The four truths of the principle of contradiction

The Maverick Philosopher responds to this scientific experiment, which someone advanced as an empirical refutation of the principle of contradiction:

A team of scientists has succeeded in putting an object large enough to be visible to the naked eye into a mixed quantum state of moving and not moving.

Andrew Cleland at the University of California, Santa Barbara, and his team cooled a tiny metal paddle until it reached its quantum mechanical “ground state”– the lowest-energy state permitted by quantum mechanics. They then used the weird rules of quantum mechanics to simultaneously set the paddle moving while leaving it standing still.

Most of what I’m going to say about this overlaps with what MP says. (I won’t expand on his thought that journalists writing about science should be read cum grano salis, though it is important to keep in mind whenever one reads claims about what QM teaches us about reality.  Guys need to get grants and feed their kinds, and you don’t sell books with modest claims. This sort of PR stuff goes way back- Galileo was an underappreciated master of it)

St. Thomas usually explains the principle of contradiction as the inability to simultaneously affirm and deny. Note that this is principally a truth that we verify within ourselves. It is principally a statement about how we think. We attain truth by a judgment that consists, minimally, in affirming or denying some form of a subject; and the repugnance of affirmation and denial of the same thing to one subject is the principle of contradiction. Thus, the POC is a consequence of our knowing with a subject. But what is it to know with a subject? A subject is clearly imperfectly known, since it requires that we add something to it in thought. The first things we know are not truths, still less are they “beliefs” (justified or otherwise) they are subjects that we must add to in thought- even when these additions express identities on the side of the thing. It is the fundamental imperfection of our thought that makes the POC necessary to us. We know by gathering together a multitude of diverse thoughts that each perfect an imperfectly known subject. In the measure that the first thing known is perfect the POC becomes less and less necessary. The divine Word in no way requires an addition in thought in order to perfect some initial subject-thought. The angels have something only analogous to the POC (they do not add to an imperfect thought discursively, but they do have a multitude of thoughts), but this analogous principle is less important the higher the angel gets, since, as his starting point becomes more perfect, he has less need for a multitude of concepts.

On the one hand, the experiment is a clear case of an affirmation, sc. “a quantum state can be induced in larger scale objects”. The whole point of the experiment is to cease being indifferent to affirming or denying this, and to come down definitively on one side or another. In the face of this experiment, it’s not only that we can’t affirm and deny simultaneously (whatever that would mean) but it is unreasonable to do so. When you see the terms “quantum state” and “inducible in large-scale objects”, you’re supposed to affirm the latter of the former, and never deny it. Another unremarkable use of the POC.

But there is also a clear difficulty in what we do when we encounter the terms “paddle” and “in motion”. Apparently, we are supposed to affirm and deny the latter of the former.  Here it’s important to notice the exact character of a denial that is contrary to an affirmation. It is true that something is either moving or not. “Not-moving”, however, can be taken in more than one way: a bolted-down anvil, God, the intellect, and last Tuesday are all not moving, but the “not” does not signify exactly the same thing in each case. The bolted down anvil doesn’t move in the sense that it is at rest; God does not move because he utterly transcends the difference between motion or rest; and last Tuesday fails to meet a minimum requirement for either being at rest or being in motion (so does “the present King of France”). Whenever we use the principle of contradiction, therefore, we have at least four possibilities! For any proposition SP, we can be absolutely certain that if SP is false, S~P is true: but it can be true in three ways: . ~P must be analyzed into:

1.)   ~P simply speaking,

2.) ~P because transcends P and ~P,

3.) ~P because it fails to meet a minimum condition for being P or ~P.

(2 and 3 are different ways in which something can fail to be P simply speaking or simpliciter–  and in this sense they fall after a prior logical division.  I divide them up here into two options because the distinction between the simpliciter and secundum quid is less well known to us contemporary persons.)

The principle of contradiction founds our means of knowing, and is therefore proportioned and therefore limited to certain objects. At the same time, our mind is open to being as such, and so it has no limitation in what it knows. The elegant solution to this paradox is the principle of contradiction. In order to use it correctly, however, we have to recognize that our openness to all being (which is still somehow limited) requires that a negation can be understood in three ways. The quantum paddle is therefore not moving, but because it fails to meet some minimum condition to be in motion or rest. It is not the first entity to do so: What Aristotle understood by the term “matter” was neither in motion simply speaking nor at rest simply speaking- since it failed to meet a minimum requirement for either. Motion itself was neither a being nor a non-being simpliciter- for it failed to meet a minimal requirement of what exists simply speaking.

