Superposition and Non-Contradiction

Factual Questions
1

Occasionally I see comments to the effect that the law of non-contradiction is shown by quantum physics to be invalid. The idea is that the phenomenon of superposition is a counterexample to LNC.

Are there smart people who take this view seriously, or is this just a pop-science/pop-philosophy view?

A quick read of the wikipedia article on quantum superposition didn’t give me any reason to think there are counterexamples to LNC. Take Schroedinger’s cat, for example. Sure, we might say it’s neither alive nor dead, but it would be a false dichotomy to insist that it must be either alive or dead. It can be alive, or dead, or–superposed between the two.

Right? Or not?

-FrL-

2

I don’t think you even need go as far as quantum physics to find situations where the LNC is inapplicable. What about the old chestnut concerning whether or not you have stopped beating your wife?

It’s obvious enough that the LNC can’t be universally applicable. Given how mathematics has explored so many axiomatic systems over the last 150 years, including multi-valued logics, it wouldn’t surprise me if there’s a 1000-page tome out there somewhere explaining exactly how the LNC is meant to be used.

Now I’ve got me wondering, can we partition the set of all propositions into those for which the LNC IS valid, and those for which it ISN’T valid? Or, is that a misapplication of the LNC? :smiley: From this point I typically start gibbering and drooling. Hopefully Indistinguishable or Mathochist or Chronos will be around shortly. Or not. But definitely one of those.

3

You don’t need QM for this. Long before the discovery of electrons, Plato wrote of both the bigness and smallness of fingers, for example, each of which is determined by perception. Also hardness and softness, etc. (Cite: Plato’s Republic, Book 7). LNC applies only in systems of logic where Not (p and not p).

4

Either I have or I haven’t stopped beating my wife. In my own case, I have not stopped. I have not stopped because I never started, and in order to stop, one must first have started.

A multi-valued logic doesn’t violate LNC, rather, it “violates” (probably not the right word for me to use) the law of the excluded middle. In other words, it admits of truth values other than “true” and “false”. But it doesn’t thereby say some propositions have more than one truth value.

Having said that, there are indeed logicians (and maybe even mathematicians?) who discuss inconsistent systems. The logician Graham Priest, for example, says that the way we should deal with things like the Liar paradox is simply to admit that there are some true contradictions. Most people think that if there are any true contradictions, then all contradictions are true. (Logic as it is usually done implies this.) Priest (and a few others) have tried to show that there can be ways of doing logic where contradictions don’t “explode” like this.

But the great majority of logicians don’t accept this. LNC is thought of as pretty much foundational.

Still, it wouldn’t suprise me, and it would be interesting to me, if there were some logician or mathematician (or physicist even) who does take superposition to be relevant to the question of the validity of LNC. Hence my OP.
-FrL-

5

It’s not clear to me how Plato’s observations about bigness and smallness (and other such adjectives) are supposed to show counterexamples to LNC.

I guess it’s supposed to go like this? I am both tall and not tall, since I am tall for a graduate student, and not tall for a basketball player?

But every claim that X is tall is a claim that X is tall for a Y. “Tall” can’t be defined without mentioning reference classes like this. So the word “tall” when used by itself is just used as a shortcut for what is actually meant–“tall for a Y.” And I am not both tall and not tall for a basketball player.

-FrL-

6

Plato’s example just seems like a case of equivocation. ‘Big’ and ‘small’ have different meanings in different contexts. Different meaning = no contradiction.

I don’t think quantum physics provides a counterexample to LNC. You have logical rules and material rules of inference. If an object is both here and there at time t, you can derive a contradiction *only if * you also have the material rule of inference, “An object cannot be in two places at the same time.” This material rule may be false at the quantum level, and so you can derive no contradiction from a subatomic object’s being in two places at one time.

7

Well, Hilary Putnam at one point argued that the principles of quantum mechanics gave reason to support a non-classical logic as the “correct” logic; however, to the best of my knowledge, the law of noncontradiction wasn’t one of the things he advocated revising. Rather, the main classical law people reject in quantum logic is the distributive law: that p AND (q OR r) = (p AND q) OR (p AND r). Instead of working with the logic of Boolean algebras, one can work with the logic of orthocomplemented lattices more generally; but even in orthocomplemented lattices, the law of noncontradiction (essentially) is guaranteed to hold.

