About 0:50 in his video “Libertarianism and Property Rights from First Principles”, Shane Killian states that the Law of Non-Contradiction is “the foundational principle of all logic and reason.” That’s going a bit far. To be sure, the LNC is one of the first principles of all logic and reason, but so is the Law of Identity, the Law of Difference (aka the Law of the Excluded Middle), and the Law of Ground (aka Principle of Sufficient Reason). However, this is a minor point, so let’s let it go.
The problem is that Shane gets the LNC wrong. He states it as follows: “Something cannot be A and not-A at the same time.”
He then adds “Also, if something is A, it cannot be B, if A and B are mutually exclusive.” This latter part is not necessary and is actually an argument:
1 ~(A & ~A) [LNC]
2 A [Posit]
3 B ⇒ ~A [Posit]
4 B [Posit]
5 ~A [3, 4]
6 A & ~A [2, 5]
7 ∅ [1, 6]
This too is a minor side issue. Here he’s just trying to add something that follows from the LNC as part of it. But since it does really follow, we can let that go too. I point out these two small errors because they tend to show that Shane doesn’t actually have a very good grasp of logic. And that’s a problem when you are trying to make a purely logical argument.
So here is the real problem with Shane’s statement of the LNC. He states it as “Something cannot be both A and not-A at the same time.”
But the actual Law of Non-Contradiction runs “A thing cannot be both A and not-A at the same time and in the same respect.”
Let’s do some quick sources:
“It is not possible for the same thing at the same time both to belong and not belong to the same thing in the same respect.” – Aristotle, Metaphysics Γ 4, 1005b 18ff.
“[The] law of non-contradiction states that the same property cannot at the same time both belong and not belong to the same object in the same respect. So “S is P” and “S is not P” cannot both be true at the same time – unless we take “S” or “P” differently in the two statements.” – Harry J. Gensler, Introduction to Logic
“[The LNC] states that contradictory statements cannot both be true in the same sense at the same time.” –Wikipedia
We could go on (and on), but we need not. The point is clear. Shane has left out the crucial clause of the LNC “in the same respect.” And it isn’t hard to see why this clause is necessary. For example, I am charged with a crime, and found to be not guilty; however, I really did commit the crime. Thus I am both guilty of the crime, and not guilty of the crime at the same time. If Shane’s formulation were correct, this would be impossible. But here the “respect” clause kicks in: I am, at the same time, guilty of the crime in respect to my having done it, but not guilty of the crime in respect to my legal status of having been found not guilty.” We could multiply examples at will.
Now, at about 1:55, Shane attempts to apply his incorrect version of the LNC to the propositions “Consistency is preferable” and “Consistency is not preferable.” He states “One or the other must be true, but both cannot be.” But unfortunately he is wrong. And he is wrong because of the respect clause of the LNC. For one thing, he uses the term “preferable” as if it were an absolute, and not a relative, term. But “preferable” is relative to the preferences of beings capable of having them. One person may prefer something which another does not.
But even worse, there are situations in which consistency is manifestly NOT preferable—e.g. when one has bad, evil, or insane principles, inconsistency might be much preferable to consistency. If one lived in a society with a particularly unjust law, it would be preferable if this law were applied inconsistently, because it would do less harm overall than a thorough and totally consistent application of the law would.
We may also cite Emerson’s famous remark:
Clearly, Emerson means something like “Consistency is not preferable, when it is foolish.” I take him to be describing those people who adopt one or more principles which they then hold to be completely incorrigible regardless of the amount of evidence produced against them. They will reject any argument and any evidence that contradicts the previously accepted principle—in the name of “consistency is preferable.”
Or, to take a rather different example, in Lewis Carroll’s Alice in Wonderland and Through the Looking Glass every character, as far as I can tell, behaves with complete logical consistency. And they are all (as the Cheshire Cat notes) quite mad. The charm of these books is in part this juxtaposition of total logical consistency with complete madness.
Or, again, to take another wildly different example, General Relativity and Quantum Mechanics are mutually inconsistent. Each one seems to logically entail that the other is false. However, we simply don’t know which one is false, if both are false, in what way one or both are false, or if there is some over-arching theory that could reconcile the two. Given our limited knowledge and the fact that without General Relativity we cannot deal with large-scale cosmic phenomena at all, and without Quantum Mechanics, we cannot deal with small-scale subatomic phenomena at all—it seems that our (provisional) acceptance of these two mutually inconsistent theories is preferable to throwing out one or both of them on the simple ground that they are inconsistent. So in this case “Consistency is preferable” may be true in respect to our wanting to reconcile Relativity and Quantium theory, but “Consistency is preferable” may be false if it were taken as a directive to throw out one or both of the theories in order to have “consistency.”
On the other hand, it is obvious, I take it, that one ought to have consistent beliefs, and if one’s beliefs can be shown to contain a contradiction, then one needs no reevaluate one’s beliefs, because at least one of them is false.
So Shane is wrong to say that “Consistency is preferable” and “Consistency is not preferable” is a “true dichotomy” (to use his term) that one could decide on the basis of a priori reasoning. Shane attempts to prove that “Consistency is not preferable” is itself a consistent principle, and therefore self-defeating, since if it were true, we should reject it, since it is consistent.
Now, I’m the last person to object to principle being shown to be false by way of retortion (that is, applying the principle to itself in such a way that shows it defeats itself). But what Shane has actually done is shown that “Consistency is never preferable in any respect” is a self-defeating principle. He has not shown that “Consistency is not preferable at some times and in some respects.” (Note that he hasn’t even shown that “Consistency is not preferable” is self-defeating by his own incorrect statement of the LNC, since he should have stated it “Consistency is never preferable at any time.” Even by his own misstatement of the LNC, he has not shown that consistency isn’t preferable at some times and not at others—he simply dropped the time clause, as if “preferable” could be treated as an eternal, invariant property (without any argument for such).
Shane’s argument is meant to be a logical step by step argument, starting with Principle 1: The Law of Non-Contradiction. That would be, I admit, a good place to start. You couldn’t find a better first principle. But Shane gets the Law of Non-Contradiction wrong. So his Principle 1, on which his entire argument rests, as a misstatement of the Law of Non-Contradiction, is false. So we really don’t need to see anything more to know that the rest of his argument is worthless, since it rests entirely on a false first principle.
We have seen how he attempts to derive Principle 2: The Principle of Consistency from his Principle 1, but his argument fails both because he has got Principle 1 wrong (by omitting the in the same respect clause) and even gets the argument wrong with respect to his own incorrect statement of it (by omitting the at the same time clause)—therefore his Principle 2: The Principle of Consistency is not established.
I didn’t go further than this. There is no need to do so. Shane is next going to attempt to establish a Principle 3: The Burden of Proof on the basis of Principle 2, but since he has not established Principle 2, and Principle 1 is false anyhow, there’s really no point in going further with his argument.