The Grammar of Logic

Well-Formed Formulas & Recursive Logic

Fundamentals of Propositional Logic | Lesson 5

What is a WFF?

A Well-Formed Formula (WFF) is a string of symbols that is grammatically correct according to the rules of our formal language.

(P → Q) ∧ R

Valid WFF

P ∧ ∨ Q

Invalid (Syntax Error)

The Recursive Recipe

Rule 1: Base Case

Any atomic proposition (P, Q, R...) is a WFF.

Rule 2: Negation

If α is a WFF, then ¬α is a WFF.

Rule 3: Binary

If α and β are WFFs, then (α * β) is a WFF (where * is ∧, ∨, →, ↔).

"Infinite complexity from finite rules."

Game: Syntax Sentry

Identify the illegal string in each set:

A. ¬¬P

B. (P ∧ Q) ∨ R

C. P ∧ Q ∨ R

D. ((P → Q))

E. P ↔ (¬Q)

F. ∧ P Q

HINT: Check for missing parentheses or misplaced operators!

Peer Argument Formalization

Take a paragraph from your thesis draft (or a peer's). Formalize the central claim into a Well-Formed Formula.

Draft Paragraph

"While the primary data suggests a correlation, if we account for the outlier variables, the result is statistically insignificant unless we re-calibrate the model."

(D ∧ (O → ¬S)) ∨ R

WFF Slides

WFF Slides