OR by "v". The symbol "=" together with "Df" is used to indicate "is defined as", whereas in sections ✸13 and following, "=" is defined as (mathematically) "identical with", i.e., contemporary mathematical "equality" (cf. discussion in section ✸13). Logical equivalence is represented by "≡" (contemporary "if and only if"); "elementary" propositional functions are written in the customary way, e.g., "f(p)", but later the function sign appears directly before the variable without parenthesis e.g., "φx", "χx", etc. Example, PM introduces the definition of "logical product" as follows:✸3.01. p . q .=. ~(~p v ~q) Df. where "p . q" is the logical product of p and q.✸3.02. p ⊃ q ⊃ r .=. p ⊃ q . q ⊃ r Df. This definition serves merely to abbreviate proofs.Translation of the formulas into contemporary symbols: Various authors use alternate symbols, so no definitive translation can be given. However, because of criticisms such as that of Kurt Gödel below, the best contemporary treatments will be very precise with respect to the "formation rules" (the syntax) of the formulas. The first formula might be converted into modern symbolism as follows: (p & q) =df (~(~p v ~q)) alternately (p & q) =df (¬(¬p v ¬q)) alternately (p ∧ q) =df (¬(¬p v ¬q)) etc. The second formula might be converted as follows: (p → q → r) =df (p → q) & (q → r) But note that this is not (logically) equivalent to (p → (q → r)) nor to ((p → q) → r), and these two are not logically equivalent either. An introduction to the notation of "Section B Theory of Apparent Variables" (formulas ✸8–✸14.34) These sections concern what is now known as predicate logic, and predicate logic with identity (equality). NB: As a result of criticism and advances, the second edition of PM (1927) replaces ✸9 with a new ✸8 (Appendix A). This new section eliminates the first edition's distinction between real and apparent variables, and it eliminates "the primitive idea 'assertion of a propositional function'. To add to the complexity of the treatment, ✸8 introduces the notion of substituting a "matrix", and the Sheffer stroke: Matrix: In contemporary usage, PMs matrix is (at least for propositional functions), a truth table, i.e., all truth-values of a propositional or predicate function. Sheffer stroke: Is the contemporary logical NAND (NOT-AND), i.e., "incompatibility", meaning: "Given two propositions p and q, then ' p