Again, our language and thought is attuned to a particular bandwidth of reality, even while it is open to all being. As a result, we have things in our bandwidth that we can name with simple names and understand by direct concepts, but we can also use these things to speak of what exceeds the reality of our bandwidth, and that which falls short of it. Here is the crucial aid that QM can give to us: it shows us new ways in which our bandwidth is limited. It seems that it is not merely bodies that we are proportioned to understanding, but bodies in “normal” environments. The word “normal”, of course, is perfectly obscure and perhaps not that helpful, but we know at least that there will be some times when we will encounter bodies that, under certain conditions, will fail to fall within the bandwidth of things we understand well. We will have to understand them by negation. If some one asks “is the paddle moving? ” The correct answer is no. The yes claim is false. BUT to say that the paddle is “not moving” does not mean we are saying it is at rest.


  1. Woppodie said,

    March 23, 2010 at 10:11 am

    Good post. Any attempt to refute the PoC necessarily assumes it, or else one could just say that the PoC is simultaneously true and not true. It amazes me that people actually argue that it is not true. What was it that Aristotle called his pupil that asked him for a proof of it? A vegetable?

  2. March 23, 2010 at 11:13 am

    That’s right, but it’s still important to note that if one thought that, in a statement “S~P” that there was only one way to be ~P, that there would be all sorts of statements from both theology, Aristotle’s Physics, and Quantum physics that actually violate the PoC.

  3. Baduin said,

    March 23, 2010 at 1:04 pm

    I suggest Aristotle’s passage on the sea battle in De Interpretatione.

    The future is neither true nor false – at present we can only say that something “either will be or will be not, but not both”. The sentences “it (necessarily) will be” AND “it will (necessarily) not be” are BOTH false.

    The same applies to the quantum wave before collapse, including the experiment in question.

    And so, it is not necessary to employ Cusanus in such a simple matter.

  4. March 23, 2010 at 2:38 pm

    I don’t get the Cusa reference. I’m certainly not advocating the coincidence of contradictories. To the meat of your comment.

    The future is neither true nor false – at present we can only say that something “either will be or will be not, but not both”. The sentences “it (necessarily) will be” AND “it will (necessarily) not be” are BOTH false.

    The same applies to the quantum wave before collapse, including the experiment in question.

    True- the contingent is not necessary. But I didn’t read the paper that way- this isn’t my field, and I get lost in it quite easily, but I don’t think it’s a matter of “given X will happen in the future, what is it now?” but of “what sort of entity can get the results we see?” It is not a matter of future and present, but of effect to cause. Even if it were a matter of future and present, the scientist would still strive to know the future in its causes, which is legitimate (true, QM causes have special considerations). Aristotle isn’t speaking there of the future so far as it is in its causes.

    More simply, Aristotle’s claim is about singular futures, but science is about universals.

    The argument in the paper was a helpful referent, but I don’t take it as central to the post. The boring old double slit experiment, wave-particle duality, or the observation of superposition (here and not here) would probably do the trick. I only took this one because there was some blogger buzz about it.

  5. Dave Kustra said,

    March 25, 2010 at 5:36 am

    Other researchers have successfully demonstrated the feasibility of superconducting circuits, which are electromagnetic oscillators. Cleland’s anomaly could have been the result of an induced state in the measuring equipment itself, due to the exceptionally low absolute temperature under which the tests were conducted. Similar measuring errors occur when the measuring equipment is magnetized by what it is measuring or when time appears to be relatively slowed down because the measuring device is in motion (atomic clock experiment aboard a jet plane). Moreover, Cleland’s oscillator may appear to be at rest relative to a human observer, whereas it is in motion relative to itself. This is because the circuit and the observer are in different states. This explanation would be in line with general relativity theory. I, for one, do not think that relativity actually overthrows the POC, but a phenomenologist might argue otherwise.

  6. March 25, 2010 at 5:46 am

    These are helpful comments. We are still trying to figure out how to transition from empirical findings to ordinary language descriptions.

