A natural order is the result of evolution—whether economic, linguistic, biological, or any other kind. For example, to argue that money is a natural order is to argue that money evolved, that nobody designed money—see Murray Rothbard on that here. That is, the order of money came about naturally. Nobody had to come up with the idea consciously.

If a scholar with a background in logic, mathematics, and linguistics—to give another example—thinks consciously about how logical thought and communication work and then designs a language for that purpose, then the scholar is working not with natural but artificial order, not with natural but artificial language. Anarchism in economic policy would keep the economy as laissez faire as possible, and by analogy, anarchism in linguistic policy would keep the language as laissez faire as possible. The scholar in the hypothetical, then, is going against the anarchic spirit. He’s interfering with the natural order.

Linguistics, being the study of natural language, is the study of the natural order of language. The most fundamental insight of linguistics, which also justifies the field, is that most people have order in their linguistic action, order worth studying, that’s evolved and undesigned, that’s natural and unconscious. Linguistics makes that implicit order explicit.

It may be useful to make a distinction between formalism and substantivism. When doing formalism, you push the substance out of your mind and keep only the form. And when doing substantivism, you do the opposite: You push the form out of your mind and keep only the substance. For example, doing formalism in mathematics means ignoring the semantics of the notation and paying attention only to the syntax.

Depending on the historical context, the zeitgeist may be either more formalist or more substantivist. For example, British empiricism came about immediately after English and the other vernaculars of Europe replaced Latin in scholarship. That may not be a coincidence: As Morris Cohen argued in Reason and Nature (1931), that change in language led scholars to try to take the old insights preserved in Latin and translate them into English, French, etc., and the failure to do so in many cases shows that much of Latin-based scholarship was form without substance. British empiricism, being radically substantivist in spirit, was one of the most important reactions to that.

When we put our thoughts into words, we risk being influenced without even noticing that we’re being influenced. Some of the mechanisms for that:

1. Each language has its own way of categorizing phenomena, and how a language categorizes phenomena presupposes in a hidden way certain beliefs and values.
2. Some words have valence.
3. Some words are associated with other words. If you believe a proposition made out of words, and then that word-proposition brings to mind another word-proposition, then the feeling of belief in the former proposition will add something of a feeling of belief to the latter proposition too.

On the question of what logic is about, what it’s supposed to help out with, I’ve often thought that the most fundamental answer is probably integration. Logic is the study of how to integrate your beliefs, how to be consistent, how to not contradict yourself. Although it’s not always the most useful social move to enforce integration in yourself and others—the social world often rewards inconsistency, contradiction—the tools of logic are there for that purpose. When you do want to integrate your beliefs, logic is there to help.

That way of explaining what logic is about, though—that logic is supposed to help out with something sometimes useful, sometimes not—is a way of zooming out and explaining what logic’s place is in life. If instead we just take for granted, as most scientists and philosophers do, that we shouldn’t contradict ourselves, then there’s perhaps a better way of explaining what logic is about:

Consider the distinction between the linguistic form of a word and its logical substance. In Japanese, along with Japanese and all of the other natural languages of civilization, there are at least two forms: (a) the spoken form, and (b) the written form. For logic, though, the linguistic form doesn’t matter. What matters is the logical substance. For example: Whether the word “moon” is spoken or written, the literal meaning is the same: 🌙. And in the same way, for logic it doesn’t matter whether we write “moon” or 月. Spoken or written English, spoken or written Japanese, all of those are just different linguistic forms associated with the same logical substance.

But it doesn’t stop with vocabulary. Consider grammar too:

1. “English men and women”
2. “English men and English women”

Famously, logic is about such equivalencies. Why? For the same reason: because here again the linguistic form is different but the logical substance is the same.

Consider also:

1. “English men and women, together, make up more than half of everybody here.”
2. “English men and English women, together, make up more than half of everybody here.”

