, etc.; you’ll run out of elements before you run out of natural numbers). Some infinite sets are countable, like the natural numbers themselves. Even “bigger” sets, like the rational numbers, can be counted. (This should be surprising the first time you learn it–ask Dan to draw you a picture some time.) But some infinite sets are so big they can’t be counted, like the real numbers. (Again, Dan can draw you an intuitive proof.) The size (“cardinality”) of some infinite sets is countable (=the same size as the natural numbers). The cardinality of others is not. (There is another notion to account for the intuition that the rationals are strictly larger than the naturals, despite being countable. Dan can tell you about that too.)

    So, “countably many” means “finite or the size of the natural numbers”, but usually when people say “countably many” they mean (as Dan did here) “countably many (but not finite)”.

  14. jmsytsma says:


    As I said at the start, the opponent of non-occurrent belief could object to premises (1) and (2): the problem with them is that they equivocate about “knows.” Specifically, the argument doesn’t clearly distinguish between occurrent and non-occurrent knowledge and beliefs. Once that is done, however, the argument either falls apart or else the revised first or second premise can be denied for one of the reasons that I gave. (My main reason for chiming in was that I thought one could reasonably challenge the argument in a different way than hitting premise (3).)

    The main issue is that the type of knowledge at issue in the “known proposition” should be the same, so we get two versions of the argument (I’m putting “belief” in quotes, for now, because the next issue is to decide what sort of belief follows in each case):

    — occurrent —

    (1a) Anyone who (occurrently or non-occurrently) knows basic maths occurrently knows that 2+2=4.

    (2a) If someone occurrently knows that 2+2=4, then that person “believes” that 2+2=4.

    (3a) The occurrently known proposition in premises 1a and 2a (i.e. that 2+2=4) can be replaced by a very large number of other propositions (e.g. that 2+3=5 or that 5-1=4) while maintaining the truth of the premises.

    (4a) Therefore, anyone who occurrently knows basic maths has a very large number of “beliefs” (countably-many?).

    (5a) Regular people do not have a large number of occurrent beliefs.

    (6a) Therefore, many of the beliefs of regular people are non-occurrent.

    — non-occurrent —

    (1b) Anyone who (occurrently or non-occurrently) knows basic maths non-occurrently knows that 2+2=4.

    (2b) If someone has the non-occurrent knowledge that 2+2=4, then that person has the “belief” that 2+2=4.

    (3b) The non-occurrently known proposition in premises 1b and 2b (i.e. that 2+2=4) can be replaced by a very large number of other propositions (e.g. that 2+3=5 or that 5-1=4) while maintaining the truth of the premises.

    (4b) Therefore, anyone who non-occurrently knows basic maths has a very large number of “beliefs” (countably-many?).

    (5b) Regular people do not have a large number of occurrent beliefs.

    (6b) Therefore, many of the beliefs of regular people are non-occurrent.


    Let’s look at the second version first. The “believes” in (2b) seems like it must be non-occurrent belief that is at issue (as it doesn’t seem that non-occurrent knowledge implies occurrent belief). Making that change, however, means that premises (4b) and (5b) aren’t really needed. The updated argument is:

    (1b) Anyone who (occurrently or non-occurrently) knows basic maths non-occurrently knows that 2+2=4.

    (2b) If someone has the non-occurrent knowledge that 2+2=4, then that person has the non-occurrent belief that 2+2=4.

    (3b) The non-occurrently known proposition in premises 1b and 2b (i.e. that 2+2=4) can be replaced by a very large number of other propositions (e.g. that 2+3=5 or that 5-1=4) while maintaining the truth of the premises.

    (6b) Therefore, many of the beliefs of regular people are non-occurrent.

    With regard to this argument, the opponent of non-occurrent beliefs would clearly want to deny (2b), as I indicated, and perhaps might want to deny (1b).

    Now for the first version. The “believes” in (2a) seems like it must be occurrent belief that is at issue (as it doesn’t seem that occurrent knowledge implies non-occurrent belief). Therefore, the “beliefs” in premise (4a) should also be replaced with occurrent belief. Making those changes, however, we have a problem: (4a) and (5a) together suggest the conclusion that nobody occurrently knows basic math. I think that is true, but it isn’t the conclusion you wanted from the argument and (6a) no longer follows. So, this version of the argument falls apart.

