Ever tried reading logical texts from the 20s or before (e.g., C. I. Lewis’s Symbolic Logic)? Confused by the absence of parentheses and all the dots and colons? Here’s Carnap’s explanation of the notation (from Abriss der Logistik):

4 c. The Dot Rules

The dot symbols (. : :. :: etc.) replace the bracketing of propositions. The dot signs fall into three distinct levels, depending on whether they occur

- between two propositions in a conjunction,
- after an operator (x), (∃ x), [(ιx)(φx)],
- after |-, before and after the sign ⊃, ≡ ∨, |, =
_{Df}.Dot rules for reading: The scope of a dot symbol (for 1, to the left and to the right, for 2 to the right, for 3 to left or right, depending) extends either to the end of the proposition or to a dot symbol with more dots or to a symbol of the same or a higher level with the same number of dots.

Dot rules for writing: If the scope of a dot sybol is to extend beyond that of another, it must, if it is of a higher level than the latter, contain at least as many dots, and otherwise more dots.

Examples:

p ∨ . q . r means p ∨ (q . r) |- : (p, q) : p ∨ q . ⊃ . q ∨ p “ |- {(p, q) . [(p ∨ q) ⊃ (q ∨ p)]} p : ∨ : q . ⊃ . q ∨ p “ p ∨ [q ⊃ (q ∨ p)] (x) . φx . ⊃ . p ∨ q “ [(x) . φx] ⊃ (p ∨ q) (x) : φx . ⊃ . p ∨ q “ (x) . [φx ⊃ (p ∨ q)] (x) : φx ⊃ p . ∨ q “ (x) . [(φx ⊃ p) ∨ q] (x) : φx ⊃ p : ∨ q “ [(x) . (φx ⊃ p)] ∨ q