Thanks for the many replies to Challenge #66! I include the original problem, followed by a representative sample of thoughtful responses which are published below.
Keep supporting Zeno's, which is totally for critical thinking and is never going commercial!!!
Ron Barnette

Believers at the Coffeehouse?

Philosophy night at Zeno's: As Maggie and Charles were discussing with some Coffeehouse patrons the notion of belief , Charles asserted that to believe something, one must at least assent to some proposition or to think that something so characterized is true. (It was observed around the Coffeehouse that he's an avid reader of the philosopher Quine.) "True or false as the belief might actually be, one must at least entertain the thought that such-and-such is true," he maintained confidently, as he added, "Otherwise, one could be said to believe something that one has never even thought about, which is crazy!"  Maggie nodded, then reminded Charles somewhat impatiently, "Yes, of course, but that's obvious my dear...please offer something more profound about the nature of belief, reasons and grounds for belief, how belief is different from knowledge, etc. Please!" Charles was a bit embarrassed.

"Hold on," interrupted Carl, a faithful patron.  "Charles and Maggie, do you realize that you each have fewer eyebrows than you have hairs on your head, and that if you each subtracted the number of your own eyes from the number of your own fingers, this would equal eight?" 

"Come on, Carl, what does this have to do with our philosophical discussion!," exclaimed Charles. "Of course we know this, but what's your point," added frustrated Maggie with Charles' nodding approval.

"Simply this," Carl added, "I dare say that you have never even thought about these facts before, have you? And yet you state emphatically that you know these and thus believe these facts to be true. My mentioning them didn't produce your beliefs, nor did you undergo any change of beliefs when you heard me. Thus, I conclude that you must have held these beliefs beforehand without ever having entertained such thoughts, no?" 

"No, of course we never thought about these precise facts you describe, but....." 

BUT WHAT, Charles and Maggie? Is Carl's conclusion correct? Is Carl's reasoning flawed? Can we believe matters that have never even occurred to us?

 Here are samples of the many replies to this Zeno's Challenge. Many thoughts of appreciation are extended to those who continue to support and drop by Zeno's!!!  Keep up your support as you submit your replies and discussions! Thank you much...Ron Barnette

From Mark Young in Canada:

So how does Carl know that the beliefs aren't new?  I, for one, had to re-read each claim before I accepted it was true.  If I had believed them already, I would have agreed to them right away, wouldn't I? 

Let's try another one (I'll call it P): 

  Do you realize that the number of days in the week multiplied
  by the number of planets in the solar system is six more than
  the number of stars on the American flag? 

A little bit of thot -- and remembering about poor Pluto's demotion -- will lead you to conclude that P is true.  But did you already believe it?  You already believed that there are seven days in the week, that there are eight planets, and that there are fifty stars on the US flag.
Did you already believe that eight times seven is fifty-six, and that fifty-six is six more than fifty?  Perhaps you did.  All those beliefs together entail P.  Does that mean that you already believed P? 

I find P hard to countenance as something I already believed -- definitely harder than the examples Carl gave.  But if each of us believes everything that's entailed by our own beliefs, then each of us believes P.  In fact, each of us believes some things that we probably wouldn't even be able to identify as our own beliefs.  For example, anyone who knows how to multiply and what a square root is would already believe that the square root of 7739045 is
2781.91390952344174082886918409632 (more or less :-).  Seems highly counter-intuitive to me! 

But Carl being wrong doesn't mean that Charles is right, let alone obviously so.  One of the earlier challenges (#54) raised a question about infinite beliefs by observing (as Hintikka did) that, if you believe something, then you believe that you believe it. 

  Bap > BaBap

"The seemingly obvious idea is that one can't believe that something is the case without at least believing that one believes it. 'Charles believes that Maggie is at the store, but Charles doesn't believe that he believes this' does appear to be plainly false, to be sure, if not bordering on an inconsistency."

Yet that is contrary to Charles' claim, for we cannot consciously entertain every one of the infinite chain of "beliefs" that results from this axiom (BaBap > BaBaBap > BaBaBaBap > ...).  If Hintikka's axiom above is true, then Charles' (Quine's?) is false, and vice versa. 

But the fact is that there is no generally accepted set of necessary and sufficient conditions for saying that someone believes P (or any other proposition).  We each have belief-ascription practices, and it's not necessarily the case that all our practices can be gathered under a single theoretical umbrella.  (It's not even necessarily the case that all one person's belief-ascription practices can be explained in a coherent way.)  Some may be happy to ascribe someone a belief they've never entertained (because it follows in a relevant way from some beliefs they have entertained); others not so much.  I'm in the not-so-much crowd.

Since there's no fixed definition of what a belief is, there are few axioms that apply to all the variant (and largely implicit) definitions out there.  Hintikka's axiom of iterated belief and Charles' axiom of entertained belief both fall afoul of someone's conception of belief -- each other's, for example.  That's just something we have to put up with
-- unless we want to stay in the realm of pure math. 

