Contradiction and Impossibility

Metaphysics and Epistemology
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Contradiction and Impossibility

Contradictions are impossibilities of reality. One will not be able to build a self-contradictory device and should not give effort to it. But does the fact that existence is identity imply that a thing impossible of existence has self-contradictions in its posited identity? I argue for denial of this thesis.

In Galt’s speech, after setting out the fundamental relation of consciousness to existence, Rand goes on to maintain:

“To exist is to be something, as distinguished from the nothing of non-existence, it is to be an entity of a specific nature made of specific attributes. . . . Existence is identity. . . .

“Whatever you choose to consider, be it an object an attribute or an action, the law of identity remains the same. A leaf cannot be a stone at the same time, it cannot be all red and all green at the same time, it cannot freeze and burn at the same time” (1016).

As footnote we might add that different times cannot be same times, where we are asserting a character of time (not merely displaying the meaning of our terms “same” and “different”).

Does it follow that things not possible of existence are either lacking in identity or have contradictory identities in their specifications? I say No, what comes after the “or” is too restricted. 

That a thing impossible of existence have all the incompatibilities within its identity-specifications be contradiction-incompatibilities would seem to imply that its impossibility of existence can be shown by exhibiting self-contradictions in its formulation. 

But there are mathematical impossibilities whose impossibilities are not shown by showing they would be self-contradictory. We instead show conclusively their impossibility by mathematical induction. An example would be that there is no counting number n for which n2 - 1 is not evenly divisible by 4. That is, for which n2 - 1 does not equal 4m, where m is some whole number. 

Furthermore, I do not see anywhere a proof showing by else-contradiction that there must be a proof by else-contradiction for any occasion of truth.

Rand’s illustrations of non-contradiction concordant with physical reality are cases from common experience. For illustration of wider scope, I should look to the 2nd Law of thermodynamics. One would be ill advised to try to cross it in one’s inventions. If you’ve an idea for a device that would be in fact a perfect refrigerator, then take seriously its heavy likelihood of unworkability due to being an exception to the 2nd Law. But is the idea of a perfect refrigerator shown to be an unreachable device by a showing of self-contradiction in the concept of a perfect refrigerator? Not that I know of, and it would be odd to find contradiction in better and better refrigerators only when they reach the line of being perfect ones (ones transferring heat from a low-temperature area to a high-temperature area and requiring no work to do so).

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5 hours ago, Boydstun said:

An example would be that there is no counting number n for which n2 - 1 is not evenly divisible by 4. That is, for which n2 - 1 does not equal 4m, where m is some whole number. 

Do  you mean no odd counting number?

 

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I don’t know maths , but doesn’t that open you up to the square root of 2 conundrum, if there is one , like I said no maths here.

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Just now, tadmjones said:

1 hour ago, Doug Morris said:

Do  you mean no odd counting number?

 

 

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This seems like an echo of the analytic-synthetic dichotomy. If something is false by contradicting itself or a general principle, that would make it “analytically” false, but if it is false by contradicting some fact of empirical reality, that would make it “synthetically” false.

Peikoff wrote a paper about the analytic-synthetic dichotomy and I think it’s part of ITOE as an appendix or something…

Objectivism rejects the analytic-synthetic dichotomy. I can’t remember what Peikoff wrote in his paper about why, but my thinking is that general principles themselves come from empirical reality, so that it’s ultimately nonsensical to distinguish between “true from general principles” and “true from empirical reality.” The former (if the principles are accurate) would be a special case of the latter, more like “true from empirical reality via general principles.”

I suppose the same thing would be true of false statements, that if they are false it is always because of empirical reality, though sometimes this can be shown by means of general principles.

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17 hours ago, Doug Morris said:

Do  you mean no odd counting number?

 

No. Put any counting number at all in the formula, and you will see it works.

15 hours ago, necrovore said:

This seems like an echo of the analytic-synthetic dichotomy. If something is false by contradicting itself or a general principle, that would make it “analytically” false, but if it is false by contradicting some fact of empirical reality, that would make it “synthetically” false.

Peikoff wrote a paper about the analytic-synthetic dichotomy and I think it’s part of ITOE as an appendix or something…

Objectivism rejects the analytic-synthetic dichotomy. I can’t remember what Peikoff wrote in his paper about why, but my thinking is that general principles themselves come from empirical reality, so that it’s ultimately nonsensical to distinguish between “true from general principles” and “true from empirical reality.” The former (if the principles are accurate) would be a special case of the latter, more like “true from empirical reality via general principles.”