  15. Re: Jonathan Livengood
    (Hey there, yea first names are totally cool. On the internet, we’re all peers, right? I take it that we’re actually peers as well (Google believes you’re in Pitt HPS). That said:)

    Thanks for the feedback again. I did take your cardinality argument as fun. I was intrigued by it, which is what prompted that response.

    I do now think the argument works, too, and that it may even boarder on trivial (as you point out). I was trying to use premises that seem obviously true such as 1. I was concerned that using premises like the banalities that you suggest might prompt this response: Well, S doesn’t really believe that stuff; it’s just that she would believe it had she thought about it for a second or took the time to remember it. So the role that 1 and 2 play is to motivate that she does actually know/believe.

    I have an email out to the main person I know who objects to non-occurent/implicit beliefs, so we’ll see what he has to say. I’m betting that it’s a complaint with a particular way of understand non-occurent/implicit beliefs.

    Thanks again for the feedback!

    Re: James Gray
    Hey there,

    As Jon points out below, I do think this argument brings out a quite intuitive way in which we have these types of beliefs.

    Re: Jonny boy
    Thanks for putting it all out on the table. I think I may get punched if I prove one more logicy/mathy result in the office. It’s recently been a logicfest in there, and I think people are getting tired of logic. Wanna come back and be my mathematical logic friend?

    Re: jmsytsma
    Ah yes! I see now! Very interesting! Thanks for clarifying that. Humm .. that’s definitely a very nice way to think about it. I need to run to a reading group now, but I’m going to think about this more later. Thanks! (BTW, what’s up with all the HPS people? I know y’all are very smart and all, but how do you have this much time? I need to switch programs apparently …) I’m thankful that you spent the time here though!

  16. jmsytsma says:


    I’m not sure what explains Jonathan and Bryan’s (cyber) presence, but my (cyber) presence is easily explained by my habit of following them around. They typically lead me to interesting discussions. I’m not sure what to say about having so much time: I know I’m capable of working 80 hours a week at some kinds of jobs (because I used to do it), but I can’t devote anywhere near that many hours to my philosophical work and have most of the time be productive. I find thinking about other problems (like the argument you posed) to be a beneficial diversion.


    I’m a little surprised that you are so quick to accept that the denial of non-occurrent beliefs is silly. I don’t know that I’m really committed to the denial, but I feel some real tug here. And, depending on how you spell out what occurrent belief amounts to I probably do want to deny that the same sort of thing would exist non-occurrently. The basic motivation here, for me, would be the feeling that *beliefs* aren’t things that could be stored in the brain; rather, one *believes* this or that (or maybe better, one implicitly or explicitly assents to something). If what we mean by “belief” in the occurrent case is to pick out something about this episode of assent, then I’m not sure why we would want to use the same term in the non-occurrent case when there is no episode of assent. Rather than say I have a non-occurrent belief that diamonds are hard, for example, it seems better to say that I am disposed to believe that diamonds are hard (and will so believe if I am put in a situation where I have reason to do so). Of course, I have no problem with the general practice of just calling these beliefs… but, philosophically I feel some urge to caution here.

  17. Jonathan Livengood says:


    Very nice. But how does your point generalize to the long list argument for non-occurrent beliefs? I don’t know how few trivial sentences one could get by with, but I’m confident that given a few minutes, I could generate a list of fifty sentences that every sane adult of average intelligence would understand and assent to without deliberation. (We could set it up as a psychology experiment if need be: give people a “yes” button and a “no” button, present them with sentences under the instruction, “Press the ‘yes’ button if the sentence is true and the ‘no’ button if it is false,” and time their responses. Pick a time too short to count as deliberating and show that there are a good number of sentences that fit.) Anyway, fifty seems like it would be enough. So, I’d like to know if there is something more general hiding behind your argument that I’m just not seeing, yet. I’m going to suggest something below that might be what you are up to, but I kind of doubt it. If it is, sorry for being so thick. If it isn’t, then let me know what you think.

