| | Paul,
You make an interesting analogy between black and white pixels and true and false statements. You write, Suppose we dispense with the Mandelbrot set for the time being. Imagine that we can represent a true statement, for instance, “existence exists,” by a black pixel and a false statement as a white pixel. Knowledge is inexhaustible — we will never be able to know everything about everything, thus there are an infinite number of true statements. I would say "a potentially infinite number of true statements"; there is always at any time a finite number of true statements, because nothing, including propositional knowledge, is infinite. As well, the world is infinitely complex, as was succinctly expressed by a programming guru: “Everything is deeply intertwingled.” The world is highly complex, but not infinitely so. Thus, all true statements can be linked to one another and they must be mutually supportive. Could you give an example of what you mean by "mutually supportive"? How, for example, would the true statement, "Ayn Rand was a novelist" be supported by the true statement, "The grass outside my window is green."? All true statements can be derived from “existence exists.” You're saying that the proposition "existence exists" implies all true statements. To be sure, the proposition "existence exists" is consistent with all true statements, but I don't think you can derive all true statements from existence exists. Don't you have to verify that a statement is true empirically -- by direct observation?
You wrote, This graphically demonstrates Rand’s dictum, “If you discern contradictions, examine your premises.” There is no such thing as a grey area with respect to truth and falsehood — you just haven’t examined the logic at a deep enough level. Grey areas are called grey areas because contradictions exist. Merlin replied, "She surely objected to grey areas with underlying contradictions, but did allow for grey areas in another sense. She called them 'borderline cases'."
But there are no borderline cases between truth and falsehood. By the law of excluded middle, a meaningful proposition is either true false. Truth is "the correspondence of a proposition to the facts of reality." Either the proposition corresponds or it doesn't correspond; there is no third alternative. A person may not know whether or not it corresponds, but lack of knowledge about its correspondence has no bearing on the fact of its correspondence or lack thereof, which exists independently of one's knowledge.
In Objectivism: The Philosophy of Ayn Rand (pp. 163-171), Peikoff talks about so-called "arbitrary" statements, which he says are neither true nor false, because they lack sufficient content enabling them to be cognitively processed -- to be related to the facts of reality. He gives the example of the following statement: "Your fate will be determined by your birth on the cusp of Capricorn and Aquarius." This statement, he says, is neither true nor false, because since there is no evidence to support it, there is no way of verifying or falsifying it. Accordingly, Peikoff states, "An arbitrary idea must be given the exact treatment its nature demands. One must treat it as though nothing had been said. The reason is that, cognitively speaking, nothing has been said."
In his refutation of agnosticism in the April 1963 issue of The Objectivist Newsletter, Nathaniel Branden makes a similar point about the arbitrary assertion of God's existence. He writes, "When a person makes an assertion for which no rational grounds are given, his statement is -- epistemologically -- without cognitive content. It is as though nothing had been said." So, the Objectivist position on statements such as God exists for which no evidence is provided is to treat them as arbitrary and therefore as neither true nor false.
I think this view is, well, false, because a statement for which no evidence is provided can still be a meaningful statement, and as such is either true or false, even if we don't know which it is and perhaps can never know. Take the statement, "There is life on Mars." If such a statement were asserted arbitrarily with no supporting evidence, would it follow that it is, therefore, neither true nor false? No, for either there is life on Mars or there isn't. If there is life on Mars, then the statement is true (even if we don't know it), and if there isn't life on Mars, then the statement is false (even if we don't know it). The presence or absence of Martian life exhausts the possibilities. A meaningful statement -- and the statement "There is life on Mars" is certainly meaningful -- is either true or false. There is no third alternative.
Of course, neither Peikoff nor Branden is saying that arbitrary statements are "grey" in Paul's sense of the term; they're saying that such statements lack meaning and cannot, therefore, be assessed as either true or false. But to say that, because a statement is arbitrary it lacks meaning is incorrect. Arbitrary statements can still be meaningful. In fact, to call something a "statement" implies that it's meaningful; otherwise, it wouldn't state anything. If it states something, then despite its arbitrariness, it is meaningful and is therefore either true or false. Moreover, to say that, because a statement lacks supporting evidence it cannot be assessed as true or false is not to say that it cannot be true or false. Even if one cannot determined whether or not a meaningful statement corresponds to reality, it either does or it doesn't. There is no third alternative.
- Bill
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