What is the O'ist view on the Liar's Paradox?

[Tom Rexton]

Proposition A: "This statement is false."

There are other variations of the statement above, but essentiatlly they are all the same. The conclusion that follows is that if proposition A is true, then it is false. But if it is false, then it is true.

But proposition A cannot be both true and false.

According to the Law of Bivalence a statement is either true or false. According to the Law of the Excluded Middle a proposition or its negation is true. And according to the Law of Non-contradiction a proposition and its negation cannot be true at the same time and in the same respect. But proposition A violates all three laws.

How does Objectivism address this paradox?

[DavidV]

"Invisible pink elephants are very smart."

Is that true or false? Neither -- only statements that about something that exists can be true or false.

[Fred]

I'm glad you brought this up, Tom. The liar's paradox warrants special attention.

A proposition cannot be both true and false at the same time, because a contradiction cannot exist. But, as David's example demonstrates, there's such a thing as a bad proposition.

The proposition in question is invalid because it is detatched from reality. To what does "this statement" refer?

Coming to grips with self-reference is crucial for my research into the foundations of mathematics and computer science. For instance, the liar's paradox is the basis for Goedel's incompleteness theorem.

[Godless Capitalist]

Basically the laws you cite only apply to meaningful statements, which this paradox isn't.

Fred, can you explain more about how this applies to Godel's Incompleteness Theorem?

[MinorityOfOne]

Harry Binswanger talks about this in one of his lectures. I don't remember which one. Basically, the problem with the sentence is that it's purely self-referential. It doesn't refer to a fact of reality, it just kind of pretends it's doing so by taking over the grammatical structure of a referring sentence. Since it's purely self-referential, the issue of truth or falsehood can't arise.

(So, actually, this is kind of a fourth category, aside from truth, falsehood, and arbitrariness. But nobody except philosopers would worry about it, so it's not too important, except insofar as it's useful to understand so they don't confuse you.)

[MinorityOfOne]

p.s., the law of bivalence is wrong too. It assumes that the arbitrary can be subsumed by either truth or falsehood, but it can't.

[msb]

The problem is that you can't solely refer to your refering (apart from refering to something). It becomes more clear when you think of the simplified version:

"This sentence is true."

Exact same problem. It's saying about its saying; it says nothing. I think Matt is right that it's different from the arbitrary, but it has the same effect of being completely cognitively empty.

EDIT: The lecture where HB discusses this is "The Metaphysics of Consciousness." He's got a good name for it to, something like the Fallacy of Pure Self-reference. He illuminatingly shows how the Kantian theory of perception commits this fallacy by assuming that man perceives his form of perception, instead of perceiving objects.

[MinorityOfOne]

That's a good way to put it. You can't refer to your referring, because then what are you referring to? If you think through it, it ends up being an infinite regress. You're referring to your referring to your referring to your referring... and so on, with no connection to anything in reality.

One of my professors wore a shirt to class like that. The front said: "The sentence on the back of this shirt is true." The back said: "The sentence on the front of this shirt is false." The shirt was from the American Philosophical Association's yearly conference. (And no, the reference to the shirt doesn't save it. :-) )

[Tom Rexton]

OK, let me see if I understand what you guys mean: a proposition can only be true or false if and only if it makes a statement about the facts of reality and can be verified or falsified by reference to the same?

I never thought it would be that simple. But then again I never really gave it as much thought as I should.

[Godless Capitalist]

agreed

[Capitalism Forever]

can you explain more about how this applies to Godel's Incompleteness Theorem?

The Incompleteness Theorem says that, given a sufficiently sophisticated, consistent set of axioms and a valid set of deduction rules, there will be at least one proposition which cannot be deduced from the axioms, but is still an inevitable consequence of the axioms; namely, a proposition which effectively says, "This proposition cannot be deduced from the axioms." It cannot be false (you cannot deduce any false propositions from consistent axioms using valid deduction rules), therefore it is true.

Self-referentiality is a key here, as well as being able to refer to the set of propositions that can be deduced from the axioms. Both are made possible by the "sophistication" of the axioms. As far as I know, no system of axioms can be sophisticated enough to enable referring the set of all propositions which are inevitable consequences of the axioms, therefore it isn't possible to have a proposition that says "This proposition is false."

[Fred]

CF's explanation of Goedel's theorem is good. More concisely, the incompleteness theorem (as strengthened by Rosser) states that any formal system including first-order arithmetic is either inconsistent or incomplete.

CF is also correct to point out that the Goedel sentence does not actually say "I am false," as in the liar's paradox, but rather: "I am not a theorem of the present formal system." Or: "I am not a consequence of the axioms and rules of inference you have chosen."

Because of the clever machinery he introduces to accomplish the self-reference, Goedel seems to sidestep our objections to the liar's paradox.

I studied this theorem two years ago as the culmination of a course in mathematical logic -- before I was familiar with the Objectivist epistemology. Due to its importance for the theory of computation, I plan to revisit it in the near future. (I am currently scrutinizing a closely related topic, the undecidability of the halting problem.)

