Let’s use PL to stand for “Propositional Logic”.

Here are the key points I want to emphasize:

1. Formal languages are very different from natural languages

Natural languages exist in the real world, in flesh-and-blood communities of language users.

The grammar of a natural language like English is incredibly complex. We discover the grammar of natural language through empirical investigation.

PL is a formal language, an artificial language.

The grammar of an artificial language like PL is incredibly simple. We don’t discover the grammar of an artificial language, we stipulate it — we define it however we want.

The properties of formal languages are mathematically stipulated. Formal languages exist as mathematical abstractions. Metaphysically, they have more in common with computer code than real-world spoken language.

2. We use formal languages as simplified models of natural language

English has conjunctions, disjunctions and conditionals.

“John and Mary went to the store” expresses an English language conjunction.

“Mary will walk the dog if John agrees to maker dinner” expresses an English language conditional.

But the semantics of natural language is often so complex and varied that it’s not always clear how the words “and” or “if-then” are actually being used, or what semantic and syntactic rules they’re actually following.

Is the “and” functioning as a logical conjunction, or some other kind of conjunction?

What do we even mean by a “logical conjunction”?

This is why we invent formal languages like PL — to help us answer these kinds of questions.

Any given formal language is designed to represent or model the logical behavior of a select few natural language words. When we use it we abstract away from all other features of natural language sentences.

PL, for example, was designed to model the logical behavior of conjunctions (“and”), disjunctions (“or”), conditionals (“if-then”) and negations (“not-“).

That’s it. The whole apparatus of PL exists in order to model the logical properties of a small handful of natural language words.

3. Symbols help us to distinguish the model from what is being modeled.

So what we do is create an artificial language and stipulate a syntax and a semantics for the logical connectives “and”, “or”, “if-then” and so on.

We stipulate, for example, that the semantics of PL is truth-functional, and we give precise rules for determining when any compound sentence within the language is true or false, for all possible truth values of the component sentences.

That’s how we define the logical connectives. That’s how we define what “logical conjunction” means, what “logical disjunction” means, and so on.

This is why it’s helpful to have a separate formalism for distinguishing the logical connectives of PL from natural language expressions. They remind is that

• “A ∧ B”, the logical conjunction of PL, is not identical to the English language conjunction.
• “A ∨ B”, the logical disjunction of PL, is not identical to the English language disjunction.
• “A ⊃ B”, the logical conditional of PL, is not identical to the English language conditional.

The logical connectives are formal abstractions defined by mathematical rules. The English language connectives are part of the real world of semantically complex linguistic practices, deployed by real-world language users trying to communicate with one another. PL is as different from natural language as a system of differential equations is from the real-world system that the equations are designed to model.

However, just as abstract mathematical representations can be used to model, to represent aspects of the behavior of real-world physical systems, so can abstract formal languages be used to model or represent aspects of the behavior of real-world languages and language users.

Hence we can ask: is the syntax and semantics of this fragment of natural language well-described by the syntax and semantics of the artificial language PL?

Formal languages are useful precisely because in many cases the answer to this question is “YES”.

When the behavior of a physical system is accurately described by the behavior of a mathematical model, this tells us something about the physical system.

Similarly, when the behavior of natural language is accurately described by the behavior of a formal language, this tells us something about natural language.

Linguists and philosophers can argue over exactly what this “something” is, but really that’s a higher-level interpretive question.

4. The expressive capacity of natural language far outstrips what any formal language can represent

By now this shouldn’t be a surprise, but it’s worth saying up front.

Think of natural human language by analogy with a functioning human brain. It’s incredibly complex, it does all sorts of things that we can describe and catalogue but our understanding of how it does these things is very incomplete.

We can construct models of neural function, models of information processing within brain regions, models of higher level computational functioning, models of how symbolic representations might be instantiated and manipulated in the brain, and so on.

All of this contributes to our growing understanding of how brains function, but it doesn’t add up to anything like a single overarching theory of the brain. It’s a collection of models and theories designed to describe different aspects of brain function and behavior. It gives us insight into aspects of brain function, but the complexity of brain function far outstrips these models.

Language is like this. Linguists study different aspects of natural language from many different theoretical perspectives, but our understanding is still very partial, very incomplete. Formal languages help us to understand certain aspects of natural language, but we should not be surprised to learn that natural language transcends the simple rules described by our formal, artificial toy languages.