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Propositional logic syntax defines the exact grammar for constructing well-formed formulas (WFFs) from atomic propositions and logical connectives (¬, ∧, ∨, →, ↔). A WFF is built inductively: every atomic proposition is a WFF, and if φ and ψ are WFFs then so are ¬φ, (φ ∧ ψ), and so on. This purely syntactic definition makes no reference to meaning — a formula is either grammatically valid or it is not. The distinction between syntax (shape of formulas) and semantics (meaning of formulas) is one of the deepest ideas in formal logic.
Write out the inductive grammar rule explicitly, then practice identifying which strings are WFFs and which are not. Parse formulas into syntax trees to build intuition for structure before worrying about meaning.
When you first learned about logical connectives — AND, OR, NOT, if-then — you probably wrote them in English or with informal symbols. Propositional logic syntax is the step of making this completely precise: defining a formal grammar that specifies, with no ambiguity, exactly which strings of symbols count as legal formulas. This matters because a proof system needs to know unambiguously what it is operating on.
The grammar is defined *inductively*. Start with a set of atomic propositions — call them p, q, r, or p₁, p₂, ... — which are the simplest WFFs. Then, if φ and ψ are already WFFs, the following are also WFFs: ¬φ (negation), (φ ∧ ψ) (conjunction), (φ ∨ ψ) (disjunction), (φ → ψ) (conditional), and (φ ↔ ψ) (biconditional). Nothing else is a WFF. This inductive definition means every WFF has a unique *parse tree* — a tree showing how the formula was built up from atoms — and that tree is the formula's structure.
The most important idea in this topic is the distinction between *syntax* and *semantics*. Syntax is about the shape of formulas: is this string grammatically valid? Semantics is about meaning: what does this formula *say*, and is it true? These are completely separate questions. The string "p ∧ ¬p" is syntactically well-formed — it satisfies the grammar — but semantically it is a contradiction, always false. The string "p ∧ ∧ q" is syntactically invalid and has no semantic value at all. A good logician keeps these layers cleanly separated.
Precedence conventions exist to reduce parentheses. Without them, even a simple formula like ¬p ∧ q → r would require full parenthesization: ((¬p) ∧ q) → r. The standard precedence ordering is: ¬ binds tightest, then ∧, then ∨, then →, then ↔. So ¬p ∧ q → r parses as ((¬p) ∧ q) → r. A very common error is treating ¬p ∧ q as ¬(p ∧ q); always apply the precedence rules before interpreting a formula.
Once you can reliably parse WFFs and draw their syntax trees, you are ready for propositional semantics — the study of which truth-value assignments make WFFs true. The clean separation of syntax from semantics is what lets formal logic give precise answers to questions like "Is this argument valid?" and "Is there any truth assignment that makes both φ and ψ true simultaneously?" You will use this syntactic foundation throughout natural deduction, model theory, and first-order logic.