    Daniel (mostly to you, anyway, since it’s your post),

    I’m worried now that what is at stake is figuring out the best way to use language about beliefs. Here are three things that might be meant by “belief” (all three are roughly pragmatist in character, so there are probably many more versions that I just don’t care about):

    (1) A belief is a habit of action,

    (2) A belief is a habit of action that could be reflected on, reasoned about, or changed,

    (3) A belief is a habit of action that is being reflected on, reasoned about, or changed.

    Versions (1) and (2) allow for non-occurrent beliefs, but (3) does not. (1) makes beliefs commonplace everywhere. My television, computer, and thermostat all have beliefs on this version. So do even dumber physical systems, like an electron in a magnetic field. (2) applies restrictions to get a little closer to animal beliefs, but I don’t know how far it succeeds. Anyway, that aside, one might endorse (3) in order to separate habits of action, which can endure over time, from the reflective activation of a habit, which is typically a momentary event. Or one might endorse (3) because one has scruples against propensities or counterfactuals or some such. (I don’t, so I don’t have any reason to adopt (3).)

    I’d be interested to know what account of “belief” you are working with–when the qualifiers “occurrent” or “non-occurrent” are not in play. That is, what do you tell someone who simply asks, “What is a belief?”?

    As to why there are so many (three?) HPSers on here, it’s probably a combination of (a) our spring break is this week, (b) the posts here are interesting, and (c) we don’t have enough sense to stay off the blogo-tubes.

  18. Jon S. says:

    Oh, y’know, I’ll come back some day. What started this anyway? I think the notion of non-occurrent attitudes (as one encounters them in, say, Ken Walton’s stuff) is (supposed to be) pretty non-controversial. There should be a theory-neutral sense of belief relevant here, which more or less provides a constraint on what your account of beliefs is. (I.e., if you can’t make room for non-occurrent beliefs, you don’t have an adequate account of belief.) The sense of belief in play is the one in sentences like “Kripke believes that Wittgenstein wrote _Philosophical Investigations_.” It’s prima facie true, and (we may assume) Kripke isn’t thinking about it right now.

    (Aside from the first two sentences, which are addressed to Dan, that’s intended for all of you still taking part.)


  19. jmsytsma says:


    So, I think you are right to say that what is at stake is how we use belief language. Let me be very clear that I really know nothing about the relevant literature: But, I assume that the initial argument fits within a dialectic in which beliefs are being understood in such a way that the primary example of a belief is consciously assenting to a proposition: That is, one thinks “2+2=4” and thinks that it is true. That notion of a belief is very different from what you suggest. (I’m sure it will be little surprise to you that I prefer the type of direction that your suggestions run in.) But, if your exemplar of a belief is thinking to oneself that “2+2=4” is true, then the occurrent/non-occurrent distinction with regard to beliefs is pretty natural: The exemplar is an occurrent belief, but it also seems like we want to say that I believe that “2+2=4” whether I am currently thinking that or not. (Do we want to say that I believe that “2+2=4” when in a deep dreamless sleep? When in a coma?) Common sense seems to tell us that we all believe all sorts of things (or have all sorts of knowledge) that we aren’t thinking about at the moment; heck, I seem to have beliefs that I have never even thought about before—for example, that “pirates are less likely to have a pension plan than United States senators.” So, starting from our exemplar, and recognizing what common sense tells us here, it seems natural to treat the belief like a thing (for convenience, let’s say we treat it as a string like “‘2+2=4’ is true”) that is stored in the mind/brain and that can be brought before the mind (and is then an occurrent belief) but need not be (and is then a non-occurrent belief). Now, if something more or less like that is what is at issue, then it seems pretty reasonable to deny that there are non-occurrent beliefs: One can accept that one can think to oneself that “2+2=4” is true, and yet deny that the content of that thought is stored in the mind/brain when you aren’t thinking about it and is sometimes trotted out and placed before the mind’s gaze. Of course, we could replace “2+2=4” with “Snow is Cold,” “Diamonds are Hard,” are any other of the banalities you suggest. The objection is that “if you mean *that* by belief, then there are no non-occurrent beliefs”; the specific examples don’t really matter for that objection. Of course, if you mean something else by belief—and mean something by belief on which it would be natural to make the occurrent/non-occurrent distinction—then the objection doesn’t apply.