    The simplest answer might be to try to discover some respect in which there is no contradiction. I am skeptical about whether this approach will work, and I think it can be proven that this approach will not always work- in fact it does not work answer questions like “do motion/ potential being/ categorical relations/ God/ immaterial existence / forms pre-existing in the human intellect…etc exist?” We need a more expansive notion of the negation of what is proportionate to understanding.

  7. March 25, 2010 at 5:50 am

    Again, the more important point is the paradox of human intelligence: it can understand all thing even while it is limited to on very narrow class of things that it can understand. All outside of that class is understood with some negation of what is within it. This is the reason there are three negations, and not just one.

  8. Ben Espen said,

    March 26, 2010 at 6:49 am

    The suspicion that terminology fails us is probably a good one here. What are the researchers actually doing? I can translate what they mean in this passage:

    Cleland and his team took a more direct measure of quantum weirdness at the large scale. They began with a a tiny mechanical paddle, or “quantum drum,” around 30 micrometers long that vibrates when set in motion at a particular range of frequencies. Next they connected the paddle to a superconducting electrical circuit that obeyed the laws of quantum mechanics. They then cooled the system down to temperatures below one-tenth of a kelvin.

    At this temperature, the paddle slipped into its quantum mechanical ground state. Using the quantum circuit, Cleland and his team verified that the paddle had no vibrational energy whatsoever. They then used the circuit to give the paddle a push and saw it wiggle at a very specific energy.

    Next, the researchers put the quantum circuit into a superposition of “push” and “don’t push,” and connected it to the paddle. Through a series of careful measurements, they were able to show that the paddle was both vibrating and not vibrating simultaneously.

    When they say the circuit obeys the laws of quantum dynamics, they mean that the circuit can be described in terms of the Schrödinger equation, which is a linear wave equation. Being a wave equation, it gives us the superposition principle on purely mathematical grounds. When the researchers say, we used the circuit to give the paddle a push, and saw it wiggle at a specific energy, they mean we applied a wave to our circuit, and measured a wave on the object.

    So when they say they simultaneously pushed and didn’t push, they mean they applied two different waves at the same time, and then they measured two corresponding waves in the object. Since waves of the type applied are separable in thought and in practice, you can say that the two waves are each doing their thing at the same time.

    However, this is really not quite the same thing as “not vibrating” and “vibrating”, because we probably cannot visually observe the object because light would make it too hot and ruin the experimental condition. These words are used as shorthand for what is actually measured, probably electrical voltages.

  9. Onus Probandi said,

    March 28, 2010 at 8:26 pm

    Dr. Chastek:

    It would appear that you and others may have misinterpreted the findings. It’s not necessarily that the object is moving and, at the same time, not moving. From what’s been reported thus far, it’s simply that the object is essentially appearing at 2 places at the same time, as the following citations (see below) would explain. So, in the Catholic world, if you consider that to be a defiance of the Law of Non-Contradiction, then bilocation of certain saints would happen to defy said law likewise.

    Quantum Entanglement

    In a related experiment, they placed the mechanical resonator in a quantum superposition, a state in which it simultaneously had zero and one quantum of excitation. This is the energetic equivalent of an object being in two places at the same time. The researchers showed that the resonator again behaved as expected by quantum theory.”

    NPR Interview with Dr. Cleland

    Andrew Cleland of the University of California-Santa Barbara says some of the laws are … well, the word “weird” comes to mind.

    “One of the most striking is quantum mechanics says that an object can be in two places at the same time. Or two configurations at the same time,” he says.

  10. Onus Probandi said,

    March 28, 2010 at 9:17 pm


    Team’s quantum object is biggest by factor of billions

    Researchers have created a “quantum state” in the largest object yet.

    Such states, in which an object is effectively in two places at once, have until now only been accomplished with single particles, atoms and molecules.


    In two places at once: Strange world of quantum mechanics shown to work in visible world for first time

    The mind-bending laws of quantum mechanics, where tiny atoms and molecules can effectively be in two places at once, have been applied to a visible object for the first time.

    A U.S team managed to create a quantum state in an object billions of times larger than any previously tested.

    Read more:

    So, thus taken, does this mean that your great Padre Pio, who was said to be able to appear in two places at once, was the first huge object to defy the Law of Contradiction?

    • March 29, 2010 at 5:34 am

      This comment makes a point similar to ones above. I’ll write to it, and then I’ll have to clock out of the discussion. Others, of course, can respond if they want.

