How would logic represent this idea?

Great Debates
41

Let me try my hand in this. olanv’s argument (as I understand it, anyway) hasn’t been directly addressed in my (admittedly brief) reading of this thread, so I’ll try to address it specifically.

olanv, you seem to have a fundamental misunderstanding of what Goedel’s Incompleteness Theorem (GIT) says. Your argument, paraphrased, seems to be the following:

  1. GIT claims that all propositions in any (“strong” enough) formal system are undecidable–we are unable to determine the truth value of any statement.

  2. Since GIT itself is one of those statements, we, in fact, cannot even determine that GIT itself is true.

  3. Therefore, GIT collapses under its own undecidability.

In fact, your argument (assuming this is your intended argument), is perfectly valid.

The flaw, however, is in your first premise. You have misunderstood/misrepresented GIT.

GIT does not claim that all propositions are undecidable, only that there are some undecidable propositions. In fact, using GIT, we can conclude that any proposition falls into one of three categories:

A. Those propositions that are provably true within the formal system.

B. Those that are provably false within the formal system, and

C. Those that can neither be proven true nor proven false within the system (the undecidable propositions that GIT guarantees).

GIT only claims that there are certain propositions that are undecidable, not that, as a general rule, all propositions are undecidable. There still remain propositions that we can prove are true (such as GIT itself), as well as propositions we can prove are false.

How do we know which of the three categories a given proposition may fall in to? In general, it’s not at all easy.

In certain cases, however, it’s really quite simple. If we can construct a proof of the truth of a proposition, that’s all we need to establish that that proposition belongs in category A. GIT itself falls into this category, simply because of the fact that it has been proven. Similarly, if we can prove that a given proposition is false, we have demonstrated that it lives in category B.

Unfortunately, in general, if we’re given some arbitrary proposition, there’s no systematic way of demonstrating what type it is (i.e., which of the above categories it falls into). It can often be done (it’s been done many, many times, as evidenced simply by the sheer number of theorems throughout mathematics), but, in general, it can’t be done.

Demonstrating that a proposition falls into category C is often the most difficult to establish. A standard approach is the following. Given a proposition P:

  1. Construct a model M which does two things: a. Satisfies the axioms of the formal system, and b. Satisfies P (in other words, P is true in this model).

  2. Also, construct a model M’ which does two things: a. Satisfies the axioms of the formal system, and b. Satisfies (not P).

  3. demonstrates that P is consistent with the formal system; 2. demonstrates that (not P) is consistent with the formal system. Together, 1. and 2. demonstrate that P is undecidable in the formal system–one way to think of it, in some sense, is that the formal system is not “strong” enough to distinguish between, for example, the models M and M’, and is therefore not strong enough to determine the truth value of P.

In fact, to give a specific example, this is exactly the method used by Goedel and Cohen to prove the undecidability of the Continuum Hypothesis (CH) in Zermelo-Fraenkel-Choice (ZFC) set theory. In the 30’s (I believe), Goedel showed that CH was consistent with ZFC; in the 60’s, Cohen showed that (not CH) was consistent with ZFC. Together, these two results demonstrate the undecidability of CH in ZFC.

42

You’re absolutely correct! That’s my claim. However, my claim is also that it’s impossible to conclude undecidability without referencing the concept of undeterminability. Undecidability is a decidedly anthropomorphic sense of undeterminability. The begged question is, does undeterminability even exist as a concept that doesn’t always and instantly refute itself? Which also directly impacts the question of undecidability (what does undecidability then mean if not defined as “an accepted measure of undeterminability within the scope of thought, reason or logic?”)

This is my point. If you even suggest that undeterminability exists in some way that is not self refuting… then you have effectively opened up a logical vaccuum in the universe of logic that can arbitrarily dump rules that effect not only strong systems but even the weakest ones in a manner that refutes your claim of the existence of undeterminability. What this ultimately suggests is that undecidability as a concept is bunk. Someone’s making a mistake, somewhere.

What I’m saying, is that when playing ‘games’ with a concept like undeterminability, there is no “acceptable” measure of allowance for this concept in the assertation of truth.
Do I know exactly where someone screwed up? No. But through this reasoning, I can conclude that someone must have screwed up their logic somewhere… because per my observation, undecidability implies undeterminability and undeterminability is self refuting.

43

Well, I had thought that maybe (just maybe) I was following your argument, however, your last post has shattered that thought. I do not follow your “explanations” at all. This statement, in particular:

makes no sense to me whatsoever. What, exactly, are you attempting to say?

Several posts ago, you asked others to, “Define: Undeterminability, Non-Determinability, Adeterminability, Anti-determinability, Indeterminability”, however, it’s you that have been tossing those words around (along with others, such as undecidability) as if each of them has some distinct, distinguishing meaning. What do you mean by them?

Anyway, and perhaps more to the point, here’s a fairly simple (and informal) example. Say we have the following axioms:

  1. All boojums are snarks.
  2. Lewis is a boojum.
  3. Carroll is not a snark.
  4. Dodgson is a snark.

Now consider the following propositions:

A. Lewis is a snark.
B. Carroll is a boojum.
C. Dodgson is a boojum.

We wish to determine the truth value of each of the propositions A, B, and C.

I can prove A:

Lewis is a boojum. (Axiom 2)
All boojums are snarks. (Axiom 1)

Therefore, Lewis is a snark.

I can disprove B:

Suppose Carroll is a boojum.
All boojums are snarks. (Axiom 1)
Therefore, Carroll is a snark.
However, Carroll is not a snark. (Axiom 3)
Contradiction, hence our original supposition is false, and therefore Carroll is not a boojum.

So A is True, B is False. What about C?

I’ll leave C as an exercise for you, olanv. Surely, according to you, C must be True or False within our system (since Undecidability doesn’t seem to be an option according to you).

Which is it? True or False? Prove your answer.

44

The question is not begged. If you are going to hurl stones at the towers of logic you migt want to at least acquaint yourself with the terminology of the field. It might mitigate, somewhat, the impression that you are entirely lacking in the erudition necessary to develop your arguments in a rational and compelling manner.

Setting aside issues of word choice, I will go ahead and answer your question. Yes. Undeterminablilty (or indeterminism, if you prefer) does exist as a concept that does not “always and instantly refute itself”.

In fact, several of us have given specific examples of systems containing indeterminism. Thus far, you have exhibited an impressive prediliction for avoiding comment upon even the simplest of such systems. But let’s give you another chance, eh? Would you please demonstrate how the inability of my trivial little system above to assign a truth value to R “instantly refutes itself”.

I shall await the results of this test of intellectual character with great anticipation.