narrowest scope. Here it is replaced by the modern symbol for conjunction "∧", thus ⊢ (p ∧ q . ⊃ . p ⊃ q). The two remaining single dots pick out the main connective of the whole formula. They illustrate the utility of the dot notation in picking out those connectives which are relatively more important than the ones which surround them. The one to the left of the "⊃" is replaced by a pair of parentheses, the right one goes where the dot is and the left one goes as far to the left as it can without crossing a group of dots of greater force, in this case the two dots which follow the assertion-sign, thus ⊢ ((p ∧ q) ⊃ . p ⊃ q) The dot to the right of the "⊃" is replaced by a left parenthesis which goes where the dot is and a right parenthesis which goes as far to the right as it can without going beyond the scope already established by a group of dots of greater force (in this case the two dots which followed the assertion-sign). So the right parenthesis which replaces the dot to the right of the "⊃" is placed in front of the right parenthesis which replaced the two dots following the assertion-sign, thus ⊢ ((p ∧ q) ⊃ (p ⊃ q)). Example 2, with double, triple, and quadruple dots:✸9.521. ⊢ : : (∃x). φx . ⊃ . q : ⊃ : . (∃x). φx . v . r : ⊃ . q v r stands for ((((∃x)(φx)) ⊃ (q)) ⊃ ((((∃x) (φx)) v (r)) ⊃ (q v r))) Example 3, with a double dot indicating a logical symbol (from volume 1, page 10): p⊃q:q⊃r.⊃.p⊃r stands for (p⊃q) ∧ ((q⊃r)⊃(p⊃r)) where the double dot represents the logical symbol ∧ and can be viewed as having the higher priority as a non-logical single dot. Later in section ✸14, brackets "[ ]" appear, and in sections ✸20 and following, braces "{ }" appear. Whether these symbols have specific meanings or are just for visual clarification is unclear. Unfortunately the single dot (but also ":", ":.", "::", etc.) is also used to symbolise "logical product" (contemporary logical AND often symbolised by "&" or "∧"). Logical implication is represented by Peano's "Ɔ" simplified to "⊃", logical negation is symbolised by an elongated tilde, i.e., "~" (contemporary "~" or "¬"), the logical