    With regard to your suggestion that a belief is a habit of action (perhaps with some further stipulations), the objection doesn’t apply: If that is what you mean by belief, then the occurrent/non-occurrent distinction doesn’t seem natural (as it doesn’t seem that a habit of action is itself something that is sometimes before the mind and most of the time not). If you mean something like *that* by belief, then I think you should deny that there are non-occurrent beliefs because you should deny that beliefs are the type of things that can be occurrent or non-occurrent. (And that means you should reject the initial argument because you reject the hidden premise operating in (4), (5), and (6) that beliefs are either occurrent or non-occurrent.)

    Your three suggestions raise another interesting point, though, in that it is easy to put too much emphasis on language in some discussions of beliefs; but, when we start thinking of non-human agents, we probably want to think of belief more functionally in terms behaviors (inner or outer).

    I accept that we say things like “Kripke believes that Wittgenstein wrote PI” and that clearly this is true. But what exactly is meant by it? It isn’t prima facia obvious to me that it is asserting that Kripke has a non-occurrent belief. It might be that the common understanding of “believes” is such that the occurrent/non-occurrent distinction doesn’t make sense; that the term is understood in terms of something like a habit of action. Or, it might be that such sentences are understood to be leaving something out, and that our intuition that such a statement is true reflects our understanding of it as saying something like, “If Kripke were to think about it, then he would say that he believes that Wittgenstein wrote PI.”

    Let me give a really quick reason to think that the ordinary understanding of belief statements that might suggest the acceptance of a non-occurrent belief might really be doing something different. Consider the first definition for “believe” on Dictionary.com: “To accept as true or real.” And, the relevant sense of “accept” is defined as: “To agree or consent to.” But, it wouldn’t seem like we mean to be asserting that there is non-occurrent agreement or non-occurrent consent.

  20. Jon S. says:

    Several letters,

    I see. It was kind of mysterious to me what ground people thought they had to stand on in denying non-occurrent beliefs. Now it seems like we’re probably operating with different concepts. I think “Kripke believes that Wittgenstein wrote PI” implies something like “If Kripke were to think about it, then he would say that he believes that Wittgenstein wrote PI.” I don’t think the implication works the other way around. As for the habits of action thing, it seems to me that habits of action are typically not before the mind (“conscious”, “occurrent,” etc.), unless circumstances obtain in which the habitual action is taken. So they’d be non-occurrent by default.

    As for the dictionary, I think that’s not plausibly the central sense of the term (as I use it). (It’s also not the first entry on Dictionary.com, unless the page loads differently for me than it does for you. My first entry is: “to have confidence in the truth, the existence, or the reliability of something, although without absolute proof that one is right in doing so: Only if one believes in something can one act purposefully.”)

    So, I take it we just have different concepts. Of course, I think mine’s the folk concept, but that’s a question for xphil types.

  21. Andrew says:

    “I accept that we say things like “Kripke believes that Wittgenstein wrote PI” and that clearly this is true.”

    So, is the idea that you deny the following principle?

    (B) For all x and all p, x has a belief that p iff x believes that p.

    If (B) is true, it follows pretty quickly that Kripke has a non-occurrent belief.

  22. jmsytsma says:


    (Feel free to call me Justin, by the way.)

    Interesting, different entries come up for “believes” versus “believe.” Go figure!

    My point was not that the two sentences imply each other, but that it is plausible to think that the way in which people understand the first sentence is more in line with the latter. If so, then the intuition that the first sentence is true would not imply commitment to the existence of non-occurrent beliefs.