...mark young

From Harry Evans:

I believe this paradox to be one of language more than one of philosophy. We consider the idea of belief to be, in this case, very general: one of simply understanding what is apparently a fact; i.e understanding that there are more hairs on Charles' head than there are over his eyes. If we are to take that as the definition of belief, the sole definition of it, then I would postulate that the process of understanding (whether this understanding be true or false) a concept can occur in a split second. Belief in this context is an active state which cannot occur until the thought is thought.

However, the far more interesting way of looking at this paradox is to assume that our language is limited.
To illustrate, let's take a look at the Ancient Greek for love: eros. Or was it philia, or storge.. or agape? The answer of course is all four, each representing a different aspect of what we call 'love'. In a more modern and cultural example, we see the difference between 'I love you' and 'I love ya'; both being types of love in their own right, and both being entirely different.

The definition of belief that is alluded to in the problem is not clear, and it is necessary for analysis of this. The 'belief' that we consider to be absurd to occur whilst we have never entertained the thought of it is not the same 'belief' that occurs when we think of how many hairs are in our eyebrows compared to our head.The second of these 'belief's is in fact something very different from what the loose attribution of the word 'belief' suggests to us. It is a belief of some kind, that is very difficult to doubt, but it is not a belief in the statement 'I have more hairs in my eyebrows than on my head'. It is, instead, a belief in our own comparative skills, and our own senses: and that is a belief that has itself come from a general rule which we build from experiences in the past.

What is hard to believe is that any singular belief can exist without first contemplating it. Our experiences build from the past our beliefs in the future: these may turn out to be untrue, but are generally rules to live by, until they are proven to be false. Beliefs are more fundamental than simple statements such as 'the table has as many legs as a spider split down the middle'. The beliefs in that statement are the belief in our senses (or, more accurately, what Bertrand Russell called 'sense-data') and the belief in the concepts of splitting and a middle (both of which are abstract and not observable through the senses). Finally, there is the belief that what you believe is true (which could be argued as a paradox in itself, but that's a problem for another day).

In conclusion, the problem is not with the nature of belief, but the nature of what you define belief to be. Belief must be broken down to the lowest denominator before you can truly understand what's going on in any given statement. A statement does not have to have been previously thought about to be assumed, but it is not the statement which is believed, it is the principles that underlie it.

Harry Evans

From Topi Linkala in Finland:

Carl's examples were all about natural numbers.

If we take the Peano axioms of natural numbers:

1. For every natural number x, x = x.
2. For all natural numbers x and y, if x = y, then y = x.
3. For all natural numbers x, y and z, if x = y and y = z, then
   x = z.
4. For all a and b, if a is a natural number and a = b, then b is
   also a natural number.
5. 0 is a natural number.
6. For every natural number n, S(n) is a natural number.
7. For every natural number n, S(n) != 0.
8. For all natural numbers m and n, if S(m) = S(n), then m = n.
9. If K is a set such that:
      0 is in K, and
      for every natural number n, if n is in K, then S(n) is in K,
      then K contains every natural number.

And then define notation:

S(0)=1, S(1)=2, S(2)=3, S(3)=4, S(4)=5, S(5)=6, S(6)=7, S(7)=8, S(8)=9, S(9)=10... (This notation schema can be fully described when addition and multiplication is defined for natural numbers, but I'm leaving it as an excersise.)

addition (+):

A1: a+0 = a
A2: S(a+b) = a+S(b)

This leads to the conclusion that:

S(a) = (A1) S(a+0) = (A2) a+S(0) = a+1

Using the 9th axiom it can be proved that a+b = b+a, (a+b)+c = a+(b+c) and a+b = a+c => b = c for all natural numbers, but once again I leave all but one as an excersise.

S1: a+(b+c) = (a+b)+c


(a+b)+0 = (A1) a+b = (A1) a+(b+0)

If I: (a+b)+c = a+(b+c) then
(a+b)+S(c) = (A2) S((a+b)+c) = (I) S(a+(b+c)) = (A2) a+S(b+c) =
(A2) a+(b+S(c))

multiplication (*):

M1: a*0 = 0
M2: a*S(b) = a*b+a

Using the 9th axiom it can be proved that a*b = b*a, (a*b)*c = a*(b*c), a*(b+c) = a*b+a*c and ab = ac & a != 0 => b = c for all natural numbers. Proving is left as an excersise.

Order (<=):

O1: a <= b if there exist a natural number c so that a+c = b.

With this we can define other order relations:

O2: a < b iff a<=b & a != b
O3: a >= b iff b <= a
O4: a > b iff b < a

Substraction (-):

S1: c = a-b iff a > b & a = b+c

With these axioms and definitions I can believe in any addition, substraction, multiplication or ordering clause even if I've never thought those particular numbers that must exist for the calculation or comparasion. I can even belive in any natural number even if I havent thought of it ever.

We can expand this to any conseptual system where one can believe in any actual instance of those concepts without any prior knowledge of that instance.

Topi Linkala