It’s impossible for #1 to be true without #2 also being true, and it’s impossible for #1 to be false without #2 also being false—logically impossible. That’s because logically speaking, #1 and #2 are the same. They’re different linguistic forms associated with the same proposition.

Logic maps the complexity of the surface forms of natural language to something simpler and deeper: the underlying logic, the underlying literal meaning. The surface-grammatical form “[adjective X] [noun X] and [noun Y],” exemplified in the phrase “English men and women,” is logically equivalent, equivalent in its literal meaning, to the surface-grammatical form “[adjective X] [noun X] and [adjective X] [noun Y],” exemplified in the phrase “English men and English women.”

To follow George Boole in adapting the traditional notation of algebra to logic:

1. “English men and women.” x(y + z)
2. “English men and English women.” xy + xz

Logic, then, (a) takes as an axiom that we shouldn’t contradict ourselves, presupposes that we shouldn’t contradict ourselves, and then (b) ignores everything except literal meaning. That’s what logic is about.

Categories such as that associated with the word “man” are actually categories of categories. On the phenomenalist level, which is the ultimate foundation of knowledge, even the phrase “Noam Chomsky,” which isn’t associated with a category of people but with a person, isn’t associated with a grouping of individuals but with an individual, and thus isn’t a category on the physicalist level—there’s only one physically existing Noam Chomsky—is associated with a category nonetheless, just a category not of physically existing people but of sensory complexes. There are myriad sensory complexes that count as Chomsky: his face young and old, his face from one angle and another, etc.

The word “man,” then, which includes in its denotation Noam Chomsky, Michael Tomasello, etc., is associated with a physicalist category of phenomenalist categories. It’s associated with a grouping of individuals, the individuals themselves being groupings of sensory complexes.

Just as there are myriad people who count as men—that’s what makes it a category—there are myriad sensory complexes that count as Chomsky.

In logic, the connotation of a word is what’s shared among all of the referents of the word. For example, the connotation of the word “food” is anything edible. What’s essential to being food is being edible. The denotation of a word, then, is all of the referents of the word. For example, the denotation of the word “food” includes the broccoli in my refrigerator.

The connotation and the denotation are often in synchrony. Knowing that the word “food” connotes being edible lets you figure out what that word denotes—the broccoli in my refrigerator is one of countless examples—and knowing what that word denotes lets you figure out what it connotes. But the connotation and the denotation are also often not in synchrony. That comes about when the referents change. Imagine, for instance, a small but well-known Christian denomination that changes from monotheism (which is the standard Christian doctrine) to polytheism. The individuals in the denomination, long well-known as “Christians,” all change from professing monotheism to professing polytheism. The connotation and the denotation fall out of step with each other. The word “Christian,” which traditionally includes belief in monotheism in its connotation (along with belief in Jesus of Nazareth as the Messiah, belief in the New Testament as scripture, etc., as the distinguishing criteria of being Christian), also traditionally includes the individuals in that denomination in its denotation.

We must decide whether to:

1. Either change the familiar connotation of the word “Christian,” the connotation that we’re used to, such that it newly accepts polytheism in its connotational range, which preserves the familiar denotational range. [the inclusive move]
2. Or change the familiar denotation of that word such that it newly rejects the individuals in the aforementioned denomination in its denotational range, which preserves the familiar connotational range. [the exclusive move]

That is, we must decide whether to:

1. Either pin down what the word refers to and then change what it means when what it refers to changes.
2. Or pin down what the word means and then change what it refers to when the world changes.

Interestingly, pinning down or fixing the denotation of the subject of a proposition makes for a synthetic proposition (because the proposition “Christians believe in monotheism” is thereby made to be non-tautologically true) and pinning down or fixing the connotation instead makes for an analytic proposition (because the same proposition is thereby made to be tautologically true).

Also interestingly, the question of whether to either denotation-fix or connotation-fix is social. A social negotiation must happen between the inclusive and the exclusive: between the people who prefer the denotation-fix-inclusive move and the people who prefer the connotation-fix-exclusive move.