    With regard to habits of action, my claim is that on this understanding of beliefs, beliefs are not the sort of thing that can be present to the mind. Certainly one can think about a habit of action, as I can think about a person long dead, a building in another city, and so on; but, that doesn’t seem to imply that the habit of action is present to the mind. But, if beliefs are not the sort of thing that can be present to the mind, it strikes me as odd to talk of them as non-occurrent or to say that they are non-occurrent by default. This strikes me like reasoning that because a rock isn’t alive, it is dead by default. My claim is that if we understand beliefs, following Jonathan’s suggestion, as being something like habits of action (perhaps with some further stipulations), then we want to say that the occurrent/non-occurrent distinction doesn’t apply and beliefs are neither occurrent nor non-occurrent. Instead, we might want talk about them currently being exercised or not, or maybe just talk about the action being taken or not.

    I fully agree with the last point: In so far as your claim is that there is a folk understanding of belief and that this should place some constraints on our account of belief, I take it that we need to investigate the folk notion. My point is that the shape of the folk notion might well surprise you and that we should be cautious here. I don’t claim that one of my suggestions is right, but that they are plausible given the evidence we are working with. (As an x-phi type, I would offer to run a study to look at it, but I’m running half a dozen studies this semester and have already given something to just about every subject I have access to!)

    Let me note, though, that I’m really not sure that we should put much weight on the folk intuitions here. Say that it turns out that the folk notion of belief treats it as a sort of thing that is sometimes before the mind and sometimes not. I guess I would want to say that that is too bad and argue that they are mistaken.

  23. jmsytsma says:


    No, I don’t mean to be denying (B); rather, I am suggesting that the sentence can be understood in ways such that its truth does not imply that Kripke has a non-occurrent *belief* (where he has something that could be brought before the mind, but currently isn’t before his mind). And, I suggested two possibilities: (1) the sentence is basically short hand for “if Kripke were to think about it, he would believe that W wrote PI”; (2) the term “believes” does not refer to something that could be brought before the mind.

  24. Jon S. says:

    Ah, so that’s your name. Ok. So, Justin,

    “My point was not that the two sentences imply each other, but that it is plausible to think that the way in which people understand the first sentence is more in line with the latter. If so, then the intuition that the first sentence is true would not imply commitment to the existence of non-occurrent beliefs.”

    Of course, insofar as one accepts that the implication goes only the direction I’ve suggested, one understands the former as strictly stronger than the latter. So if we are to understand it as “more in line with the latter”, we need some formulation that is strictly stronger than the latter but somehow “more in line with” it. To be clear: my point is basically that, because the implication goes only one direction, the former is not to be understood as saying roughly what the latter says. If you have a formulation of the sort of suggested, kudos; but I don’t expect you to have one. (Or, if you do, I don’t expect it to be one that I would accept as the way to understand the claim about what Kripke believes.)

    I agree that, if beliefs are habits of action (perhaps with some further stipulations), the distinction starts to look weird. But I don’t think it’s wrong to say that then beliefs are non-occurrent by default. The proper analogy is to say that rocks are non-alive by default, which is true (if a bit strange to say). Of course, if all beliefs are non-occurrent, then there’s no point in mentioning it, or in talking about occurrent beliefs at all. But I think that construing beliefs as habits of action isn’t what people have in mind when they talk about (non-)occurrent beliefs to begin with. When Dan tells us what started all this, I hope that the reasons for the distinction will become clearer.

    I’m actually pretty Gricey (cf. “Post-War Oxford Philosophy”) when it comes to x-phi and folk concepts. So, even though I think my concept is the folk concept, I don’t really care if I’m wrong about that. (I didn’t realize you were an x-phi guy, or I wouldn’t have brought it up as a way of tabling the issue.) Regardless, I wouldn’t want to be mistaken about what beliefs are, or make a useless distinction, so I will vaguely gesture in the direction of what I think beliefs are: they’re one of a number of attitudes that people can have towards propositions. I don’t have a further story about attitudes, but I would think that they’re things that can sometimes be before the mind, and other times not. If anything, I would think that the thing to deny would be the possibility of having attitudes not before the mind, and so the possibility of non-occurrent beliefs (rather than occurrent ones). But I don’t really know. Is any of this inconsistent with some prevailing view on what attitudes are? (If the habits-of-action thing is supposed to be an analysis of belief qua propositional attitude, then you have more of a story than I do, so I hope Dan has something to say to help out.)


