Keep supporting Zeno's, which is totally for critical thinking and is never going commercial!!!

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.

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:

(http://en.wikipedia.org/wiki/Peano_axioms)

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

Proof:

(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

From Topi Linkala in Finland:

Carl's examples were all about natural numbers.

If we take the Peano axioms of natural numbers:

(http://en.wikipedia.org/wiki/Peano_axioms)

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

Proof:

(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