Logic and the World

SelfAwarePatterns is an excellent blog if you’re interested in science, philosophy and similar topics (which covers pretty much everything). Earlier this week, its author, a self-aware pattern named Michael Smith, wrote about the nature of logic. He quoted several brief definitions of logic, including one by Gottlob Frege (1848-1925), one of history’s greatest logicians. According to Frege, logic is “the science of the most general laws of truth”, to which Mike Smith responded:

Gottlob Frege’s definition seems closest to my own current personal intuition about it, namely that logic represents the most fundamental relationships in our universe. These relationships are so fundamental, that we can take them and extrapolate truths using them, and often we’ll be right.

After reading this, I began writing a comment but quickly saw that my comment was turning into a post of my own. And since I need to keep this blog going in order to continue raking in the big money, here it is: 

Whenever I try to understand what logic is and how it relates to the world, I end up thinking about the status of Aristotle’s three fundamental axioms of logic: the Law of Identity (A = A); the Law of Non-Contradiction (it is not the case that A and not A), and the Law of the Excluded Middle (either A or not A), where “A” represents a statement like “Snow is white” or “I’ve never made a single penny writing this blog”.

The Law of Identity seems to reflect how the world is without question, partly because it’s supremely uninformative. As Bishop Butler said: “Everything is what it is, and not another thing”. I’m not sure the Law of Identity states a fundamental relationship, since self-identity isn’t much of a relationship. There is only one party involved. But it seems undeniable that A equals A, whatever A happens to be.

The Law of Non-Contradiction seems to reflect how the world is too. It’s exceedingly hard to imagine how things could be otherwise in our universe or any other universe (e.g., “Vitamin C is ascorbic acid and yet it isn’t.”). Despite this difficulty, some enterprising logicians have accepted dialetheism: the view that the very same proposition can be both true and false. That seems plainly wrong. Can we step into the same river twice? Well, yes, we can (“It’s the mighty Mississippi”) and no, we can’t (“The Mississippi had different water in it yesterday”). But which answer is correct depends on what you mean by “same river”. It’s the same river it was yesterday in one sense, although it’s not the same in another sense.

How about a self-referential statement like “This statement is false”? To be fair, that’s the kind of sentence dialetheist logicians are interested in. If “this statement is false” is true, it’s false. But if it’s false, it’s true. That is certainly weird, but is the sentence in question really both true and false? I don’t think so. It seems to me that it’s a badly-formed sentence. Its apparent meaning contradicts our natural presumption as speakers of a language that speakers don’t undermine their own claims (i.e., give with one hand and take back with the other). In this case, it seems best to follow the doctor’s advice when the patient said “It hurts when I do this”. The doctor, of course, answered: “Don’t do that”. Or in this case, don’t say stuff like “This statement is false”. Just because we can put certain words together doesn’t make it a proper sentence.

Then there’s subatomic physics. Light is a field of waves and also a stream of particles! The evidence indicates that light acts as if it’s a wave in some cases and as if it’s a particle in others, but saying that it acts the same way at the same time makes no sense. To me anyway. It’s better in this case to infer that our everyday concepts of “wave” and “particle” aren’t adequate to describe the nature of light. But that doesn’t mean light is a counterexample to the Law of Non-Contradiction.

So far, so good for classical logic accurately representing the universe. Things get more complicated, however, when we consider the Law of the Excluded Middle. Personally, I don’t buy it at all. The idea is that every proposition is either true or false. Unless we define “proposition” as “a bearer of truth or falsity”, there are lots of propositions that aren’t clearly true or false. There are vague propositions, for example. Has George lost enough hair to be considered bald? What if he lost one more hair, or 500 more, or 50,000 more? Where is the line between being bald and being hairy? And there is the matter of probability. For example, according to the principle of quantum superposition, “a physical system – such as an electron – exists partly in all its particular theoretically possible states simultaneously”. Is an electron here or there? Most physicists think it’s a matter of probability. An electron could be here and it could be there, but it’s not definitely anywhere until it’s measured or otherwise interfered with.