  25. jmsytsma says:


    Maybe I’m not understanding your implication response; but, let me give it another try. Let’s just set the issue of how the folk might understand the statement at issue aside (as it will be easier if I just say what I think). If you asked me, “Does Kripke believe that W wrote PI?” I would say, “Yes, of course.” If you then responded – “Hah! Kripke surely isn’t thinking about that right now, so you are admitting that there are non-occurrent beliefs!” – I would want to respond: “No, I took it as understood in the question that you were asking about what Kripke would say if asked who wrote the PI.”

    That is, I share your intuition that “Kripke believes that W wrote PI” is true; but, that intuition reflects my understanding of the sentence. Now, I take it that you are noting that these sentences don’t literally say the same thing (as the first implies the second, while the second doesn’t imply the first). Fair enough. But, my intuition that the first sentence is true reflects my reading it as expressing roughly the same sentiment as the second sentence. If I read it as implying that Kripke has a non-occurrent belief, then of course (insofar as I am denying that there are non-occurrent beliefs for purposes of this argument) I deny that the sentence is true.

    On your second point, I think that is right: Whether or not we want to say that something that can never be occurrent is non-occurrent, the point would seem to be that those engaged in this debate must understand belief in some other way. If one understands beliefs in a way that (at best) they could only be non-occurrent, then it would be very strange to accept the existence of occurrent beliefs and argue about whether there are non-occurrent beliefs. But, accepting all of that, I think my point still holds: It is plausible, I hold, that “Kripke believes that W wrote PI” seems prima facia true because our standard understanding of belief is not one on which it makes much sense to make the occurrent/non-occurrent distinction. Now, maybe we want to say that on this understanding all beliefs are technically non-occurrent. Still, that would hardly seem to support in the debate over whether there are non-occurrent beliefs *in addition* to the occurrent beliefs that we all accept.

    I don’t think I have any issue with the attitude view of beliefs (actually, I think it is pretty close to what I suggested several posts above, if we take assent as the attitude typical of belief). I would then want to argue (for purposes of this discussion), however, that one does not have a non-occurrent attitude toward a proposition; rather, one has a disposition to take that attitude toward that proposition in certain circumstances.

    Alright, this has been fun, but it is past my bedtime!

  26. Jon S. says:


    I’m pretty convinced that we just have different understandings of the sentence “K believes that W wrote PI”. I take it to mean that, at some point in the past, Kripke adopted a stance towards the proposition that W wrote PI (say, a stance of acceptance, in line with the definition you offered earlier). And since nothing has happened in the mean time to change that stance, other than the question of what stance to take having passed out of Kripke’s consciousness, it seems reasonable to me to say that he still has it. I.e., that he still has the same attitude towards the proposition, viz., one of belief. While it’s true that he would say that he believes that W wrote PI if asked, that’s only because he would remember the stance he previously adopted, see no reason to depart from it, and reconfirm it.

    Maybe you think it doesn’t make sense to say that Kripke has a stance towards a proposition when he’s not thinking about it. Or, (more likely) maybe you think that having a stance towards a proposition is having a particular habit of action with respect to it.

    What worries me about reconstruing claims like “K believes that W wrote PI” as claims about what Kripke would do is the likelihood that the latter kinds of claims might be true when the former aren’t. E.g., Kripke would say that “Wittgenstein” has 12 letters if you asked him, but (we can assume) it wouldn’t be true to say that “K believes that “Wittgenstein” has 12 letters in it” ahead of time, because he might not have ever thought about it. Now, maybe that’s unfair, because you can add some conditions to what Kripke would have to do to qualify as having the belief (e.g., he can’t count the letters before answering). But I’m a bit skeptical that succinct conditions about what Kripke would do can track exactly the truths about what beliefs he has. More ideologically: it strikes me as too behaviorist-y to want them.


  27. jmsytsma says:


    Not the first time I’ve been accused of being behaviorist-y!