8

Eh, it’s all in how you view the other truth values. Some people view Kleene’s three-valued logic or Belnap’s four-valued logic as just having new truth values between “false” and “true”, while others make a point of going further to view those new values as particular combinations of “false” and “true” (e.g., “neither” or “both”); i.e., as names for having more or less than one truth value. At any rate, depending on what we take the law of noncontradiction to be, we could consider it “violated” by these logics just as well as its dual, the law of the excluded middle, could be. For example, if the law of noncontradiction was meant to be “p AND (NOT p) entails FALSE”, then this fails to hold whenever p is one of the intermediate values in the above mentioned logics.

Even without using (things easily analyzable as) multi-valued logics, of course, we can still get rid of the law of noncontradiction; using the dual of intuitionistic logic, for example (I wish I knew a good standard name for it; I’ve heard it called Brazilian or subtractive logic), one could not prove falsity from p & ~p, for the same reason one can’t prove p v ~p from truth in intuitionistic logic. Of course, since you’ve referred to Graham Priest and the paraconsistent camp, you must have some familiarity with this already; I understand your question really is as to whether anyone has tried motivating paraconsistent logic via the empirical discovery of quantum mechanics specifically.

9

While the cat is in the box, it is not correct to say “the cat is dead”, nor is it correct to say “the cat is alive”, and it certainly is not correct to say “the cat is dead and the cat is alive”.

For a surprisingly close analogy, If I draw this
_
on the board, it is correct to say that “the line is horizontal”. Likewise, I can draw this
|
and say “the line is vertical”. But if I draw this
/
it is not correct to say either one. I could say something like “the line is not completely non-horizontal”, or “the line is not completely non-vertical”, or “the line is in a state intermediate between vertical and horizontal”, or any of a variety of other such statements, but I can’t say “the line is both horizontal and vertical”.

10

Indeed, almost everything that one might want to “pop science”-ishly say about quantum mechanics and its implications for logic really seems to just be about superpositions and nothing more, and thus to apply just as well to, say, the situation where we think of a superposition as a pair of states of the universe (and thus take our logic to be that of the topos of pairs of objects). In this case, our logic is still classical: truth values just happen to fall in the four-element Boolean algebra instead of the two-element one.

To illustrate: suppose lines can come in several flavors. Definitely vertical <|, |>, definitely horizontal <-, ->, and two “ambiguous” superpositions <|, -> and <-, |>. Corresponding to these would be various truth values; if I ask “Is <|, -> vertical?”, the answer will be “<yes, no>”, and so forth. But these new truth values, just pairs of classical values, have all the same algebraic properties as classical values themselves; it’ll still be the case that p & ~p always equals false (or, rather, <false, false>), and so forth. So even though there are new truth values, the logic at play doesn’t change, in some sense. The same principle at play as in the fact that pairs of reals manipulated component-wise don’t come close to forming a field with 2-valued logic (for example, <3, 0> * <0, 3> = 0 (as <0, 0>), without either factor being zero), but they do form a field with the appropriate (still classical) 4-valued logic (since (<3, 0> is zero OR <0, 3> is zero) is (<false, true> OR <true, false>), which is definitely true (i.e., <true, true>)).

Hm, that may not have been very clear at all. Well, anyway, I’m posting it for now, as I hurry out the door, and perhaps I’ll be able to explain it better later. (It’s not meant to illustrate the principles of quantum mechanics at all; it’s specifically meant to illustrate the pre-existing intuition of “superpositions” in itself)

11

However, you could say, “The line is both not-horizontal and not-vertical.” In addition, a reasonable definition of “vertical” and of “horizontal” would enable you to say, “Every line is either horizontal or not horizontal” and “Every line is either vertical or not vertical.”

As to the poor cat, I guess we assume that dead = not-alive and alive = not-dead, i.e., there’s no third possibility. (Of course, in practice, death is a process, which starts with an animal being alive, and ends with an animal being dead, and there’s no exact dividing line between the two). But we know that the cat is either alive or dead – just not which of the two states the cat in in. Similarly, I know that a person that I met 20 years ago, but have not heard of since, is either alive or dead, but I don’t know which. Ignorance in a particular case does no harm to the law of the excluded middle.