Thursday, October 8, 2015

And Now Something Completely Different

Here is a little paper I wrote, inspired by a conversation with Robert Audi at the Society of Christian Philosophers conference at Azusa Pacific in spring 2015.  Of course, Audi is not to be blamed for my ideas.

Beliefs are not Conjunctive

I.          One of the easiest and most obvious inference rules in logic is the rule of conjunction.  If A is true and B is true, then A and B is true.  We make students create the truth table for this inference mostly so they can practice building truth tables.  No one doubts that propositions are conjunctive.
1.     A
2.     B             /  therefore A+B
3.     A+B          1, 2, conjunction

II.        It may seem counterintuitive, but the rule of conjunction does not apply to beliefs.  It is axiomatic that if a person believes something, she believes that it is true.  At first guess, we might think that if two beliefs are held to be true, the conjunction of those beliefs must also be believed.  But this is not so. 

III.       The best way to prove this is to assume the opposite and see what happens.  We will assume that beliefs are conjunctive.  On this assumption, if a person considers proposition D and finds that she believes it, and then considers proposition E and finds that she believes it, then she should believe a new proposition: D + E.
            Let us imagine that our subject considers all her beliefs sequentially.  We will let B1 stand for her first belief, B2 for her second, and so on.  After considering B1 and B2, she should find that she believes B1 + B2.  Soon after, she should also discover that she also believes B1 + B2 + B3.
            Now, human beings hold an indefinitely large number of beliefs, but the number of beliefs a person holds may be finite.  Let’s assume that it is.  (If people hold an infinite number of beliefs, my overall argument still holds.)  Given enough time, our subject should find that she believes the following proposition, which we will call proposition omega:
            Bω = {B1, B2, B3, B4, …Bω} where Bω is the last of her beliefs. 
            It will interest some people that Bω is one of the beliefs that our subject believes; that is, the subject’s class of beliefs is self-referential.  The argument does not depend on this fact.
            The problem is this.  No reflective person believes Bω.  To the contrary: it is safe to say that all reflective people believe the opposite, i.e. they believe  ~Bω.  To see that this is so, consider the meaning of ~ Bω in ordinary English: "It is not true that all of my beliefs are true," which is the same as saying, “At least one of my beliefs is false.”

IV.  All of us know that among our indefinitely large class of beliefs, there are almost certainly some that are wrong.  If it were possible to consider individually each one of our beliefs (neuropsychologically speaking, this may be impossible) we would find that we think each one is true.  After all, that’s what it means to believe something; you think that it is true.  And yet, we do not believe that all our beliefs are true.  We have very great confidence that at least one of our beliefs is false.  But if conjunction rules over beliefs, we ought to believe Bω.  The fact that we do not believe Bω shows that conjunction does not rule over our beliefs.

           Now--what possible significance can this little essay have for readers of Story and Meaning?

            I'm not entirely sure.  But here's one thought.  Human beings, though we like to praise ourselves as being rational creatures, don't obey logic.  Our minds are complex enough that we are unable to simultaneously consider all the things we believe.  Considered one by one, a person will find that she believes an indefinitely large number of things.  
           [Actually, we all believe an infinite number of things; we just don't take time to consider them.  Let's consider only my beliefs about basic arithmetic.  I believe that 1 + 1 =2 and that 1 + 2 = 3 and that 1+ 3 = 4 and that 1 + 4 = 5, and so on ad infinitum.]
           Someone might try to defuse my little argument by distinguishing "actual" beliefs from "potential" beliefs.  Perhaps the rule of conjunction should only be applied to the beliefs I am actually holding before my attentive mind right now.  After all, it's pretty rare for anyone to actively believe more than a few things at a time.  If I am currently believing only three things, A, B, and C, then it seems plausible that I could also believe A+B+C.  The reason reflective people believe ~ Bω is that they know Bω contains an enormous number of beliefs.
          But that won't do.  We often criticize other people's statements because they conflict with things those people said at other times.  We honor consistency in others and in ourselves.  It naturally produces discomfort when someone discovers that what he believes now is inconsistent with what he believed yesterday.
           Human beings are not logical sometimes.  That is, we are inconsistent.  But we want to live lives of faithfulness and integrity.  When we discover inconsistencies in ourselves--in our beliefs or in the beliefs implied by our actions--we want to repair them.  Of course, sometimes recognizing an internal contradiction is so uncomfortable that we hide from it.
           And now we see the connection to stories.  Human beings are narrative creatures.  We find our identity in the stories we tell ourselves.


  1. I followed you, Phil, down to the last paragraph. How does the rest of the essay demonstrate that we tell ourselves stories? Is it to hide from our inconsistencies? I had thought our stories are mostly to trace (or imagine) strands of order in what otherwise seem to be strings of random events.

  2. You're right. There's a leap there. It's not like the only alternative to being a logic machine is to be a narrator.

    I think we ARE narrative beings, tying the events in our lives together via the stories we tell. We get more insight into character through good storytelling than through Kantian-style imperatives.