    I actually think I’m ok with your reading of “K believes W wrote PI”; but, where you take it to be reasonable to say that K still has the attitude he adopted toward the proposition, as an opponent of non-occurrent beliefs I would want to say that K is still disposed to take that attitude toward the proposition. This isn’t quite what you suggest in the second paragraph: Insofar as we think that the occurrent/non-occurrent distinction is a good one to make with regard to belief (and I take it that we are both playing devil’s advocate to some extent on this), and accepting that belief (for present purposes at least) is to adopt a stance of acceptance toward a proposition, I would want to say that what is going on in K when he is actually taking a stance of acceptance toward a proposition is rather different from when he is not (but would if the relevant situation arose); different enough that we shouldn’t say that they are of the same type or call both of them beliefs.

    Now, of course, as your worry suggests, being “disposed to adopt a stance of acceptance toward a proposition” would seem to cover different cases than “has adopted a stance of acceptance toward a proposition.” That seems correct, but that just reflects that you are building some history into your notion of belief. So, if that is acceptable, then I should be able to call on some history in specifying the relevant type of disposition: So, “K believes W wrote PI” means “K once adopted a stance of acceptance toward the proposition ‘W wrote PI’, has not subsequently revised that stance, and is disposed to adopt that same stance toward the proposition if it were to arise.” The point is that in either case we are calling on some assumptions about K’s history of stance taking with regard to this proposition to decide on either what K “non-occurrently believes” or is “disposed to believe”; but our debate shouldn’t turn on the use of the historical assumptions, but on whether what is going on in K now should be thought of as a kind of belief (that happens to be non-occurrent) or merely a disposition to a belief.


  28. Jonathan Livengood says:


    I think I’m happy either way on this, but here is what I was thinking on habits of action vis-a-vis the occurrent/non-occurrent distinction. Habits are propensities or something like that. They lie around not doing much until poked and then spring into life. So, I thought it made sense to think of occurrent beliefs as beliefs being manifested. If we are operating with your first version of “belief,” then manifesting is basically like saying a sentence in one’s head or to oneself. But I don’t really like that picture, because it doesn’t help one see the connection between beliefs and actions at all. On the habit view, beliefs lie around quiet for most of their lives. But every once in a while, they manifest themselves in action. These actions are evidence for the beliefs. But when the actions cease, the beliefs are still there, just not manifestly. Maybe it is inappropriate to think that a belief so understood is occurrent when it is manifest, but it didn’t seem like much of a stretch yesterday.


    Behaviorism has come in for a lot of undeserved abuse in the last twenty years or so. I think this, mostly, has nothing to do with the underpinnings of the behaviorist approach to the mind, which is to require experiments in which a behavior of some kind distinguishes between psychological theories. The big behaviorist mistake was thinking that the brain (and related neural structures) were irrelevant to psychology. This leads to the very odd view that the brain need not have any particular structure to account for our actions; it could be an amorphous gel or homogeneous goo. But when one takes the brain seriously, one still looks for behaviors. It’s just that one looks for behaviors both outside and inside the brain.

    It might very well be that “habit of action” is too crude. The identity step might be wrong, for starters. It might be that beliefs merely cause or support habits of action. I admit that I tend to obscure the distinction between signs and their objects. More so (and less viciously, I hope) when the objects are forever out of sight. Or it may be that beliefs are habits of action plus something–whatever it is that makes your equivalence come out all right. (Although, I’m not so sure that I don’t just want to bite the bullet and say that Kripke *does* know that “Wittgenstein” has twelve letters. I’m just as happy to say that as to say that he knows 153*27=4131, for instance. Just where the folk stand on this sort of thing would be an interesting x-phi project, if anyone is interested.) Anyway, I guess I just want to urge that you need to tell us enough to let us do some experiments to pick out the something more that constitutes belief.

