equality ("=") sign:✸10.02 φx ⊃x ψx .=. (x). φx ⊃ ψx Df Contemporary notation: ∀x(φ(x) → ψ(x)) (or a variant)✸10.03 φx ≡x ψx .=. (x). φx ≡ ψx Df Contemporary notation: ∀x(φ(x) ↔ ψ(x)) (or a variant) PM attributes the first symbolism to Peano. Section ✸11 applies this symbolism to two variables. Thus the following notations: ⊃x, ⊃y, ⊃x, y could all appear in a single formula. Section ✸12 reintroduces the notion of "matrix" (contemporary truth table), the notion of logical types, and in particular the notions of first-order and second-order functions and propositions. New symbolism "φ ! x" represents any value of a first-order function. If a circumflex "＾" is placed over a variable, then this is an "individual" value of y, meaning that "ŷ" indicates "individuals" (e.g., a row in a truth table); this distinction is necessary because of the matrix/extensional nature of propositional functions. Now equipped with the matrix notion, PM can assert its controversial axiom of reducibility: a function of one or two variables (two being sufficient for PMs use) where all its values are given (i.e., in its matrix) is (logically) equivalent ("≡") to some "predicative" function of the same variables. The one-variable definition is given below as an illustration of the notation (PM 1962:166–167):✸12.1 ⊢: (Ǝ f): φx .≡x. f ! x Pp; Pp is a "Primitive proposition" ("Propositions assumed without proof") (PM 1962:12, i.e., contemporary "axioms"), adding to the 7 defined in section ✸1 (starting with ✸1.1 modus ponens). These are to be distinguished from the "primitive ideas" that include the assertion sign "⊢", negation "~", logical OR "V", the notions of "elementary proposition" and "elementary propositional function"; these are as close as PM comes to rules of notational formation, i.e., syntax. This means: "We assert the truth of the following: There exists a function f with the property that: given all values of x, their evaluations in function φ (i.e., resulting their matrix) is logically equivalent to some f evaluated at those same values of x. (and vice versa, hence logical equivalence)". In other words: given a matrix determined by property φ applied to variable x, there exists a function f that, when applied to the x is logically equivalent to the matrix. Or: every matrix φx can be represented by a function f applied to x, and vice versa.✸13: The identity operator "=" : This is a definition that uses the