On Philosophy

June 25, 2007

I Don’t Believe That 25 – 11 = 14

Filed under: Mind — Peter @ 12:00 am

I am, however, willing to claim that it is true*. How can this be? Surely if I am willing to assent that something is the case than that is a sign that I believe it. Traditionally this has been one way of defining what someone believes. But I think that is a bad definition, because it obscures an important distinction between statements such as “25 – 11 = 14” and “my name is Peter”, both of which I would assent to. And this distinction is important because it can allow a straightforward treatment of some otherwise problematic cases.

Let me just start by making the distinctions that I think are important. Statements such as “my name is Peter” I call beliefs. A belief, thus understood, is a proposition that is directly integrated into the person’s mental operations. This doesn’t mean that the belief is conscious, but rather that it is immediately available when needed. If someone asks me “what is your name?” I will immediately be able to respond. And beliefs, by hypothesis, thus also guide how we think; there is no chance that I will naturally entertain the idea that my name is not Peter unless I make a special effort. In contrast I call statements such as “25 – 11 = 14” derived beliefs. A derived belief, so defined, is a proposition that the person will conclude is true (has the disposition to conclude is true) on the basis of propositions that they believe or other derived beliefs (in the case of 25 – 11 it follows from a plain belief I have about the algorithm that subtraction operates by). Unlike plain beliefs, derived beliefs are not ever-present in our minds. It is natural to say that plain beliefs are always in mind, although usually unconscious, but derived beliefs are not usually present in any way, except when they are consciously derived from plain beliefs. Thus it is quite possible to entertain the belief that 25 – 11 is some value other than 14 without some conscious effort, and when asked the result of 25 – 11 is not immediately present.** And of course this naturally implies a third and forth category, un-derived beliefs and mis-derived beliefs, which are, respectively, propositions that logically follow from a subject’s beliefs or derived beliefs but which the subject isn’t able to actually derive from them because of limits on their intellectual capacities, and, similarly, mis-derived beliefs are propositions that do not logically follow from the subject’s beliefs and derived beliefs, but which the subject consistently believes due to limits on their intellectual capacities. Mis-derived beliefs act much like derived beliefs, psychologically, and so I won’t mention them for the rest of the post, but you can assume that much of what I have to say about derived beliefs includes them as well.

The primary purpose in making all these distinctions is to understand how beliefs, in all their flavors, work psychologically, with an eye towards the nature of contradictory beliefs and logical thinking. Now it seems natural to assert that no one (at least no human) can believe both a statement and its negation simultaneously. Simply try it; the best you can do is entertain the negation of something you believe in, not believe it as well. But accepting that psychological fact raises an apparent dilemma, because people can believe propositions that conflict with each other indirectly, by implying facts that that are the negation of each other (“thou shall not kill” and “the death penalty can be a just punishment” imply “you should not kill prisoner X” and “you should kill prisoner X”). And if we accept the additional premise that people believe the logical consequences of their beliefs (actually we should say that people believe the consequences of their beliefs and whatever derivation system they believe) this contradicts the claim that people cannot have contradictory beliefs. Here we can apply the distinction I have developed above to resolve the apparent contradiction. We can admit that plain beliefs cannot contradict each other while allowing derived beliefs to potentially contradict plain beliefs. Remember, derived beliefs aren’t always present like plain beliefs are. Thus there is no psychological tension between a plain belief and a derived belief, so long as that derived belief isn’t currently being entertained.

Of course this leads us to consider what exactly happens when a derived belief contradicts a plain belief. For the same reasons that we are unable to have two plain beliefs that are negations of each other it is equally impossible to entertain a beliefs of any kind that contradict each other. In a sense then we only partially solved the puzzle above, we solved it in the case where the contradictory derived beliefs were “dormant”, i.e. not currently in mind as propositions that should be believed. The obvious conclusion is that one of the plain beliefs that the contradictory derived belief follows from must be rejected and replaced with its negation in order to resolve the tension. I call this the optimistic theory about belief revision. The optimistic theory implies that we can change people’s minds, and lead them to agree with us, by showing that their plain belief that “X” contradicts their other plain beliefs (perhaps several of them) in some way, and thus lead them to believe “~X” instead. That would be great, but it happens only rarely. What the optimistic theory neglects are that there are other ways to revise our beliefs in light of contradiction. One way is to reject instead the derivation process that led to the contradictory derived belief. To return to the parenthetical example I used above often the tension between “thou shall not kill” and “the death penalty can be a just punishment” is resolved by denying that “thou shall not kill” is a principle that applies in all cases, despite the fact that its wording implies that it does. That is perhaps the most common way in which a tension between derived beliefs is resolved. If the person doesn’t revise their beliefs in that way then they probably revise them by slightly redefining them. This very post is an example of that method in action. Here we are resolving a contradiction between several intuitively true theses about belief by substituting for our old conception of belief a very similar but more nuanced conception that allows us to dodge the apparent contradiction. Another example of this method of belief revision would be someone who resolved the contradiction between “thou shall not kill” and “the death penalty can be a just punishment” by saying that “kill” in the first case implies malicious intent, and that no such intent need be present while carrying out the death penalty (this is a response I have a certain amount of respect for). So by pointing out the contradiction between “X” and their other beliefs we don’t motivate them to accept “~X”, but rather “Y”, a proposition very very similar to “X” (which in terms of trying to convince someone to change their mind isn’t really any progress at all). So the optimistic theory of belief revision is overly optimistic because it fails to take into account the many other ways beliefs can be revised in the light of contradiction that don’t involve substantial changes in what the person believes.

As usual I have strayed from my intended topic, which was to show how the belief/derived belief distinction could be explanatorily useful. I suppose I will leave further applications as an exercise for the reader.