Concerns about vagueness and probability have led to the creation of alternative logics. So-called “many-valued” logics reject the Law of the Excluded Middle. “Fuzzy” logic replaces it with a continuum of values, ranging from true to false and allowing points in between. We might instead reject the Law of Contradiction and accept that some well-formed declarative sentences, like “George is bald”, are both true and false. “Paraconsistent” logics do that. As Mike Smith pointed out in his post, there is even “quantum” logic, which tries to deal with the peculiar laws of quantum physics.

There is good reason, therefore, to believe that Aristotle’s three axioms are somewhat misleading if they’re taken as an attempt to state fundamental features of the world or even relationships between the world and language (or thought). We should agree that the Law of Identity applies to the world (in fact, it applies to every possible world). After that, we’re in a gray area. There is no denying that the world is what it is (as that annoying phrase “it is what it is” seems to call into question – after all, what isn’t what it is?). Furthermore, we learn logic by paying attention to the world and use logic to navigate the world, but logic, I think, is better understood as “the science of the laws of discursive thought” (James McCosh, 1811-1888) than as a general description of how things are.

Bet on Achilles to Beat the Tortoise

Many are the times I’ve thought about Achilles and the tortoise:

Zeno concluded from this and other paradoxes that motion is an illusion, so it’s important to show why Achilles will beat the tortoise. Otherwise, we’ll all have to sit very still.

Zeno’s argument goes something like this:

1) When Achilles starts running at point A, the tortoise is already at B.
2) By the time Achilles reaches point B, the tortoise is at point C.
3) By the time Achilles reaches point C, the tortoise is at point D.
4) This series can be extended forever.
5) In order to catch up, Achilles will have to perform an infinite series of tasks in a finite amount of time.
6) It’s impossible to perform an infinite series of tasks in a finite amount of time.
7) Therefore, Achilles can never catch up to the tortoise.

But we know that mighty Achilles is going to catch up to the tortoise and win the race. Fast runners beat slow walkers. Hence, the paradox.

In his book A Brief History of the Philosophy of Time, Adrian Bardon suggests that Aristotle had a good response to Zeno. Aristotle apparently argued that Zeno’s paradox rests on confusing “an abstract value (i.e. time), which is mathematically divisible into instants, with actual change, which is not literally composed of infinitesimal units of change”. In other words, Aristotle distinguished between “the rules for time (as a mere abstraction) and those for change (as a real phenomenon)”. Maybe Aristotle was right, but I’m not sure Zeno would have been convinced.

Bardon also discusses the mathematical concept of a “limit”, which “allows for an infinite number of finite quantities to add up to a finite sum”. That’s the idea mentioned at the end of the video. Many have concluded that Zeno can be answered using this concept, although Bardon asks: “Can a limit be a real endpoint to a real process, or is it just a new mathematical convention that disregards the metaphysical question about time and change with which Aristotle and Zeno are struggling? Does it really help matters to say that [motion or change] represents convergence on a limit? That wouldn’t have sounded like real motion to either Zeno or Aristotle”.

Maybe Zeno’s argument does require a subtle and sophisticated response. But what struck me as weak about his argument was one of the premises listed above:

6) It’s impossible to perform an infinite series of tasks in a finite amount of time.

Really? Who says? Isn’t that what we do every time we move from point A to point B? Having arrived at point B, haven’t we also traveled 1/2 of the distance, 1/3 of the distance, 1/4 of the distance, 1/5 of the distance and so on? Isn’t this a clear example of performing an infinite series of tasks in a finite amount of time? Granted that the tasks overlap, but it seems fair (albeit boring) to describe what we’ve done this way, without having to explain what the mathematical concept of a “limit” is or draw a distinction between the rules for speaking about time and the rules for speaking about change.

Perhaps this is mere sophistry and Zeno would have considered it such, since he and the Sophists were contemporaries. I think it’s a simple truth that moving around involves doing many little things by doing one big thing. Take that, Zeno!

On a similar note, the English philosopher G. E. Moore wrote a famous article called “Proof of an External World”. In that article, he said he could prove the existence of the world outside our minds by drawing our attention to his two hands: “by doing this, ipso facto, I have proved the existence of external things”. Whether that’s a great argument or not is an open question. Moore defended his argument, however, by pointing out that skeptical philosophical arguments (such as proving motion to be an illusion) often rely on philosophical intuitions or generalities (such as premise 6 above) that we have much less reason to accept than the common sense beliefs they supposedly refute.

All right, it’s safe to start moving again.