The laws of thought are three:
1. The law of non-contradiction (LNC), which states that the statements p and not-p cannot both be true. In symbolic notation: ~ (p • ~p)
2. The law of identity, which says that everything is identical with itself. In symbolic notation, a = a
3. The law of excluded middle (LEM), which states that either p or not-p is true. In symbolic notation: p V ~p
As philosophers often point out, the laws can be stated either in logical terms (i.e. in terms of propositions and their logical relationships) or in ontological terms (i.e. in terms of the things that propositions are about and their metaphysical relationships). But the difference is irrelevant to the points I will be making, so I’ll ignore it for present purposes.
The reason these are characterized as laws of thought is that reason, it is claimed, would not be possible at all if they were not true. They are first principles of rationality in the sense that they are so basic to it that they are more obviously correct than any argument that could be given either for or against them. Hence, it is claimed, even someone who claims to have reason to doubt or deny any of them must implicitly presuppose them in the very effort to question them.
Take LNC, which is commonly taken to be the most fundamental of the laws. The traditional defense against would-be skeptics is that it simply cannot coherently be denied. As Aristotle points out in the Metaphysics, to assert anything at all is to put it forward as true, and therefore not false. But that includes the skeptic’s own statement that LNC is false. In making this assertion, the skeptic is claiming that it is true, and therefore not false, that LNC is false. If he weren’t, there’d be no disagreement between him and the defender of LNC. But this itself presupposes LNC, so that the assertion is self-undermining.
Note that it misses the point to allege that the defender is begging the question by presupposing what is at issue. For the point isn’t that the defender is presupposing what is at issue. The point is that the critic himself is presupposing what is at issue. It isn’t that the critic can coherently deny LNC even though the defender affirms it. It is rather than the critic himself, no less than the defender, cannot avoid commitment to LNC.
Another way to see the incoherence of denying LNC is via the principle that from a contradiction, anything follows. Here’s a common way to explain how. Suppose that LNC is false, so that two propositions p and ~p are both true. Then, by the rule of addition in propositional logic, we can infer from p that either p or q (i.e. p V q), where q can be any proposition at all (including the proposition that the denial of LNC is false). By the rule of disjunctive syllogism, p V q and ~p will then together give us q. Hence, from the denial of LNC, you can derive the falsity of the denial of LNC. You will thereby have shown that the skeptic can be refuted from his own premise. Again, skepticism about LNC is incoherent.
Things are a bit trickier with LEM, and there are also technical arguments by which some have nevertheless tried to challenge LNC. I’m not going to get into all of that here. Having given a sense of the traditional approach to defending the laws of thought, I’ll focus just on the objections from quantum mechanics, specifically.
Consider the wave-particle duality phenomena exhibited in the famous double-slit experiment. The same particles, the experiment shows, behave in both a wave-like manner and a particle-like manner. But doesn’t this violate LNC? In particular, doesn’t it show that something can be both a particle and at the same time a non-particle (because it’s also a wave)? Or doesn’t it violate LEM, insofar as it shows that it is not true that something is either a particle or not a particle? Or consider the famous thought experiment involving Schrödinger’s cat. Doesn’t it violate LNC insofar as it shows that a cat can be both alive and dead at the same time? Or doesn’t it violate LEM insofar as it implies that it is false to maintain that either the cat is alive or it is not alive?
Thus, it is claimed, modern physics shows that we need to revise classical logic, insofar as quantum mechanics has refuted these laws of thought.
Well, not so fast. The first point to make in response is that, here as in other contexts, people speak quite sloppily when they assert that “Quantum mechanics shows that...” As philosopher of physics Peter Lewis notes in his book , when discussing quantum mechanics and its implications, we need to distinguish between (1) quantum phenomena, (2) quantum theory, and (3) alternative possible interpretations of quantum theory. The phenomena observed in the two-slit experiment would be an example of quantum phenomena. The mathematical representation of the physical systems central to quantum phenomena together with the laws said to govern those systems and the way their states are measured comprise quantum theory. And Bohr’s Copenhagen interpretation, Bohm’s pilot wave interpretation, Everett’s many worlds interpretation, etc. would be alternative possible interpretations of quantum theory and quantum phenomena.
Now, if we’re talking about what quantum mechanics can actually be said to have shown, that is confined to categories (1) and (2). We know that there are these odd phenomena, and the mathematical representation quantum theory gives us is the best description we have of the systems associated with those phenomena. But the breathless pop philosophy claims made in the name of quantum mechanics typically appeal instead to ideas in category (3) – all of which are controversial at best. None of them can be said to have been shown, or proved, or established by physics.
That includes claims to the effect that quantum mechanics has refuted the laws of thought. It has done no such thing. There is nothing in either quantum phenomena or quantum theory that entails that. Rather, what has happened is that some people have proposed interpreting quantum phenomena and quantum theory in a way that gives up one or more of the laws of thought. That’s all. And even then, the interpretation is not a purely “scientific” interpretation of quantum mechanics, because none of the competing interpretations in category (3) is purely scientific. All of them involve bringing certain philosophical assumptions to bear on the interpretation of quantum mechanics. (For example, Bohr’s interpretation famously takes for granted an instrumentalist philosophy of science.)
The upshot of this is that revisions to the laws of thought can be read out of quantum mechanics only if they are first read into it. And thus quantum mechanics itself does nothing at all to establish the plausibility of such a revision. For any proposed interpretation in category (3), we need to ask: What philosophical assumptions are independently known to be the most plausible, and thus suitable to guide us in deciding how to interpret quantum mechanics? The traditional metaphysician would answer that the laws of thought are among these assumptions. Those who would reject one or more of these laws would disagree, but the point is that they cannot appeal to quantum mechanics as a reason for doing so without begging the question. Hence arguments from quantum mechanics for rejecting the laws of thought are ultimately circular.
To address the specific examples cited, consider first Schrödinger’s famous thought experiment. The first thing to say is that it is only a thought experiment, intended to call attention to some puzzling questions raised by the notion of a quantum superposition. That’s it. Anyone who says “Quantum mechanics shows that a cat can be alive and dead at the same time!” doesn’t know what he’s talking about. You could try to argue that the cat would be both alive and dead at the same time. Or you could try to argue that it would be neither alive nor dead. But you could instead, with no less justification (indeed, I would say with far greater justification), argue that neither of these interpretations makes any sense. And that is exactly what the traditional metaphysician argues, on the grounds that LNC and LEM cannot coherently be denied. Absolutely nothing in “the science” itself shows otherwise.
Same with the two-slit experiment. What we can say with confidence is simply that there are some weird phenomena here. But how to interpret them is another story. Yes, there is something here that in some respects behaves in a wavelike way and in other respects in a particle-like way. But by no means does that entail that it is, say, both a particle and not a particle at the same time, or that it is neither a particle nor a non-particle. Here too, you could try to make the case that we should interpret what is going on in a way that rejects either LNC or LEM. But you could with no less justification (indeed, I would say with far greater justification) hold instead that any such oddball interpretation is just a non-starter. And once again, “the science” itself gives no reason whatsoever to doubt the soundness of this traditional metaphysical judgment. (I discuss the philosophy of quantum mechanics in more detail in chapter 5 of Aristotle’s Revenge.)