The syntax tree of a logical formula records its recursive construction. A propositional variable is a leaf, negation has one child, and binary implication has an ordered pair of children. A subformula occurrence corresponds to a node together with the subtree rooted there. Different occurrences may have identical formula strings.
The unique readability theorem asserts uniqueness of the outer formation rule and its immediate components in a logical formula. For the fully parenthesized language with negation and implication, every formula is uniquely a variable, a negation of one formula, or an implication between an ordered pair of formulas. It gives a well-defined syntax tree and licenses recursive definitions and structural induction on formulas.
In the grammar , no logical formula is a proper initial segment of another. A recursive parser reads one variable, or a negation followed by one formula, or a parenthesized pair of formulas separated by implication. Structural induction shows that this parser consumes precisely one whole formula. Its deterministic stopping position proves prefix-freeness. Consequently any formula substring in a valid formula coincides with the formula of a syntax subtree.
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