  29. Justin says:


    I think I am also happy to go either way on this. I think that the language sounds a bit strange to say that a propensity being exercised is occurrent, but I have no real problem with it. (I do not think that there is much reason to think that such a propensity comes before the mind in being exercised, however.) In so far as I feel some tug for denying non-occurrent beliefs, however, it is because, like you, I lean toward a propensity view of beliefs…. and it seems to me that the debate over non-occurrent beliefs assumes a different view of beliefs, under which I am inclined to deny that the thing at issue exists when not occurrent. If what is meant by belief is thinking to oneself that a proposition is true, for example, then it strikes me as rather silly to think that *that* exists non-occurrently (but I don’t really want to deny that one sometimes thinks such things to oneself).

  30. Jon S. says:

    I think I’m basically happy now. Some quick thoughts:


    When you add the historical thing in there, then I don’t think I have any problems with your restatement of what is asserted by someone who says that K believes that W wrote PI. I also think what we call the thing now going on in K becomes a merely terminological question, the answer to which can just be stipulated in whatever way is found to be most suggestive in a given case. What is hopefully clear, at least, is what non-occurrent beliefs are supposed to be, viz., things like what is going on in K now with respect to the proposition towards which he once adopted a stance of acceptance, which he never since revised. If the objection is now an objection to calling that thing, whatever it is, a non-occurrent belief, well, ok.


    Re: “The identity step might be wrong, for starters. It might be that beliefs merely cause or support habits of action. I admit that I tend to obscure the distinction between signs and their objects. More so (and less viciously, I hope) when the objects are forever out of sight.”

    Obscuring that distinction is exactly what I find objectionable. But, assuming you haven’t been following other discussions, it will be news to you that my intuitions are generally *very* Kantian. (Even more so in metaphysics than in ethics.) So, I assume we can be happy to agree to disagree here.


    (This seems like too clean of a resolution for blog comments. Somebody object angrily.)

  31. dtlocke says:

    Yo Dan:

    Yo Dustin, I’m sure you meant this jokingly given your presentation of it, but some may actually go with this option, e.g. David Lewis. In that case, I think the puzzle could be reformulated in terms of non-occurent knowledge to have a similar conclusion.

    Care to do the reformulation for me?

  32. jmsytsma says:


    Sounds about right to me. I’m not sure if it resolves the larger issue (as I take it that each of us suspects that the concern over non-occurrent beliefs behind the initial post understands them somewhat differently than what we have arrived at), but even so it isn’t bad for the blogonets!

  33. James Gray says:


    I guess the argument isn’t as counter-intuitive as I thought. More importantly, I wanted to discuss one strategy for argument that occurrent beliefs really exist.

    Yes, if we assume that 2+2=4 is a non-occurrent belief, then it does make sense that we have a lot of non-occurrent beliefs. 1+1=2, for example. However, that in and of itself isn’t evidence that occurrent beliefs really exist.

    I suppose someone could also argue that non-occurrent beliefs don’t “exist,” but they are still implied by our psychology somehow. Isn’t there metaphysical implications to non-occurrent beliefs? This should be discussed in order to know what kind of ontology we are referring to.

  34. Jon S. says:

    Hi James,

    I agree that it would be nice to know what the metaphysical/ontological(/neurological) implications of non-occurrent beliefs are. I hope Dan tells us what brought this up to begin with soon!

    (I have a fairly strong suspicion that it grew out of Ken Walton’s aesthetics survey last term, but Ken just talks about occurrent/non-occurrent propositional attitudes without giving an account of what propositional attitudes themselves are. I think usually non-occurrent just means dispositional, but not in Ken’s world: see, e.g., the quote from Currie in section iv of http://dingo.sbs.arizona.edu/~snichols/Papers/imaginingandbelievingJAACFinal.pdf –and fn. 34 for Ken being wily.)

    But I really think our valiant poster should fill us in on the background to this particular post before we try to find an answer to your question.


  35. Ninjay says:

    If premise #3 allows for an infinite number of possibilities, all of which are considered beliefs under premise #4, wouldn’t premise #6 state that they are nonoccurrent unless they are being used/applied, and that the number of occurrent and nonoccurrent beliefs is constantly changing? For instance, if I know that 2+2=4, then the number of nonoccurrent beliefs is infinity-1. I cannot count 13579+24680 because I do not know it yet, or can I since I know that it will be true?

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