The syntax of SL

Use and mention

Most of the time we use words to talk about things. Sometimes we need to talk about the words themselves, and then we mention them. Compare “Boston is in Massachusetts” (about a city) with “‘Boston’ has six letters” (about a name).

The book marks mention in two ways: by putting the expression in single quotes, or by displaying it on its own indented line. You’ll see both throughout this site.

A related pair of terms: when we use one language to talk about another, the language being discussed is the object language and the language doing the talking is the metalanguage. In this course, SL (and later PL) is the object language, and English is the metalanguage.

The vocabulary of SL

Sentence letters are the capital letters A through Z, with or without a positive-integer subscript: A, B, …, Z, A₁, B₁, …, A₂, … That gives infinitely many, which is the point: there’s no limit to how many different claims you might want to symbolize.

The only punctuation marks are the parentheses ‘(’ and ‘)’. A unary connective attaches to one sentence; a binary connective joins two.

Metavariables

To state general rules we need variables that range over sentences of SL, the way a mathematician’s x ranges over numbers. The book uses boldface P, Q, R, S (with subscripts if needed) for this. They are part of English, the metalanguage, not part of SL. Boldface lowercase p, q, r, s play the same role for English sentences.

Watch for this: P & Q is not a sentence of SL. It’s a pattern. A & B, (C ∨ D) & ∼E, and H & H are sentences that fit it.

The recursive definition of “sentence of SL”

1. Every sentence letter is a sentence of SL.
2. If P is a sentence of SL, so is ∼P.
3. If P and Q are sentences of SL, so is (P & Q).
4. If P and Q are sentences of SL, so is (P ∨ Q).
5. If P and Q are sentences of SL, so is (P ⊃ Q).
6. If P and Q are sentences of SL, so is (P ≡ Q).
7. Nothing else is a sentence of SL. (Everything must be built by clauses 1–6.)

This is a recursive definition. Clause 1 gives the starting items outright. Clauses 2–6 say how to make new sentences from ones you already have. Clause 7, the closure clause, rules out everything else.

Notice that every binary clause wraps its result in parentheses. That’s how SL avoids ambiguity: each pair of parentheses shows exactly which two sentences a connective joins.

Try it: build each target sentence from sentence letters, one clause at a time.

Is it a sentence?

To test a string, ask whether it could have been built by clauses 1–6. If you get stuck, the place you get stuck tells you why it fails. The common failures:

• A binary connective without a sentence on both sides.
• Two sentences side by side with no binary connective between them.
• More than one binary connective with nothing to show the grouping.
• Parentheses that don’t balance, or that surround a single sentence like (A).
• A symbol that isn’t in SL’s vocabulary, including boldface metavariables.

Informal conventions apply here. The outer parentheses of a sentence may be dropped, and square brackets may replace parentheses. So B & D counts as a sentence.

Main connectives and immediate components

Sentence letters are the atomic sentences of SL. Everything else is a compound sentence and contains at least one connective. Each compound has one main connective, the one added last when the sentence was built, and its scope covers the whole sentence. The main connective decides what type of sentence it is.

A quick way to find it: if the sentence starts with ∼ and that tilde’s scope runs to the very end, the tilde is the main connective. Otherwise, it’s the one binary connective that isn’t inside any parentheses (after outer parentheses are dropped).

Components

The components of a sentence P are P itself, its immediate components, the immediate components of those, and so on down to the sentence letters. Put simply: every sentence you can find inside P, plus P itself.

Type any sentence into the explorer to see its components as a tree, and the order in which the definition builds it.

Two conventions

• Outer parentheses may be dropped. A sentence that begins with ‘(’ and ends with the matching ‘)’ can lose that pair: A ⊃ (B & ∼D) instead of (A ⊃ (B & ∼D)). A negation of a compound has no outer parentheses: it starts with ∼. So ∼(A ∨ B) can’t become ∼A ∨ B.
• Square brackets may replace parentheses to make nesting easier to read: ∼[(A ∨ B) ⊃ (C ≡ D)].