      First: I’m not saying anything occurs in “defiance” of the principle of contradiction! The post explains exactly why this is not possible! It also strives to make the point that things that are contradictory to us can sometimes be used to explain things that exceed or fall short of the reality we are proportioned to.

      Thanks for the links.They were quite helpful.

      The PoC does not allow us to say, without some verification of natural science or metaphysics, that any existent being must be either moving or resting. It does allow us to say that an object is moving or not moving, but “non moving” means three things in the real order. To speak to superposition: the same applies to the set of contradictories : one substance is in one place and one substance is not in one place. “Not in one place” means more than one thing, and unless one is clear about this, he will be forced to admit that things “violate” the principle of contradiction. This is the whole point of the post, a point that can still be valid even if the description of the experiment in question is false. Again, the post shows that if we affirm that something is both moving and not moving, it need not violate PoC, which can be true, and be demonstrated, even if the antecedent is false. When the object in question transcends what we are proportionate to, or falls short of it, its very being will be described as both P and ~P. Things in lower and higher orders are described in ways that would be contradictory if said of things in the order we are proportionate to. It’s not just a matter of saying “they don’t contradict if we consider that we are speaking in different respects!” This is true, but we need to say more. We also must notice that we understand them best by affirming and denying of them things that contradict in our own order. St. Thomas figured this our about spiritual existence long ago: God is best understood if we approach him by two paths: as both abstract and concrete; personal and essential; incarnate and transcendent; etc. Our mind can never reduce these concepts to unity- we simply use two approaches to study one thing. QM is starting to get some idea that it has to use the same trick when things fall short of what is proportionate to our intellect.

      Again, contradiction is mind-relative. What is contradictory to us in our own order is not contradictory to a higher intellect, and nothing is contradictory to God. This is not to advance a doctrine of two truths, it is simply to point out that contradiction occurs in the means of knowing- It is a principle of thought and not of things. What contradicts in the order of things we are proportioned to is a tool to understand the singular natures of things above and below this order. If God or electrons existed among the things we understand directly, we would have to deny their existence (this is another true consequence with a false antecedent.)

    • Dave Kustra said,

      March 30, 2010 at 6:28 am

      Thanks for the clarification and the useful links, which show how our understanding of the created universe is constantly evolving as we develop new physical models and set aside older ones. As to Padre Pio, if the bilocations did in fact occur as widely reported by witnesses, then it seems that the PoC was violated insofar as the laws of physics, as we currently understand them, were violated. However, although physical laws may have been violated, moral laws were not, for every documented instance of bilocation was for some good purpose, e.g., to come to the aid of someone in need and to be present for a penitent in confession at the same time. We believe that the Creator of the laws of nature can change them, as any lawmaker can make exceptions to laws he promulgates. This is not something empirical science should feel obliged to prove. It is a matter of faith for each individual to decide upon using his or her free will. Faith without doubt is not faith, but certain knowledge. So, every matter dealing with the faith has an element of doubt, else it would not come under the category of faith. As to the possibility of bilocation, it is not specifically a matter of faith, except insofar as we believe that Creator is omnipotent over His creation, and hence, is capable of effecting such things. However, it seems reasonable, if one believes that God exists, that God might allow instances of this physical phenomenon for the sake of a higher moral purpose. But, to what higher purpose does Cleland’s paddle aspire? One cannot absolutely disprove Cleland any more than one can disprove Padre Pio, but I can more readily doubt Cleland’s tentative conclusions than the eyewitness accounts concerning the bilocations. It seems that subatomic particles and waves are shrouded in much the same sort of mystery as matters of faith. Both the science of theology and the physical sciences prove some things, but other things require a measure of faith on the part of their respective participants.

  11. DM said,

    September 18, 2010 at 4:34 pm

    To boil down the question further: what does it mean for a particle, such as a photon, to *be* in a superposition of spin-up and spin-down? How is it “both”?

    You draw a distinction between act and being. Does it apply here? Can spin-up and spin-down be seen as acts of the photon which can somehow co-occur? Because the only way I can think of permitting a photon in “be-ing” spin-up and spin-down is we discard the idea that spin is the genus, and spin-up and spin-down the species, and elevate spin-up and spin-down to genera each of which contain only one species (themselves), thus breaking spin into two aspects to avoid contradiction. However, it is not immediately clear to me, without giving it some more thought and a little research, that such a division of spin into two parts is permitted by QM.

%d bloggers like this: