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