Like Heisenberg's "uncertainty principle," Goedel's infamous theorem may well be a true statement about reality which is merely widely misinterpreted due to bad philosophy.

[Capitalism Forever]

Goedel's infamous theorem may well be a true statement about reality which is merely widely misinterpreted due to bad philosophy.

I think that's pretty much the case. The skeptics jump on it as an opportunity to prove their point, when in fact it has a rather limited application. Essentially, it says "you can't prove everything that's true using this specific method," but the skeptics act as if it said, "you can't prove anything."

[NIJamesHughes]

Some say that Goedel's theorum disproves Ojectivism as a complete sysytem, as Dr. Peikoff states. Does anyone know anything about that?

[Capitalism Forever]

Some say that Goedel's theorum disproves Ojectivism as a complete sysytem, as Dr. Peikoff states. Does anyone know anything about that?

Skeptics use Godel's theorem to attack Objectivism in two ways:

• They claim that the theorem invalidates logic; therefore, they say, Objectivism is wrong in advocating reason as the means of achieving one's goals.
  The truth is, of course, that Godel's theorem doesn't invalidate any method of reasoning at all; it just shows a limitation of the deductive method of reasoning (namely, that you can't deductively enumerate all the logical consequences of a set of axioms that are similar to the axioms of number theory--only most of them). In fact, Godel used logic quite extensively to prove his theorem!
  
  
• They point to the axioms of Objectivism, postulate that Godel's theorem applies to those axioms, and declare that this invalidates all the conclusions that Objectivism reaches from the axioms. Again, they base their "argument" on the frivolous idea that Godel's theorem invalidates everything it comes into contact with.
  

[NIJamesHughes]

Ayn Rand has said that all knowledge is hiearchial, and that the three axioms of existence are the fundamentals of all knowledge. Does it follow then that all knowledge can be deduced to the axiom of existence?

wouldn't contradict the assumption that you can't deduce all the ideas in a system to the axioms of the system?

Again, I don't know much about Goedel's theorum, so if I'm being assumptive, please forgive me.

[Capitalism Forever]

Does it follow then that all knowledge can be deduced to the axiom of existence?

You deduce things from axioms (or, more generally, from premises). For example, given the two premises:

• John is a friend of mine.
  
• I don't have any Canadian friends.
  

you can deduce that John is not Canadian.

Clearly, we don't form all our knowledge by taking the three axioms and applying the rules of decution. That really wouldn't get us very far. We use our senses to observe things; then we induce, integrate, evaluate, and deduce.

[NIJamesHughes]

doesn't all knowledge rest on those three axioms?

[AshRyan]

doesn't all knowledge rest on those three axioms?

Yes, but that is not the same thing as saying that all knowledge can be arrived at deductively beginning with the axioms.

For example, your knowledge that apples grow on trees depends on the facts that existence exists (since apples and trees must exist for any relationship to hold between them), that A is A (since apples and trees must exist for that relationship to hold), and that your consciousness is conscious of that fact (since we are concerned here with your knowledge that apples grow on trees). In that way, all knowledge indeed rests on the axioms.

But that's not the same as saying that you can start with those three statements ("existence exists," "A is A," and "consciousness is conscious") as premises in a deductive argument, and arrive at the conclusion that apples grow on trees simply by applying the rules of logic.

[Tom Rexton]

Some say that Goedel's theorum disproves Ojectivism as a complete sysytem, as Dr. Peikoff states. Does anyone know anything about that?

I'd also like to add that Gödel's incompleteness theorem applies only to sufficienty strong axiomatic systems (formal systems of mathematics), from which mathematical theorems are deductively derived. It is NOT universally applicable, and certainly not to Objectivism, because, as AshRyan said, Objectivism is not a formal axiomatic system. Its principles are not deductively derived from the axioms of existence, identity and consciousness, nor are they proven by the said axioms.

[Capitalism Forever]

That's right; moreover, even if Godel's theorem applied and all knowledge were decuded from those axioms, all that knowledge would still be perfectly valid. Godel's theorem only means that omniscience (knowing everything) would be impossible--but of course omniscience is impossible anyway.

[NIJamesHughes]

Wouldn't it be correct to say that No Knowledge could be deduced from the axioms? The 3 axioms ("EIC" for shortness) are primaries, the base of all concepts, so it is logical that nothing is deduced from them; the only way to travel would be up.

[msb]

What about all the metaphysical corollaries like causality, or the fact that the universe is a plenum? That's not a rhetorical question. I really don't know if you would call them deductions or not.

[NIJamesHughes]

causality can be deduced back to identity and then to existence. Causality is an axiom but it isn't fundamental in the sense that EIC are, because it can be reduced (deduced) but existence, identity and consciousness, can not.

[msb]

So then you can deduce (some) knowledge from the axioms? (This was my question.)