* Ignoring, for the rest of this post, that truth means something different in a mathematical and non-mathematical context, as making that distinction would add unnecessary complications.

** Sad fact: for about a minute the title of this post was “I Don’t Believe That 25 – 11 = 13” while I was writing it, until I realized that that claim, while true, didn’t quite have the attention grabbing appeal that I wanted it to.

Terminology:
Belief = plain belief = basic belief: always present, usually unconscious, cannot contradict each other
Derived belief: occasionally present consciously, can contradict each other, but not plain beliefs
Un-derived belief: never present, consciously or unconsciously
Mis-derived belief: occasionally present consciously, can contradict each other (and derived beliefs), but not plain beliefs

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8 Comments

  1. I’m a bit unnerved by the idea of a belief that is “always present, usually unconscious”. How would we tell the difference between an unconscious basic belief and a non-basic belief that is very quickly derived?

    I wonder whether this is really a difference of kind, or just of degree. I agree that ‘My name is Peter’ is immediately available (for assent, input into decision-making, or whatever), while ’25-11=14′ is not immediately available but can be derived after a moment’s consideration. But maybe these are just different points along a continuum of ‘availability’? To continue with the maths examples, what about ’10-9=1′? I also have to derive that, but I can do it much more quickly than the previous sum. What about ‘2+2=4’? That seems to be pretty much immediately available, but is it as available as ‘My name is Peter’?

    Perhaps ‘My name is Peter’ is just very quickly derived because of my familiarity with it. If I hadn’t heard or used my name for ten years, perhaps it would take me ten seconds of mental legwork to derive it again. If I hadn’t heard it for three years, perhaps it would take three seconds. And so on. There seems to be a continuum of accessibility of beliefs, rather than a sharp distinction between plain and derived beliefs.

    By the way, it’s immediately obvious to me, without any calculation, that ’25-11=14′ and ’25-11=13′ can’t both be true, but it may not be immediately obvious which of them should be rejected. So it seems that I can accept or reject the conjunction of two non-plain beliefs without having derived or considered either of them.

    Comment by Toby — June 25, 2007 @ 2:02 am

  2. Yes, it is probably possible to establish a more complicated continum of belief. But there is no need for it explanitorily (it doesn’t add anything useful), so it’s best not to deal with it unless absolutely necessary.

    As for your last point, they aren’t direct negations so the belief that only one of them can be true must be derived from a belief that artihmatical sums can only have one result. And in any case “I can accept or reject the conjunction of two non-plain beliefs without having derived or considered either of them” is a point unconnected to anything I have claimed; all I discuss here is what happens when plain beliefs can be shown to imply contradictions via derived beliefs, other contradictions are undiscussed.

    Comment by Peter — June 25, 2007 @ 2:09 am

  3. Yes, it is probably possible to establish a more complicated continum of belief. But there is no need for it explanitorily (it doesn’t add anything useful), so it’s best not to deal with it unless absolutely necessary.

    I’m not sure I agree with this. You’ve proposed a schema in which there are (at least) two distinct types of belief: plain ones and derived ones. My suggestion is that there’s only one type of belief really, and the differences you highlight are simply due to how quickly accessible different beliefs are to the conscious mind, not to a fundamental difference. I think one category is simpler than two categories, and I think my schema is also closer to intuition. If I’m right, then we need only resort to your schema if there is something we can’t explain with mine, and I don’t think the dilemma you raise in your post fits the bill.

    And in any case “I can accept or reject the conjunction of two non-plain beliefs without having derived or considered either of them” is a point unconnected to anything I have claimed;

    Agreed. It was a parenthetical observation, not directed at anything in particular in your post.

    Comment by Toby — June 25, 2007 @ 3:06 am

  4. That’s like arguing that 1 and 2 are the same number because there is a continuous spectrum of values between them.

    Comment by Peter — June 25, 2007 @ 3:40 am

  5. That’s like arguing that 1 and 2 are the same number because there is a continuous spectrum of values between them.

    How so? I agree with you that the two descriptions of beliefs that you present are different. I simply disagree with you on the nature of the difference. You think it is a difference of kind; I think it is a difference of degree.

    Comment by Toby — June 25, 2007 @ 4:42 am

  6. Specifically I think that your argument that there is only one kind of belief is commiting the continuum fallacy http://en.wikipedia.org/wiki/Continuum_fallacy .

    Secondly: Surely you admit that derived beliefs exist (they must, the question is whether plain beliefs can be collapsed into them). Derived beliefs must come from some other beliefs, by defintion. If there were only derived beliefs then belief could never get started and we would believe nothing. We believe some things. ∴ there exist beliefs that are not some kind of derived belief.

    Comment by Peter — June 25, 2007 @ 4:50 am

  7. Specifically I think that your argument that there is only one kind of belief is commiting the continuum fallacy

    Let me say again that I do not think that the two sorts of belief you describe are the same (that is, I don’t think that there are no differences between them). I simply disagree on the nature of the difference. I don’t think it is explanatorily useful to draw a distinction and then characterise the beliefs on either side as being of different kinds.

    An analogy: If I said that a human baby and a human adult are the same because there’s a continuum of different stages of development between them, then I’d be committing the continuum fallacy. But I wouldn’t be committing the continuum fallacy if I simply pointed out that a human baby and a human adult were of the same species despite the differences in their respective levels of development. Their levels of development are different in degree, not kind.

    Comment by Toby — June 25, 2007 @ 5:47 am

  8. Then you recognize that we can validly explain circumstances by appealing to the child-adult distinction (he couldn’t go on the ride or to the movies because he was a child). Similiarly the derived / plain distinction may play a useful explanatory role, as it does here.

    Comment by Peter — June 25, 2007 @ 12:49 pm


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