Questioner :

When you leave points unjustified in your argument, aren't you admitting that you haven't thought things out as well as you should have? And when you leave terms undefined, aren't you admitting that you don't even have a precise notion of what it is that you're trying to say?

I've admitted nothing of the sort. Would you agree, that at any one time, only finitely many statements may have been made in a given argument, or even by the person offering the argument, during the entirety of his past? Or, for that matter, in the entire description of his philosophy?

Questioner :

What does this have to do with ...?

Answerer :

Bear with me, please, I am going somewhere with this.

Questioner :

Yes, of course I agree that only finitely many comments would have been made by him. Even your little spiels don't go on forever. They just seem to.

Do you know what a partial ordering of a set is?

Questioner :

Well, yes. I took Math classes, too. We take a relation "<<" on a set, such that

1. for no member "a" of the set, do we have a << a
2. if a << b and b << c, then a << c.

Then the relation "_<<", where

a _<< b if and only

1. if a << b ... or
2. a and b are the same element of the set

is a partial ordering. "<<" would be considered a strict partial ordering. Is there a point in this?

Answerer :

And would you agree that any finite partially ordered set must have at least one member "a" such that there is no member "b" of the set such that b << a? Because otherwise, we could form an infinite string of distinct elements of the set, by first

1. Listing all of the elements of the set, in some order.



2. Taking the first element of the list, in the order given in step one, as the first element of the string.



3. At each step, given the n-th element of the set in the string, which we'll denote s(n), choose s(n+1) to be the first element "b" of the set, from the ordering in step one, such that

   b << s(n)

We could never have s(m) = s(n) for any m > n. We know that if m > n, then s(m) << s(n). If this were not so, then there would be a first positive whole number m1 greater than n, such that s(m1) << s(n) failed to hold. (This we see, as follows. Suppose that there there was at least one such number m2 greater than n, such that s(m2) << s(n) failed to hold. Then, if we applied the procedure

1. let c = m2, m3 = m2,



2. if s(c) << s(n) fails to hold, then replace the current value of m3 with c,



3. reduce the value of c, by one,



4. if c > n, then return to step ii, if not, move on to step v,


5. let m1 = m3.

Our procedure must terminate in fewer than m2 steps, so we know that it will produce a value for m1, in step v (which will be reached). By the technique used, we know that m1 will be a member of the set of positive whole numbers c for which the relationship s(c) << s(n) fails to hold, and that there can be no member of that set smaller than m1. (If there was, it would have replaced m1 in some repetition of step ii).

By construction, we know that

s(m1) << s(m1-1) and s(m1-1) << s(n), so s(m1) << s(n),

contrary to assumption. This contradiction indicates that our assumption had to be mistaken.

But to get back to our prior point, if s(m) = s(n), then since s(m1) << s(n), we would have that s(n) << s(n), contrary to the definition of a strict partial ordering.

Therefore, if there is no element "a" of the set, such that for no element "b" of the set,
is b << a, then we have an infinite chain of distinct elements, taken from a finite set.

But, this is a contradiction, because there aren't infinitely many distinct elements to be found in the set.

Questioner :

And how does this justify your failure to support your points?

You've asked why I couldn't simply prove the truth of all of my points. But, as a matter of first principles, this is exactly what nobody can ever do without resorting to circular argumentation, and hoping that nobody notices what it is, that one has done.

In order to apply the above argument to our situation, here, we could begin by ordering the assertions made in our system of arguments (our philosophical system) in alphabetical order, with blank spaces following the first character treated as a character preceding "a" in alphabetical order, and treating the various punctuation marks as other "letters" each occupying some place in the roster of letters. Say, for example, by having "z" followed by the period, comma, semicolon, colon, left panathesis, right paranthesis, quote mark, question mark, exclamation mark, and dash in that order, with other symbols spelled out, for example.

Having thus established a linear order on our set of statements, we could then construct an application of the preceding argument.

We could then place a strict partial order on the set of statements, by saying that statement "A" precedes statement "B" (A << B) if statement "A" is directly used in the demonstration of statement "B", or precedes a statement, which precedes "B". As we saw above, given such a partial ordering, we must have a first statement, left unpreceded. An element "a", for which there is no element "b" for which b << a.

That is to say, a first principle, left unproved.

Questioner :

How do you know that "precession", to use your term, will be a strict partial ordering? (Precession being the relation in question, when you spoke of statement A preceding statement B, in your argument).

This follows from the method of construction, IF we have abstained from circular argumentation, in the system of arguments under examination. "Transitivity" (the property holding that if a << b and b << c, then a << c) is immediate from the definition of precession. In order to have A preceding itself, we would have to have a chain of arguments, in which A is used to prove a statement B, which, as we follow the chain of arguments back, is used to justify something, which is used to justify something, which ... after a while, leads us back to statement A. If such a chain exists, then circular reasoning is present in the system of arguments under examination.

There's no escaping it. You can't prove everything, any more than you can define everything. (The same argument, above, would apply, if we take the relation of precession to be defined, by saying that term A precedes term B, if term A appears in the definition of term B. Thus, the observation that one can't define everything).

Or, to use the traditional terms, we can't dispense with the necessity of having axioms (statements left unproved) and primitive concepts (terms left undefined, which are used to define all other terms).

Questioner :

A little overkill, don't you think?

The less that is given, the more that has to be proved. Don't ask fundamental questions, if you don't have the patience to hear the unavoidably long answers.

Your options :

• continue
• go straight to the next argument