Editing Principia Mathematica (section)

=== Introduction to the notation of the theory of classes and relations ===

The text leaps from section '''✱14''' directly to the foundational sections '''✱20 GENERAL THEORY OF CLASSES''' and '''✱21 GENERAL THEORY OF RELATIONS'''. "Relations" are what is known in contemporary [[set theory]] as sets of [[ordered pair]]s. Sections '''✱20''' and '''✱22''' introduce many of the symbols still in contemporary usage. These include the symbols "ε", "⊂", "∩", "∪", "–", "Λ", and "V": "ε" signifies "is an element of" (''PM'' 1962:188); "⊂" ('''✱22.01''') signifies "is contained in", "is a subset of"; "∩" ('''✱22.02''') signifies the intersection (logical product) of classes (sets); "∪" ('''✱22.03''') signifies the union (logical sum) of classes (sets); "–" ('''✱22.03''') signifies negation of a class (set); "Λ" signifies the null class; and "V" signifies the universal class or universe of discourse.

Small Greek letters (other than "ε", "ι", "π", "φ", "ψ", "χ", and "θ") represent classes (e.g., "α", "β", "γ", "δ", etc.) (''PM'' 1962:188):
: ''x'' ε α
:: "The use of single letter in place of symbols such as ''ẑ''(φ''z'') or ''ẑ''(φ '''!''' ''z'') is practically almost indispensable, since otherwise the notation rapidly becomes intolerably cumbrous. Thus ' ''x'' ε α' will mean ' ''x'' is a member of the class α'". (''PM'' 1962:188)
:α ∪ –α = V
::The union of a set and its inverse is the universal (completed) set.<ref>See the ten postulates of Huntington, in particular postulates IIa and IIb at ''PM'' 1962:205 and discussion at p. 206.</ref>
:α ∩ –α = Λ
::The intersection of a set and its inverse is the null (empty) set.

When applied to relations in section '''✱23 CALCULUS OF RELATIONS''', the symbols "⊂", "∩", "∪", and "–" acquire a dot: for example: "⊍", "∸".<ref>The "⊂" sign has a dot inside it, and the intersection sign "∩" has a dot above it; these are not available in the "Arial Unicode MS" font.</ref>

'''The notion, and notation, of "a class" (set)''': In the first edition ''PM'' asserts that no new primitive ideas are necessary to define what is meant by "a class", and only two new "primitive propositions" called the [[Axiom of reducibility|axioms of reducibility]] for classes and relations respectively (''PM'' 1962:25).<ref>Wiener 1914 "A simplification of the logic of relations" (van Heijenoort 1967:224ff) disposed of the second of these when he showed how to reduce the theory of relations to that of classes</ref> But before this notion can be defined, ''PM'' feels it necessary to create a peculiar notation "''ẑ''(φ''z'')" that it calls a "fictitious object". (''PM'' 1962:188)
: ⊢''':''' ''x'' ε ''ẑ''(φ''z'') '''.'''≡'''.''' (φ''x'')
:: "i.e., ' ''x'' is a member of the class determined by (φ''ẑ'')' is [logically] equivalent to ' ''x'' satisfies (φ''ẑ''),' or to '(φ''x'') is true.'". (''PM'' 1962:25)

At least ''PM'' can tell the reader how these fictitious objects behave, because "A class is wholly determinate when its membership is known, that is, there cannot be two different classes having the same membership" (''PM'' 1962:26). This is symbolised by the following equality (similar to '''✱13.01''' above:
: ''ẑ''(φ''z'') = ''ẑ''(ψ''z'') '''.''' ≡ ''':''' (''x'')''':''' φ''x'' '''.'''≡'''.''' ψ''x''
::"This last is the distinguishing characteristic of classes, and justifies us in treating ''ẑ''(ψ''z'') as the class determined by [the function] ψ''ẑ''." (''PM'' 1962:188)

Perhaps the above can be made clearer by the discussion of classes in ''Introduction to the Second Edition'', which disposes of the ''Axiom of Reducibility'' and replaces it with the notion: "All functions of functions are extensional" (''PM'' 1962:xxxix), i.e.,
: φ''x'' ≡<sub>''x''</sub> ψ''x'' '''.'''⊃'''.''' (''x'')''':''' ƒ(φ''ẑ'') ≡ ƒ(ψ''ẑ'') (''PM'' 1962:xxxix)

This has the reasonable meaning that "IF for all values of ''x'' the ''truth-values'' of the functions φ and ψ of ''x'' are [logically] equivalent, THEN the function ƒ of a given φ''ẑ'' and ƒ of ψ''ẑ'' are [logically] equivalent." ''PM'' asserts this is "obvious":
: "This is obvious, since φ can only occur in ƒ(φ''ẑ'') by the substitution of values of φ for ''p, q, r, ...'' in a [logical-] function, and, if φ''x'' ≡ ψ''x'', the substitution of φ''x'' for ''p'' in a [logical-] function gives the same truth-value to the truth-function as the substitution of ψ''x''. Consequently there is no longer any reason to distinguish between functions classes, for we have, in virtue of the above,
: φ''x'' ≡<sub>''x''</sub> ψ''x'' '''.'''⊃'''.''' (''x'')'''.''' φ''ẑ'' = '''.''' ψ''ẑ''".
Observe the change to the equality "=" sign on the right. ''PM'' goes on to state that will continue to hang onto the notation "''ẑ''(φ''z'')", but this is merely equivalent to φ''ẑ'', and this is a class. (all quotes: ''PM'' 1962:xxxix).