I suppose the same thing would be true of false statements, that if they are false it is always because of empirical reality, though sometimes this can be shown by means of general principles.

Yes, I am familiar with that fine paper of Peikoff’s, and I have serial work in progress on it here.* I don’t think Peikoff leaned towards thinking that self-contradiction in concept was the only way to show a concept or thesis to be false in principle. Which is the idea I was arguing against here. I do not accept any version of the analytic-synthetic distinction, and I do not rely on it in this little essay.

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The statement “contradictions are impossibilities of reality” is more accurately stated as “contradictions are not possible in reality”, avoiding the implication that this is a bi-conditional relation. The leaves open to consideration that “impossibilities are impossible because then depend on a contradiction”. A device (other than an AI) cannot contradict itself, but a person will not succeed in building a device that embodies a contradiction. It is irrational to hold that A & -A are simultaneously true of the same thing. Just as “Dogs are mammals” does not entail that “Mammals are dogs”, the law that “contradictions are not possible in reality” does not entail that “the impossible-in-reality is because of a contradiction”. Perhaps that is the case, but it is a separate question requiring its own proof.

The shame of argumentum ad bombum requires us to carefully scrutinize claims of impossibility, to at least substitute argumentum ad ignorantiam when countering a claim of impossibility. Scientific claims – theories of actions which can exist – state necessary and sufficient conditions for an action, and can be validated by observation. The law “conservation of charge” entails the impossibility of certain actions such as the particle decay e⁻ →  e⁺, where we have a principled “impossibility” and we can say that “the event and the law cannot both be true”, a recognition of a contradiction. The decay e⁻ →  e⁺ is not a “self-contradiction”, in fact I do not see the usefulness of “self-contradiction” except as a casual way to reject errors such as the stolen concept. The forbidden event does not “argue against itself”, it argues against (contradicts) a separate established truth.

There can be erroneous judgments, apparent contradictions in a thing belonging “at the same time to the same thing and in the same respect”, because of an incorrect identification of “same respect”, for example. A leaf can be a stone – I have one on my desk – but it is not both at the same time, it is now a stone.

Here is a starter for an argument that anything which is actually impossible is because the event entails a contradiction. The concept “impossible” derives from the premise of a wholly deterministic universe populated by an infinite set of monkeys. Eventually, the monkeys will bring about all precursor conditions, which in a fully deterministic universe yields all possible actions (therefore all possible entities). We cannot directly cause a given particle to decay, but we have a good idea of the laws limiting such decays, and we empirically derive those laws by a mix of observation (a conspicuous gap in existents) and rationalization (mathematical statements that encode the gap). Prior to the postulation of a law, the gap is an inexplicable mystery, but given the law, we can point to a specific judgment that contradicts the possibility of the forbidden event.

In the deterministic infinite monkey universe without God, every imaginable event will occur, except those forbidden by law. But we have to discover those laws. We did not know of the conservation law of color charge 70 years ago. Perhaps an actually unobserved event is highly unlikely rather than actually impossible. Those laws do not have to be very fancy and mathematical. If you have a 6-sided die whose faces are painted red, white, black, green, blue, yellow, it is impossible that rolling the die will ever yield gold-side up. This seems obvious, as long as we also make explicit why rolling a die does not cause a massive mutation in the EM absorption properties of a paint.

If something is impossible but cannot be explained by appeal to contradiction, that means that there are certain things that are in principle unknowable. Not “currently unknown”, but “actually unknowable”. Hard-core unknowability is generally reserved for God, who in fact does not exist.

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1 hour ago, Boydstun said:

No. Put any counting number at all in the formula, and you will see it works

If you want n^2-1 to be divisible by four, it is necessary and sufficient that n is odd. Plug in any even number and what you get will be odd (and in particular not divisible by four). For example, 2^2-1=3. 

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David,

The thesis I oppose is not that all impossibilities are explicable by appeal to contradiction, but that all are explicable by appeal to self-contradiction. The example I gave concerning a character of integers shows that a mathematical fact is explicable (and certified) by reasoning not relying on self-contradiction upon denial.

Sidebar: Not every imaginable event will occur, except those forbidden by law. Because the infinity of those events is larger than the infinity of infinite time.

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3 minutes ago, Jon C said:

If you want n^2-1 to be divisible by four, it is necessary and sufficient that n is odd. Plug in any even number and what you get will be odd (and in particular not divisible by four). For example, 2^2-1=3. 

Jon, thanks.

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Yes, all odd counting numbers <—in my 1991 essay (p. 46).